diff --git "a/data/records.jsonl" "b/data/records.jsonl"
--- "a/data/records.jsonl"
+++ "b/data/records.jsonl"
@@ -1,842 +1,842 @@
-{"case": "cirincir@1", "density": 0.9999989965161072, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc1.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "1", "friedman_metric_symbol": "r", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "r", "metric_value": 1.0, "reference_text": "r = 1 | Trivial.", "source_image": "ccc1.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc1.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 1 | Trivial.", "friedman_source_image": "ccc1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "d8dddd0385fd067c", "improvement": -5.017423239106478e-07, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1979053238242443e-07, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00000", "metric_symbol": "r", "metric_value": 1.000000501742324, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0, "setup": "cirincir", "side": 2.000001003484648, "solution_path": "data/solutions/generated/cirincir_001_d8dddd0385fd067c.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d8dddd0385fd067c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@2", "density": 0.4999800974887553, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc2.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "2", "friedman_metric_symbol": "r", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "r", "metric_value": 2.0, "reference_text": "r = 2 | Trivial.", "source_image": "ccc2.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc2.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 2 | Trivial.", "friedman_source_image": "ccc2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "ab9a33067266df01", "improvement": -3.9806210858639446e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.204627911923552e-06, "max_pair_overlap_depth": -5.010172176866945e-05, "metric_expression": "2.00004", "metric_symbol": "r", "metric_value": 2.0000398062108586, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "cirincir", "side": 4.000079612421717, "solution_path": "data/solutions/generated/cirincir_002_ab9a33067266df01.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/ab9a33067266df01.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@3", "density": 0.646133359064875, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc3.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "1 + 2 / \u221a3 = 2.154+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.154, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1 + 2 / \u221a3 = 2.154+", "metric_symbol": "r", "metric_value": 2.154, "reference_text": "r = 1 + 2 / \u221a3 = 2.154+ | Trivial.", "source_image": "ccc3.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc3.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 1 + 2 / \u221a3 = 2.154+ | Trivial.", "friedman_source_image": "ccc3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "fe973ff9a23dc34c", "improvement": -0.000763178385080554, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.918381111833185e-06, "max_pair_overlap_depth": -9.100356622360195e-05, "metric_expression": "2.15476", "metric_symbol": "r", "metric_value": 2.1547631783850805, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.154, "setup": "cirincir", "side": 4.309526356770161, "solution_path": "data/solutions/generated/cirincir_003_fe973ff9a23dc34c.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fe973ff9a23dc34c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@4", "density": 0.6862704996379871, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc4.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "1 + \u221a2 = 2.414+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.414, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1 + \u221a2 = 2.414+", "metric_symbol": "r", "metric_value": 2.414, "reference_text": "r = 1 + \u221a2 = 2.414+ | Trivial.", "source_image": "ccc4.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc4.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 1 + \u221a2 = 2.414+ | Trivial.", "friedman_source_image": "ccc4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "9d3e9e7ea6ba298b", "improvement": -0.00025050219633326876, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.985176337373275e-06, "max_pair_overlap_depth": -4.660490859054178e-05, "metric_expression": "2.41425", "metric_symbol": "r", "metric_value": 2.4142505021963334, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.414, "setup": "cirincir", "side": 4.828501004392667, "solution_path": "data/solutions/generated/cirincir_004_9d3e9e7ea6ba298b.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9d3e9e7ea6ba298b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@5", "density": 0.6851849889863537, "friedman_analytical_or_proved": true, "friedman_credit": "Graham", "friedman_date": "1968", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc5.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "2.701+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.701, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@5", "credit": "Graham", "date": "1968", "family_updated": "", "metric_expression": "2.701+", "metric_symbol": "r", "metric_value": 2.701, "reference_text": "r = 2.701+ | Proved by Graham in 1968.", "source_image": "ccc5.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc5.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 2.701+ | Proved by Graham in 1968.", "friedman_source_image": "ccc5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "9c2ab1f687c06eec", "improvement": -0.0003514001027240532, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.911145805306603e-06, "max_pair_overlap_depth": -4.497481208476373e-05, "metric_expression": "2.70135", "metric_symbol": "r", "metric_value": 2.701351400102724, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.701, "setup": "cirincir", "side": 5.402702800205448, "solution_path": "data/solutions/generated/cirincir_005_9c2ab1f687c06eec.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Graham", "svg_path": "data/svg/9c2ab1f687c06eec.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@6", "density": 0.6666416612940322, "friedman_analytical_or_proved": true, "friedman_credit": "Graham", "friedman_date": "1968", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc6.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3", "friedman_metric_symbol": "r", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@6", "credit": "Graham", "date": "1968", "family_updated": "", "metric_expression": "3", "metric_symbol": "r", "metric_value": 3.0, "reference_text": "r = 3 | Proved by Graham in 1968.", "source_image": "ccc6.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc6.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3 | Proved by Graham in 1968.", "friedman_source_image": "ccc6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "5d3d5beee3c5f33c", "improvement": -5.626367118827247e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.011848064282674e-06, "max_pair_overlap_depth": -4.5345095143201064e-05, "metric_expression": "3.00006", "metric_symbol": "r", "metric_value": 3.0000562636711883, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "cirincir", "side": 6.0001125273423765, "solution_path": "data/solutions/generated/cirincir_006_5d3d5beee3c5f33c.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Graham", "svg_path": "data/svg/5d3d5beee3c5f33c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@7", "density": 0.77774147574625, "friedman_analytical_or_proved": true, "friedman_credit": "Graham", "friedman_date": "1968", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc7.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3", "friedman_metric_symbol": "r", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@7", "credit": "Graham", "date": "1968", "family_updated": "", "metric_expression": "3", "metric_symbol": "r", "metric_value": 3.0, "reference_text": "r = 3 | Proved by Graham in 1968.", "source_image": "ccc7.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc7.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3 | Proved by Graham in 1968.", "friedman_source_image": "ccc7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "2b1985e4094e6893", "improvement": -7.001351167312819e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.2947164267690425e-06, "max_pair_overlap_depth": -6.231680049051391e-05, "metric_expression": "3.00007", "metric_symbol": "r", "metric_value": 3.000070013511673, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "cirincir", "side": 6.000140027023346, "solution_path": "data/solutions/generated/cirincir_007_2b1985e4094e6893.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Graham", "svg_path": "data/svg/2b1985e4094e6893.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@8", "density": 0.7324681263291527, "friedman_analytical_or_proved": true, "friedman_credit": "Braaksma", "friedman_date": "1963", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc8.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3.304+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.304, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@8", "credit": "Braaksma", "date": "1963", "family_updated": "", "metric_expression": "3.304+", "metric_symbol": "r", "metric_value": 3.304, "reference_text": "r = 3.304+ | Proved by Braaksma in 1963.", "source_image": "ccc8.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc8.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3.304+ | Proved by Braaksma in 1963.", "friedman_source_image": "ccc8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "0193661647fd3854", "improvement": -0.0008414425517750956, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.96872891495326e-06, "max_pair_overlap_depth": -4.3191348294779885e-05, "metric_expression": "3.30484", "metric_symbol": "r", "metric_value": 3.304841442551775, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.304, "setup": "cirincir", "side": 6.60968288510355, "solution_path": "data/solutions/generated/cirincir_008_0193661647fd3854.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Braaksma", "svg_path": "data/svg/0193661647fd3854.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@9", "density": 0.6893878045655164, "friedman_analytical_or_proved": true, "friedman_credit": "Pirl", "friedman_date": "1969", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc9.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3.613+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.613, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@9", "credit": "Pirl", "date": "1969", "family_updated": "", "metric_expression": "3.613+", "metric_symbol": "r", "metric_value": 3.613, "reference_text": "r = 3.613+ | Proved by Pirl in 1969.", "source_image": "ccc9.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc9.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3.613+ | Proved by Pirl in 1969.", "friedman_source_image": "ccc9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "b51df67181f6833b", "improvement": -0.00017882622267562098, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.656082665643879e-06, "max_pair_overlap_depth": -2.396342999144707e-05, "metric_expression": "3.61318", "metric_symbol": "r", "metric_value": 3.6131788262226756, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.613, "setup": "cirincir", "side": 7.226357652445351, "solution_path": "data/solutions/generated/cirincir_009_b51df67181f6833b.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Pirl", "svg_path": "data/svg/b51df67181f6833b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@10", "density": 0.6877649990272218, "friedman_analytical_or_proved": true, "friedman_credit": "Pirl", "friedman_date": "1969", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc10.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3.813+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.813, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@10", "credit": "Pirl", "date": "1969", "family_updated": "", "metric_expression": "3.813+", "metric_symbol": "r", "metric_value": 3.813, "reference_text": "r = 3.813+ | Proved by Pirl in 1969.", "source_image": "ccc10.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc10.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3.813+ | Proved by Pirl in 1969.", "friedman_source_image": "ccc10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "be14c55679a54b28", "improvement": -0.00011553912424266954, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.081326946625552e-06, "max_pair_overlap_depth": -3.967607896537828e-05, "metric_expression": "3.81312", "metric_symbol": "r", "metric_value": 3.813115539124243, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.813, "setup": "cirincir", "side": 7.626231078248486, "solution_path": "data/solutions/generated/cirincir_10_be14c55679a54b28.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Pirl", "svg_path": "data/svg/be14c55679a54b28.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@11", "density": 0.714418364520267, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1994", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc11.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3.923+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.923, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@11", "credit": "Melissen", "date": "1994", "family_updated": "", "metric_expression": "3.923+", "metric_symbol": "r", "metric_value": 3.923, "reference_text": "r = 3.923+ | Proved by Melissen in 1994.", "source_image": "ccc11.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc11.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3.923+ | Proved by Melissen in 1994.", "friedman_source_image": "ccc11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "e624f391927921fb", "improvement": -0.0009190348380849755, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.92009600810772e-06, "max_pair_overlap_depth": -4.346382940223492e-05, "metric_expression": "3.92392", "metric_symbol": "r", "metric_value": 3.923919034838085, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.923, "setup": "cirincir", "side": 7.84783806967617, "solution_path": "data/solutions/generated/cirincir_011_e624f391927921fb.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Melissen", "svg_path": "data/svg/e624f391927921fb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@12", "density": 0.7389828254275508, "friedman_analytical_or_proved": false, "friedman_credit": "Kravitz", "friedman_date": "1967", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc12.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "4.029+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.029, "friedman_reference": {"analytical_or_proved": false, "case": "cirincir@12", "credit": "Kravitz", "date": "1967", "family_updated": "", "metric_expression": "4.029+", "metric_symbol": "r", "metric_value": 4.029, "reference_text": "r = 4.029+ | Found by Kravitz in 1967.", "source_image": "ccc12.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc12.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 4.029+ | Found by Kravitz in 1967.", "friedman_source_image": "ccc12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "0d7fc8a14b1c4e0e", "improvement": -0.0007068210362959348, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.338373973515445e-06, "max_pair_overlap_depth": -5.019209633871924e-05, "metric_expression": "4.02971", "metric_symbol": "r", "metric_value": 4.029706821036296, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.029, "setup": "cirincir", "side": 8.059413642072592, "solution_path": "data/solutions/generated/cirincir_012_0d7fc8a14b1c4e0e.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Kravitz", "svg_path": "data/svg/0d7fc8a14b1c4e0e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@13", "density": 0.7244255598642273, "friedman_analytical_or_proved": true, "friedman_credit": "F. Fodor", "friedman_date": "2003", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc13.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "2 + \u221a5 = 4.236+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.236, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@13", "credit": "F. Fodor", "date": "2003", "family_updated": "", "metric_expression": "2 + \u221a5 = 4.236+", "metric_symbol": "r", "metric_value": 4.236, "reference_text": "r = 2 + \u221a5 = 4.236+ | Proved by F. Fodor in 2003.", "source_image": "ccc13.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc13.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 2 + \u221a5 = 4.236+ | Proved by F. Fodor in 2003.", "friedman_source_image": "ccc13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "a118f33872e0cb30", "improvement": -0.00018378579659206196, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.324667153277062e-06, "max_pair_overlap_depth": -4.7467215169572086e-05, "metric_expression": "4.23618", "metric_symbol": "r", "metric_value": 4.236183785796592, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.236, "setup": "cirincir", "side": 8.472367571593184, "solution_path": "data/solutions/generated/cirincir_013_a118f33872e0cb30.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "F. Fodor", "svg_path": "data/svg/a118f33872e0cb30.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@14", "density": 0.7472109217319929, "friedman_analytical_or_proved": false, "friedman_credit": "Goldberg", "friedman_date": "1971", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc14.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "4.328+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.328, "friedman_reference": {"analytical_or_proved": false, "case": "cirincir@14", "credit": "Goldberg", "date": "1971", "family_updated": "", "metric_expression": "4.328+", "metric_symbol": "r", "metric_value": 4.328, "reference_text": "r = 4.328+ | Found by Goldberg in 1971.", "source_image": "ccc14.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc14.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 4.328+ | Found by Goldberg in 1971.", "friedman_source_image": "ccc14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "b2ec44e582e5ab93", "improvement": -0.000549738133919675, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.732062089120916e-06, "max_pair_overlap_depth": -4.9839299053555663e-05, "metric_expression": "4.32855", "metric_symbol": "r", "metric_value": 4.32854973813392, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.328, "setup": "cirincir", "side": 8.65709947626784, "solution_path": "data/solutions/generated/cirincir_014_b2ec44e582e5ab93.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Goldberg", "svg_path": "data/svg/b2ec44e582e5ab93.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@15", "density": 0.7334069336541974, "friedman_analytical_or_proved": false, "friedman_credit": "Kravitz", "friedman_date": "1967", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc15.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "4.521+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.521, "friedman_reference": {"analytical_or_proved": false, "case": "cirincir@15", "credit": "Kravitz", "date": "1967", "family_updated": "", "metric_expression": "4.521+", "metric_symbol": "r", "metric_value": 4.521, "reference_text": "r = 4.521+ | Found by Kravitz in 1967.", "source_image": "ccc15.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc15.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 4.521+ | Found by Kravitz in 1967.", "friedman_source_image": "ccc15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "eeff7b918e6d6618", "improvement": -0.00144322894722837, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.2291583220711573e-06, "max_pair_overlap_depth": -2.9276985216153406e-05, "metric_expression": "4.52244", "metric_symbol": "r", "metric_value": 4.522443228947228, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.521, "setup": "cirincir", "side": 4.522443228947228, "solution_path": "data/solutions/generated/cirincir_015_eeff7b918e6d6618.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Kravitz", "svg_path": "data/svg/eeff7b918e6d6618.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@16", "density": 0.7510554296860762, "friedman_analytical_or_proved": false, "friedman_credit": "Goldberg", "friedman_date": "1971", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc16.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "4.615+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.615, "friedman_reference": {"analytical_or_proved": false, "case": "cirincir@16", "credit": "Goldberg", "date": "1971", "family_updated": "", "metric_expression": "4.615+", "metric_symbol": "r", "metric_value": 4.615, "reference_text": "r = 4.615+ | Found by Goldberg in 1971.", "source_image": "ccc16.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc16.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 4.615+ | Found by Goldberg in 1971.", "friedman_source_image": "ccc16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "7824261a02b5ed90", "improvement": -0.0005556989154120728, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.219226596077988e-06, "max_pair_overlap_depth": -4.8324434488655754e-05, "metric_expression": "4.61556", "metric_symbol": "r", "metric_value": 4.615555698915412, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.615, "setup": "cirincir", "side": 9.231111397830825, "solution_path": "data/solutions/generated/cirincir_016_7824261a02b5ed90.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Goldberg", "svg_path": "data/svg/7824261a02b5ed90.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@17", "density": 0.7402604737017356, "friedman_analytical_or_proved": false, "friedman_credit": "Reis", "friedman_date": "1975", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc17.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "4.792+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.792, "friedman_reference": {"analytical_or_proved": false, "case": "cirincir@17", "credit": "Reis", "date": "1975", "family_updated": "", "metric_expression": "4.792+", "metric_symbol": "r", "metric_value": 4.792, "reference_text": "r = 4.792+ | Found by Reis in 1975.", "source_image": "ccc17.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc17.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 4.792+ | Found by Reis in 1975.", "friedman_source_image": "ccc17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "68c41c427924b99a", "improvement": -0.00016960544150546042, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.357123401303568e-06, "max_pair_overlap_depth": -5.05938755308577e-05, "metric_expression": "4.79217", "metric_symbol": "r", "metric_value": 4.792169605441505, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.792, "setup": "cirincir", "side": 9.58433921088301, "solution_path": "data/solutions/generated/cirincir_017_68c41c427924b99a.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Reis", "svg_path": "data/svg/68c41c427924b99a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@18", "density": 0.7608872603215449, "friedman_analytical_or_proved": true, "friedman_credit": "Kravitz", "friedman_date": "1967", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc18.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "1 + \u221a2 + \u221a6 = 4.863+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.863, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@18", "credit": "Kravitz", "date": "1967", "family_updated": "", "metric_expression": "1 + \u221a2 + \u221a6 = 4.863+", "metric_symbol": "r", "metric_value": 4.863, "reference_text": "r = 1 + \u221a2 + \u221a6 = 4.863+ | Found by Kravitz in 1967.", "source_image": "ccc18.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc18.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 1 + \u221a2 + \u221a6 = 4.863+ | Found by Kravitz in 1967.", "friedman_source_image": "ccc18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "6acc78721c0b4ca1", "improvement": -0.000804343250861983, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.509170524220508e-07, "max_pair_overlap_depth": -3.6364882113559815e-05, "metric_expression": "4.86380", "metric_symbol": "r", "metric_value": 4.863804343250862, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.863, "setup": "cirincir", "side": 9.727608686501725, "solution_path": "data/solutions/generated/cirincir_018_6acc78721c0b4ca1.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Kravitz", "svg_path": "data/svg/6acc78721c0b4ca1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@19", "density": 0.8031578398507155, "friedman_analytical_or_proved": true, "friedman_credit": "F. Fodor", "friedman_date": "1999", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc19.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "1 + \u221a2 + \u221a6 = 4.863+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.863, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@19", "credit": "F. Fodor", "date": "1999", "family_updated": "", "metric_expression": "1 + \u221a2 + \u221a6 = 4.863+", "metric_symbol": "r", "metric_value": 4.863, "reference_text": "r = 1 + \u221a2 + \u221a6 = 4.863+ | Proved by F. Fodor in 1999.", "source_image": "ccc19.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc19.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 1 + \u221a2 + \u221a6 = 4.863+ | Proved by F. Fodor in 1999.", "friedman_source_image": "ccc19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "8325b5bff319f74f", "improvement": -0.0008071741580044645, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.170379222960378e-06, "max_pair_overlap_depth": -3.89864445309307e-05, "metric_expression": "4.86381", "metric_symbol": "r", "metric_value": 4.863807174158005, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.863, "setup": "cirincir", "side": 9.72761434831601, "solution_path": "data/solutions/generated/cirincir_019_8325b5bff319f74f.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "F. Fodor", "svg_path": "data/svg/8325b5bff319f74f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirincir@20", "density": 0.7622179661920169, "friedman_analytical_or_proved": false, "friedman_credit": "Goldberg", "friedman_date": "1971", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc20.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "5.122+", "friedman_metric_symbol": "r", "friedman_metric_value": 5.122, "friedman_reference": {"analytical_or_proved": false, "case": "cirincir@20", "credit": "Goldberg", "date": "1971", "family_updated": "", "metric_expression": "5.122+", "metric_symbol": "r", "metric_value": 5.122, "reference_text": "r = 5.122+ | Found by Goldberg in 1971.", "source_image": "ccc20.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc20.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 5.122+ | Found by Goldberg in 1971.", "friedman_source_image": "ccc20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "f0718319f5a05220", "improvement": -0.0004226268079490225, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.474954062416714e-06, "max_pair_overlap_depth": -3.4642111702343215e-05, "metric_expression": "5.12242", "metric_symbol": "r", "metric_value": 5.122422626807949, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.122, "setup": "cirincir", "side": 10.244845253615898, "solution_path": "data/solutions/generated/cirincir_020_f0718319f5a05220.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Goldberg", "svg_path": "data/svg/f0718319f5a05220.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@1", "density": 0.9068802327278791, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc1.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2/\u221a3 = 1.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.154, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2/\u221a3 = 1.154+", "metric_symbol": "s", "metric_value": 1.154, "reference_text": "s = 2/\u221a3 = 1.154+ | Trivial.", "source_image": "hc1.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc1.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2/\u221a3 = 1.154+ | Trivial.", "friedman_source_image": "hc1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "37d86bed2252b572", "improvement": -0.0007129204441553849, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.894559686663413e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.15471", "metric_symbol": "s", "metric_value": 1.1547129204441553, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.154, "setup": "cirinhex", "side": 1.1547129204441553, "solution_path": "data/solutions/generated/cirinhex_001_37d86bed2252b572.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/37d86bed2252b572.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@2", "density": 0.5208916187605724, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc2.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "1 + 2/\u221a3 = 2.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.154, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1 + 2/\u221a3 = 2.154+", "metric_symbol": "s", "metric_value": 2.154, "reference_text": "s = 1 + 2/\u221a3 = 2.154+ | Trivial.", "source_image": "hc2.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc2.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 1 + 2/\u221a3 = 2.154+ | Trivial.", "friedman_source_image": "hc2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "4ba45a45587fd527", "improvement": -0.0007173376388385222, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.535667492477913e-06, "max_pair_overlap_depth": -1.2900139699567603e-05, "metric_expression": "2.15472", "metric_symbol": "s", "metric_value": 2.1547173376388384, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.154, "setup": "cirinhex", "side": 2.1547173376388384, "solution_path": "data/solutions/generated/cirinhex_002_4ba45a45587fd527.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4ba45a45587fd527.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@3", "density": 0.6801656739683151, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc3.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "4/\u221a3 = 2.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.309, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4/\u221a3 = 2.309+", "metric_symbol": "s", "metric_value": 2.309, "reference_text": "s = 4/\u221a3 = 2.309+ | Trivial.", "source_image": "hc3.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc3.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 4/\u221a3 = 2.309+ | Trivial.", "friedman_source_image": "hc3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "fa4a26f729d80904", "improvement": -0.00041650453514563424, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3364075199806535e-06, "max_pair_overlap_depth": -1.818559246702378e-05, "metric_expression": "2.30942", "metric_symbol": "s", "metric_value": 2.309416504535146, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.309, "setup": "cirinhex", "side": 2.309416504535146, "solution_path": "data/solutions/generated/cirinhex_003_fa4a26f729d80904.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fa4a26f729d80904.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@4", "density": 0.6802066538449848, "friedman_analytical_or_proved": true, "friedman_credit": "Shahriar Manzoor", "friedman_date": "April 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc4.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2/\u221a3 + 4/\u221a7 = 2.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.666, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@4", "credit": "Shahriar Manzoor", "date": "April 2002", "family_updated": "", "metric_expression": "2/\u221a3 + 4/\u221a7 = 2.666+", "metric_symbol": "s", "metric_value": 2.666, "reference_text": "s = 2/\u221a3 + 4/\u221a7 = 2.666+ | Found by Shahriar Manzoor | in April 2002.", "source_image": "hc4.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc4.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2/\u221a3 + 4/\u221a7 = 2.666+ | Found by Shahriar Manzoor | in April 2002.", "friedman_source_image": "hc4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "4ea68db180a8d4c4", "improvement": -0.0006041510947536466, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3940250920849735e-07, "max_pair_overlap_depth": -4.6158989302380604e-05, "metric_expression": "2.66660", "metric_symbol": "s", "metric_value": 2.6666041510947536, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.666, "setup": "cirinhex", "side": 2.6666041510947536, "solution_path": "data/solutions/generated/cirinhex_004_4ea68db180a8d4c4.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Shahriar Manzoor", "svg_path": "data/svg/4ea68db180a8d4c4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@5", "density": 0.672041103262025, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1999", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc5.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2.999+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.999, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@5", "credit": "Erich Friedman", "date": "1999", "family_updated": "", "metric_expression": "2.999+", "metric_symbol": "s", "metric_value": 2.999, "reference_text": "s = 2.999+ | Found by Erich Friedman | in 1999.", "source_image": "hc5.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc5.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2.999+ | Found by Erich Friedman | in 1999.", "friedman_source_image": "hc5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "eb88c54ef20a6355", "improvement": -0.00041167259487728813, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6369105032841844e-07, "max_pair_overlap_depth": -3.862655436481788e-05, "metric_expression": "2.99941", "metric_symbol": "s", "metric_value": 2.9994116725948774, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.999, "setup": "cirinhex", "side": 2.9994116725948774, "solution_path": "data/solutions/generated/cirinhex_005_eb88c54ef20a6355.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/eb88c54ef20a6355.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@6", "density": 0.7289750468062609, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc6.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2 + 2/\u221a3 = 3.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.154, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 2/\u221a3 = 3.154+", "metric_symbol": "s", "metric_value": 3.154, "reference_text": "s = 2 + 2/\u221a3 = 3.154+ | Trivial. |", "source_image": "hc6.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc6.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2 + 2/\u221a3 = 3.154+ | Trivial. |", "friedman_source_image": "hc6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "9b7724fab73c49c1", "improvement": -0.0007742481949115643, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.7088474341983044e-06, "max_pair_overlap_depth": -4.5158069021944414e-05, "metric_expression": "3.15477", "metric_symbol": "s", "metric_value": 3.1547742481949115, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.154, "setup": "cirinhex", "side": 3.1547742481949115, "solution_path": "data/solutions/generated/cirinhex_006_9b7724fab73c49c1.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9b7724fab73c49c1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@7", "density": 0.8504825176668179, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc7.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2 + 2/\u221a3 = 3.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.154, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 2/\u221a3 = 3.154+", "metric_symbol": "s", "metric_value": 3.154, "reference_text": "s = 2 + 2/\u221a3 = 3.154+ | Trivial. |", "source_image": "hc7.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc7.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2 + 2/\u221a3 = 3.154+ | Trivial. |", "friedman_source_image": "hc7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "7368090471750227", "improvement": -0.0007526785063349806, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.857578518338343e-06, "max_pair_overlap_depth": -3.893260533160969e-05, "metric_expression": "3.15475", "metric_symbol": "s", "metric_value": 3.154752678506335, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.154, "setup": "cirinhex", "side": 3.154752678506335, "solution_path": "data/solutions/generated/cirinhex_007_7368090471750227.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7368090471750227.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@8", "density": 0.7030218969047355, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc8.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "3.709+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.709, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@8", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "3.709+", "metric_symbol": "s", "metric_value": 3.709, "reference_text": "s = 3.709+ | Found by David W. Cantrell | in July 2002.", "source_image": "hc8.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc8.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 3.709+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "hc8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "afebd04e4720460b", "improvement": -0.0004503331818734857, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1028444985304873e-08, "max_pair_overlap_depth": -2.2199913737530608e-05, "metric_expression": "3.70945", "metric_symbol": "s", "metric_value": 3.7094503331818736, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.709, "setup": "cirinhex", "side": 3.7094503331818736, "solution_path": "data/solutions/generated/cirinhex_008_afebd04e4720460b.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/afebd04e4720460b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@9", "density": 0.6761453195206264, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc9.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "20/7 + 2/\u221a3 = 4.011+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.011, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@9", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "20/7 + 2/\u221a3 = 4.011+", "metric_symbol": "s", "metric_value": 4.011, "reference_text": "s = 20/7 + 2/\u221a3 = 4.011+ | Found by David W. Cantrell | in July 2002.", "source_image": "hc9.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc9.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 20/7 + 2/\u221a3 = 4.011+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "hc9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "64b59a5605599f8e", "improvement": -0.0009011593404411045, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.491846259144452e-06, "max_pair_overlap_depth": -2.6999063512000987e-05, "metric_expression": "4.01190", "metric_symbol": "s", "metric_value": 4.011901159340441, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.011, "setup": "cirinhex", "side": 4.011901159340441, "solution_path": "data/solutions/generated/cirinhex_009_64b59a5605599f8e.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/64b59a5605599f8e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@10", "density": 0.7125686135385136, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc10.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "4.119+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.119, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@10", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "4.119+", "metric_symbol": "s", "metric_value": 4.119, "reference_text": "s = 4.119+ | Found by David W. Cantrell | in July 2002.", "source_image": "hc10.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc10.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 4.119+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "hc10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "cd79fdfeb9e3f24b", "improvement": -0.00041597258721015606, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.240617525155585e-06, "max_pair_overlap_depth": -1.610926851469685e-05, "metric_expression": "4.11942", "metric_symbol": "s", "metric_value": 4.11941597258721, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.119, "setup": "cirinhex", "side": 4.11941597258721, "solution_path": "data/solutions/generated/cirinhex_010_cd79fdfeb9e3f24b.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/cd79fdfeb9e3f24b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@11", "density": 0.7162109607848701, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1999", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc11.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2 + 4/\u221a3 = 4.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.309, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@11", "credit": "Erich Friedman", "date": "1999", "family_updated": "", "metric_expression": "2 + 4/\u221a3 = 4.309+", "metric_symbol": "s", "metric_value": 4.309, "reference_text": "s = 2 + 4/\u221a3 = 4.309+ | Found by Erich Friedman | in 1999.", "source_image": "hc11.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc11.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2 + 4/\u221a3 = 4.309+ | Found by Erich Friedman | in 1999.", "friedman_source_image": "hc11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "cda2ccbbf4de67aa", "improvement": -0.00047984763128994246, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.3670191174924895e-06, "max_pair_overlap_depth": -2.2677850788799248e-05, "metric_expression": "4.30948", "metric_symbol": "s", "metric_value": 4.30947984763129, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.309, "setup": "cirinhex", "side": 4.30947984763129, "solution_path": "data/solutions/generated/cirinhex_011_cda2ccbbf4de67aa.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/cda2ccbbf4de67aa.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@12", "density": 0.7813203208949625, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1999", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc12.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2 + 4/\u221a3 = 4.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.309, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@12", "credit": "Erich Friedman", "date": "1999", "family_updated": "", "metric_expression": "2 + 4/\u221a3 = 4.309+", "metric_symbol": "s", "metric_value": 4.309, "reference_text": "s = 2 + 4/\u221a3 = 4.309+ | Found by Erich Friedman | in 1999.", "source_image": "hc12.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc12.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2 + 4/\u221a3 = 4.309+ | Found by Erich Friedman | in 1999.", "friedman_source_image": "hc12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "c3d795864cfeb74c", "improvement": -0.0004818532104344797, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.009897945360223e-06, "max_pair_overlap_depth": -3.7913329796079864e-05, "metric_expression": "4.30948", "metric_symbol": "s", "metric_value": 4.309481853210435, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.309, "setup": "cirinhex", "side": 4.309481853210435, "solution_path": "data/solutions/generated/cirinhex_012_c3d795864cfeb74c.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/c3d795864cfeb74c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@13", "density": 0.7368031353920613, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1999", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc13.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "8/\u221a3 = 4.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.618, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@13", "credit": "Erich Friedman", "date": "1999", "family_updated": "", "metric_expression": "8/\u221a3 = 4.618+", "metric_symbol": "s", "metric_value": 4.618, "reference_text": "s = 8/\u221a3 = 4.618+ | Found by Erich Friedman | in 1999.", "source_image": "hc13.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc13.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 8/\u221a3 = 4.618+ | Found by Erich Friedman | in 1999.", "friedman_source_image": "hc13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "62c003d981fbe524", "improvement": -0.0009678209334929022, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.589858579651107e-06, "max_pair_overlap_depth": -5.483667702588235e-05, "metric_expression": "4.61897", "metric_symbol": "s", "metric_value": 4.618967820933493, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.618, "setup": "cirinhex", "side": 4.618967820933493, "solution_path": "data/solutions/generated/cirinhex_013_62c003d981fbe524.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/62c003d981fbe524.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@14", "density": 0.7773468150841377, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc14.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2 + 2/\u221a3 + 4/\u221a7 = 4.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.666, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@14", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "2 + 2/\u221a3 + 4/\u221a7 = 4.666+", "metric_symbol": "s", "metric_value": 4.666, "reference_text": "s = 2 + 2/\u221a3 + 4/\u221a7 = 4.666+ | Found by David W. Cantrell | in July 2002.", "source_image": "hc14.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc14.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2 + 2/\u221a3 + 4/\u221a7 = 4.666+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "hc14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "d3d8d6f246feeaad", "improvement": -0.0006539683763504556, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4490171600378687e-06, "max_pair_overlap_depth": -3.940553381864831e-05, "metric_expression": "4.66665", "metric_symbol": "s", "metric_value": 4.666653968376351, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.666, "setup": "cirinhex", "side": 4.666653968376351, "solution_path": "data/solutions/generated/cirinhex_014_d3d8d6f246feeaad.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d3d8d6f246feeaad.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@15", "density": 0.7368006709671345, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "January 2006", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc15.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "4.961+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.961, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@15", "credit": "David W. Cantrell", "date": "January 2006", "family_updated": "", "metric_expression": "4.961+", "metric_symbol": "s", "metric_value": 4.961, "reference_text": "s = 4.961+ | Found by David W. Cantrell | in January 2006.", "source_image": "hc15.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc15.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 4.961+ | Found by David W. Cantrell | in January 2006.", "friedman_source_image": "hc15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "d923ba7fbcb56e93", "improvement": -0.0005756365508533534, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.177000685334775e-07, "max_pair_overlap_depth": -3.4253106946380285e-05, "metric_expression": "4.96158", "metric_symbol": "s", "metric_value": 4.961575636550854, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.961, "setup": "cirinhex", "side": 4.961575636550854, "solution_path": "data/solutions/generated/cirinhex_015_d923ba7fbcb56e93.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d923ba7fbcb56e93.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@16", "density": 0.774039756730763, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc16.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "4.999+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.999, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@16", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "4.999+", "metric_symbol": "s", "metric_value": 4.999, "reference_text": "s = 4.999+ | Found by David W. Cantrell | in July 2002.", "source_image": "hc16.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc16.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 4.999+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "hc16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "b6a042d1d6eee5a4", "improvement": -0.0005089546268814971, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.665337648678829e-06, "max_pair_overlap_depth": -4.998313684900779e-05, "metric_expression": "4.99951", "metric_symbol": "s", "metric_value": 4.999508954626881, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.999, "setup": "cirinhex", "side": 4.999508954626881, "solution_path": "data/solutions/generated/cirinhex_016_b6a042d1d6eee5a4.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/b6a042d1d6eee5a4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@17", "density": 0.7801137203230791, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "January 2006", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc17.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "5.133+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.133, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@17", "credit": "David W. Cantrell", "date": "January 2006", "family_updated": "", "metric_expression": "5.133+", "metric_symbol": "s", "metric_value": 5.133, "reference_text": "s = 5.133+ | Found by David W. Cantrell | in January 2006.", "source_image": "hc17.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc17.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 5.133+ | Found by David W. Cantrell | in January 2006.", "friedman_source_image": "hc17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "4927714d5d50ab2b", "improvement": -0.00027458420964432406, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.495950815863736e-06, "max_pair_overlap_depth": -5.684433644859155e-05, "metric_expression": "5.13327", "metric_symbol": "s", "metric_value": 5.133274584209644, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.133, "setup": "cirinhex", "side": 5.133274584209644, "solution_path": "data/solutions/generated/cirinhex_017_4927714d5d50ab2b.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/4927714d5d50ab2b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@18", "density": 0.81913146509039, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc18.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "4 + 2/\u221a3 = 5.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.154, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@18", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4 + 2/\u221a3 = 5.154+", "metric_symbol": "s", "metric_value": 5.154, "reference_text": "s = 4 + 2/\u221a3 = 5.154+ | Trivial. |", "source_image": "hc18.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc18.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 4 + 2/\u221a3 = 5.154+ | Trivial. |", "friedman_source_image": "hc18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "223b1a3f7b7d0e67", "improvement": -0.000759899456760138, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.33856736717064e-06, "max_pair_overlap_depth": -1.657857602177515e-05, "metric_expression": "5.15476", "metric_symbol": "s", "metric_value": 5.15475989945676, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.154, "setup": "cirinhex", "side": 5.15475989945676, "solution_path": "data/solutions/generated/cirinhex_018_223b1a3f7b7d0e67.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/223b1a3f7b7d0e67.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@19", "density": 0.8646358262071194, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc19.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "4 + 2/\u221a3 = 5.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.154, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@19", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4 + 2/\u221a3 = 5.154+", "metric_symbol": "s", "metric_value": 5.154, "reference_text": "s = 4 + 2/\u221a3 = 5.154+ | Trivial. |", "source_image": "hc19.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc19.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 4 + 2/\u221a3 = 5.154+ | Trivial. |", "friedman_source_image": "hc19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "78f9327e80d8ab20", "improvement": -0.0007686707015803762, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.035946609195264e-07, "max_pair_overlap_depth": -1.6136694912294303e-05, "metric_expression": "5.15477", "metric_symbol": "s", "metric_value": 5.15476867070158, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.154, "setup": "cirinhex", "side": 5.15476867070158, "solution_path": "data/solutions/generated/cirinhex_019_78f9327e80d8ab20.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/78f9327e80d8ab20.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@20", "density": 0.7503080962551325, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc20.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "5.676+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.676, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@20", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "5.676+", "metric_symbol": "s", "metric_value": 5.676, "reference_text": "s = 5.676+ | Found by David W. Cantrell | in July 2012.", "source_image": "hc20.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc20.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 5.676+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "hc20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "247c6c06779a7d8b", "improvement": -0.0013304688728235092, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.66649728101143e-06, "max_pair_overlap_depth": -2.983410910761819e-05, "metric_expression": "5.67733", "metric_symbol": "s", "metric_value": 5.677330468872824, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.676, "setup": "cirinhex", "side": 5.677330468872824, "solution_path": "data/solutions/generated/cirinhex_020_247c6c06779a7d8b.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/247c6c06779a7d8b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@21", "density": 0.7789652581067265, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc21.gif", "friedman_metric_expression": "5.709+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.709, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@21", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "5.709+", "metric_symbol": "s", "metric_value": 5.709, "reference_text": "s = 5.709+ | Found by David W. Cantrell | in July 2012.", "source_image": "hc21.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc21.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 5.709+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "hc21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "38040eceeb63c1e3", "improvement": -0.0005199689632231497, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.315374129847953e-06, "max_pair_overlap_depth": -3.751451804223649e-05, "metric_expression": "5.70952", "metric_symbol": "s", "metric_value": 5.709519968963223, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.709, "setup": "cirinhex", "side": 5.709519968963223, "solution_path": "data/solutions/generated/cirinhex_021_38040eceeb63c1e3.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/38040eceeb63c1e3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@22", "density": 0.7509530082141859, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc22.gif", "friedman_metric_expression": "5.952+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.952, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@22", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "5.952+", "metric_symbol": "s", "metric_value": 5.952, "reference_text": "s = 5.952+ | Found by David W. Cantrell | in July 2012.", "source_image": "hc22.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc22.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 5.952+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "hc22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "cd7262f9f7d0d279", "improvement": 0.0001229278734493633, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.663138574891647e-06, "max_pair_overlap_depth": -3.268885699680979e-05, "metric_expression": "5.95188", "metric_symbol": "s", "metric_value": 5.951877072126551, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.952, "setup": "cirinhex", "side": 5.951877072126551, "solution_path": "data/solutions/generated/cirinhex_022_cd7262f9f7d0d279.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/cd7262f9f7d0d279.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@23", "density": 0.7694641578789073, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc23.gif", "friedman_metric_expression": "34/7 + 2/\u221a3 = 6.011+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.011, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@23", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "34/7 + 2/\u221a3 = 6.011+", "metric_symbol": "s", "metric_value": 6.011, "reference_text": "s = 34/7 + 2/\u221a3 = 6.011+ | Found by David W. Cantrell | in July 2012.", "source_image": "hc23.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc23.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 34/7 + 2/\u221a3 = 6.011+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "hc23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "de2d2aabdc650ad2", "improvement": -0.0009964252593954726, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.1931154748950945e-06, "max_pair_overlap_depth": -3.920500316656472e-05, "metric_expression": "6.01200", "metric_symbol": "s", "metric_value": 6.011996425259396, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.011, "setup": "cirinhex", "side": 6.011996425259396, "solution_path": "data/solutions/generated/cirinhex_023_de2d2aabdc650ad2.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/de2d2aabdc650ad2.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinhex@24", "density": 0.7747882438352293, "friedman_analytical_or_proved": false, "friedman_credit": "Rob Pratt", "friedman_date": "March 2020", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc24.gif", "friedman_metric_expression": "6.117+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.117, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@24", "credit": "Rob Pratt", "date": "March 2020", "family_updated": "", "metric_expression": "6.117+", "metric_symbol": "s", "metric_value": 6.117, "reference_text": "s = 6.117+ | Found by Rob Pratt | in March 2020.", "source_image": "hc24.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc24.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 6.117+ | Found by Rob Pratt | in March 2020.", "friedman_source_image": "hc24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "61a86ae12328c1ca", "improvement": -0.003164603994216364, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4542492504119338e-06, "max_pair_overlap_depth": -2.04098607499148e-05, "metric_expression": "6.12016", "metric_symbol": "s", "metric_value": 6.120164603994216, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.117, "setup": "cirinhex", "side": 6.120164603994216, "solution_path": "data/solutions/generated/cirinhex_024_61a86ae12328c1ca.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Rob Pratt", "svg_path": "data/svg/61a86ae12328c1ca.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@1", "density": 0.8647965566225578, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.1.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "1.453+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.453, "friedman_reference": {"analytical_or_proved": true, "case": "cirinpen@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1.453+", "metric_symbol": "s", "metric_value": 1.453, "reference_text": "s = 2tan(36\u00b0) | s = 1.453+ | Trivial.", "source_image": "pent.1.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.1.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 2tan(36\u00b0) | s = 1.453+ | Trivial.", "friedman_source_image": "pent.1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "2249439da521383e", "improvement": -9.321311914245456e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.798939718275719e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.45309", "metric_symbol": "s", "metric_value": 1.4530932131191425, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.453, "setup": "cirinpen", "side": 1.4530932131191425, "solution_path": "data/solutions/generated/cirinpen_001_2249439da521383e.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2249439da521383e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@2", "density": 0.5049856179230741, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.2.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "2.689+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.689, "friedman_reference": {"analytical_or_proved": true, "case": "cirinpen@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2.689+", "metric_symbol": "s", "metric_value": 2.689, "reference_text": "s = 2tan(36\u00b0) + \u221a5 - 1 | s = 2.689+ | Trivial.", "source_image": "pent.2.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.2.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 2tan(36\u00b0) + \u221a5 - 1 | s = 2.689+ | Trivial.", "friedman_source_image": "pent.2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "042df6c460b769f5", "improvement": -0.00021758992828191381, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.904067621218203e-06, "max_pair_overlap_depth": -5.9422400474407056e-05, "metric_expression": "2.68922", "metric_symbol": "s", "metric_value": 2.689217589928282, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.689, "setup": "cirinpen", "side": 2.689217589928282, "solution_path": "data/solutions/generated/cirinpen_002_042df6c460b769f5.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/042df6c460b769f5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@3", "density": 0.6022625992350803, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.3.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "3.015+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.015, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@3", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "3.015+", "metric_symbol": "s", "metric_value": 3.015, "reference_text": "s = 3.015+ | Found by George Sicherman | in July 2012.", "source_image": "pent.3.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.3.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 3.015+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "7afe63c4f78d1c3c", "improvement": -0.0009082861357154748, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.348511423333477e-06, "max_pair_overlap_depth": -3.760319804513301e-05, "metric_expression": "3.01591", "metric_symbol": "s", "metric_value": 3.0159082861357156, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.015, "setup": "cirinpen", "side": 3.0159082861357156, "solution_path": "data/solutions/generated/cirinpen_003_7afe63c4f78d1c3c.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/7afe63c4f78d1c3c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@4", "density": 0.6648380416307991, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.4.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "3.314+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.314, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@4", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "3.314+", "metric_symbol": "s", "metric_value": 3.314, "reference_text": "s = 3.314+ | Found by George Sicherman | in July 2012.", "source_image": "pent.4.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.4.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 3.314+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "3b6f81450d56967c", "improvement": -0.0005342806983965609, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4291241632899983e-06, "max_pair_overlap_depth": -5.729132607923404e-05, "metric_expression": "3.31453", "metric_symbol": "s", "metric_value": 3.3145342806983966, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.314, "setup": "cirinpen", "side": 3.3145342806983966, "solution_path": "data/solutions/generated/cirinpen_004_3b6f81450d56967c.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/3b6f81450d56967c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@5", "density": 0.7656755676390014, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.5.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "3.453+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.453, "friedman_reference": {"analytical_or_proved": true, "case": "cirinpen@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3.453+", "metric_symbol": "s", "metric_value": 3.453, "reference_text": "s = 2tan(36\u00b0) + 2 | s = 3.453+ | Trivial.", "source_image": "pent.5.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.5.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 2tan(36\u00b0) + 2 | s = 3.453+ | Trivial.", "friedman_source_image": "pent.5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "6a788c9a1ab05f1f", "improvement": -0.00013102756244531832, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4842628671150067e-07, "max_pair_overlap_depth": -2.3617760550376232e-05, "metric_expression": "3.45313", "metric_symbol": "s", "metric_value": 3.453131027562445, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.453, "setup": "cirinpen", "side": 3.453131027562445, "solution_path": "data/solutions/generated/cirinpen_005_6a788c9a1ab05f1f.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6a788c9a1ab05f1f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@6", "density": 0.7570230558933833, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.6.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "3.804+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.804, "friedman_reference": {"analytical_or_proved": true, "case": "cirinpen@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3.804+", "metric_symbol": "s", "metric_value": 3.804, "reference_text": "s = 2tan(36\u00b0) + 4sin(36\u00b0) | s = 3.804+ | Trivial.", "source_image": "pent.6.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.6.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 2tan(36\u00b0) + 4sin(36\u00b0) | s = 3.804+ | Trivial.", "friedman_source_image": "pent.6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "a73fe466a153c6dc", "improvement": -0.0002716609440795992, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.235659036235575e-06, "max_pair_overlap_depth": -1.860019961119619e-05, "metric_expression": "3.80427", "metric_symbol": "s", "metric_value": 3.8042716609440794, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.804, "setup": "cirinpen", "side": 3.8042716609440794, "solution_path": "data/solutions/generated/cirinpen_006_a73fe466a153c6dc.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a73fe466a153c6dc.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@7", "density": 0.7122554139396908, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.7.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "4.236+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.236, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@7", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "4.236+", "metric_symbol": "s", "metric_value": 4.236, "reference_text": "s = 4.236+ | Found by George Sicherman | in July 2012.", "source_image": "pent.7.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.7.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 4.236+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "f2fb4964063d721f", "improvement": -0.0002500861768837126, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.256168874145928e-06, "max_pair_overlap_depth": -7.389596920859276e-05, "metric_expression": "4.23625", "metric_symbol": "s", "metric_value": 4.2362500861768835, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.236, "setup": "cirinpen", "side": 4.2362500861768835, "solution_path": "data/solutions/generated/cirinpen_007_f2fb4964063d721f.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/f2fb4964063d721f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@8", "density": 0.7010960627351681, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.8.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "4.564+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.564, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@8", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "4.564+", "metric_symbol": "s", "metric_value": 4.564, "reference_text": "s = 4.564+ | Found by George Sicherman | in July 2012.", "source_image": "pent.8.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.8.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 4.564+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "7550c7a4a15ba971", "improvement": -0.0006415548993858877, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.759741664666194e-07, "max_pair_overlap_depth": -2.5138721591932978e-05, "metric_expression": "4.56464", "metric_symbol": "s", "metric_value": 4.564641554899386, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.564, "setup": "cirinpen", "side": 4.564641554899386, "solution_path": "data/solutions/generated/cirinpen_008_7550c7a4a15ba971.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/7550c7a4a15ba971.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@9", "density": 0.7058543729480167, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.9.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "4.825+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.825, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@9", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "4.825+", "metric_symbol": "s", "metric_value": 4.825, "reference_text": "s = 4.825+ | Found by George Sicherman | in July 2012.", "source_image": "pent.9.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.9.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 4.825+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "44a9c06211e4a669", "improvement": -0.0001870114230673181, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.33965970525874e-06, "max_pair_overlap_depth": -6.87249875483964e-05, "metric_expression": "4.82519", "metric_symbol": "s", "metric_value": 4.8251870114230675, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.825, "setup": "cirinpen", "side": 4.8251870114230675, "solution_path": "data/solutions/generated/cirinpen_009_44a9c06211e4a669.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/44a9c06211e4a669.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@10", "density": 0.7033183837462447, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.10.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "5.095+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.095, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@10", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "5.095+", "metric_symbol": "s", "metric_value": 5.095, "reference_text": "s = 5.095+ | Found by George Sicherman | in July 2012.", "source_image": "pent.10.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.10.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 5.095+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "38c046e395ee0204", "improvement": -0.0003552139806179966, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.918153382491795e-06, "max_pair_overlap_depth": -5.2545257565039805e-05, "metric_expression": "5.09536", "metric_symbol": "s", "metric_value": 5.095355213980618, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.095, "setup": "cirinpen", "side": 5.095355213980618, "solution_path": "data/solutions/generated/cirinpen_010_38c046e395ee0204.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/38c046e395ee0204.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@11", "density": 0.7314145264425335, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.11.gif", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "5.240+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.24, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@11", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "5.240+", "metric_symbol": "s", "metric_value": 5.24, "reference_text": "s = 5.240+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.11.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.11.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 5.240+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "3c4a20ceb03b166b", "improvement": -0.0004068061718305316, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.25538255039082e-06, "max_pair_overlap_depth": -4.98370515433777e-05, "metric_expression": "5.24041", "metric_symbol": "s", "metric_value": 5.240406806171831, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.24, "setup": "cirinpen", "side": 5.240406806171831, "solution_path": "data/solutions/generated/cirinpen_011_3c4a20ceb03b166b.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/3c4a20ceb03b166b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@12", "density": 0.7492511392398852, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.12.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "5.407+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.407, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@12", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "5.407+", "metric_symbol": "s", "metric_value": 5.407, "reference_text": "s = 5.407+ | Found by George Sicherman | in July 2012.", "source_image": "pent.12.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.12.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 5.407+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "1156bc3b50d815fe", "improvement": -0.0008839986719992865, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7054584944851143e-06, "max_pair_overlap_depth": -3.0020938044383882e-05, "metric_expression": "5.40788", "metric_symbol": "s", "metric_value": 5.407883998671999, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.407, "setup": "cirinpen", "side": 5.407883998671999, "solution_path": "data/solutions/generated/cirinpen_012_1156bc3b50d815fe.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/1156bc3b50d815fe.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@13", "density": 0.7430341365754948, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.13.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "5.651+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.651, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@13", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "5.651+", "metric_symbol": "s", "metric_value": 5.651, "reference_text": "s = 5.651+ | Found by George Sicherman | in July 2012.", "source_image": "pent.13.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.13.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 5.651+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "213cdbfbbdb494b1", "improvement": -0.0012028933276413412, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.063538795673338e-06, "max_pair_overlap_depth": -3.936589445130423e-05, "metric_expression": "5.65220", "metric_symbol": "s", "metric_value": 5.652202893327641, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.651, "setup": "cirinpen", "side": 5.652202893327641, "solution_path": "data/solutions/generated/cirinpen_013_213cdbfbbdb494b1.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/213cdbfbbdb494b1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@14", "density": 0.7576628256268029, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.14.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "5.808+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.808, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@14", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "5.808+", "metric_symbol": "s", "metric_value": 5.808, "reference_text": "s = 5.808+ | Found by George Sicherman | in July 2012.", "source_image": "pent.14.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.14.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 5.808+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "44805be876ac7787", "improvement": -0.0006669796446772125, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.282572655998294e-06, "max_pair_overlap_depth": -2.37953924338008e-05, "metric_expression": "5.80867", "metric_symbol": "s", "metric_value": 5.808666979644677, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.808, "setup": "cirinpen", "side": 5.808666979644677, "solution_path": "data/solutions/generated/cirinpen_014_44805be876ac7787.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/44805be876ac7787.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@15", "density": 0.7748926471130506, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.15.gif", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "5.945+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.945, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@15", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "5.945+", "metric_symbol": "s", "metric_value": 5.945, "reference_text": "s = 5.945+ | Found by George Sicherman | in July 2012.", "source_image": "pent.15.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.15.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 5.945+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "36fcf02cb6fd9f92", "improvement": -0.0003210236521935528, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.123112133491702e-06, "max_pair_overlap_depth": -5.711461050461608e-05, "metric_expression": "5.94532", "metric_symbol": "s", "metric_value": 5.945321023652194, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.945, "setup": "cirinpen", "side": 5.945321023652194, "solution_path": "data/solutions/generated/cirinpen_015_36fcf02cb6fd9f92.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/36fcf02cb6fd9f92.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@16", "density": 0.7775664873268765, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.16.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "6.129+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.129, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@16", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "6.129+", "metric_symbol": "s", "metric_value": 6.129, "reference_text": "s = 6.129+ | Found by George Sicherman | in July 2012.", "source_image": "pent.16.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.16.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 6.129+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "39b6519b6fb7eef4", "improvement": -0.0007346433642512196, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.515317749508867e-06, "max_pair_overlap_depth": -2.965957000666819e-05, "metric_expression": "6.12973", "metric_symbol": "s", "metric_value": 6.129734643364251, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.129, "setup": "cirinpen", "side": 6.129734643364251, "solution_path": "data/solutions/generated/cirinpen_016_39b6519b6fb7eef4.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/39b6519b6fb7eef4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@17", "density": 0.7527556084558986, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.17.gif", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "6.419+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.419, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@17", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "6.419+", "metric_symbol": "s", "metric_value": 6.419, "reference_text": "s = 6.419+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.17.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.17.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 6.419+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "5e677c639e143e85", "improvement": -0.0026689184768269314, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.2234530213948176e-06, "max_pair_overlap_depth": -4.317787105012627e-05, "metric_expression": "6.42167", "metric_symbol": "s", "metric_value": 6.4216689184768265, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.419, "setup": "cirinpen", "side": 6.4216689184768265, "solution_path": "data/solutions/generated/cirinpen_017_5e677c639e143e85.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/5e677c639e143e85.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@18", "density": 0.7574722105185439, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.18.gif", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "6.586+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.586, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@18", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "6.586+", "metric_symbol": "s", "metric_value": 6.586, "reference_text": "s = 6.586+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.18.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.18.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 6.586+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "34fda06736bbaaf7", "improvement": -0.0012379322999080955, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.803338162222119e-06, "max_pair_overlap_depth": -4.5158559001112764e-05, "metric_expression": "6.58724", "metric_symbol": "s", "metric_value": 6.587237932299908, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.586, "setup": "cirinpen", "side": 6.587237932299908, "solution_path": "data/solutions/generated/cirinpen_018_34fda06736bbaaf7.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/34fda06736bbaaf7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@19", "density": 0.7551620068495595, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.19.gif", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "6.777+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.777, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@19", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "6.777+", "metric_symbol": "s", "metric_value": 6.777, "reference_text": "s = 6.777+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.19.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.19.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 6.777+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "dfaba7cef14e53f5", "improvement": -0.0010877146457399789, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.911905938958071e-06, "max_pair_overlap_depth": -5.528697090007029e-05, "metric_expression": "6.77809", "metric_symbol": "s", "metric_value": 6.77808771464574, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.777, "setup": "cirinpen", "side": 6.77808771464574, "solution_path": "data/solutions/generated/cirinpen_019_dfaba7cef14e53f5.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/dfaba7cef14e53f5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@20", "density": 0.7572204985441312, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.20.gif", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "6.944+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.944, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@20", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "6.944+", "metric_symbol": "s", "metric_value": 6.944, "reference_text": "s = 6.944+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.20.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.20.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 6.944+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "d513055d785b23f7", "improvement": -0.0007124298362715109, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4085836122521584e-06, "max_pair_overlap_depth": -2.3984136221066876e-05, "metric_expression": "6.94471", "metric_symbol": "s", "metric_value": 6.9447124298362715, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.944, "setup": "cirinpen", "side": 6.9447124298362715, "solution_path": "data/solutions/generated/cirinpen_020_d513055d785b23f7.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d513055d785b23f7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@21", "density": 0.7590915818467144, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.21.gif", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "7.107+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.107, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@21", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "7.107+", "metric_symbol": "s", "metric_value": 7.107, "reference_text": "s = 7.107+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.21.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.21.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 7.107+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "71add6eacc6c4fb9", "improvement": -0.0004368549852129533, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.952484924405695e-06, "max_pair_overlap_depth": -3.840822292655588e-05, "metric_expression": "7.10744", "metric_symbol": "s", "metric_value": 7.107436854985213, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.107, "setup": "cirinpen", "side": 7.107436854985213, "solution_path": "data/solutions/generated/cirinpen_021_71add6eacc6c4fb9.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/71add6eacc6c4fb9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@22", "density": 0.7595290345154431, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.22.gif", "friedman_metric_expression": "7.271+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.271, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@22", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "7.271+", "metric_symbol": "s", "metric_value": 7.271, "reference_text": "s = 7.271+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.22.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.22.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 7.271+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "ebdb05232e56526e", "improvement": -0.0015983061562394596, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.413438632810411e-06, "max_pair_overlap_depth": -6.856237147889743e-05, "metric_expression": "7.27260", "metric_symbol": "s", "metric_value": 7.272598306156239, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.271, "setup": "cirinpen", "side": 7.272598306156239, "solution_path": "data/solutions/generated/cirinpen_022_ebdb05232e56526e.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/ebdb05232e56526e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@23", "density": 0.7675010590620791, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.23.gif", "friedman_metric_expression": "7.397+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.397, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@23", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "7.397+", "metric_symbol": "s", "metric_value": 7.397, "reference_text": "s = 7.397+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.23.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.23.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 7.397+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "0115bd21477c01e6", "improvement": -0.0003280102549396702, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.583943801847681e-07, "max_pair_overlap_depth": -3.4243400256706025e-05, "metric_expression": "7.39733", "metric_symbol": "s", "metric_value": 7.39732801025494, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.397, "setup": "cirinpen", "side": 7.39732801025494, "solution_path": "data/solutions/generated/cirinpen_023_0115bd21477c01e6.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/0115bd21477c01e6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinpen@24", "density": 0.7719374015643087, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.24.gif", "friedman_metric_expression": "7.534+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.534, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@24", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "7.534+", "metric_symbol": "s", "metric_value": 7.534, "reference_text": "s = 7.534+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.24.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.24.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 7.534+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "9eb26a95274f1d1d", "improvement": -0.0006837609794780874, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.468927518428643e-06, "max_pair_overlap_depth": -2.3713548386705696e-05, "metric_expression": "7.53468", "metric_symbol": "s", "metric_value": 7.534683760979478, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.534, "setup": "cirinpen", "side": 7.534683760979478, "solution_path": "data/solutions/generated/cirinpen_024_9eb26a95274f1d1d.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/9eb26a95274f1d1d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@1", "density": 0.7853971453379307, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs1.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "cs1.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs1.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "cs1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "329c377d412c468f", "improvement": -1.2962348967526793e-06, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0795596527835016e-07, "max_pair_overlap_depth": 0.0, "metric_expression": "2.00000", "metric_symbol": "s", "metric_value": 2.0000012962348968, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "cirinsqu", "side": 2.0000012962348968, "solution_path": "data/solutions/generated/cirinsqu_001_329c377d412c468f.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/329c377d412c468f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@2", "density": 0.5390070210000131, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs2.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + \u221a2 = 3.414+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.414, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + \u221a2 = 3.414+", "metric_symbol": "s", "metric_value": 3.414, "reference_text": "s = 2 + \u221a2 = 3.414+ | Trivial.", "source_image": "cs2.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs2.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + \u221a2 = 3.414+ | Trivial.", "friedman_source_image": "cs2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "e3bab8445b304ebe", "improvement": -0.00022959896202801033, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1788679802625666e-06, "max_pair_overlap_depth": -1.6252235923275293e-05, "metric_expression": "3.41423", "metric_symbol": "s", "metric_value": 3.414229598962028, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.414, "setup": "cirinsqu", "side": 3.414229598962028, "solution_path": "data/solutions/generated/cirinsqu_002_e3bab8445b304ebe.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/e3bab8445b304ebe.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@3", "density": 0.6096071430698954, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs3.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + 1/\u221a2 + \u221a6/2 = 3.931+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.931, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 1/\u221a2 + \u221a6/2 = 3.931+", "metric_symbol": "s", "metric_value": 3.931, "reference_text": "s = 2 + 1/\u221a2 + \u221a6/2 = 3.931+ | Trivial.", "source_image": "cs3.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs3.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + 1/\u221a2 + \u221a6/2 = 3.931+ | Trivial.", "friedman_source_image": "cs3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "63c314af1da3741c", "improvement": -0.0009731189575252763, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.540614135605807e-06, "max_pair_overlap_depth": -7.777950709542125e-05, "metric_expression": "3.93197", "metric_symbol": "s", "metric_value": 3.9319731189575253, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.931, "setup": "cirinsqu", "side": 3.9319731189575253, "solution_path": "data/solutions/generated/cirinsqu_003_63c314af1da3741c.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/63c314af1da3741c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@4", "density": 0.7853846601471004, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs4.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial.", "source_image": "cs4.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs4.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 4 | Trivial.", "friedman_source_image": "cs4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "9917d423eeeb95cb", "improvement": -3.438618804896265e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3561316251363706e-06, "max_pair_overlap_depth": -2.9461600919766795e-05, "metric_expression": "4.00003", "metric_symbol": "s", "metric_value": 4.000034386188049, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "cirinsqu", "side": 4.000034386188049, "solution_path": "data/solutions/generated/cirinsqu_004_9917d423eeeb95cb.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9917d423eeeb95cb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@5", "density": 0.6737513675455145, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs5.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + 2\u221a2 = 4.828+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.828, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 2\u221a2 = 4.828+", "metric_symbol": "s", "metric_value": 4.828, "reference_text": "s = 2 + 2\u221a2 = 4.828+ | Trivial.", "source_image": "cs5.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs5.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + 2\u221a2 = 4.828+ | Trivial.", "friedman_source_image": "cs5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "f91bd37c36f9320b", "improvement": -0.000476351133615438, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7157366511154635e-07, "max_pair_overlap_depth": -2.294469225416762e-05, "metric_expression": "4.82848", "metric_symbol": "s", "metric_value": 4.828476351133616, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.828, "setup": "cirinsqu", "side": 4.828476351133616, "solution_path": "data/solutions/generated/cirinsqu_005_f91bd37c36f9320b.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/f91bd37c36f9320b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@6", "density": 0.6639170340529372, "friedman_analytical_or_proved": true, "friedman_credit": "Graham", "friedman_date": "1963", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs6.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + 12/\u221a13 = 5.328+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.328, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@6", "credit": "Graham", "date": "1963", "family_updated": "", "metric_expression": "2 + 12/\u221a13 = 5.328+", "metric_symbol": "s", "metric_value": 5.328, "reference_text": "s = 2 + 12/\u221a13 = 5.328+ | Proved by Graham in 1963.", "source_image": "cs6.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs6.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + 12/\u221a13 = 5.328+ | Proved by Graham in 1963.", "friedman_source_image": "cs6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "96dda57e28868925", "improvement": -0.0003611830538092775, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.928735258591388e-06, "max_pair_overlap_depth": -7.200734794432506e-05, "metric_expression": "5.32836", "metric_symbol": "s", "metric_value": 5.32836118305381, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.328, "setup": "cirinsqu", "side": 5.32836118305381, "solution_path": "data/solutions/generated/cirinsqu_006_96dda57e28868925.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Graham", "svg_path": "data/svg/96dda57e28868925.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@7", "density": 0.669268877528072, "friedman_analytical_or_proved": true, "friedman_credit": "Schaer", "friedman_date": "1964", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs7.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "4 + \u221a3 = 5.732+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.732, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@7", "credit": "Schaer", "date": "1964", "family_updated": "", "metric_expression": "4 + \u221a3 = 5.732+", "metric_symbol": "s", "metric_value": 5.732, "reference_text": "s = 4 + \u221a3 = 5.732+ | Proved by Schaer in 1964.", "source_image": "cs7.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs7.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 4 + \u221a3 = 5.732+ | Proved by Schaer in 1964.", "friedman_source_image": "cs7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "7e4d3361af467e5d", "improvement": -0.0002304452542292168, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.33238067148811e-06, "max_pair_overlap_depth": -6.633595249638802e-05, "metric_expression": "5.73223", "metric_symbol": "s", "metric_value": 5.732230445254229, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.732, "setup": "cirinsqu", "side": 5.732230445254229, "solution_path": "data/solutions/generated/cirinsqu_007_7e4d3361af467e5d.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Schaer", "svg_path": "data/svg/7e4d3361af467e5d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@8", "density": 0.7309263604724501, "friedman_analytical_or_proved": true, "friedman_credit": "Schaer/Meir", "friedman_date": "1964", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs8.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + \u221a2 + \u221a6 = 5.863+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.863, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@8", "credit": "Schaer/Meir", "date": "1964", "family_updated": "", "metric_expression": "2 + \u221a2 + \u221a6 = 5.863+", "metric_symbol": "s", "metric_value": 5.863, "reference_text": "s = 2 + \u221a2 + \u221a6 = 5.863+ | Proved by Schaer/Meir in 1964.", "source_image": "cs8.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs8.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + \u221a2 + \u221a6 = 5.863+ | Proved by Schaer/Meir in 1964.", "friedman_source_image": "cs8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "67f6750a9af94d2c", "improvement": -0.0008535799033406732, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.247320970871769e-06, "max_pair_overlap_depth": -5.202375940349668e-05, "metric_expression": "5.86385", "metric_symbol": "s", "metric_value": 5.863853579903341, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.863, "setup": "cirinsqu", "side": 5.863853579903341, "solution_path": "data/solutions/generated/cirinsqu_008_67f6750a9af94d2c.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Schaer/Meir", "svg_path": "data/svg/67f6750a9af94d2c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@9", "density": 0.7853720614609764, "friedman_analytical_or_proved": true, "friedman_credit": "Schaer", "friedman_date": "1964", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs9.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@9", "credit": "Schaer", "date": "1964", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Proved by Schaer in 1964.", "source_image": "cs9.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs9.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 6 | Proved by Schaer in 1964.", "friedman_source_image": "cs9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "a35561fa5a518c8b", "improvement": -9.970453832330861e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.26657331825875e-06, "max_pair_overlap_depth": -3.06706256441025e-05, "metric_expression": "6.00010", "metric_symbol": "s", "metric_value": 6.000099704538323, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.0, "setup": "cirinsqu", "side": 6.000099704538323, "solution_path": "data/solutions/generated/cirinsqu_009_a35561fa5a518c8b.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Schaer", "svg_path": "data/svg/a35561fa5a518c8b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@10", "density": 0.6900075339157021, "friedman_analytical_or_proved": true, "friedman_credit": "De Groot", "friedman_date": "1990", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs10.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "6.747+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.747, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@10", "credit": "De Groot", "date": "1990", "family_updated": "", "metric_expression": "6.747+", "metric_symbol": "s", "metric_value": 6.747, "reference_text": "s = 6.747+ | Proved by De Groot in 1990.", "source_image": "cs10.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs10.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 6.747+ | Proved by De Groot in 1990.", "friedman_source_image": "cs10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "522948d614943781", "improvement": -0.0005796538825402209, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6772848518975536e-06, "max_pair_overlap_depth": -4.0928067794876455e-05, "metric_expression": "6.74758", "metric_symbol": "s", "metric_value": 6.74757965388254, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.747, "setup": "cirinsqu", "side": 6.74757965388254, "solution_path": "data/solutions/generated/cirinsqu_10_522948d614943781.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "De Groot", "svg_path": "data/svg/522948d614943781.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@11", "density": 0.7006956988559802, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs11.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "7.022+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.022, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@11", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "7.022+", "metric_symbol": "s", "metric_value": 7.022, "reference_text": "s = 7.022+ | Proved by Peikert in 1991.", "source_image": "cs11.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs11.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 7.022+ | Proved by Peikert in 1991.", "friedman_source_image": "cs11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "b0ae38086de8cdf9", "improvement": -0.0007394033277021705, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.667419736983149e-06, "max_pair_overlap_depth": -2.585792898823769e-05, "metric_expression": "7.02274", "metric_symbol": "s", "metric_value": 7.022739403327702, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.022, "setup": "cirinsqu", "side": 7.022739403327702, "solution_path": "data/solutions/generated/cirinsqu_011_b0ae38086de8cdf9.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/b0ae38086de8cdf9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@12", "density": 0.7384288108172983, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs12.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + 15\u221a(2/17) = 7.144+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.144, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@12", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "2 + 15\u221a(2/17) = 7.144+", "metric_symbol": "s", "metric_value": 7.144, "reference_text": "s = 2 + 15\u221a(2/17) = 7.144+ | Proved by Peikert in 1991.", "source_image": "cs12.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs12.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + 15\u221a(2/17) = 7.144+ | Proved by Peikert in 1991.", "friedman_source_image": "cs12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "e58733a8646795ab", "improvement": -0.0011482300231175913, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.939770909988098e-06, "max_pair_overlap_depth": -5.93949466169974e-05, "metric_expression": "7.14515", "metric_symbol": "s", "metric_value": 7.145148230023118, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.144, "setup": "cirinsqu", "side": 7.145148230023118, "solution_path": "data/solutions/generated/cirinsqu_012_e58733a8646795ab.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/e58733a8646795ab.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@13", "density": 0.7332320892098975, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs13.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "7.463+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.463, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@13", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "7.463+", "metric_symbol": "s", "metric_value": 7.463, "reference_text": "s = 7.463+ | Proved by Peikert in 1991.", "source_image": "cs13.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs13.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 7.463+ | Proved by Peikert in 1991.", "friedman_source_image": "cs13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "60997371c312edb4", "improvement": -0.0002137620977080701, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.323292483705444e-06, "max_pair_overlap_depth": -3.8715255959331785e-05, "metric_expression": "7.46321", "metric_symbol": "s", "metric_value": 7.463213762097708, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.463, "setup": "cirinsqu", "side": 7.463213762097708, "solution_path": "data/solutions/generated/cirinsqu_013_60997371c312edb4.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/60997371c312edb4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@14", "density": 0.7356508336971848, "friedman_analytical_or_proved": true, "friedman_credit": "Wengerodt", "friedman_date": "1987", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs14.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "6 + \u221a3 = 7.732+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.732, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@14", "credit": "Wengerodt", "date": "1987", "family_updated": "", "metric_expression": "6 + \u221a3 = 7.732+", "metric_symbol": "s", "metric_value": 7.732, "reference_text": "s = 6 + \u221a3 = 7.732+ | Proved by Wengerodt in 1987.", "source_image": "cs14.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs14.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 6 + \u221a3 = 7.732+ | Proved by Wengerodt in 1987.", "friedman_source_image": "cs14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "2ccdf5f3d4bee77b", "improvement": -0.000200169889159163, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.480532338595026e-07, "max_pair_overlap_depth": -3.0206326999682176e-05, "metric_expression": "7.73220", "metric_symbol": "s", "metric_value": 7.732200169889159, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.732, "setup": "cirinsqu", "side": 7.732200169889159, "solution_path": "data/solutions/generated/cirinsqu_014_2ccdf5f3d4bee77b.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Wengerodt", "svg_path": "data/svg/2ccdf5f3d4bee77b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@15", "density": 0.7620259985870776, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs15.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "4 + \u221a2 + \u221a6 = 7.863+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.863, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@15", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "4 + \u221a2 + \u221a6 = 7.863+", "metric_symbol": "s", "metric_value": 7.863, "reference_text": "s = 4 + \u221a2 + \u221a6 = 7.863+ | Proved by Peikert in 1991.", "source_image": "cs15.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs15.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 4 + \u221a2 + \u221a6 = 7.863+ | Proved by Peikert in 1991.", "friedman_source_image": "cs15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "edc16b42d64e4d82", "improvement": -0.0008581593284961997, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.610873706565542e-06, "max_pair_overlap_depth": -3.564256081922679e-05, "metric_expression": "7.86386", "metric_symbol": "s", "metric_value": 7.863858159328497, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.863, "setup": "cirinsqu", "side": 7.863858159328497, "solution_path": "data/solutions/generated/cirinsqu_015_edc16b42d64e4d82.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/edc16b42d64e4d82.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@16", "density": 0.7853654831135446, "friedman_analytical_or_proved": true, "friedman_credit": "Wengerodt", "friedman_date": "1983", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs16.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "8", "friedman_metric_symbol": "s", "friedman_metric_value": 8.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@16", "credit": "Wengerodt", "date": "1983", "family_updated": "", "metric_expression": "8", "metric_symbol": "s", "metric_value": 8.0, "reference_text": "s = 8 | Proved by Wengerodt in 1983.", "source_image": "cs16.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs16.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 8 | Proved by Wengerodt in 1983.", "friedman_source_image": "cs16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "6ef91613d0db44e8", "improvement": -0.00016644451351055523, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.383802090528576e-06, "max_pair_overlap_depth": -3.451016213729119e-05, "metric_expression": "8.00017", "metric_symbol": "s", "metric_value": 8.00016644451351, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.0, "setup": "cirinsqu", "side": 8.00016644451351, "solution_path": "data/solutions/generated/cirinsqu_016_6ef91613d0db44e8.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Wengerodt", "svg_path": "data/svg/6ef91613d0db44e8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@17", "density": 0.7333955245290497, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs17.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "8.532+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.532, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@17", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "8.532+", "metric_symbol": "s", "metric_value": 8.532, "reference_text": "s = 8.532+ | Proved by Peikert in 1991.", "source_image": "cs17.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs17.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 8.532+ | Proved by Peikert in 1991.", "friedman_source_image": "cs17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "9211255584acb7a4", "improvement": -0.001560450998908891, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6768059695481128e-06, "max_pair_overlap_depth": -3.5588125128604986e-05, "metric_expression": "8.53356", "metric_symbol": "s", "metric_value": 8.533560450998909, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.532, "setup": "cirinsqu", "side": 8.533560450998909, "solution_path": "data/solutions/generated/cirinsqu_017_9211255584acb7a4.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/9211255584acb7a4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@18", "density": 0.7546080214690231, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs18.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + 24/\u221a13 = 8.656+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.656, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@18", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "2 + 24/\u221a13 = 8.656+", "metric_symbol": "s", "metric_value": 8.656, "reference_text": "s = 2 + 24/\u221a13 = 8.656+ | Proved by Peikert in 1991.", "source_image": "cs18.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs18.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + 24/\u221a13 = 8.656+ | Proved by Peikert in 1991.", "friedman_source_image": "cs18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "4872462eca76de82", "improvement": -0.0006623865813519814, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.585640835636951e-06, "max_pair_overlap_depth": -5.7251545610981225e-05, "metric_expression": "8.65666", "metric_symbol": "s", "metric_value": 8.656662386581353, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.656, "setup": "cirinsqu", "side": 8.656662386581353, "solution_path": "data/solutions/generated/cirinsqu_018_4872462eca76de82.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/4872462eca76de82.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@19", "density": 0.752224113091453, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs19.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "8.907+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.907, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@19", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "8.907+", "metric_symbol": "s", "metric_value": 8.907, "reference_text": "s = 8.907+ | Proved by Peikert in 1991.", "source_image": "cs19.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs19.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 8.907+ | Proved by Peikert in 1991.", "friedman_source_image": "cs19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "72fa8cf8e074fd32", "improvement": -0.0009569875134012307, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.019564082723974e-06, "max_pair_overlap_depth": -6.564483433546542e-05, "metric_expression": "8.90796", "metric_symbol": "s", "metric_value": 8.907956987513401, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.907, "setup": "cirinsqu", "side": 8.907956987513401, "solution_path": "data/solutions/generated/cirinsqu_019_72fa8cf8e074fd32.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/72fa8cf8e074fd32.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@20", "density": 0.7794478091956536, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs20.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "130/17 + 16/17\u221a2 = 8.978+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.978, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@20", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "130/17 + 16/17\u221a2 = 8.978+", "metric_symbol": "s", "metric_value": 8.978, "reference_text": "s = 130/17 + 16/17\u221a2 = 8.978+ | Proved by Peikert in 1991.", "source_image": "cs20.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs20.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 130/17 + 16/17\u221a2 = 8.978+ | Proved by Peikert in 1991.", "friedman_source_image": "cs20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "ed16784370181dab", "improvement": -0.0003475703211961445, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3975782322539487e-06, "max_pair_overlap_depth": -5.140241392753708e-05, "metric_expression": "8.97835", "metric_symbol": "s", "metric_value": 8.978347570321196, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.978, "setup": "cirinsqu", "side": 8.978347570321196, "solution_path": "data/solutions/generated/cirinsqu_020_ed16784370181dab.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/ed16784370181dab.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@21", "density": 0.7531827030031223, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman Packing Center", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs21.gif", "friedman_metric_expression": "9.358+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.358, "friedman_reference": {"analytical_or_proved": false, "case": "cirinsqu@21", "credit": "Erich Friedman Packing Center", "date": "", "family_updated": "", "metric_expression": "9.358+", "metric_symbol": "s", "metric_value": 9.358, "reference_text": "s = 9.358+", "source_image": "cs21.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs21.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 9.358+", "friedman_source_image": "cs21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "65afc927a4d3cb10", "improvement": -0.0011070505635846217, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.857662265602471e-06, "max_pair_overlap_depth": -3.291297220409106e-05, "metric_expression": "9.35911", "metric_symbol": "s", "metric_value": 9.359107050563585, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.358, "setup": "cirinsqu", "side": 9.359107050563585, "solution_path": "data/solutions/generated/cirinsqu_021_65afc927a4d3cb10.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman Packing Center", "svg_path": "data/svg/65afc927a4d3cb10.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@22", "density": 0.7716499915329439, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman Packing Center", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs22.gif", "friedman_metric_expression": "9.463+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.463, "friedman_reference": {"analytical_or_proved": false, "case": "cirinsqu@22", "credit": "Erich Friedman Packing Center", "date": "", "family_updated": "", "metric_expression": "9.463+", "metric_symbol": "s", "metric_value": 9.463, "reference_text": "s = 9.463+", "source_image": "cs22.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs22.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 9.463+", "friedman_source_image": "cs22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "27d48d864449f174", "improvement": -0.0010297949122062278, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1647620365984324e-06, "max_pair_overlap_depth": -2.796224374757017e-05, "metric_expression": "9.46403", "metric_symbol": "s", "metric_value": 9.464029794912205, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.463, "setup": "cirinsqu", "side": 9.464029794912205, "solution_path": "data/solutions/generated/cirinsqu_022_27d48d864449f174.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman Packing Center", "svg_path": "data/svg/27d48d864449f174.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@23", "density": 0.7628623175184639, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman Packing Center", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs23.gif", "friedman_metric_expression": "2 + 2\u221a2 + 2\u221a6 = 9.727+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.727, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@23", "credit": "Erich Friedman Packing Center", "date": "", "family_updated": "", "metric_expression": "2 + 2\u221a2 + 2\u221a6 = 9.727+", "metric_symbol": "s", "metric_value": 9.727, "reference_text": "s = 2 + 2\u221a2 + 2\u221a6 = 9.727+", "source_image": "cs23.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs23.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + 2\u221a2 + 2\u221a6 = 9.727+", "friedman_source_image": "cs23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "cfa99060ee2203a1", "improvement": -0.005306391434194424, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8298221013957061e-06, "max_pair_overlap_depth": -2.6081243190478887e-05, "metric_expression": "9.73231", "metric_symbol": "s", "metric_value": 9.732306391434195, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.727, "setup": "cirinsqu", "side": 9.732306391434195, "solution_path": "data/solutions/generated/cirinsqu_023_cfa99060ee2203a1.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman Packing Center", "svg_path": "data/svg/cfa99060ee2203a1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirinsqu@24", "density": 0.7749295071503051, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman Packing Center", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs24.gif", "friedman_metric_expression": "6 + \u221a2 + \u221a6 = 9.863+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.863, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@24", "credit": "Erich Friedman Packing Center", "date": "", "family_updated": "", "metric_expression": "6 + \u221a2 + \u221a6 = 9.863+", "metric_symbol": "s", "metric_value": 9.863, "reference_text": "s = 6 + \u221a2 + \u221a6 = 9.863+", "source_image": "cs24.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs24.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 6 + \u221a2 + \u221a6 = 9.863+", "friedman_source_image": "cs24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "e2eca298edbdb88c", "improvement": -0.0009181131343805049, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.506144878683841e-06, "max_pair_overlap_depth": -4.0021533562306644e-05, "metric_expression": "9.86392", "metric_symbol": "s", "metric_value": 9.86391811313438, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.863, "setup": "cirinsqu", "side": 9.86391811313438, "solution_path": "data/solutions/generated/cirinsqu_024_e2eca298edbdb88c.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman Packing Center", "svg_path": "data/svg/e2eca298edbdb88c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirintri@1", "density": 0.6045962978262086, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct1.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "2\u221a3 = 3.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2\u221a3 = 3.464+", "metric_symbol": "s", "metric_value": 3.464, "reference_text": "s = 2\u221a3 = 3.464+ | Trivial.", "source_image": "ct1.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct1.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 2\u221a3 = 3.464+ | Trivial.", "friedman_source_image": "ct1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "10a7a380c7f5376a", "improvement": -0.00011161401610815247, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7091087705090047e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "3.46411", "metric_symbol": "s", "metric_value": 3.464111614016108, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.464, "setup": "cirintri", "side": 3.464111614016108, "solution_path": "data/solutions/generated/cirintri_001_10a7a380c7f5376a.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/10a7a380c7f5376a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirintri@2", "density": 0.48598475821399, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct2.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "2 + 2\u221a3 = 5.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 2\u221a3 = 5.464+", "metric_symbol": "s", "metric_value": 5.464, "reference_text": "s = 2 + 2\u221a3 = 5.464+ | Trivial.", "source_image": "ct2.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct2.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 2 + 2\u221a3 = 5.464+ | Trivial.", "friedman_source_image": "ct2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "f5cde1b8295da3ad", "improvement": -0.00022144929157175142, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7587277048611867e-06, "max_pair_overlap_depth": -7.068795172848752e-05, "metric_expression": "5.46422", "metric_symbol": "s", "metric_value": 5.464221449291572, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.464, "setup": "cirintri", "side": 5.464221449291572, "solution_path": "data/solutions/generated/cirintri_002_f5cde1b8295da3ad.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/f5cde1b8295da3ad.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirintri@3", "density": 0.7289907317557431, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct3.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "2 + 2\u221a3 = 5.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 2\u221a3 = 5.464+", "metric_symbol": "s", "metric_value": 5.464, "reference_text": "s = 2 + 2\u221a3 = 5.464+ | Trivial.", "source_image": "ct3.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct3.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 2 + 2\u221a3 = 5.464+ | Trivial.", "friedman_source_image": "ct3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "fcee9d6e8b7c5ce0", "improvement": -0.000170499845251193, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0755666179917966e-07, "max_pair_overlap_depth": -5.0474041760395494e-05, "metric_expression": "5.46417", "metric_symbol": "s", "metric_value": 5.464170499845252, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.464, "setup": "cirintri", "side": 5.464170499845252, "solution_path": "data/solutions/generated/cirintri_003_fcee9d6e8b7c5ce0.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fcee9d6e8b7c5ce0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirintri@4", "density": 0.6045810458812538, "friedman_analytical_or_proved": true, "friedman_credit": "Milano", "friedman_date": "1987", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct4.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "4\u221a3 = 6.928+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.928, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@4", "credit": "Milano", "date": "1987", "family_updated": "", "metric_expression": "4\u221a3 = 6.928+", "metric_symbol": "s", "metric_value": 6.928, "reference_text": "s = 4\u221a3 = 6.928+ | Proved by Milano in 1987.", "source_image": "ct4.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct4.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 4\u221a3 = 6.928+ | Proved by Milano in 1987.", "friedman_source_image": "ct4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "9e0c39beb3cc8aa7", "improvement": -0.0003106176498706148, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7330840119143005e-06, "max_pair_overlap_depth": -4.49553106092182e-05, "metric_expression": "6.92831", "metric_symbol": "s", "metric_value": 6.9283106176498706, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.928, "setup": "cirintri", "side": 6.9283106176498706, "solution_path": "data/solutions/generated/cirintri_004_9e0c39beb3cc8aa7.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Milano", "svg_path": "data/svg/9e0c39beb3cc8aa7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirintri@5", "density": 0.6510985676665063, "friedman_analytical_or_proved": true, "friedman_credit": "Milano", "friedman_date": "1987", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct5.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "4 + 2\u221a3 = 7.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@5", "credit": "Milano", "date": "1987", "family_updated": "", "metric_expression": "4 + 2\u221a3 = 7.464+", "metric_symbol": "s", "metric_value": 7.464, "reference_text": "s = 4 + 2\u221a3 = 7.464+ | Proved by Milano in 1987.", "source_image": "ct5.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct5.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 4 + 2\u221a3 = 7.464+ | Proved by Milano in 1987.", "friedman_source_image": "ct5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "ad32c3b8e4c714b5", "improvement": -0.00025126705611278055, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.12146637612598e-06, "max_pair_overlap_depth": -3.650753517803196e-05, "metric_expression": "7.46425", "metric_symbol": "s", "metric_value": 7.464251267056113, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.464, "setup": "cirintri", "side": 7.464251267056113, "solution_path": "data/solutions/generated/cirintri_005_ad32c3b8e4c714b5.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Milano", "svg_path": "data/svg/ad32c3b8e4c714b5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirintri@6", "density": 0.7813295579764934, "friedman_analytical_or_proved": true, "friedman_credit": "Oler/Groemer", "friedman_date": "1961", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct6.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "4 + 2\u221a3 = 7.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@6", "credit": "Oler/Groemer", "date": "1961", "family_updated": "", "metric_expression": "4 + 2\u221a3 = 7.464+", "metric_symbol": "s", "metric_value": 7.464, "reference_text": "s = 4 + 2\u221a3 = 7.464+ | Proved by Oler/Groemer in 1961.", "source_image": "ct6.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct6.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 4 + 2\u221a3 = 7.464+ | Proved by Oler/Groemer in 1961.", "friedman_source_image": "ct6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "09bddcb97a85d368", "improvement": -0.00019740182247307558, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6105130511423482e-06, "max_pair_overlap_depth": -3.3410124574029254e-05, "metric_expression": "7.46420", "metric_symbol": "s", "metric_value": 7.4641974018224735, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.464, "setup": "cirintri", "side": 7.4641974018224735, "solution_path": "data/solutions/generated/cirintri_006_09bddcb97a85d368.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Oler/Groemer", "svg_path": "data/svg/09bddcb97a85d368.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirintri@7", "density": 0.6370913423934993, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1993", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct7.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "2 + 4\u221a3 = 8.928+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.928, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@7", "credit": "Melissen", "date": "1993", "family_updated": "", "metric_expression": "2 + 4\u221a3 = 8.928+", "metric_symbol": "s", "metric_value": 8.928, "reference_text": "s = 2 + 4\u221a3 = 8.928+ | Proved by Melissen in 1993.", "source_image": "ct7.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct7.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 2 + 4\u221a3 = 8.928+ | Proved by Melissen in 1993.", "friedman_source_image": "ct7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "1d479488f6f25560", "improvement": -0.0003824812590771387, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.6242968528307244e-06, "max_pair_overlap_depth": -4.768068922000879e-05, "metric_expression": "8.92838", "metric_symbol": "s", "metric_value": 8.928382481259078, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.928, "setup": "cirintri", "side": 8.928382481259078, "solution_path": "data/solutions/generated/cirintri_007_1d479488f6f25560.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Melissen", "svg_path": "data/svg/1d479488f6f25560.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirintri@8", "density": 0.6718185539703104, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1993", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct8.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "2 + 2\u221a3 + 2\u221a33/3 = 9.293+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.293, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@8", "credit": "Melissen", "date": "1993", "family_updated": "", "metric_expression": "2 + 2\u221a3 + 2\u221a33/3 = 9.293+", "metric_symbol": "s", "metric_value": 9.293, "reference_text": "s = 2 + 2\u221a3 + 2\u221a33/3 = 9.293+ | Proved by Melissen in 1993.", "source_image": "ct8.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct8.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 2 + 2\u221a3 + 2\u221a33/3 = 9.293+ | Proved by Melissen in 1993.", "friedman_source_image": "ct8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "ad6e4ca149e6364f", "improvement": -0.0018763116025901638, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.090428418316975e-06, "max_pair_overlap_depth": -4.461887232398709e-05, "metric_expression": "9.29488", "metric_symbol": "s", "metric_value": 9.29487631160259, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.293, "setup": "cirintri", "side": 9.29487631160259, "solution_path": "data/solutions/generated/cirintri_008_ad6e4ca149e6364f.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Melissen", "svg_path": "data/svg/ad6e4ca149e6364f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirintri@9", "density": 0.728985395977573, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1993", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct9.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "6 + 2\u221a3 = 9.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@9", "credit": "Melissen", "date": "1993", "family_updated": "", "metric_expression": "6 + 2\u221a3 = 9.464+", "metric_symbol": "s", "metric_value": 9.464, "reference_text": "s = 6 + 2\u221a3 = 9.464+ | Proved by Melissen in 1993.", "source_image": "ct9.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct9.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 6 + 2\u221a3 = 9.464+ | Proved by Melissen in 1993.", "friedman_source_image": "ct9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "108d2156f1819ef5", "improvement": -0.0002555633725904727, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.43897889048894e-06, "max_pair_overlap_depth": -2.63301951002326e-05, "metric_expression": "9.46426", "metric_symbol": "s", "metric_value": 9.46425556337259, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.464, "setup": "cirintri", "side": 9.46425556337259, "solution_path": "data/solutions/generated/cirintri_009_108d2156f1819ef5.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Melissen", "svg_path": "data/svg/108d2156f1819ef5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirintri@10", "density": 0.8099824382101458, "friedman_analytical_or_proved": true, "friedman_credit": "Oler/Groemer", "friedman_date": "1961", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct10.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "6 + 2\u221a3 = 9.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@10", "credit": "Oler/Groemer", "date": "1961", "family_updated": "", "metric_expression": "6 + 2\u221a3 = 9.464+", "metric_symbol": "s", "metric_value": 9.464, "reference_text": "s = 6 + 2\u221a3 = 9.464+ | Proved by Oler/Groemer in 1961.", "source_image": "ct10.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct10.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 6 + 2\u221a3 = 9.464+ | Proved by Oler/Groemer in 1961.", "friedman_source_image": "ct10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "abf88e494059d012", "improvement": -0.0002633633602293628, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.885918628489861e-06, "max_pair_overlap_depth": -4.501428401626484e-05, "metric_expression": "9.46426", "metric_symbol": "s", "metric_value": 9.46426336336023, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.464, "setup": "cirintri", "side": 9.46426336336023, "solution_path": "data/solutions/generated/cirintri_010_abf88e494059d012.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Oler/Groemer", "svg_path": "data/svg/abf88e494059d012.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirintri@11", "density": 0.693141460882906, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1993", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct11.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "4 + 2\u221a3 + 4\u221a6/3 = 10.730+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.73, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@11", "credit": "Melissen", "date": "1993", "family_updated": "", "metric_expression": "4 + 2\u221a3 + 4\u221a6/3 = 10.730+", "metric_symbol": "s", "metric_value": 10.73, "reference_text": "s = 4 + 2\u221a3 + 4\u221a6/3 = 10.730+ | Proved by Melissen in 1993.", "source_image": "ct11.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct11.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 4 + 2\u221a3 + 4\u221a6/3 = 10.730+ | Proved by Melissen in 1993.", "friedman_source_image": "ct11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "64f835afcfaa2245", "improvement": -0.00025448918442982915, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7228747402064926e-06, "max_pair_overlap_depth": -3.113241300978231e-05, "metric_expression": "10.73025", "metric_symbol": "s", "metric_value": 10.73025448918443, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.73, "setup": "cirintri", "side": 10.73025448918443, "solution_path": "data/solutions/generated/cirintri_011_64f835afcfaa2245.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Melissen", "svg_path": "data/svg/64f835afcfaa2245.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirintri@12", "density": 0.7289832647467578, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1994", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct12.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "4 + 4\u221a3 = 10.928+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.928, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@12", "credit": "Melissen", "date": "1994", "family_updated": "", "metric_expression": "4 + 4\u221a3 = 10.928+", "metric_symbol": "s", "metric_value": 10.928, "reference_text": "s = 4 + 4\u221a3 = 10.928+ | Proved by Melissen in 1994.", "source_image": "ct12.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct12.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 4 + 4\u221a3 = 10.928+ | Proved by Melissen in 1994.", "friedman_source_image": "ct12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "c94cd40d17cad41e", "improvement": -0.00039696929599308817, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.790000236816084e-06, "max_pair_overlap_depth": -3.824380327710486e-05, "metric_expression": "10.92840", "metric_symbol": "s", "metric_value": 10.928396969295994, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.928, "setup": "cirintri", "side": 10.928396969295994, "solution_path": "data/solutions/generated/cirintri_012_c94cd40d17cad41e.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Melissen", "svg_path": "data/svg/c94cd40d17cad41e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirintri@13", "density": 0.7248761583522253, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1993", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct13.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "4 + 2\u221a6/3 + 10\u221a3/3 = 11.406+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.406, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@13", "credit": "Melissen", "date": "1993", "family_updated": "", "metric_expression": "4 + 2\u221a6/3 + 10\u221a3/3 = 11.406+", "metric_symbol": "s", "metric_value": 11.406, "reference_text": "s = 4 + 2\u221a6/3 + 10\u221a3/3 = 11.406+ | Found by Melissen in 1993.", "source_image": "ct13.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct13.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 4 + 2\u221a6/3 + 10\u221a3/3 = 11.406+ | Found by Melissen in 1993.", "friedman_source_image": "ct13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "4ca74ad9707b79ad", "improvement": -0.0008146992920448071, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.27136224545599e-06, "max_pair_overlap_depth": -4.371332125208127e-05, "metric_expression": "11.40681", "metric_symbol": "s", "metric_value": 11.406814699292045, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 11.406, "setup": "cirintri", "side": 11.406814699292045, "solution_path": "data/solutions/generated/cirintri_013_4ca74ad9707b79ad.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Melissen", "svg_path": "data/svg/4ca74ad9707b79ad.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirintri@14", "density": 0.7728234005113342, "friedman_analytical_or_proved": true, "friedman_credit": "Erd\u0151s/Oler", "friedman_date": "1961", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct14.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "8 + 2\u221a3 = 11.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@14", "credit": "Erd\u0151s/Oler", "date": "1961", "family_updated": "", "metric_expression": "8 + 2\u221a3 = 11.464+", "metric_symbol": "s", "metric_value": 11.464, "reference_text": "s = 8 + 2\u221a3 = 11.464+ | Found by Erd\u0151s/Oler in 1961.", "source_image": "ct14.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct14.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 8 + 2\u221a3 = 11.464+ | Found by Erd\u0151s/Oler in 1961.", "friedman_source_image": "ct14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "67743153538612b8", "improvement": -0.00032539636501560665, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.0042126467086945e-06, "max_pair_overlap_depth": -3.7763457817252544e-05, "metric_expression": "11.46433", "metric_symbol": "s", "metric_value": 11.464325396365016, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 11.464, "setup": "cirintri", "side": 11.464325396365016, "solution_path": "data/solutions/generated/cirintri_014_67743153538612b8.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erd\u0151s/Oler", "svg_path": "data/svg/67743153538612b8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "cirintri@15", "density": 0.82802371463932, "friedman_analytical_or_proved": true, "friedman_credit": "Erd\u0151s/Groemer", "friedman_date": "1961", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct15.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "8 + 2\u221a3 = 11.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@15", "credit": "Erd\u0151s/Groemer", "date": "1961", "family_updated": "", "metric_expression": "8 + 2\u221a3 = 11.464+", "metric_symbol": "s", "metric_value": 11.464, "reference_text": "s = 8 + 2\u221a3 = 11.464+ | Proved by Erd\u0151s/Groemer in 1961.", "source_image": "ct15.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct15.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 8 + 2\u221a3 = 11.464+ | Proved by Erd\u0151s/Groemer in 1961.", "friedman_source_image": "ct15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "2a26f9d337f635e0", "improvement": -0.00033479280357262553, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.377565219179871e-06, "max_pair_overlap_depth": -4.0761680023315705e-05, "metric_expression": "11.46433", "metric_symbol": "s", "metric_value": 11.464334792803573, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 11.464, "setup": "cirintri", "side": 11.464334792803573, "solution_path": "data/solutions/generated/cirintri_015_2a26f9d337f635e0.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erd\u0151s/Groemer", "svg_path": "data/svg/2a26f9d337f635e0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@1", "density": 0.9999892980292665, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/1.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "1. | | s = 1 | Trivial.", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/1.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "1. | | s = 1 | Trivial.", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "6b50d5741feee5dc", "improvement": -5.351028316669115e-06, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6755141583345576e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00001", "metric_symbol": "s", "metric_value": 1.0000053510283167, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0, "setup": "domindom", "side": 2.0000107020566333, "solution_path": "data/solutions/generated/domindom_001_6b50d5741feee5dc.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6b50d5741feee5dc.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@2", "density": 0.5146978614404703, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/2.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "(2+\u221a2)/\u221a3 = 1.971+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.971, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@2", "credit": "Erich Friedman", "date": "April 2026", "family_updated": "", "metric_expression": "(2+\u221a2)/\u221a3 = 1.971+", "metric_symbol": "s", "metric_value": 1.971, "reference_text": "2. | | s = (2+\u221a2)/\u221a3 = 1.971+ | Found by Erich Friedman in April 2026.", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/2.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "2. | | s = (2+\u221a2)/\u221a3 = 1.971+ | Found by Erich Friedman in April 2026.", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "e1760976ec64bd64", "improvement": -0.000236880650400062, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9155926735958673e-06, "max_pair_overlap_depth": -6.0125633082330765e-06, "metric_expression": "1.97124", "metric_symbol": "s", "metric_value": 1.9712368806504001, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.971, "setup": "domindom", "side": 3.9424737613008003, "solution_path": "data/solutions/generated/domindom_002_e1760976ec64bd64.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/e1760976ec64bd64.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@3", "density": 0.7499858951262421, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/4.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "3.-4. | | s = 2 | Trivial.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/4.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "3.-4. | | s = 2 | Trivial.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "5194ab3dddc343bb", "improvement": -1.8806763611323163e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3349574317089719e-06, "max_pair_overlap_depth": -6.50598393479697e-06, "metric_expression": "2.00002", "metric_symbol": "s", "metric_value": 2.0000188067636113, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "domindom", "side": 4.000037613527223, "solution_path": "data/solutions/generated/domindom_003_5194ab3dddc343bb.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/5194ab3dddc343bb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@4", "density": 0.9999632147512098, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/4.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "3.-4. | | s = 2 | Trivial.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/4.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "3.-4. | | s = 2 | Trivial.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "14a703d2c83d5b43", "improvement": -3.6786263687194065e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.672168442576563e-06, "max_pair_overlap_depth": -2.5740489017265176e-05, "metric_expression": "2.00004", "metric_symbol": "s", "metric_value": 2.000036786263687, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "domindom", "side": 4.000073572527374, "solution_path": "data/solutions/generated/domindom_004_14a703d2c83d5b43.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/14a703d2c83d5b43.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@5", "density": 0.7999239260919105, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/5.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "5/2 = 2.5", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5/2 = 2.5", "metric_symbol": "s", "metric_value": 2.5, "reference_text": "5. | | s = 5/2 = 2.5 | Trivial.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/5.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "5. | | s = 5/2 = 2.5 | Trivial.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "af0995e19f0dc8ce", "improvement": -0.00011887395946308388, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.6075584103633105e-07, "max_pair_overlap_depth": -2.297221660940707e-05, "metric_expression": "2.50012", "metric_symbol": "s", "metric_value": 2.500118873959463, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.5, "setup": "domindom", "side": 5.000237747918926, "solution_path": "data/solutions/generated/domindom_005_af0995e19f0dc8ce.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/af0995e19f0dc8ce.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@6", "density": 0.7051360525077088, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/6.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "2.916+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.916, "friedman_reference": {"analytical_or_proved": false, "case": "domindom@6", "credit": "Erich Friedman", "date": "April 2026", "family_updated": "", "metric_expression": "2.916+", "metric_symbol": "s", "metric_value": 2.916, "reference_text": "6. | | s = 2.916+ | Found by Erich Friedman in April 2026.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/6.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "6. | | s = 2.916+ | Found by Erich Friedman in April 2026.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "0f3d5defa1e042a8", "improvement": -0.001018377106842383, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.01302900812739e-06, "max_pair_overlap_depth": -7.57681649177222e-06, "metric_expression": "2.91702", "metric_symbol": "s", "metric_value": 2.9170183771068423, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.916, "setup": "domindom", "side": 2.9170183771068423, "solution_path": "data/solutions/generated/domindom_006_0f3d5defa1e042a8.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/0f3d5defa1e042a8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@7", "density": 0.777726224180711, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "7.-9. | | s = 3 | Trivial.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "7.-9. | | s = 3 | Trivial.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "7b5335d8b8c4a5db", "improvement": -9.942973726095872e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3331344280185253e-06, "max_pair_overlap_depth": -2.1961883454491016e-05, "metric_expression": "3.00010", "metric_symbol": "s", "metric_value": 3.000099429737261, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "domindom", "side": 6.000198859474522, "solution_path": "data/solutions/generated/domindom_007_7b5335d8b8c4a5db.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7b5335d8b8c4a5db.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@8", "density": 0.8888442639489805, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@8", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "7.-9. | | s = 3 | Trivial.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "7.-9. | | s = 3 | Trivial.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "953d10b3e926402a", "improvement": -7.530742160444248e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.80012014980602e-06, "max_pair_overlap_depth": -1.8965164302886528e-05, "metric_expression": "3.00008", "metric_symbol": "s", "metric_value": 3.0000753074216044, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "domindom", "side": 6.000150614843209, "solution_path": "data/solutions/generated/domindom_008_953d10b3e926402a.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/953d10b3e926402a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@9", "density": 0.9999668695349745, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@9", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "7.-9. | | s = 3 | Trivial.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "7.-9. | | s = 3 | Trivial.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "43b9630822b40990", "improvement": -4.9696932403531235e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7542337882889854e-07, "max_pair_overlap_depth": -3.0527855267026283e-06, "metric_expression": "3.00005", "metric_symbol": "s", "metric_value": 3.0000496969324035, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "domindom", "side": 6.000099393864807, "solution_path": "data/solutions/generated/domindom_009_43b9630822b40990.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/43b9630822b40990.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@10", "density": 0.8162099861183555, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/10.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "7/2 = 3.5", "friedman_metric_symbol": "s", "friedman_metric_value": 3.5, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@10", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7/2 = 3.5", "metric_symbol": "s", "metric_value": 3.5, "reference_text": "10. | | s = 7/2 = 3.5 | Trivial.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/10.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "10. | | s = 7/2 = 3.5 | Trivial.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "d7508c45c3f18a7c", "improvement": -0.0002498690138819093, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5739397918324016e-07, "max_pair_overlap_depth": -9.291197802940854e-05, "metric_expression": "3.50025", "metric_symbol": "s", "metric_value": 3.500249869013882, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.5, "setup": "domindom", "side": 7.000499738027764, "solution_path": "data/solutions/generated/domindom_010_d7508c45c3f18a7c.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d7508c45c3f18a7c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@11", "density": 0.7611589259525644, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/11.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "(5+\u221a41)/3 = 3.801+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.801, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@11", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "(5+\u221a41)/3 = 3.801+", "metric_symbol": "s", "metric_value": 3.801, "reference_text": "11. | | s = (5+\u221a41)/3 = 3.801+ | Found by Maurizio Morandi in April 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/11.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "11. | | s = (5+\u221a41)/3 = 3.801+ | Found by Maurizio Morandi in April 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "0fa9d3de612fbf03", "improvement": -0.0005321723096423092, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0757791549262947e-06, "max_pair_overlap_depth": -0.0001019023981195799, "metric_expression": "3.80153", "metric_symbol": "s", "metric_value": 3.8015321723096425, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.801, "setup": "domindom", "side": 3.8015321723096425, "solution_path": "data/solutions/generated/domindom_011_0fa9d3de612fbf03.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/0fa9d3de612fbf03.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@12", "density": 0.7822558234911755, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/12.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "5/2+\u221a2 = 3.914+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.914, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@12", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "5/2+\u221a2 = 3.914+", "metric_symbol": "s", "metric_value": 3.914, "reference_text": "12. | | s = 5/2+\u221a2 = 3.914+ | Found by Maurizio Morandi in April 2026.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/12.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "12. | | s = 5/2+\u221a2 = 3.914+ | Found by Maurizio Morandi in April 2026.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "3a75d33bd56228b0", "improvement": -0.002663137487304734, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.0673284962267076e-07, "max_pair_overlap_depth": -1.4217486933421064e-05, "metric_expression": "3.91666", "metric_symbol": "s", "metric_value": 3.916663137487305, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.914, "setup": "domindom", "side": 3.916663137487305, "solution_path": "data/solutions/generated/domindom_012_3a75d33bd56228b0.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T03:07:58+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/3a75d33bd56228b0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@14", "density": 0.8749484114386541, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@14", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "14.-16 | | s = 4 | Trivial.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "14.-16 | | s = 4 | Trivial.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "8be6cbfdbea40909", "improvement": -0.00011792192603543583, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.966859551203129e-06, "max_pair_overlap_depth": -1.688780940467538e-05, "metric_expression": "4.00012", "metric_symbol": "s", "metric_value": 4.000117921926035, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "domindom", "side": 8.00023584385207, "solution_path": "data/solutions/generated/domindom_014_8be6cbfdbea40909.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/8be6cbfdbea40909.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@15", "density": 0.9374471956142247, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@15", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "14.-16 | | s = 4 | Trivial.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "14.-16 | | s = 4 | Trivial.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "d2b8e4757dde349c", "improvement": -0.00011265411524785662, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.415030228686277e-06, "max_pair_overlap_depth": -1.0456826571081379e-05, "metric_expression": "4.00011", "metric_symbol": "s", "metric_value": 4.000112654115248, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "domindom", "side": 8.000225308230496, "solution_path": "data/solutions/generated/domindom_015_d2b8e4757dde349c.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d2b8e4757dde349c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@16", "density": 0.9999595560012681, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@16", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "14.-16 | | s = 4 | Trivial.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "14.-16 | | s = 4 | Trivial.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "5102d8436800a7ff", "improvement": -8.08904511222508e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.23688533190969e-06, "max_pair_overlap_depth": -4.621969549978289e-06, "metric_expression": "4.00008", "metric_symbol": "s", "metric_value": 4.000080890451122, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "domindom", "side": 8.000161780902245, "solution_path": "data/solutions/generated/domindom_016_5102d8436800a7ff.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/5102d8436800a7ff.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@17", "density": 0.8394293686085971, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/18.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "9/2 = 4.5", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@17", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "9/2 = 4.5", "metric_symbol": "s", "metric_value": 4.5, "reference_text": "17.-18 | | s = 9/2 = 4.5 | Trivial.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "17.-18 | | s = 9/2 = 4.5 | Trivial.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "3b388ddd068a843f", "improvement": -0.00020586075893191946, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.953267209728551e-07, "max_pair_overlap_depth": -5.000451830872166e-05, "metric_expression": "4.50021", "metric_symbol": "s", "metric_value": 4.500205860758932, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.5, "setup": "domindom", "side": 9.000411721517864, "solution_path": "data/solutions/generated/domindom_017_3b388ddd068a843f.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3b388ddd068a843f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domindom@18", "density": 0.887184224626392, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/18.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "9/2 = 4.5", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@18", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "9/2 = 4.5", "metric_symbol": "s", "metric_value": 4.5, "reference_text": "17.-18 | | s = 9/2 = 4.5 | Trivial.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "17.-18 | | s = 9/2 = 4.5 | Trivial.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "6a5fcb6ec4e55187", "improvement": -0.004321147560441219, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4781434060372476e-06, "max_pair_overlap_depth": -4.894490537976637e-05, "metric_expression": "4.50432", "metric_symbol": "s", "metric_value": 4.504321147560441, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.5, "setup": "domindom", "side": 4.504321147560441, "solution_path": "data/solutions/generated/domindom_018_6a5fcb6ec4e55187.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6a5fcb6ec4e55187.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@1", "density": 0.46408935997723705, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/1.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "2+2/\u221a3 = 3.15470+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.1547, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2+2/\u221a3 = 3.15470+", "metric_symbol": "s", "metric_value": 3.1547, "reference_text": "s = 2+2/\u221a3 = 3.15470+ | Trivial.", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/1.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 2+2/\u221a3 = 3.15470+ | Trivial.", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "6121ec989c6c813b", "improvement": -4.219103236957622e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.613940413552034e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "3.15474", "metric_symbol": "s", "metric_value": 3.1547421910323696, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.1547, "setup": "domintri", "side": 3.1547421910323696, "solution_path": "data/solutions/generated/domintri_001_6121ec989c6c813b.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6121ec989c6c813b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@2", "density": 0.4973974214603669, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/2.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "2+4/\u221a3 = 4.30940+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.3094, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2+4/\u221a3 = 4.30940+", "metric_symbol": "s", "metric_value": 4.3094, "reference_text": "s = 2+4/\u221a3 = 4.30940+ | Trivial.", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/2.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 2+4/\u221a3 = 4.30940+ | Trivial.", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "75b0ecc0320e01de", "improvement": -0.00011019921439459068, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.546312299587745e-06, "max_pair_overlap_depth": -2.318487220109855e-05, "metric_expression": "4.30951", "metric_symbol": "s", "metric_value": 4.309510199214395, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.3094, "setup": "domintri", "side": 4.309510199214395, "solution_path": "data/solutions/generated/domintri_002_75b0ecc0320e01de.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/75b0ecc0320e01de.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@3", "density": 0.746073641317824, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/3.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "2+4/\u221a3 = 4.30934+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.30934, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2+4/\u221a3 = 4.30934+", "metric_symbol": "s", "metric_value": 4.30934, "reference_text": "s = 2+4/\u221a3 = 4.30934+ | Trivial.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/3.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 2+4/\u221a3 = 4.30934+ | Trivial.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "ecf54a038a2d7184", "improvement": -0.00023515521175365706, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2511235262659097e-06, "max_pair_overlap_depth": -8.934604283755698e-05, "metric_expression": "4.30958", "metric_symbol": "s", "metric_value": 4.309575155211753, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.30934, "setup": "domintri", "side": 4.309575155211753, "solution_path": "data/solutions/generated/domintri_003_ecf54a038a2d7184.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/ecf54a038a2d7184.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@4", "density": 0.6377999539199563, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/4.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "5.38188+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.38188, "friedman_reference": {"analytical_or_proved": false, "case": "domintri@4", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.38188+", "metric_symbol": "s", "metric_value": 5.38188, "reference_text": "s = 5.38188+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/4.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 5.38188+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "6f3730492b03ce4f", "improvement": -0.0002283152146249634, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.401205447247945e-07, "max_pair_overlap_depth": -4.238554796887062e-05, "metric_expression": "5.38211", "metric_symbol": "s", "metric_value": 5.382108315214625, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.38188, "setup": "domintri", "side": 5.382108315214625, "solution_path": "data/solutions/generated/domintri_004_6f3730492b03ce4f.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/6f3730492b03ce4f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@5", "density": 0.7734497127773695, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/5.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "2+2\u221a3 = 5.46410+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.4641, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@5", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2+2\u221a3 = 5.46410+", "metric_symbol": "s", "metric_value": 5.4641, "reference_text": "s = 2+2\u221a3 = 5.46410+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/5.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 2+2\u221a3 = 5.46410+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "272a5ede0aa9a415", "improvement": -0.00018874969044357925, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.315600305266359e-07, "max_pair_overlap_depth": -2.2346020378205544e-05, "metric_expression": "5.46429", "metric_symbol": "s", "metric_value": 5.464288749690444, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.4641, "setup": "domintri", "side": 5.464288749690444, "solution_path": "data/solutions/generated/domintri_005_272a5ede0aa9a415.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/272a5ede0aa9a415.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@6", "density": 0.6961017329945272, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/6.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "4+4/\u221a3 = 6.30940+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.3094, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@6", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4+4/\u221a3 = 6.30940+", "metric_symbol": "s", "metric_value": 6.3094, "reference_text": "s = 4+4/\u221a3 = 6.30940+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/6.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 4+4/\u221a3 = 6.30940+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "5709487003259292", "improvement": -0.0002307959840699425, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1058985619703807e-06, "max_pair_overlap_depth": -5.695799924376743e-05, "metric_expression": "6.30963", "metric_symbol": "s", "metric_value": 6.30963079598407, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.3094, "setup": "domintri", "side": 6.30963079598407, "solution_path": "data/solutions/generated/domintri_006_5709487003259292.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/5709487003259292.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@7", "density": 0.7379982685276676, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/7.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "6.61409+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.61409, "friedman_reference": {"analytical_or_proved": false, "case": "domintri@7", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.61409+", "metric_symbol": "s", "metric_value": 6.61409, "reference_text": "s = 6.61409+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/7.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 6.61409+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "4f3a41ae14776742", "improvement": -0.004813180227946745, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.155481361422797e-07, "max_pair_overlap_depth": -2.185552101252597e-05, "metric_expression": "6.61890", "metric_symbol": "s", "metric_value": 6.618903180227947, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.61409, "setup": "domintri", "side": 6.618903180227947, "solution_path": "data/solutions/generated/domintri_007_4f3a41ae14776742.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/4f3a41ae14776742.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@8", "density": 0.7218039663224785, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/8.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "6+2/\u221a3 = 7.15470+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.1547, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@8", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6+2/\u221a3 = 7.15470+", "metric_symbol": "s", "metric_value": 7.1547, "reference_text": "s = 6+2/\u221a3 = 7.15470+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/8.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 6+2/\u221a3 = 7.15470+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "6c005b2a9a930ec2", "improvement": -0.00014189325334523772, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4194624753243232e-06, "max_pair_overlap_depth": -3.0722639065561275e-05, "metric_expression": "7.15484", "metric_symbol": "s", "metric_value": 7.154841893253345, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.1547, "setup": "domintri", "side": 7.154841893253345, "solution_path": "data/solutions/generated/domintri_008_6c005b2a9a930ec2.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/6c005b2a9a930ec2.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@9", "density": 0.7778159590517906, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/9.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "5+4/\u221a3 = 7.30940+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.3094, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@9", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5+4/\u221a3 = 7.30940+", "metric_symbol": "s", "metric_value": 7.3094, "reference_text": "s = 5+4/\u221a3 = 7.30940+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/9.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 5+4/\u221a3 = 7.30940+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "efd3c01a911fbe39", "improvement": -0.0011072089155117482, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4249809133477843e-06, "max_pair_overlap_depth": -9.12510211048545e-05, "metric_expression": "7.31051", "metric_symbol": "s", "metric_value": 7.310507208915512, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.3094, "setup": "domintri", "side": 7.310507208915512, "solution_path": "data/solutions/generated/domintri_009_efd3c01a911fbe39.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/efd3c01a911fbe39.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@10", "density": 0.7955429934788115, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/10.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "3+8/\u221a3 = 7.61885+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.61885, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@10", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3+8/\u221a3 = 7.61885+", "metric_symbol": "s", "metric_value": 7.61885, "reference_text": "s = 3+8/\u221a3 = 7.61885+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/10.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 3+8/\u221a3 = 7.61885+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "0ce8e56f538c019b", "improvement": -0.000761914714127343, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.832683484389887e-07, "max_pair_overlap_depth": -3.7959726496283874e-06, "metric_expression": "7.61961", "metric_symbol": "s", "metric_value": 7.6196119147141275, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.61885, "setup": "domintri", "side": 7.6196119147141275, "solution_path": "data/solutions/generated/domintri_010_0ce8e56f538c019b.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/0ce8e56f538c019b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@11", "density": 0.7639001254938724, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/11.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "7+2/\u221a3 = 8.15470+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.1547, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "7+2/\u221a3 = 8.15470+", "metric_symbol": "s", "metric_value": 8.1547, "reference_text": "s = 7+2/\u221a3 = 8.15470+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/11.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 7+2/\u221a3 = 8.15470+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "9dea260c7a23e4ab", "improvement": -0.0006523939405695955, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1114403341932665e-06, "max_pair_overlap_depth": -5.305705658820159e-06, "metric_expression": "8.15535", "metric_symbol": "s", "metric_value": 8.15535239394057, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.1547, "setup": "domintri", "side": 8.15535239394057, "solution_path": "data/solutions/generated/domintri_011_9dea260c7a23e4ab.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/9dea260c7a23e4ab.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@12", "density": 0.8025727620592626, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/12.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "6+4/\u221a3 = 8.30940+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.3094, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@12", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6+4/\u221a3 = 8.30940+", "metric_symbol": "s", "metric_value": 8.3094, "reference_text": "s = 6+4/\u221a3 = 8.30940+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/12.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 6+4/\u221a3 = 8.30940+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "c7b6e34fcb0ca6b3", "improvement": -0.0008309724073285452, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.6999836404354686e-06, "max_pair_overlap_depth": -5.386448125865861e-06, "metric_expression": "8.31023", "metric_symbol": "s", "metric_value": 8.310230972407329, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.3094, "setup": "domintri", "side": 8.310230972407329, "solution_path": "data/solutions/generated/domintri_012_c7b6e34fcb0ca6b3.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/c7b6e34fcb0ca6b3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@13", "density": 0.8080926405411509, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/13.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "4+8/\u221a3 = 8.61885+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.61885, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4+8/\u221a3 = 8.61885+", "metric_symbol": "s", "metric_value": 8.61885, "reference_text": "s = 4+8/\u221a3 = 8.61885+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/13.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 4+8/\u221a3 = 8.61885+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "9f9d19c6f117401b", "improvement": -0.0011205125576889685, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5415762652602807e-06, "max_pair_overlap_depth": -3.765847597092531e-05, "metric_expression": "8.61997", "metric_symbol": "s", "metric_value": 8.619970512557689, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.61885, "setup": "domintri", "side": 8.619970512557689, "solution_path": "data/solutions/generated/domintri_013_9f9d19c6f117401b.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/9f9d19c6f117401b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@14", "density": 0.8398945758370188, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/14.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "3+10/\u221a3 = 8.77350+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.7735, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@14", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "3+10/\u221a3 = 8.77350+", "metric_symbol": "s", "metric_value": 8.7735, "reference_text": "s = 3+10/\u221a3 = 8.77350+ | Found by Maurizio Morandi | in April 2026.", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/14.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 3+10/\u221a3 = 8.77350+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "333566d5c4b3657c", "improvement": -0.0008773842173539492, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.796172321890858e-09, "max_pair_overlap_depth": -3.2457310312317134e-05, "metric_expression": "8.77438", "metric_symbol": "s", "metric_value": 8.774377384217354, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.7735, "setup": "domintri", "side": 8.774377384217354, "solution_path": "data/solutions/generated/domintri_014_333566d5c4b3657c.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/333566d5c4b3657c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@15", "density": 0.7991843102868493, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/15.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "7+4/\u221a3 = 9.30940+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.3094, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@15", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "7+4/\u221a3 = 9.30940+", "metric_symbol": "s", "metric_value": 9.3094, "reference_text": "s = 7+4/\u221a3 = 9.30940+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/15.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 7+4/\u221a3 = 9.30940+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "2da1308626a02461", "improvement": -0.0013965021666475508, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5011317742776256e-06, "max_pair_overlap_depth": -8.262234862044249e-05, "metric_expression": "9.31080", "metric_symbol": "s", "metric_value": 9.310796502166648, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.3094, "setup": "domintri", "side": 9.310796502166648, "solution_path": "data/solutions/generated/domintri_015_2da1308626a02461.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/2da1308626a02461.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@16", "density": 0.8249213221645568, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/16.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "6+2\u221a3 = 9.46410+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.4641, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@16", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "6+2\u221a3 = 9.46410+", "metric_symbol": "s", "metric_value": 9.4641, "reference_text": "s = 6+2\u221a3 = 9.46410+ | Found by Maurizio Morandi | in April 2026.", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/16.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 6+2\u221a3 = 9.46410+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "cc2a76b6a75c7120", "improvement": -0.000851797392776632, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.428695004104409e-06, "max_pair_overlap_depth": -2.0153462060745753e-05, "metric_expression": "9.46495", "metric_symbol": "s", "metric_value": 9.464951797392777, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.4641, "setup": "domintri", "side": 9.464951797392777, "solution_path": "data/solutions/generated/domintri_016_cc2a76b6a75c7120.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/cc2a76b6a75c7120.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@17", "density": 0.8218638425440753, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/17.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "4+10/\u221a3 = 9.77350+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.7735, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4+10/\u221a3 = 9.77350+", "metric_symbol": "s", "metric_value": 9.7735, "reference_text": "s = 4+10/\u221a3 = 9.77350+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/17.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 4+10/\u221a3 = 9.77350+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "842ace76b15a5c6d", "improvement": -0.0008796443209906357, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.8870325675374086e-06, "max_pair_overlap_depth": -2.297706007370337e-05, "metric_expression": "9.77438", "metric_symbol": "s", "metric_value": 9.774379644320991, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.7735, "setup": "domintri", "side": 9.774379644320991, "solution_path": "data/solutions/generated/domintri_017_842ace76b15a5c6d.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/842ace76b15a5c6d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@18", "density": 0.8433917909368376, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/18.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "3+4\u221a3 = 9.92820+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.9282, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@18", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3+4\u221a3 = 9.92820+", "metric_symbol": "s", "metric_value": 9.9282, "reference_text": "s = 3+4\u221a3 = 9.92820+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/18.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 3+4\u221a3 = 9.92820+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "286272298adf99a4", "improvement": -0.00035960973454685075, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.814371048432239e-06, "max_pair_overlap_depth": -1.8972863250255045e-05, "metric_expression": "9.92856", "metric_symbol": "s", "metric_value": 9.928559609734547, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.9282, "setup": "domintri", "side": 9.928559609734547, "solution_path": "data/solutions/generated/domintri_018_286272298adf99a4.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/286272298adf99a4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@19", "density": 0.8255562955017095, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/19.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "8+4/\u221a3 = 10.30940+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.3094, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@19", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "8+4/\u221a3 = 10.30940+", "metric_symbol": "s", "metric_value": 10.3094, "reference_text": "s = 8+4/\u221a3 = 10.30940+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/19.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 8+4/\u221a3 = 10.30940+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "77c1eb98f0e7fed4", "improvement": -0.0008247746809697531, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.73171847830086e-06, "max_pair_overlap_depth": -2.8002623395084747e-06, "metric_expression": "10.31022", "metric_symbol": "s", "metric_value": 10.31022477468097, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.3094, "setup": "domintri", "side": 10.31022477468097, "solution_path": "data/solutions/generated/domintri_019_77c1eb98f0e7fed4.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/77c1eb98f0e7fed4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@20", "density": 0.843501980040664, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/20.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "7+2\u221a3 = 10.46410+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.4641, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@20", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "7+2\u221a3 = 10.46410+", "metric_symbol": "s", "metric_value": 10.4641, "reference_text": "s = 7+2\u221a3 = 10.46410+ | Found by Maurizio Morandi | in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/20.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 7+2\u221a3 = 10.46410+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "19214c32485a435f", "improvement": -0.0008371510213134314, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8019716271983103e-06, "max_pair_overlap_depth": -2.85508650477162e-06, "metric_expression": "10.46494", "metric_symbol": "s", "metric_value": 10.464937151021314, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.4641, "setup": "domintri", "side": 10.464937151021314, "solution_path": "data/solutions/generated/domintri_020_19214c32485a435f.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/19214c32485a435f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@21", "density": 0.8533496545178485, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/21.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "2+5\u221a3 = 10.66025+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.66025, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@21", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "2+5\u221a3 = 10.66025+", "metric_symbol": "s", "metric_value": 10.66025, "reference_text": "s = 2+5\u221a3 = 10.66025+ | Found by Maurizio Morandi | in April 2026.", "source_image": "21.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/21.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 2+5\u221a3 = 10.66025+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "21.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "797030ebf0f8c4e4", "improvement": -0.0010660665758290833, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.564559491289197e-06, "max_pair_overlap_depth": -2.4360684990565318e-05, "metric_expression": "10.66132", "metric_symbol": "s", "metric_value": 10.661316066575829, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.66025, "setup": "domintri", "side": 10.661316066575829, "solution_path": "data/solutions/generated/domintri_021_797030ebf0f8c4e4.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/797030ebf0f8c4e4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@22", "density": 0.8506833122891716, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/22.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "4(1+\u221a3) = 10.92820+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.9282, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@22", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "4(1+\u221a3) = 10.92820+", "metric_symbol": "s", "metric_value": 10.9282, "reference_text": "s = 4(1+\u221a3) = 10.92820+ | Found by Maurizio Morandi | in April 2026.", "source_image": "22.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/22.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 4(1+\u221a3) = 10.92820+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "22.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "90061b49654f08af", "improvement": -0.001092862620856394, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.631830815268069e-06, "max_pair_overlap_depth": -1.6046632975275088e-05, "metric_expression": "10.92929", "metric_symbol": "s", "metric_value": 10.929292862620857, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.9282, "setup": "domintri", "side": 10.929292862620857, "solution_path": "data/solutions/generated/domintri_022_90061b49654f08af.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/90061b49654f08af.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@23", "density": 0.8536884993646849, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/23.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "10+2/\u221a3 = 11.15470+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.1547, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@23", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "10+2/\u221a3 = 11.15470+", "metric_symbol": "s", "metric_value": 11.1547, "reference_text": "s = 10+2/\u221a3 = 11.15470+ | Found by Maurizio Morandi | in April 2026.", "source_image": "23.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/23.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 10+2/\u221a3 = 11.15470+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "23.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "8ca8e97a5947c876", "improvement": -0.0005390769738546908, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.068904299595512e-06, "max_pair_overlap_depth": -1.1497281311445562e-05, "metric_expression": "11.15524", "metric_symbol": "s", "metric_value": 11.155239076973855, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 11.1547, "setup": "domintri", "side": 11.155239076973855, "solution_path": "data/solutions/generated/domintri_023_8ca8e97a5947c876.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/8ca8e97a5947c876.svg", "tolerance": 1e-08, "verified": true}
-{"case": "domintri@24", "density": 0.8664947344465272, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/24.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "9+4/\u221a3 = 11.30940+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.3094, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@24", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "9+4/\u221a3 = 11.30940+", "metric_symbol": "s", "metric_value": 11.3094, "reference_text": "s = 9+4/\u221a3 = 11.30940+ | Found by Maurizio Morandi | in April 2026.", "source_image": "24.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/24.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 9+4/\u221a3 = 11.30940+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "24.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "f8436aa831ae57e5", "improvement": -0.0012440894174083894, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.95572835399557e-06, "max_pair_overlap_depth": -5.492290923159171e-05, "metric_expression": "11.31064", "metric_symbol": "s", "metric_value": 11.310644089417409, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 11.3094, "setup": "domintri", "side": 11.310644089417409, "solution_path": "data/solutions/generated/domintri_024_f8436aa831ae57e5.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/f8436aa831ae57e5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@1", "density": 0.826977987261634, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/1.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "r", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "r", "metric_value": 1.0, "reference_text": "r = 1 | Trivial.", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/1.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 1 | Trivial.", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "48f5aa37c90550c8", "improvement": -9.284285680699611e-06, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.547655283739573e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00001", "metric_symbol": "r", "metric_value": 1.0000092842856807, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0, "setup": "hexincir", "side": 2.0000185685713614, "solution_path": "data/solutions/generated/hexincir_001_48f5aa37c90550c8.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/48f5aa37c90550c8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@2", "density": 0.508894613048892, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/2.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a13/2 = 1.802+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.802, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a13/2 = 1.802+", "metric_symbol": "r", "metric_value": 1.802, "reference_text": "r = \u221a13/2 = 1.802+ | Trivial.", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/2.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a13/2 = 1.802+ | Trivial.", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "3b6bd2554e997adc", "improvement": -0.0008187982614582534, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4989872811899119e-06, "max_pair_overlap_depth": -3.954921521903687e-05, "metric_expression": "1.80282", "metric_symbol": "r", "metric_value": 1.8028187982614583, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.802, "setup": "hexincir", "side": 3.6056375965229166, "solution_path": "data/solutions/generated/hexincir_002_3b6bd2554e997adc.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3b6bd2554e997adc.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@3", "density": 0.6202101203168955, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/3.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "r", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "r", "metric_value": 2.0, "reference_text": "r = 2 | Trivial.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/3.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 2 | Trivial.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "194e619a74e29452", "improvement": -5.624955299809997e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.970405754344824e-06, "max_pair_overlap_depth": -7.230141190991447e-05, "metric_expression": "2.00006", "metric_symbol": "r", "metric_value": 2.000056249552998, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "hexincir", "side": 4.000112499105996, "solution_path": "data/solutions/generated/hexincir_003_194e619a74e29452.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/194e619a74e29452.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@4", "density": 0.6594633839026793, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/4.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(21\u2013\u221a145)/4 = 2.239+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.239, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@4", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "(21\u2013\u221a145)/4 = 2.239+", "metric_symbol": "r", "metric_value": 2.239, "reference_text": "r = (21\u2013\u221a145)/4 = 2.239+ | Found by Maurizio Morandi | in March 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/4.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = (21\u2013\u221a145)/4 = 2.239+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "355b11dfc363f5ba", "improvement": -0.0006783874729134887, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.544625404889757e-07, "max_pair_overlap_depth": -5.057462804924917e-05, "metric_expression": "2.23968", "metric_symbol": "r", "metric_value": 2.2396783874729134, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.239, "setup": "hexincir", "side": 4.479356774945827, "solution_path": "data/solutions/generated/hexincir_004_355b11dfc363f5ba.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/355b11dfc363f5ba.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@5", "density": 0.6766813833384198, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/5.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2.471+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.471, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@5", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "2.471+", "metric_symbol": "r", "metric_value": 2.471, "reference_text": "r = 2.471+ | Found by Maurizio Morandi | in March 2026.", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/5.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 2.471+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "9427b00c077aafc3", "improvement": -0.0009739785189006867, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.673907177263573e-06, "max_pair_overlap_depth": -5.983134966366932e-05, "metric_expression": "2.47197", "metric_symbol": "r", "metric_value": 2.4719739785189008, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.471, "setup": "hexincir", "side": 4.9439479570378015, "solution_path": "data/solutions/generated/hexincir_005_9427b00c077aafc3.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/9427b00c077aafc3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@6", "density": 0.7087991567669789, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/7.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a7 = 2.645+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.645, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a7 = 2.645+", "metric_symbol": "r", "metric_value": 2.645, "reference_text": "r = \u221a7 = 2.645+ | Trivial. |", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/7.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a7 = 2.645+ | Trivial. |", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "d7d900dd0bba32a3", "improvement": -0.0008488830398025726, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0221671226394733e-07, "max_pair_overlap_depth": -3.6674166167594e-05, "metric_expression": "2.64585", "metric_symbol": "r", "metric_value": 2.6458488830398026, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.645, "setup": "hexincir", "side": 5.291697766079605, "solution_path": "data/solutions/generated/hexincir_006_d7d900dd0bba32a3.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d7d900dd0bba32a3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@7", "density": 0.8269668154291873, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/7.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a7 = 2.645+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.645, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a7 = 2.645+", "metric_symbol": "r", "metric_value": 2.645, "reference_text": "r = \u221a7 = 2.645+ | Trivial. |", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/7.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a7 = 2.645+ | Trivial. |", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "5cd27a740cc9b2a5", "improvement": -0.0007937463486662466, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.370930279076532e-06, "max_pair_overlap_depth": -2.6143388138111234e-05, "metric_expression": "2.64579", "metric_symbol": "r", "metric_value": 2.6457937463486663, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.645, "setup": "hexincir", "side": 5.2915874926973325, "solution_path": "data/solutions/generated/hexincir_007_5cd27a740cc9b2a5.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/5cd27a740cc9b2a5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@8", "density": 0.7426247287224678, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/8.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2.984+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.984, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@8", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "2.984+", "metric_symbol": "r", "metric_value": 2.984, "reference_text": "r = 2.984+ | Found by Maurizio Morandi | in March 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/8.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 2.984+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "7d27528d78a05a43", "improvement": -0.0007729280237529501, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.6889173538057207e-06, "max_pair_overlap_depth": -1.732806256671271e-05, "metric_expression": "2.98477", "metric_symbol": "r", "metric_value": 2.984772928023753, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.984, "setup": "hexincir", "side": 5.969545856047506, "solution_path": "data/solutions/generated/hexincir_008_7d27528d78a05a43.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/7d27528d78a05a43.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@9", "density": 0.682777017123973, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/9.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3.301+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.301, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@9", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "3.301+", "metric_symbol": "r", "metric_value": 3.301, "reference_text": "r = 3.301+ | Found by Maurizio Morandi | in March 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/9.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 3.301+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "5f9496662ac0b661", "improvement": -0.000663545504778984, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.641836701819614e-06, "max_pair_overlap_depth": -2.219136600895011e-05, "metric_expression": "3.30166", "metric_symbol": "r", "metric_value": 3.301663545504779, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.301, "setup": "hexincir", "side": 6.603327091009558, "solution_path": "data/solutions/generated/hexincir_009_5f9496662ac0b661.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/5f9496662ac0b661.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@10", "density": 0.6804997422869841, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/10.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "7/2 = 3.5", "friedman_metric_symbol": "r", "friedman_metric_value": 3.5, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@10", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "7/2 = 3.5", "metric_symbol": "r", "metric_value": 3.5, "reference_text": "r = 7/2 = 3.5 | Found by Maurizio Morandi | in March 2026.", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/10.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 7/2 = 3.5 | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": false, "id": "8fccd3008565aabb", "improvement": 0.013922608209857135, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.016042661651568e-06, "max_pair_overlap_depth": -3.440098626628085e-05, "metric_expression": "3.48608", "metric_symbol": "r", "metric_value": 3.486077391790143, "n": 10, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 3.5, "setup": "hexincir", "side": 6.972154783580286, "solution_path": "data/solutions/generated/hexincir_010_8fccd3008565aabb.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/8fccd3008565aabb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@11", "density": 0.7038726397318542, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/12.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a13 = 3.605+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.605, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@11", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "\u221a13 = 3.605+", "metric_symbol": "r", "metric_value": 3.605, "reference_text": "r = \u221a13 = 3.605+ | Found by Maurizio Morandi | in March 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/12.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a13 = 3.605+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": false, "id": "2d2e76ec34c49fb5", "improvement": 0.009988305972369105, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.0630450586220945e-06, "max_pair_overlap_depth": -3.6222629933746475e-05, "metric_expression": "3.59501", "metric_symbol": "r", "metric_value": 3.595011694027631, "n": 11, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 3.605, "setup": "hexincir", "side": 7.190023388055262, "solution_path": "data/solutions/generated/hexincir_011_2d2e76ec34c49fb5.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/2d2e76ec34c49fb5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@12", "density": 0.7629156641677376, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/12.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a13 = 3.605+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.605, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@12", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "\u221a13 = 3.605+", "metric_symbol": "r", "metric_value": 3.605, "reference_text": "r = \u221a13 = 3.605+ | Found by Maurizio Morandi | in March 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/12.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a13 = 3.605+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "5515c10818a71365", "improvement": -0.001644724814655607, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.2444141653265035e-06, "max_pair_overlap_depth": -3.49998284549164e-05, "metric_expression": "3.60664", "metric_symbol": "r", "metric_value": 3.6066447248146556, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.605, "setup": "hexincir", "side": 7.213289449629311, "solution_path": "data/solutions/generated/hexincir_012_5515c10818a71365.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/5515c10818a71365.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@13", "density": 0.7347845850822048, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/13.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3.82298+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.82298, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.82298+", "metric_symbol": "r", "metric_value": 3.82298, "reference_text": "r = 3.82298+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/13.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 3.82298+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "defdebc1228b1d63", "improvement": -0.002118865563556671, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3642191814564626e-06, "max_pair_overlap_depth": -3.899276946051078e-05, "metric_expression": "3.82510", "metric_symbol": "r", "metric_value": 3.8250988655635565, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.82298, "setup": "hexincir", "side": 7.650197731127113, "solution_path": "data/solutions/generated/hexincir_013_defdebc1228b1d63.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/defdebc1228b1d63.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@14", "density": 0.7696929179154312, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/14.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a(109\u20139\u221a109) = 3.877+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.877, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@14", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "\u221a(109\u20139\u221a109) = 3.877+", "metric_symbol": "r", "metric_value": 3.877, "reference_text": "r = \u221a(109\u20139\u221a109) = 3.877+ | Found by Maurizio Morandi | in March 2026.", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/14.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a(109\u20139\u221a109) = 3.877+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "cf6592c7009ff83d", "improvement": -0.0014328751718628574, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.009541618976158e-06, "max_pair_overlap_depth": -5.416099891136783e-05, "metric_expression": "3.87843", "metric_symbol": "r", "metric_value": 3.8784328751718626, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.877, "setup": "hexincir", "side": 7.756865750343725, "solution_path": "data/solutions/generated/hexincir_014_cf6592c7009ff83d.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/cf6592c7009ff83d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@15", "density": 0.7439908773476231, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/15.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.08174+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.08174, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@15", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.08174+", "metric_symbol": "r", "metric_value": 4.08174, "reference_text": "r = 4.08174+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/15.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 4.08174+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "ea523b99f3bf81fb", "improvement": -0.001574502888137097, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.330196611616714e-06, "max_pair_overlap_depth": -5.918641434021765e-05, "metric_expression": "4.08331", "metric_symbol": "r", "metric_value": 4.083314502888137, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.08174, "setup": "hexincir", "side": 8.166629005776274, "solution_path": "data/solutions/generated/hexincir_015_ea523b99f3bf81fb.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/ea523b99f3bf81fb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@16", "density": 0.7714178714001686, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/16.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.13966+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.13966, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@16", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.13966+", "metric_symbol": "r", "metric_value": 4.13966, "reference_text": "r = 4.13966+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/16.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 4.13966+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "928ee7a851bce6e5", "improvement": -0.0019209106018989175, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.8109720468001456e-06, "max_pair_overlap_depth": -4.18854370716204e-05, "metric_expression": "4.14158", "metric_symbol": "r", "metric_value": 4.141580910601899, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.13966, "setup": "hexincir", "side": 8.283161821203798, "solution_path": "data/solutions/generated/hexincir_016_928ee7a851bce6e5.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/928ee7a851bce6e5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@17", "density": 0.7591563034967428, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/17.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.30170+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.3017, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.30170+", "metric_symbol": "r", "metric_value": 4.3017, "reference_text": "r = 4.30170+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/17.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 4.30170+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "e582b760dc061cd3", "improvement": -0.001681676380650643, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.855951803468429e-06, "max_pair_overlap_depth": -4.543663468803416e-05, "metric_expression": "4.30338", "metric_symbol": "r", "metric_value": 4.303381676380651, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.3017, "setup": "hexincir", "side": 4.303381676380651, "solution_path": "data/solutions/generated/hexincir_017_e582b760dc061cd3.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T03:07:58+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/e582b760dc061cd3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@18", "density": 0.7834006630179079, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/19.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a19 = 4.35889+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.35889, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@18", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a19 = 4.35889+", "metric_symbol": "r", "metric_value": 4.35889, "reference_text": "r = \u221a19 = 4.35889+ | Trivial.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/19.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a19 = 4.35889+ | Trivial.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "12eaac899664a007", "improvement": -0.0001945423028058002, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.696366953105269e-06, "max_pair_overlap_depth": -1.0490734119961154e-05, "metric_expression": "4.35908", "metric_symbol": "r", "metric_value": 4.3590845423028055, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.35889, "setup": "hexincir", "side": 8.718169084605611, "solution_path": "data/solutions/generated/hexincir_018_12eaac899664a007.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/12eaac899664a007.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@19", "density": 0.8269599994475847, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/19.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a19 = 4.35889+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.35889, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@19", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a19 = 4.35889+", "metric_symbol": "r", "metric_value": 4.35889, "reference_text": "r = \u221a19 = 4.35889+ | Trivial.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/19.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a19 = 4.35889+ | Trivial.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "b4bf200670caf09c", "improvement": -9.681979153608467e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.636539265552187e-06, "max_pair_overlap_depth": -6.973398227638761e-06, "metric_expression": "4.35899", "metric_symbol": "r", "metric_value": 4.358986819791536, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.35889, "setup": "hexincir", "side": 8.717973639583072, "solution_path": "data/solutions/generated/hexincir_019_b4bf200670caf09c.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b4bf200670caf09c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@20", "density": 0.7723857667872704, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/20.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.62598+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.62598, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@20", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.62598+", "metric_symbol": "r", "metric_value": 4.62598, "reference_text": "r = 4.62598+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/20.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 4.62598+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "30108dc5d1ea9a14", "improvement": -0.0015460646281777457, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8288872105642895e-06, "max_pair_overlap_depth": -4.792351042670617e-06, "metric_expression": "4.62753", "metric_symbol": "r", "metric_value": 4.627526064628178, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.62598, "setup": "hexincir", "side": 9.255052129256356, "solution_path": "data/solutions/generated/hexincir_020_30108dc5d1ea9a14.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/30108dc5d1ea9a14.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexincir@21", "density": 0.7917677222663352, "friedman_analytical_or_proved": false, "friedman_credit": "Enea Victor", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/21.png", "friedman_metric_expression": "4.68339+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.68339, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@21", "credit": "Enea Victor", "date": "April 2026", "family_updated": "", "metric_expression": "4.68339+", "metric_symbol": "r", "metric_value": 4.68339, "reference_text": "r = 4.68339+ | Found by Enea Victor | in April 2026.", "source_image": "21.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/21.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 4.68339+ | Found by Enea Victor | in April 2026.", "friedman_source_image": "21.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "49a047c29bb184c3", "improvement": -1.5457789629635954e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.498160996568856e-06, "max_pair_overlap_depth": -3.8704072840012316e-06, "metric_expression": "4.68341", "metric_symbol": "r", "metric_value": 4.68340545778963, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.68339, "setup": "hexincir", "side": 9.36681091557926, "solution_path": "data/solutions/generated/hexincir_021_49a047c29bb184c3.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Enea Victor", "svg_path": "data/svg/49a047c29bb184c3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@1", "density": 0.433010701548144, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/1.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "\u221a3 = 1.73205+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.73205, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3 = 1.73205+", "metric_symbol": "s", "metric_value": 1.73205, "reference_text": "1. | | s = \u221a3 = 1.73205+ | Trivial.", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/1.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "1. | | s = \u221a3 = 1.73205+ | Trivial.", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "899222ca3c83cd8e", "improvement": -4.8082708896668436e-06, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0003297862271054e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.73205", "metric_symbol": "s", "metric_value": 1.7320548082708898, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.73205, "setup": "hexindom", "side": 3.4641096165417795, "solution_path": "data/solutions/generated/hexindom_001_899222ca3c83cd8e.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/899222ca3c83cd8e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@2", "density": 0.7237057542591544, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/2.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "1.8946+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.8946, "friedman_reference": {"analytical_or_proved": false, "case": "hexindom@2", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.8946+", "metric_symbol": "s", "metric_value": 1.8946, "reference_text": "2. | | s = 1.8946+ | Found by Jonathan Viquerat in April 2026.", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/2.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "2. | | s = 1.8946+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "419ae881803808e0", "improvement": -0.00011952413629479075, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.154031481298006e-06, "max_pair_overlap_depth": -4.420983116036614e-05, "metric_expression": "1.89472", "metric_symbol": "s", "metric_value": 1.8947195241362949, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.8946, "setup": "hexindom", "side": 3.7894390482725897, "solution_path": "data/solutions/generated/hexindom_002_419ae881803808e0.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/419ae881803808e0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@3", "density": 0.6059845041872388, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/3.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "6\u20132\u221a3 = 2.53589+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.53589, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@3", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6\u20132\u221a3 = 2.53589+", "metric_symbol": "s", "metric_value": 2.53589, "reference_text": "3. | | s = 6\u20132\u221a3 = 2.53589+ | Found by Jonathan Viquerat in April 2026.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/3.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "3. | | s = 6\u20132\u221a3 = 2.53589+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "b8984cccc8c8714d", "improvement": -6.074031198854968e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3815485928603266e-06, "max_pair_overlap_depth": -3.5690372763075295e-05, "metric_expression": "2.53595", "metric_symbol": "s", "metric_value": 2.5359507403119887, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.53589, "setup": "hexindom", "side": 5.0719014806239775, "solution_path": "data/solutions/generated/hexindom_003_b8984cccc8c8714d.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b8984cccc8c8714d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@4", "density": 0.5981044188175139, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/4.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "22\u201311\u221a3 = 2.94744+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.94744, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@4", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "22\u201311\u221a3 = 2.94744+", "metric_symbol": "s", "metric_value": 2.94744, "reference_text": "4. | | s = 22\u201311\u221a3 = 2.94744+ | Found by Jonathan Viquerat in April 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/4.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "4. | | s = 22\u201311\u221a3 = 2.94744+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "de86d3aed3d6a439", "improvement": -5.064613954619546e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.326043343385379e-06, "max_pair_overlap_depth": -1.3607159264461544e-05, "metric_expression": "2.94749", "metric_symbol": "s", "metric_value": 2.947490646139546, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.94744, "setup": "hexindom", "side": 5.894981292279092, "solution_path": "data/solutions/generated/hexindom_004_de86d3aed3d6a439.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/de86d3aed3d6a439.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@5", "density": 0.6153950410217744, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/5.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "7(2\u221a3\u20133) = 3.24871+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.24871, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@5", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "7(2\u221a3\u20133) = 3.24871+", "metric_symbol": "s", "metric_value": 3.24871, "reference_text": "5. | | s = 7(2\u221a3\u20133) = 3.24871+ | Found by Jonathan Viquerat in April 2026.", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/5.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "5. | | s = 7(2\u221a3\u20133) = 3.24871+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "6c9e9b8c4daf8e17", "improvement": -5.988993836103518e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.879985772620616e-06, "max_pair_overlap_depth": -1.7392941288552066e-05, "metric_expression": "3.24877", "metric_symbol": "s", "metric_value": 3.248769889938361, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.24871, "setup": "hexindom", "side": 6.497539779876722, "solution_path": "data/solutions/generated/hexindom_005_6c9e9b8c4daf8e17.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/6c9e9b8c4daf8e17.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@6", "density": 0.669460522582725, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/6.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "21(15\u20132\u221a3)/71 = 3.41202+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.41202, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@6", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "21(15\u20132\u221a3)/71 = 3.41202+", "metric_symbol": "s", "metric_value": 3.41202, "reference_text": "6. | | s = 21(15\u20132\u221a3)/71 = 3.41202+ | Found by Jonathan Viquerat in April 2026.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/6.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "6. | | s = 21(15\u20132\u221a3)/71 = 3.41202+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "3331b621360b9c32", "improvement": -9.83352674857052e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.54326023255247e-06, "max_pair_overlap_depth": -5.983710998713754e-05, "metric_expression": "3.41212", "metric_symbol": "s", "metric_value": 3.4121183352674858, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.41202, "setup": "hexindom", "side": 6.8242366705349715, "solution_path": "data/solutions/generated/hexindom_006_3331b621360b9c32.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/3331b621360b9c32.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@7", "density": 0.74772405053785, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/7.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "3.4873+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.4873, "friedman_reference": {"analytical_or_proved": false, "case": "hexindom@7", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.4873+", "metric_symbol": "s", "metric_value": 3.4873, "reference_text": "7. | | s = 3.4873+ | Found by Jonathan Viquerat in April 2026.", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/7.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "7. | | s = 3.4873+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "1cf3d5c48cc9bce3", "improvement": 1.5563940758411832e-07, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.731889618803109e-06, "max_pair_overlap_depth": -2.9943125185782904e-05, "metric_expression": "3.48730", "metric_symbol": "s", "metric_value": 3.4872998443605923, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.4873, "setup": "hexindom", "side": 6.9745996887211845, "solution_path": "data/solutions/generated/hexindom_007_1cf3d5c48cc9bce3.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/1cf3d5c48cc9bce3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@8", "density": 0.7629201456996986, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/8.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "3.6906+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.6906, "friedman_reference": {"analytical_or_proved": false, "case": "hexindom@8", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.6906+", "metric_symbol": "s", "metric_value": 3.6906, "reference_text": "8. | | s = 3.6906+ | Found by Jonathan Viquerat in April 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/8.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "8. | | s = 3.6906+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "645a3046b41a06e7", "improvement": -0.00016503281621149668, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.30145200672294e-06, "max_pair_overlap_depth": -4.406733776018967e-05, "metric_expression": "3.69077", "metric_symbol": "s", "metric_value": 3.6907650328162114, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.6906, "setup": "hexindom", "side": 7.381530065632423, "solution_path": "data/solutions/generated/hexindom_008_645a3046b41a06e7.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/645a3046b41a06e7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@9", "density": 0.7092437911682054, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/9.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "(36+5\u221a3)/11 = 4.06002+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.06002, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@9", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "(36+5\u221a3)/11 = 4.06002+", "metric_symbol": "s", "metric_value": 4.06002, "reference_text": "9. | | s = (36+5\u221a3)/11 = 4.06002+ | Found by Jonathan Viquerat in April 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/9.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "9. | | s = (36+5\u221a3)/11 = 4.06002+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "412a5c4dab2378db", "improvement": -5.848340764558202e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.8730699123165095e-06, "max_pair_overlap_depth": -8.397966625661724e-06, "metric_expression": "4.06008", "metric_symbol": "s", "metric_value": 4.060078483407645, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.06002, "setup": "hexindom", "side": 8.12015696681529, "solution_path": "data/solutions/generated/hexindom_009_412a5c4dab2378db.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/412a5c4dab2378db.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@10", "density": 0.7445544573306387, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/10.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "9(2\u221a3\u20133) = 4.17691+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.17691, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@10", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "9(2\u221a3\u20133) = 4.17691+", "metric_symbol": "s", "metric_value": 4.17691, "reference_text": "10. | | s = 9(2\u221a3\u20133) = 4.17691+ | Found by Jonathan Viquerat in April 2026.", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/10.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "10. | | s = 9(2\u221a3\u20133) = 4.17691+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "b404bfce93e4b698", "improvement": -7.303556637161535e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.152317509946272e-07, "max_pair_overlap_depth": -2.262161166943777e-05, "metric_expression": "4.17698", "metric_symbol": "s", "metric_value": 4.176983035566372, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.17691, "setup": "hexindom", "side": 8.353966071132744, "solution_path": "data/solutions/generated/hexindom_010_b404bfce93e4b698.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b404bfce93e4b698.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@11", "density": 0.7041255860017571, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/11.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "4.5036+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5036, "friedman_reference": {"analytical_or_proved": false, "case": "hexindom@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.5036+", "metric_symbol": "s", "metric_value": 4.5036, "reference_text": "11. | | s = 4.5036+ | Found by Jonathan Viquerat in April 2026.", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/11.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "11. | | s = 4.5036+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "de36b527dfa03258", "improvement": -0.0012696093619926074, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.31456043970391e-06, "max_pair_overlap_depth": -1.8725366454197534e-05, "metric_expression": "4.50487", "metric_symbol": "s", "metric_value": 4.504869609361992, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.5036, "setup": "hexindom", "side": 9.009739218723984, "solution_path": "data/solutions/generated/hexindom_011_de36b527dfa03258.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/de36b527dfa03258.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@12", "density": 0.7322405469378734, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/12.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "41(18\u20135\u221a3)/83 = 4.6136+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.6136, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@12", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "41(18\u20135\u221a3)/83 = 4.6136+", "metric_symbol": "s", "metric_value": 4.6136, "reference_text": "12. | | s = 41(18\u20135\u221a3)/83 = 4.6136+ | Found by Jonathan Viquerat in April 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/12.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "12. | | s = 41(18\u20135\u221a3)/83 = 4.6136+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "32dafbf1407bbd72", "improvement": -0.0003690981500499291, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.932113721409451e-07, "max_pair_overlap_depth": -3.449528945953517e-05, "metric_expression": "4.61397", "metric_symbol": "s", "metric_value": 4.61396909815005, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.6136, "setup": "hexindom", "side": 4.61396909815005, "solution_path": "data/solutions/generated/hexindom_012_32dafbf1407bbd72.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/32dafbf1407bbd72.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@13", "density": 0.7287142254379715, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/13.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "4.8137+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.8137, "friedman_reference": {"analytical_or_proved": false, "case": "hexindom@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.8137+", "metric_symbol": "s", "metric_value": 4.8137, "reference_text": "13. | | s = 4.8137+ | Found by Jonathan Viquerat in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/13.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "13. | | s = 4.8137+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "4d59f64640558812", "improvement": -0.000276853025948931, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.288235630449776e-07, "max_pair_overlap_depth": -5.701790221179781e-05, "metric_expression": "4.81398", "metric_symbol": "s", "metric_value": 4.813976853025949, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.8137, "setup": "hexindom", "side": 9.627953706051898, "solution_path": "data/solutions/generated/hexindom_013_4d59f64640558812.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/4d59f64640558812.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@14", "density": 0.7731048613358414, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/14.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "25(6\u2013\u221a3)/22 = 4.84994+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.84994, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@14", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "25(6\u2013\u221a3)/22 = 4.84994+", "metric_symbol": "s", "metric_value": 4.84994, "reference_text": "14. | | s = 25(6\u2013\u221a3)/22 = 4.84994+ | Found by Jonathan Viquerat in April 2026.", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/14.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "14. | | s = 25(6\u2013\u221a3)/22 = 4.84994+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "1f0ed34d05c35380", "improvement": -0.00021659973266263677, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.312817282834061e-06, "max_pair_overlap_depth": -5.706330400911952e-05, "metric_expression": "4.85016", "metric_symbol": "s", "metric_value": 4.850156599732663, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.84994, "setup": "hexindom", "side": 9.700313199465326, "solution_path": "data/solutions/generated/hexindom_014_1f0ed34d05c35380.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/1f0ed34d05c35380.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@15", "density": 0.7946102881582437, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/15.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "56(27\u20134\u221a3)/227 = 4.95163+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.95163, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@15", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "56(27\u20134\u221a3)/227 = 4.95163+", "metric_symbol": "s", "metric_value": 4.95163, "reference_text": "15. | | s = 56(27\u20134\u221a3)/227 = 4.95163+ | Found by Jonathan Viquerat in April 2026.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/15.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "15. | | s = 56(27\u20134\u221a3)/227 = 4.95163+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "1b00a25adb5cc2e4", "improvement": -0.0003568532282001158, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.185835293301636e-06, "max_pair_overlap_depth": -6.286085527285046e-05, "metric_expression": "4.95199", "metric_symbol": "s", "metric_value": 4.9519868532282, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.95163, "setup": "hexindom", "side": 9.9039737064564, "solution_path": "data/solutions/generated/hexindom_015_1b00a25adb5cc2e4.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/1b00a25adb5cc2e4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@16", "density": 0.7697888241254879, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/17.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "3\u221a3 = 5.19615+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.19615, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@16", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3\u221a3 = 5.19615+", "metric_symbol": "s", "metric_value": 5.19615, "reference_text": "16.-17. | | s = 3\u221a3 = 5.19615+ | Trivial.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/17.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "16.-17. | | s = 3\u221a3 = 5.19615+ | Trivial.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "f690e0f9c2986401", "improvement": -4.135307393138277e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.00251676779817e-06, "max_pair_overlap_depth": -1.257294679668508e-05, "metric_expression": "5.19619", "metric_symbol": "s", "metric_value": 5.196191353073932, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.19615, "setup": "hexindom", "side": 10.392382706147863, "solution_path": "data/solutions/generated/hexindom_016_f690e0f9c2986401.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/f690e0f9c2986401.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@17", "density": 0.8178722029268405, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/17.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "3\u221a3 = 5.19615+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.19615, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@17", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3\u221a3 = 5.19615+", "metric_symbol": "s", "metric_value": 5.19615, "reference_text": "16.-17. | | s = 3\u221a3 = 5.19615+ | Trivial.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/17.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "16.-17. | | s = 3\u221a3 = 5.19615+ | Trivial.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "fe395a0426baa674", "improvement": -0.00013164134761289858, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.712543075160227e-06, "max_pair_overlap_depth": -2.5961203853341885e-08, "metric_expression": "5.19628", "metric_symbol": "s", "metric_value": 5.196281641347613, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.19615, "setup": "hexindom", "side": 10.392563282695226, "solution_path": "data/solutions/generated/hexindom_017_fe395a0426baa674.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fe395a0426baa674.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@18", "density": 0.7993930048940154, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/18.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "31(4\u2013\u221a3)/13 = 5.40818+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.40818, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@18", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "31(4\u2013\u221a3)/13 = 5.40818+", "metric_symbol": "s", "metric_value": 5.40818, "reference_text": "18. | | s = 31(4\u2013\u221a3)/13 = 5.40818+ | Found by Jonathan Viquerat in April 2026.", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/18.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "18. | | s = 31(4\u2013\u221a3)/13 = 5.40818+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "24be62ab2c11bd5c", "improvement": -0.0001978579530232949, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.006386418315856e-06, "max_pair_overlap_depth": -4.2927090152522496e-05, "metric_expression": "5.40838", "metric_symbol": "s", "metric_value": 5.408377857953023, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.40818, "setup": "hexindom", "side": 10.816755715906046, "solution_path": "data/solutions/generated/hexindom_018_24be62ab2c11bd5c.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/24be62ab2c11bd5c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@19", "density": 0.7722825822258855, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/19.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "5.6530+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.653, "friedman_reference": {"analytical_or_proved": false, "case": "hexindom@19", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.6530+", "metric_symbol": "s", "metric_value": 5.653, "reference_text": "19. | | s = 5.6530+ | Found by Jonathan Viquerat in April 2026.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/19.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "19. | | s = 5.6530+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "ca10088586ed125c", "improvement": -0.0002686721147275861, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.885998928396475e-06, "max_pair_overlap_depth": -2.764579666703071e-05, "metric_expression": "5.65327", "metric_symbol": "s", "metric_value": 5.653268672114727, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.653, "setup": "hexindom", "side": 11.306537344229454, "solution_path": "data/solutions/generated/hexindom_019_ca10088586ed125c.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/ca10088586ed125c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexindom@20", "density": 0.7718745532343245, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/20.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "5.8012+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.8012, "friedman_reference": {"analytical_or_proved": false, "case": "hexindom@20", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.8012+", "metric_symbol": "s", "metric_value": 5.8012, "reference_text": "20. | | s = 5.8012+ | Found by Jonathan Viquerat in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/20.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "20. | | s = 5.8012+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "49ce661db444ec25", "improvement": -0.0004641087628032281, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.575652109486185e-06, "max_pair_overlap_depth": -1.0063478294242145e-05, "metric_expression": "5.80166", "metric_symbol": "s", "metric_value": 5.801664108762803, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.8012, "setup": "hexindom", "side": 5.801664108762803, "solution_path": "data/solutions/generated/hexindom_020_49ce661db444ec25.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/49ce661db444ec25.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@1", "density": 0.9999864500659439, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/1.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "1. | | s = 1 | Trivial.", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/1.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "1. | | s = 1 | Trivial.", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "fd6c6df7b6a9fd4f", "improvement": -6.7750358792029175e-06, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.374323059598708e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00001", "metric_symbol": "s", "metric_value": 1.0000067750358792, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0, "setup": "hexinhex", "side": 1.0000067750358792, "solution_path": "data/solutions/generated/hexinhex_001_fd6c6df7b6a9fd4f.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fd6c6df7b6a9fd4f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@2", "density": 0.49998387949392886, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/2.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "2. | | s = 2 | Trivial.", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/2.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "2. | | s = 2 | Trivial.", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "4c0cc36e970c84b1", "improvement": -3.2241791775522444e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.177681809212628e-06, "max_pair_overlap_depth": -3.6384174144801684e-05, "metric_expression": "2.00003", "metric_symbol": "s", "metric_value": 2.0000322417917755, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "hexinhex", "side": 2.0000322417917755, "solution_path": "data/solutions/generated/hexinhex_002_4c0cc36e970c84b1.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4c0cc36e970c84b1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@3", "density": 0.7499882422262695, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/3.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "3. | | s = 2 | Trivial.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/3.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "3. | | s = 2 | Trivial.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "2e22e12b6d11de4d", "improvement": -1.5677215970288927e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.457810528986454e-06, "max_pair_overlap_depth": -9.916736621090457e-06, "metric_expression": "2.00002", "metric_symbol": "s", "metric_value": 2.0000156772159703, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "hexinhex", "side": 2.0000156772159703, "solution_path": "data/solutions/generated/hexinhex_003_2e22e12b6d11de4d.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2e22e12b6d11de4d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@4", "density": 0.7346808358825446, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/4.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "7/3 = 2.333+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.333, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7/3 = 2.333+", "metric_symbol": "s", "metric_value": 2.333, "reference_text": "4. | | s = 7/3 = 2.333+ | Trivial.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/4.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "4. | | s = 7/3 = 2.333+ | Trivial.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "73ddab3f27a95ffe", "improvement": -0.00035404329556776304, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.969618748468479e-06, "max_pair_overlap_depth": -1.5832978702556844e-05, "metric_expression": "2.33335", "metric_symbol": "s", "metric_value": 2.333354043295568, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.333, "setup": "hexinhex", "side": 2.333354043295568, "solution_path": "data/solutions/generated/hexinhex_004_73ddab3f27a95ffe.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/73ddab3f27a95ffe.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@5", "density": 0.7031111089487749, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "March 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/5.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "8/3 = 2.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.666, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@5", "credit": "Erich Friedman", "date": "March 2015", "family_updated": "", "metric_expression": "8/3 = 2.666+", "metric_symbol": "s", "metric_value": 2.666, "reference_text": "5. | | s = 8/3 = 2.666+ | Found by Erich Friedman | in March 2015.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/5.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "5. | | s = 8/3 = 2.666+ | Found by Erich Friedman | in March 2015.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "f45300c86d046ee7", "improvement": -0.0006930086059693252, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5309295562371306e-06, "max_pair_overlap_depth": -6.88869177611906e-06, "metric_expression": "2.66669", "metric_symbol": "s", "metric_value": 2.6666930086059693, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.666, "setup": "hexinhex", "side": 2.6666930086059693, "solution_path": "data/solutions/generated/hexinhex_005_f45300c86d046ee7.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f45300c86d046ee7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@6", "density": 0.7199337426480286, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/7.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5/\u221a3 = 2.886+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.886, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@6", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "5/\u221a3 = 2.886+", "metric_symbol": "s", "metric_value": 2.886, "reference_text": "6-7. | | s = 5/\u221a3 = 2.886+ | Found by Maurizio Morandi | in April 2015.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/7.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "6-7. | | s = 5/\u221a3 = 2.886+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "06b9c79146f8c119", "improvement": -0.0008841804633918926, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.09670685382364e-06, "max_pair_overlap_depth": -5.069396536394599e-05, "metric_expression": "2.88688", "metric_symbol": "s", "metric_value": 2.886884180463392, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.886, "setup": "hexinhex", "side": 2.886884180463392, "solution_path": "data/solutions/generated/hexinhex_006_06b9c79146f8c119.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/06b9c79146f8c119.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@7", "density": 0.8399730426517898, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/7.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5/\u221a3 = 2.886+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.886, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@7", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "5/\u221a3 = 2.886+", "metric_symbol": "s", "metric_value": 2.886, "reference_text": "6-7. | | s = 5/\u221a3 = 2.886+ | Found by Maurizio Morandi | in April 2015.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/7.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "6-7. | | s = 5/\u221a3 = 2.886+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "d2a43d4ca2836467", "improvement": -0.0007976679923635821, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.6806989039887556e-06, "max_pair_overlap_depth": -2.9189085207215193e-05, "metric_expression": "2.88680", "metric_symbol": "s", "metric_value": 2.8867976679923637, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.886, "setup": "hexinhex", "side": 2.8867976679923637, "solution_path": "data/solutions/generated/hexinhex_007_d2a43d4ca2836467.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/d2a43d4ca2836467.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@8", "density": 0.728171041468435, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/8.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "(51+5\u221a3)/18 = 3.314+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.314, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@8", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "(51+5\u221a3)/18 = 3.314+", "metric_symbol": "s", "metric_value": 3.314, "reference_text": "8. | | s = (51+5\u221a3)/18 = 3.314+ | Found by Maurizio Morandi | in April 2015.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/8.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "8. | | s = (51+5\u221a3)/18 = 3.314+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "8d9e02005a596470", "improvement": -0.0005783682856219663, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.71875974277703e-06, "max_pair_overlap_depth": -2.9528072121864923e-05, "metric_expression": "3.31458", "metric_symbol": "s", "metric_value": 3.314578368285622, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.314, "setup": "hexinhex", "side": 3.314578368285622, "solution_path": "data/solutions/generated/hexinhex_008_8d9e02005a596470.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/8d9e02005a596470.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@9", "density": 0.683335999337671, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/9.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "(24+5\u221a3)/9 = 3.628+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.628, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@9", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "(24+5\u221a3)/9 = 3.628+", "metric_symbol": "s", "metric_value": 3.628, "reference_text": "9. | | s = (24+5\u221a3)/9 = 3.628+ | Found by Maurizio Morandi | in April 2015.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/9.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "9. | | s = (24+5\u221a3)/9 = 3.628+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "21c530fbe612f27c", "improvement": -0.0011431939654920065, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.454488069704802e-06, "max_pair_overlap_depth": -2.836888994439679e-05, "metric_expression": "3.62914", "metric_symbol": "s", "metric_value": 3.629143193965492, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.628, "setup": "hexinhex", "side": 3.629143193965492, "solution_path": "data/solutions/generated/hexinhex_009_21c530fbe612f27c.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/21c530fbe612f27c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@10", "density": 0.7179574622038186, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/10.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "2 + \u221a3 = 3.732+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.732, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@10", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "2 + \u221a3 = 3.732+", "metric_symbol": "s", "metric_value": 3.732, "reference_text": "10. | | s = 2 + \u221a3 = 3.732+ | Found by Maurizio Morandi | in April 2015.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/10.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "10. | | s = 2 + \u221a3 = 3.732+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "fd5a10d8a6354174", "improvement": -7.740911509834802e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.847623197892974e-06, "max_pair_overlap_depth": -2.6880918478910942e-06, "metric_expression": "3.73208", "metric_symbol": "s", "metric_value": 3.7320774091150986, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.732, "setup": "hexinhex", "side": 3.7320774091150986, "solution_path": "data/solutions/generated/hexinhex_010_fd5a10d8a6354174.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/fd5a10d8a6354174.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@11", "density": 0.7131589068275569, "friedman_analytical_or_proved": true, "friedman_credit": "Johann Schellhorn", "friedman_date": "July 2025", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/11.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "2+10/3\u221a3 = 3.92450+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.9245, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@11", "credit": "Johann Schellhorn", "date": "July 2025", "family_updated": "", "metric_expression": "2+10/3\u221a3 = 3.92450+", "metric_symbol": "s", "metric_value": 3.9245, "reference_text": "11. | | s = 2+10/3\u221a3 = 3.92450+ | Found by Johann Schellhorn | in July 2025. | | | |", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/11.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "11. | | s = 2+10/3\u221a3 = 3.92450+ | Found by Johann Schellhorn | in July 2025. | | | |", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "cf10a9116bb04a72", "improvement": -0.0028823793751930893, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3732870796469545e-06, "max_pair_overlap_depth": -1.938912890975253e-05, "metric_expression": "3.92738", "metric_symbol": "s", "metric_value": 3.927382379375193, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.9245, "setup": "hexinhex", "side": 3.927382379375193, "solution_path": "data/solutions/generated/hexinhex_011_cf10a9116bb04a72.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Johann Schellhorn", "svg_path": "data/svg/cf10a9116bb04a72.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@12", "density": 0.7723090107470407, "friedman_analytical_or_proved": true, "friedman_credit": "Alexander Novikov et al", "friedman_date": "June 2025", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/12.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "3.94164+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.94164, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@12", "credit": "Alexander Novikov et al", "date": "June 2025", "family_updated": "", "metric_expression": "3.94164+", "metric_symbol": "s", "metric_value": 3.94164, "reference_text": "12. | | s = 3.94164+ | Found by Alexander Novikov et al | in June 2025. | Improved slightly by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/12.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "12. | | s = 3.94164+ | Found by Alexander Novikov et al | in June 2025. | Improved slightly by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "bdc671b076ad0ffe", "improvement": -0.0001644193364005453, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.69077332163481e-06, "max_pair_overlap_depth": -4.133319357890741e-07, "metric_expression": "3.94180", "metric_symbol": "s", "metric_value": 3.9418044193364006, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.94164, "setup": "hexinhex", "side": 3.9418044193364006, "solution_path": "data/solutions/generated/hexinhex_012_bdc671b076ad0ffe.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Alexander Novikov et al", "svg_path": "data/svg/bdc671b076ad0ffe.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@13", "density": 0.8124849152838151, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/13.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@13", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "13 | | s = 4 | Trivial. | | | | |", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/13.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "13 | | s = 4 | Trivial. | | | | |", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "1ac594ff5f07e3d9", "improvement": -3.7132126112560115e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.027960154271181e-06, "max_pair_overlap_depth": -5.440029557712833e-06, "metric_expression": "4.00004", "metric_symbol": "s", "metric_value": 4.000037132126113, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "hexinhex", "side": 4.000037132126113, "solution_path": "data/solutions/generated/hexinhex_013_1ac594ff5f07e3d9.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/1ac594ff5f07e3d9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@14", "density": 0.7678833113230449, "friedman_analytical_or_proved": false, "friedman_credit": "Dominik Kamp", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/14.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "4.26899+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.26899, "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@14", "credit": "Dominik Kamp", "date": "April 2026", "family_updated": "", "metric_expression": "4.26899+", "metric_symbol": "s", "metric_value": 4.26899, "reference_text": "14. | | s = 4.26899+ | Found by Dominik Kamp | in April 2026.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/14.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "14. | | s = 4.26899+ | Found by Dominik Kamp | in April 2026.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "f86af09d863723a2", "improvement": -0.0008972113481453547, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.52825412539728e-06, "max_pair_overlap_depth": -2.1736642138669993e-05, "metric_expression": "4.26989", "metric_symbol": "s", "metric_value": 4.269887211348145, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.26899, "setup": "hexinhex", "side": 4.269887211348145, "solution_path": "data/solutions/generated/hexinhex_014_f86af09d863723a2.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Dominik Kamp", "svg_path": "data/svg/f86af09d863723a2.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@15", "density": 0.7579263950656321, "friedman_analytical_or_proved": true, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/15.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "4.44727+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.44727, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@15", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "4.44727+", "metric_symbol": "s", "metric_value": 4.44727, "reference_text": "15. | | s = 4.44727+ | Found by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/15.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "15. | | s = 4.44727+ | Found by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "536c412ab623afbc", "improvement": -0.0014196922804980971, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.564398033090082e-06, "max_pair_overlap_depth": -2.5468766324809877e-06, "metric_expression": "4.44869", "metric_symbol": "s", "metric_value": 4.448689692280498, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.44727, "setup": "hexinhex", "side": 4.448689692280498, "solution_path": "data/solutions/generated/hexinhex_015_536c412ab623afbc.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/536c412ab623afbc.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@16", "density": 0.7803167098453238, "friedman_analytical_or_proved": true, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/16.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "4.52746+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.52746, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@16", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "4.52746+", "metric_symbol": "s", "metric_value": 4.52746, "reference_text": "16. | | s = 4.52746+ | Found by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/16.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "16. | | s = 4.52746+ | Found by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "17cb432379a18c18", "improvement": -0.0007289208955771898, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.541715273786282e-06, "max_pair_overlap_depth": -2.3206185233659582e-05, "metric_expression": "4.52819", "metric_symbol": "s", "metric_value": 4.528188920895577, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.52746, "setup": "hexinhex", "side": 4.528188920895577, "solution_path": "data/solutions/generated/hexinhex_016_17cb432379a18c18.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/17cb432379a18c18.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@17", "density": 0.7985097849440846, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "February 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/17.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "4.6136+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.6136, "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@17", "credit": "Timo Berthold et al", "date": "February 2026", "family_updated": "", "metric_expression": "4.6136+", "metric_symbol": "s", "metric_value": 4.6136, "reference_text": "17. | | s = 4.6136+ | Found by Timo Berthold et al | and Alex Kravatsky et al | in February 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/17.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "17. | | s = 4.6136+ | Found by Timo Berthold et al | and Alex Kravatsky et al | in February 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "df809b176b000a6d", "improvement": -0.0004717062665458016, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.290243996089003e-06, "max_pair_overlap_depth": -4.642493336670661e-05, "metric_expression": "4.61407", "metric_symbol": "s", "metric_value": 4.614071706266546, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.6136, "setup": "hexinhex", "side": 4.614071706266546, "solution_path": "data/solutions/generated/hexinhex_017_df809b176b000a6d.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/df809b176b000a6d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@18", "density": 0.8437116836701195, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/19.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "8/\u221a3 = 4.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.618, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@18", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "8/\u221a3 = 4.618+", "metric_symbol": "s", "metric_value": 4.618, "reference_text": "18-19. | | s = 8/\u221a3 = 4.618+ | Found by Maurizio Morandi | in April 2015. |", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/19.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "18-19. | | s = 8/\u221a3 = 4.618+ | Found by Maurizio Morandi | in April 2015. |", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "e320d39029a817c1", "improvement": -0.0009070314872570151, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.0443685916789865e-09, "max_pair_overlap_depth": -3.080790913889686e-05, "metric_expression": "4.61891", "metric_symbol": "s", "metric_value": 4.618907031487257, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.618, "setup": "hexinhex", "side": 4.618907031487257, "solution_path": "data/solutions/generated/hexinhex_018_e320d39029a817c1.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/e320d39029a817c1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@19", "density": 0.8905854861904853, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/19.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "8/\u221a3 = 4.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.618, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@19", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "8/\u221a3 = 4.618+", "metric_symbol": "s", "metric_value": 4.618, "reference_text": "18-19. | | s = 8/\u221a3 = 4.618+ | Found by Maurizio Morandi | in April 2015. |", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/19.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "18-19. | | s = 8/\u221a3 = 4.618+ | Found by Maurizio Morandi | in April 2015. |", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "4b11a1a09f2a5b70", "improvement": -0.0009046166982376036, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.772589960788204e-06, "max_pair_overlap_depth": -3.612300978728111e-05, "metric_expression": "4.61890", "metric_symbol": "s", "metric_value": 4.618904616698238, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.618, "setup": "hexinhex", "side": 4.618904616698238, "solution_path": "data/solutions/generated/hexinhex_019_4b11a1a09f2a5b70.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/4b11a1a09f2a5b70.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@20", "density": 0.7999496824068819, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/21.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@20", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "20-21. | | s = 5 | Trivial. | |", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/21.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "20-21. | | s = 5 | Trivial. | |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "2830e2dec38749ef", "improvement": -0.00015724989644283482, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.445035556179186e-06, "max_pair_overlap_depth": -1.6551811964360397e-06, "metric_expression": "5.00016", "metric_symbol": "s", "metric_value": 5.000157249896443, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "hexinhex", "side": 5.000157249896443, "solution_path": "data/solutions/generated/hexinhex_020_2830e2dec38749ef.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2830e2dec38749ef.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@21", "density": 0.8399773177134815, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/21.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@21", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "20-21. | | s = 5 | Trivial. | |", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/21.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "20-21. | | s = 5 | Trivial. | |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "4a431b677be571c9", "improvement": -6.750817229672634e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5389132812752791e-06, "max_pair_overlap_depth": -3.519050301936666e-06, "metric_expression": "5.00007", "metric_symbol": "s", "metric_value": 5.000067508172297, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "hexinhex", "side": 5.000067508172297, "solution_path": "data/solutions/generated/hexinhex_021_4a431b677be571c9.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4a431b677be571c9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@22", "density": 0.7873804951730353, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/22.gif", "friedman_metric_expression": "(39+8\u221a3)/10 = 5.285+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.285, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@22", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "(39+8\u221a3)/10 = 5.285+", "metric_symbol": "s", "metric_value": 5.285, "reference_text": "22. | | s = (39+8\u221a3)/10 = 5.285+ | Found by Maurizio Morandi | in April 2015.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/22.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "22. | | s = (39+8\u221a3)/10 = 5.285+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "003a713177cb171c", "improvement": -0.0009008698048820207, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.575542655386671e-06, "max_pair_overlap_depth": -1.0001604064946434e-05, "metric_expression": "5.28590", "metric_symbol": "s", "metric_value": 5.285900869804882, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.285, "setup": "hexinhex", "side": 5.285900869804882, "solution_path": "data/solutions/generated/hexinhex_022_003a713177cb171c.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/003a713177cb171c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@23", "density": 0.7886255260759908, "friedman_analytical_or_proved": false, "friedman_credit": "Alex Kravatsky et al", "friedman_date": "February 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/23.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5.400+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.4, "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@23", "credit": "Alex Kravatsky et al", "date": "February 2026", "family_updated": "", "metric_expression": "5.400+", "metric_symbol": "s", "metric_value": 5.4, "reference_text": "23. | | s = 5.400+ | Found by Alex Kravatsky et al | in February 2026.", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/23.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "23. | | s = 5.400+ | Found by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "e0a82406671652d2", "improvement": -0.00043201188015196124, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4507055603926188e-06, "max_pair_overlap_depth": -1.2502329530761358e-05, "metric_expression": "5.40043", "metric_symbol": "s", "metric_value": 5.400432011880152, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.4, "setup": "hexinhex", "side": 5.400432011880152, "solution_path": "data/solutions/generated/hexinhex_023_e0a82406671652d2.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Alex Kravatsky et al", "svg_path": "data/svg/e0a82406671652d2.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@24", "density": 0.797738483588153, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/24.gif", "friedman_metric_expression": "19\u221a3/6 = 5.484+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.484, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@24", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "19\u221a3/6 = 5.484+", "metric_symbol": "s", "metric_value": 5.484, "reference_text": "24. | | s = 19\u221a3/6 = 5.484+ | Found by Maurizio Morandi | in April 2015.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/24.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "24. | | s = 19\u221a3/6 = 5.484+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "c96649ff013a9455", "improvement": -0.00098379990619879, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.588312731744032e-06, "max_pair_overlap_depth": -4.485524381903083e-05, "metric_expression": "5.48498", "metric_symbol": "s", "metric_value": 5.484983799906199, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.484, "setup": "hexinhex", "side": 5.484983799906199, "solution_path": "data/solutions/generated/hexinhex_024_c96649ff013a9455.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/c96649ff013a9455.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@25", "density": 0.7898226009433017, "friedman_analytical_or_proved": false, "friedman_credit": "Alex Kravatsky et al", "friedman_date": "February 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/25.gif", "friedman_metric_expression": "5.62390+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.6239, "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@25", "credit": "Alex Kravatsky et al", "date": "February 2026", "family_updated": "", "metric_expression": "5.62390+", "metric_symbol": "s", "metric_value": 5.6239, "reference_text": "25. | | s = 5.62390+ | Found by Alex Kravatsky et al | in February 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/25.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "25. | | s = 5.62390+ | Found by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "4ac680367830872b", "improvement": -0.002171223468265815, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8833146044517865e-07, "max_pair_overlap_depth": -2.7316686140865798e-05, "metric_expression": "5.62607", "metric_symbol": "s", "metric_value": 5.626071223468266, "n": 25, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.6239, "setup": "hexinhex", "side": 5.626071223468266, "solution_path": "data/solutions/generated/hexinhex_025_4ac680367830872b.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Alex Kravatsky et al", "svg_path": "data/svg/4ac680367830872b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@26", "density": 0.7965297187067493, "friedman_analytical_or_proved": false, "friedman_credit": "Alex Kravatsky et al", "friedman_date": "February 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/26.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5.70972+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.70972, "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@26", "credit": "Alex Kravatsky et al", "date": "February 2026", "family_updated": "", "metric_expression": "5.70972+", "metric_symbol": "s", "metric_value": 5.70972, "reference_text": "26. | | s = 5.70972+ | Found by Alex Kravatsky et al | in February 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/26.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "26. | | s = 5.70972+ | Found by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "8ed8c9fb94f18119", "improvement": -0.0035622784284790043, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9624619447323227e-06, "max_pair_overlap_depth": -1.2105377510118753e-05, "metric_expression": "5.71328", "metric_symbol": "s", "metric_value": 5.713282278428479, "n": 26, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.70972, "setup": "hexinhex", "side": 5.713282278428479, "solution_path": "data/solutions/generated/hexinhex_026_8ed8c9fb94f18119.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Alex Kravatsky et al", "svg_path": "data/svg/8ed8c9fb94f18119.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@27", "density": 0.82621496088318, "friedman_analytical_or_proved": false, "friedman_credit": "Alex Kravatsky et al", "friedman_date": "February 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/27.gif", "friedman_metric_expression": "5.71415+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.71415, "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@27", "credit": "Alex Kravatsky et al", "date": "February 2026", "family_updated": "", "metric_expression": "5.71415+", "metric_symbol": "s", "metric_value": 5.71415, "reference_text": "27. | | s = 5.71415+ | Found by Alex Kravatsky et al | in February 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/27.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "27. | | s = 5.71415+ | Found by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "4e9628a9f49def85", "improvement": -0.0024177469948583052, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.56387472302006e-06, "max_pair_overlap_depth": -5.6693937127594296e-05, "metric_expression": "5.71657", "metric_symbol": "s", "metric_value": 5.716567746994858, "n": 27, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.71415, "setup": "hexinhex", "side": 5.716567746994858, "solution_path": "data/solutions/generated/hexinhex_027_4e9628a9f49def85.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Alex Kravatsky et al", "svg_path": "data/svg/4e9628a9f49def85.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@28", "density": 0.8011452289129106, "friedman_analytical_or_proved": false, "friedman_credit": "Alex Kravatsky et al", "friedman_date": "February 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/28.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5.90893+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.90893, "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@28", "credit": "Alex Kravatsky et al", "date": "February 2026", "family_updated": "", "metric_expression": "5.90893+", "metric_symbol": "s", "metric_value": 5.90893, "reference_text": "28. | | s = 5.90893+ | Found by Alex Kravatsky et al | in February 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/28.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "28. | | s = 5.90893+ | Found by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "ba522f97744aed1a", "improvement": -0.002919783094952777, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.424277796084301e-06, "max_pair_overlap_depth": -3.209757705180394e-05, "metric_expression": "5.91185", "metric_symbol": "s", "metric_value": 5.911849783094953, "n": 28, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.90893, "setup": "hexinhex", "side": 5.911849783094953, "solution_path": "data/solutions/generated/hexinhex_028_ba522f97744aed1a.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Alex Kravatsky et al", "svg_path": "data/svg/ba522f97744aed1a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@29", "density": 0.8055194808067645, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@29", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "29-31. | | s = 6 | Trivial. |", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "29-31. | | s = 6 | Trivial. |", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "dba25a591b8992bc", "improvement": -0.00013435185279675466, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.231638156342025e-06, "max_pair_overlap_depth": -6.307567709251671e-06, "metric_expression": "6.00013", "metric_symbol": "s", "metric_value": 6.000134351852797, "n": 29, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.0, "setup": "hexinhex", "side": 6.000134351852797, "solution_path": "data/solutions/generated/hexinhex_029_dba25a591b8992bc.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/dba25a591b8992bc.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@30", "density": 0.8333115277207207, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@30", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "29-31. | | s = 6 | Trivial. |", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "29-31. | | s = 6 | Trivial. |", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "100db5f9f28894cc", "improvement": -7.850174601031767e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7272811687230685e-07, "max_pair_overlap_depth": -4.232627726974414e-06, "metric_expression": "6.00008", "metric_symbol": "s", "metric_value": 6.00007850174601, "n": 30, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.0, "setup": "hexinhex", "side": 6.00007850174601, "solution_path": "data/solutions/generated/hexinhex_030_100db5f9f28894cc.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/100db5f9f28894cc.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinhex@31", "density": 0.8610814974425702, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@31", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "29-31. | | s = 6 | Trivial. |", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "29-31. | | s = 6 | Trivial. |", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "d503e2b08764a35e", "improvement": -0.0001031728611780025, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0312532034243658e-06, "max_pair_overlap_depth": -8.80446118189937e-06, "metric_expression": "6.00010", "metric_symbol": "s", "metric_value": 6.000103172861178, "n": 31, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.0, "setup": "hexinhex", "side": 6.000103172861178, "solution_path": "data/solutions/generated/hexinhex_031_d503e2b08764a35e.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d503e2b08764a35e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@1", "density": 0.6961402306608654, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/1.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(1+\u221a3)/\u221a2 = 1.931+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.931, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(1+\u221a3)/\u221a2 = 1.931+", "metric_symbol": "s", "metric_value": 1.931, "reference_text": "s = (1+\u221a3)/\u221a2 = 1.931+ | Trivial. |", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/1.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = (1+\u221a3)/\u221a2 = 1.931+ | Trivial. |", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "1cfff36edc20de72", "improvement": -0.000868569514983486, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8875760125229138e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.93187", "metric_symbol": "s", "metric_value": 1.9318685695149835, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.931, "setup": "hexinsqu", "side": 1.9318685695149835, "solution_path": "data/solutions/generated/hexinsqu_001_1cfff36edc20de72.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/1cfff36edc20de72.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@2", "density": 0.5214616218348717, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/2.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(1+2\u221a3)/\u221a2 = 3.156+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.156, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(1+2\u221a3)/\u221a2 = 3.156+", "metric_symbol": "s", "metric_value": 3.156, "reference_text": "s = (1+2\u221a3)/\u221a2 = 3.156+ | Trivial. |", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/2.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = (1+2\u221a3)/\u221a2 = 3.156+ | Trivial. |", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "ff43d04cda84c856", "improvement": -0.0006742348964126066, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2405599436706566e-05, "max_pair_overlap_depth": -1.629836664562411e-05, "metric_expression": "3.15667", "metric_symbol": "s", "metric_value": 3.1566742348964127, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.156, "setup": "hexinsqu", "side": 3.1566742348964127, "solution_path": "data/solutions/generated/hexinsqu_002_ff43d04cda84c856.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/ff43d04cda84c856.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@3", "density": 0.6391474518535116, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/3.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "9(6\u2013\u221a3)/11 = 3.491+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.491, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@3", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "9(6\u2013\u221a3)/11 = 3.491+", "metric_symbol": "s", "metric_value": 3.491, "reference_text": "s = 9(6\u2013\u221a3)/11 = 3.491+ | Found by Maurizio Morandi | in March 2026.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/3.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 9(6\u2013\u221a3)/11 = 3.491+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "9525d376eafe3c9c", "improvement": -0.0010949178753740796, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.084317521962771e-06, "max_pair_overlap_depth": -8.912711465047796e-05, "metric_expression": "3.49209", "metric_symbol": "s", "metric_value": 3.492094917875374, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.491, "setup": "hexinsqu", "side": 3.492094917875374, "solution_path": "data/solutions/generated/hexinsqu_003_9525d376eafe3c9c.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/9525d376eafe3c9c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@4", "density": 0.748920266367969, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/4.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(7\u2013\u221a3)/\u221a2 = 3.725+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.725, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@4", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "(7\u2013\u221a3)/\u221a2 = 3.725+", "metric_symbol": "s", "metric_value": 3.725, "reference_text": "s = (7\u2013\u221a3)/\u221a2 = 3.725+ | Found by Maurizio Morandi | in March 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/4.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = (7\u2013\u221a3)/\u221a2 = 3.725+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "33ddf042dd02d191", "improvement": -0.00010181398140396425, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.577436704193743e-06, "max_pair_overlap_depth": -5.3695636117151446e-05, "metric_expression": "3.72510", "metric_symbol": "s", "metric_value": 3.725101813981404, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.725, "setup": "hexinsqu", "side": 3.725101813981404, "solution_path": "data/solutions/generated/hexinsqu_004_33ddf042dd02d191.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/33ddf042dd02d191.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@5", "density": 0.6595778437263831, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/5.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "7(3\u2013\u221a3)/2 = 4.437+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.437, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@5", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "7(3\u2013\u221a3)/2 = 4.437+", "metric_symbol": "s", "metric_value": 4.437, "reference_text": "s = 7(3\u2013\u221a3)/2 = 4.437+ | Found by Maurizio Morandi | in March 2026.", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/5.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 7(3\u2013\u221a3)/2 = 4.437+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "38f2dc66ba941978", "improvement": -0.0009041295527705273, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.5429089870357805e-06, "max_pair_overlap_depth": -1.5874388830661168e-05, "metric_expression": "4.43790", "metric_symbol": "s", "metric_value": 4.437904129552771, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.437, "setup": "hexinsqu", "side": 4.437904129552771, "solution_path": "data/solutions/generated/hexinsqu_005_38f2dc66ba941978.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/38f2dc66ba941978.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@6", "density": 0.6726426228424899, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/6.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "20(9\u20132\u221a3)/23 = 4.813+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.813, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@6", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "20(9\u20132\u221a3)/23 = 4.813+", "metric_symbol": "s", "metric_value": 4.813, "reference_text": "s = 20(9\u20132\u221a3)/23 = 4.813+ | Found by Maurizio Morandi | in March 2026.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/6.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 20(9\u20132\u221a3)/23 = 4.813+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "31291850c6887128", "improvement": -0.0010364766922137392, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6841191118288626e-06, "max_pair_overlap_depth": -6.696342223433405e-05, "metric_expression": "4.81404", "metric_symbol": "s", "metric_value": 4.8140364766922135, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.813, "setup": "hexinsqu", "side": 4.8140364766922135, "solution_path": "data/solutions/generated/hexinsqu_006_31291850c6887128.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/31291850c6887128.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@7", "density": 0.6735519886722757, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/8.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3\u221a3 = 5.196+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.196, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3\u221a3 = 5.196+", "metric_symbol": "s", "metric_value": 5.196, "reference_text": "s = 3\u221a3 = 5.196+ | Trivial. |", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/8.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 3\u221a3 = 5.196+ | Trivial. |", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "a4c696c014daab3c", "improvement": -0.00024239437505713113, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.667089398456013e-06, "max_pair_overlap_depth": -2.5433886154724306e-05, "metric_expression": "5.19624", "metric_symbol": "s", "metric_value": 5.196242394375057, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.196, "setup": "hexinsqu", "side": 5.196242394375057, "solution_path": "data/solutions/generated/hexinsqu_007_a4c696c014daab3c.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a4c696c014daab3c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@8", "density": 0.7697897324076285, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/8.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3\u221a3 = 5.196+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.196, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@8", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3\u221a3 = 5.196+", "metric_symbol": "s", "metric_value": 5.196, "reference_text": "s = 3\u221a3 = 5.196+ | Trivial. |", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/8.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 3\u221a3 = 5.196+ | Trivial. |", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "b1252a56565d2ec0", "improvement": -0.00018828755551858478, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.033193436423119e-06, "max_pair_overlap_depth": -1.0846799477937097e-05, "metric_expression": "5.19619", "metric_symbol": "s", "metric_value": 5.196188287555518, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.196, "setup": "hexinsqu", "side": 5.196188287555518, "solution_path": "data/solutions/generated/hexinsqu_008_b1252a56565d2ec0.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b1252a56565d2ec0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@9", "density": 0.7678652278206223, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/9.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(13\u20133\u221a3)/\u221a2 = 5.518+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.518, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@9", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "(13\u20133\u221a3)/\u221a2 = 5.518+", "metric_symbol": "s", "metric_value": 5.518, "reference_text": "s = (13\u20133\u221a3)/\u221a2 = 5.518+ | Found by Maurizio Morandi | in March 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/9.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = (13\u20133\u221a3)/\u221a2 = 5.518+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "5b4ef4c36dcfba05", "improvement": -0.0002922521392276778, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.874395903155008e-06, "max_pair_overlap_depth": -4.408050364324012e-05, "metric_expression": "5.51829", "metric_symbol": "s", "metric_value": 5.5182922521392275, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.518, "setup": "hexinsqu", "side": 5.5182922521392275, "solution_path": "data/solutions/generated/hexinsqu_009_5b4ef4c36dcfba05.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/5b4ef4c36dcfba05.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@10", "density": 0.6970336920701942, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/10.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.10469+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.10469, "friedman_reference": {"analytical_or_proved": false, "case": "hexinsqu@10", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.10469+", "metric_symbol": "s", "metric_value": 6.10469, "reference_text": "s = 6.10469+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/10.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 6.10469+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "031f63d864cce10c", "improvement": -0.0004982231227232958, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5872663706407347e-08, "max_pair_overlap_depth": -2.0702587337384504e-05, "metric_expression": "6.10519", "metric_symbol": "s", "metric_value": 6.105188223122723, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.10469, "setup": "hexinsqu", "side": 6.105188223122723, "solution_path": "data/solutions/generated/hexinsqu_010_031f63d864cce10c.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/031f63d864cce10c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@11", "density": 0.7126826634184014, "friedman_analytical_or_proved": false, "friedman_credit": "Maksymilian Jankowski", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/11.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.33218+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.33218, "friedman_reference": {"analytical_or_proved": false, "case": "hexinsqu@11", "credit": "Maksymilian Jankowski", "date": "April 2026", "family_updated": "", "metric_expression": "6.33218+", "metric_symbol": "s", "metric_value": 6.33218, "reference_text": "s = 6.33218+ | Found by Maksymilian Jankowski | in April 2026.", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/11.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 6.33218+ | Found by Maksymilian Jankowski | in April 2026.", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "f0fe574f6ed3708e", "improvement": -0.0003052751802421483, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.734623780299186e-06, "max_pair_overlap_depth": -4.342994577477555e-05, "metric_expression": "6.33249", "metric_symbol": "s", "metric_value": 6.332485275180242, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.33218, "setup": "hexinsqu", "side": 6.332485275180242, "solution_path": "data/solutions/generated/hexinsqu_011_f0fe574f6ed3708e.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maksymilian Jankowski", "svg_path": "data/svg/f0fe574f6ed3708e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@12", "density": 0.7640646951117188, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/12.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "43(5\u2013\u221a3)/22 = 6.387+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.387, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@12", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "43(5\u2013\u221a3)/22 = 6.387+", "metric_symbol": "s", "metric_value": 6.387, "reference_text": "s = 43(5\u2013\u221a3)/22 = 6.387+ | Found by Maurizio Morandi | in March 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/12.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 43(5\u2013\u221a3)/22 = 6.387+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "70f372d7c2d3b859", "improvement": -0.0008027909827861279, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6841159701879252e-07, "max_pair_overlap_depth": -5.12694605685482e-05, "metric_expression": "6.38780", "metric_symbol": "s", "metric_value": 6.387802790982786, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.387, "setup": "hexinsqu", "side": 6.387802790982786, "solution_path": "data/solutions/generated/hexinsqu_012_70f372d7c2d3b859.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/70f372d7c2d3b859.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@13", "density": 0.7399851108638961, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/13.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.75558+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.75558, "friedman_reference": {"analytical_or_proved": false, "case": "hexinsqu@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.75558+", "metric_symbol": "s", "metric_value": 6.75558, "reference_text": "s = 6.75558+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/13.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 6.75558+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "96118c364cb8b05d", "improvement": -0.0003653626017898759, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.852098428599495e-06, "max_pair_overlap_depth": -2.731772979069902e-05, "metric_expression": "6.75595", "metric_symbol": "s", "metric_value": 6.75594536260179, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.75558, "setup": "hexinsqu", "side": 6.75594536260179, "solution_path": "data/solutions/generated/hexinsqu_013_96118c364cb8b05d.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/96118c364cb8b05d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@14", "density": 0.7577523835410583, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/14.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4\u221a3 = 6.928+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.928, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@14", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4\u221a3 = 6.928+", "metric_symbol": "s", "metric_value": 6.928, "reference_text": "s = 4\u221a3 = 6.928+ | Trivial. |", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/14.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 4\u221a3 = 6.928+ | Trivial. |", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "8b7d0912d343dc8b", "improvement": -0.00029395100744089575, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.277327734003356e-06, "max_pair_overlap_depth": -1.698575172759398e-05, "metric_expression": "6.92829", "metric_symbol": "s", "metric_value": 6.928293951007441, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.928, "setup": "hexinsqu", "side": 6.928293951007441, "solution_path": "data/solutions/generated/hexinsqu_014_8b7d0912d343dc8b.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/8b7d0912d343dc8b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@15", "density": 0.8035851614122442, "friedman_analytical_or_proved": false, "friedman_credit": "Maksymilian Jankowski", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/15.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.96367+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.96367, "friedman_reference": {"analytical_or_proved": false, "case": "hexinsqu@15", "credit": "Maksymilian Jankowski", "date": "April 2026", "family_updated": "", "metric_expression": "6.96367+", "metric_symbol": "s", "metric_value": 6.96367, "reference_text": "s = 6.96367+ | Found by Maksymilian Jankowski | in April 2026.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/15.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 6.96367+ | Found by Maksymilian Jankowski | in April 2026.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "0691341c684bd112", "improvement": -0.00027958460374311045, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.156990403080692e-07, "max_pair_overlap_depth": -5.813914172492218e-05, "metric_expression": "6.96395", "metric_symbol": "s", "metric_value": 6.963949584603743, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.96367, "setup": "hexinsqu", "side": 6.963949584603743, "solution_path": "data/solutions/generated/hexinsqu_015_0691341c684bd112.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maksymilian Jankowski", "svg_path": "data/svg/0691341c684bd112.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@16", "density": 0.7776171365327941, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/16.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(19\u20135\u221a3)/\u221a2 = 7.311+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.311, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@16", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "(19\u20135\u221a3)/\u221a2 = 7.311+", "metric_symbol": "s", "metric_value": 7.311, "reference_text": "s = (19\u20135\u221a3)/\u221a2 = 7.311+ | Found by Maurizio Morandi | in April 2026.", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/16.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = (19\u20135\u221a3)/\u221a2 = 7.311+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "03465bdf660197b5", "improvement": -0.000441730863443901, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8577837974051795e-06, "max_pair_overlap_depth": -3.069264188360066e-05, "metric_expression": "7.31144", "metric_symbol": "s", "metric_value": 7.311441730863444, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.311, "setup": "hexinsqu", "side": 7.311441730863444, "solution_path": "data/solutions/generated/hexinsqu_016_03465bdf660197b5.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/03465bdf660197b5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@17", "density": 0.7624212140097384, "friedman_analytical_or_proved": true, "friedman_credit": "Maksymilian Jankowski", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/17.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6(3\u2013\u221a3) = 7.60383+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.60383, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@17", "credit": "Maksymilian Jankowski", "date": "April 2026", "family_updated": "", "metric_expression": "6(3\u2013\u221a3) = 7.60383+", "metric_symbol": "s", "metric_value": 7.60383, "reference_text": "s = 6(3\u2013\u221a3) = 7.60383+ | Found by Maksymilian Jankowski | in April 2026.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/17.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 6(3\u2013\u221a3) = 7.60383+ | Found by Maksymilian Jankowski | in April 2026.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "080714e7afd39346", "improvement": -0.007366212799500005, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.378937853940429e-07, "max_pair_overlap_depth": -2.071443269358042e-05, "metric_expression": "7.61120", "metric_symbol": "s", "metric_value": 7.6111962127995, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.60383, "setup": "hexinsqu", "side": 7.6111962127995, "solution_path": "data/solutions/generated/hexinsqu_017_080714e7afd39346.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maksymilian Jankowski", "svg_path": "data/svg/080714e7afd39346.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@18", "density": 0.7882207738949305, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/18.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "32(9\u20132\u221a3)/23 = 7.702+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.702, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@18", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "32(9\u20132\u221a3)/23 = 7.702+", "metric_symbol": "s", "metric_value": 7.702, "reference_text": "s = 32(9\u20132\u221a3)/23 = 7.702+ | Found by Maurizio Morandi | in April 2026.", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/18.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 32(9\u20132\u221a3)/23 = 7.702+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "39a9a99a50ac7cba", "improvement": -0.0006161769446162424, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5800172874233264e-06, "max_pair_overlap_depth": -3.4078889656230515e-05, "metric_expression": "7.70262", "metric_symbol": "s", "metric_value": 7.702616176944616, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.702, "setup": "hexinsqu", "side": 7.702616176944616, "solution_path": "data/solutions/generated/hexinsqu_018_39a9a99a50ac7cba.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/39a9a99a50ac7cba.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@19", "density": 0.7859807737367643, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/19.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "25(3\u2013\u221a3)/4 = 7.924+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.924, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@19", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "25(3\u2013\u221a3)/4 = 7.924+", "metric_symbol": "s", "metric_value": 7.924, "reference_text": "s = 25(3\u2013\u221a3)/4 = 7.924+ | Found by Maurizio Morandi | in April 2026.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/19.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 25(3\u2013\u221a3)/4 = 7.924+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "10d9f5dde53d25ba", "improvement": -0.0009545892016307178, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.893895229540556e-06, "max_pair_overlap_depth": -3.6221248433054853e-05, "metric_expression": "7.92495", "metric_symbol": "s", "metric_value": 7.924954589201631, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.924, "setup": "hexinsqu", "side": 7.924954589201631, "solution_path": "data/solutions/generated/hexinsqu_019_10d9f5dde53d25ba.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/10d9f5dde53d25ba.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@20", "density": 0.8222377903171622, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/20.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "74(21\u20134\u221a3)/131 = 7.948+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.948, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@20", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "74(21\u20134\u221a3)/131 = 7.948+", "metric_symbol": "s", "metric_value": 7.948, "reference_text": "s = 74(21\u20134\u221a3)/131 = 7.948+ | Found by Maurizio Morandi | in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/20.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 74(21\u20134\u221a3)/131 = 7.948+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "3d530c0ac6e7e3b1", "improvement": -0.0015441274741458955, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.121785608292394e-06, "max_pair_overlap_depth": -4.480534023287852e-05, "metric_expression": "7.94954", "metric_symbol": "s", "metric_value": 7.949544127474146, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.948, "setup": "hexinsqu", "side": 7.949544127474146, "solution_path": "data/solutions/generated/hexinsqu_020_3d530c0ac6e7e3b1.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/3d530c0ac6e7e3b1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@21", "density": 0.7762795555424945, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/21.png", "friedman_metric_expression": "8.381+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.381, "friedman_reference": {"analytical_or_proved": false, "case": "hexinsqu@21", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "8.381+", "metric_symbol": "s", "metric_value": 8.381, "reference_text": "s = 8.381+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "21.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/21.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 8.381+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "21.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "5d369a2740fde606", "improvement": -0.002522153681731254, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.85221561134309e-06, "max_pair_overlap_depth": -3.989714052887727e-05, "metric_expression": "8.38352", "metric_symbol": "s", "metric_value": 8.383522153681731, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.381, "setup": "hexinsqu", "side": 8.383522153681731, "solution_path": "data/solutions/generated/hexinsqu_021_5d369a2740fde606.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/5d369a2740fde606.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@22", "density": 0.7770815845496467, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/22.png", "friedman_metric_expression": "8.568+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.568, "friedman_reference": {"analytical_or_proved": false, "case": "hexinsqu@22", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "8.568+", "metric_symbol": "s", "metric_value": 8.568, "reference_text": "s = 8.568+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "22.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/22.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 8.568+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "22.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "2f0b0033f0c66708", "improvement": -0.008379204155273712, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.913077650512531e-06, "max_pair_overlap_depth": -1.9097490104869053e-05, "metric_expression": "8.57638", "metric_symbol": "s", "metric_value": 8.576379204155273, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.568, "setup": "hexinsqu", "side": 8.576379204155273, "solution_path": "data/solutions/generated/hexinsqu_022_2f0b0033f0c66708.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/2f0b0033f0c66708.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@23", "density": 0.7967168154089415, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/23.png", "friedman_metric_expression": "5\u221a3 = 8.660+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.66, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@23", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5\u221a3 = 8.660+", "metric_symbol": "s", "metric_value": 8.66, "reference_text": "s = 5\u221a3 = 8.660+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "23.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/23.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 5\u221a3 = 8.660+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "23.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "31cc17034bd25d7b", "improvement": -0.0003983679346664104, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.345963802137002e-06, "max_pair_overlap_depth": -2.1349502021816846e-07, "metric_expression": "8.66040", "metric_symbol": "s", "metric_value": 8.660398367934667, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.66, "setup": "hexinsqu", "side": 8.660398367934667, "solution_path": "data/solutions/generated/hexinsqu_023_31cc17034bd25d7b.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/31cc17034bd25d7b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@24", "density": 0.8058676683577897, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/24.png", "friedman_metric_expression": "8.795+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.795, "friedman_reference": {"analytical_or_proved": false, "case": "hexinsqu@24", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "8.795+", "metric_symbol": "s", "metric_value": 8.795, "reference_text": "s = 8.795+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "24.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/24.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 8.795+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "24.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "941fbce4f50492c6", "improvement": -0.0012932219182459903, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0969884292392119e-07, "max_pair_overlap_depth": -1.611027152126354e-05, "metric_expression": "8.79629", "metric_symbol": "s", "metric_value": 8.796293221918246, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.795, "setup": "hexinsqu", "side": 8.796293221918246, "solution_path": "data/solutions/generated/hexinsqu_024_941fbce4f50492c6.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/941fbce4f50492c6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexinsqu@25", "density": 0.7898968080720588, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/25.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "30(4\u221a3\u20133)/13 = 9.065+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.065, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@25", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "30(4\u221a3\u20133)/13 = 9.065+", "metric_symbol": "s", "metric_value": 9.065, "reference_text": "s = 30(4\u221a3\u20133)/13 = 9.065+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "25.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/25.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 30(4\u221a3\u20133)/13 = 9.065+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "25.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "a62e2508884336f7", "improvement": -0.00298447907218069, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.02442885507961e-06, "max_pair_overlap_depth": -5.091125714784539e-06, "metric_expression": "9.06798", "metric_symbol": "s", "metric_value": 9.06798447907218, "n": 25, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.065, "setup": "hexinsqu", "side": 9.06798447907218, "solution_path": "data/solutions/generated/hexinsqu_025_a62e2508884336f7.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/a62e2508884336f7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@1", "density": 0.6666490309389511, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/1.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/1.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "c2fff75544b88239", "improvement": -3.9681174644368156e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.67464904574664e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "3.00004", "metric_symbol": "s", "metric_value": 3.0000396811746444, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "hexintri", "side": 3.0000396811746444, "solution_path": "data/solutions/generated/hexintri_001_c2fff75544b88239.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/c2fff75544b88239.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@2", "density": 0.4799802064426444, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/3.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/3.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "b77426ecb97f3ce6", "improvement": -0.00010309463302426991, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3591084715906373e-06, "max_pair_overlap_depth": -4.2178351187804886e-05, "metric_expression": "5.00010", "metric_symbol": "s", "metric_value": 5.000103094633024, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "hexintri", "side": 5.000103094633024, "solution_path": "data/solutions/generated/hexintri_002_b77426ecb97f3ce6.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b77426ecb97f3ce6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@3", "density": 0.7199659208954582, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/3.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/3.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "748bfe1ef47ea18c", "improvement": -0.00011833442488207169, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3474135610435156e-06, "max_pair_overlap_depth": -4.4955482604913666e-05, "metric_expression": "5.00012", "metric_symbol": "s", "metric_value": 5.000118334424882, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "hexintri", "side": 5.000118334424882, "solution_path": "data/solutions/generated/hexintri_003_748bfe1ef47ea18c.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/748bfe1ef47ea18c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@4", "density": 0.6666571046827545, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/4.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/4.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 6 | Trivial.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "4bf531bb9fa044b6", "improvement": -4.302939048272947e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.615149234551692e-07, "max_pair_overlap_depth": -1.1132001730862484e-05, "metric_expression": "6.00004", "metric_symbol": "s", "metric_value": 6.000043029390483, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.0, "setup": "hexintri", "side": 6.000043029390483, "solution_path": "data/solutions/generated/hexintri_004_4bf531bb9fa044b6.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4bf531bb9fa044b6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@5", "density": 0.6430466587435534, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/5.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "5(1+\u221a3)/2 = 6.8301+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.8301, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@5", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "5(1+\u221a3)/2 = 6.8301+", "metric_symbol": "s", "metric_value": 6.8301, "reference_text": "s = 5(1+\u221a3)/2 = 6.8301+ | Found by Maurizio Morandi | in April 2015.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/5.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 5(1+\u221a3)/2 = 6.8301+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "92331a16a75cf6c4", "improvement": -0.00019379102171246387, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4626103815685519e-06, "max_pair_overlap_depth": -3.302865088408366e-05, "metric_expression": "6.83029", "metric_symbol": "s", "metric_value": 6.830293791021712, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.8301, "setup": "hexintri", "side": 6.830293791021712, "solution_path": "data/solutions/generated/hexintri_005_92331a16a75cf6c4.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/92331a16a75cf6c4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@6", "density": 0.7560505607557761, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/6.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "2(1+\u221a6) = 6.89897+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.89897, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@6", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2(1+\u221a6) = 6.89897+", "metric_symbol": "s", "metric_value": 6.89897, "reference_text": "s = 2(1+\u221a6) = 6.89897+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/6.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 2(1+\u221a6) = 6.89897+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "f3621ba5529eef9a", "improvement": -0.0014548494669490708, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.849296155859392e-06, "max_pair_overlap_depth": -8.276209533908396e-05, "metric_expression": "6.90042", "metric_symbol": "s", "metric_value": 6.900424849466949, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.89897, "setup": "hexintri", "side": 6.900424849466949, "solution_path": "data/solutions/generated/hexintri_006_f3621ba5529eef9a.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/f3621ba5529eef9a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@7", "density": 0.6562369627570366, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/7.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "8", "friedman_metric_symbol": "s", "friedman_metric_value": 8.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "8", "metric_symbol": "s", "metric_value": 8.0, "reference_text": "s = 8 | Trivial. |", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/7.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 8 | Trivial. |", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "635b0cc9d15ec460", "improvement": -7.946628399579936e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.24931067292516e-06, "max_pair_overlap_depth": -1.4685785256318695e-05, "metric_expression": "8.00008", "metric_symbol": "s", "metric_value": 8.000079466283996, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.0, "setup": "hexintri", "side": 8.000079466283996, "solution_path": "data/solutions/generated/hexintri_007_635b0cc9d15ec460.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/635b0cc9d15ec460.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@8", "density": 0.6907428320047944, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/8.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "25/3 = 8.3333+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.3333, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@8", "credit": "Erich Friedman", "date": "April 2015", "family_updated": "", "metric_expression": "25/3 = 8.3333+", "metric_symbol": "s", "metric_value": 8.3333, "reference_text": "s = 25/3 = 8.3333+ | Found by Erich Friedman | in April 2015.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/8.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 25/3 = 8.3333+ | Found by Erich Friedman | in April 2015.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "adf439e6f7b2afb5", "improvement": -0.002790584626882975, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.227185763431761e-07, "max_pair_overlap_depth": -7.882545446008526e-05, "metric_expression": "8.33609", "metric_symbol": "s", "metric_value": 8.336090584626882, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.3333, "setup": "hexintri", "side": 8.336090584626882, "solution_path": "data/solutions/generated/hexintri_008_adf439e6f7b2afb5.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/adf439e6f7b2afb5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@9", "density": 0.7214446032247255, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/10.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "5\u221a3 = 8.6602+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.6602, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@9", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5\u221a3 = 8.6602+", "metric_symbol": "s", "metric_value": 8.6602, "reference_text": "s = 5\u221a3 = 8.6602+ | Trivial. |", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/10.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 5\u221a3 = 8.6602+ | Trivial. |", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "936a184a58977242", "improvement": 0.008620848542102877, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0939511862432736e-06, "max_pair_overlap_depth": -5.506633970053798e-05, "metric_expression": "8.65158", "metric_symbol": "s", "metric_value": 8.651579151457897, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.6602, "setup": "hexintri", "side": 8.651579151457897, "solution_path": "data/solutions/generated/hexintri_009_936a184a58977242.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/936a184a58977242.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@10", "density": 0.7999538247696885, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/10.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "5\u221a3 = 8.6602+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.6602, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@10", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5\u221a3 = 8.6602+", "metric_symbol": "s", "metric_value": 8.6602, "reference_text": "s = 5\u221a3 = 8.6602+ | Trivial. |", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/10.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 5\u221a3 = 8.6602+ | Trivial. |", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "5df6cec7a90f52fe", "improvement": -0.00030397942970239455, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2645686941820143e-06, "max_pair_overlap_depth": -4.384611814706174e-05, "metric_expression": "8.66050", "metric_symbol": "s", "metric_value": 8.660503979429702, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.6602, "setup": "hexintri", "side": 8.660503979429702, "solution_path": "data/solutions/generated/hexintri_010_5df6cec7a90f52fe.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/5df6cec7a90f52fe.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@11", "density": 0.7237571814877352, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/11.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "9.54823+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.54823, "friedman_reference": {"analytical_or_proved": false, "case": "hexintri@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "9.54823+", "metric_symbol": "s", "metric_value": 9.54823, "reference_text": "s = 9.54823+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/11.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 9.54823+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "d582cb032b9c575f", "improvement": -0.0011576763865779327, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.6199514834868076e-06, "max_pair_overlap_depth": -2.6419225191420814e-05, "metric_expression": "9.54939", "metric_symbol": "s", "metric_value": 9.549387676386578, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.54823, "setup": "hexintri", "side": 9.549387676386578, "solution_path": "data/solutions/generated/hexintri_011_d582cb032b9c575f.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/d582cb032b9c575f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@12", "density": 0.7199169251480841, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/12.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "10", "friedman_metric_symbol": "s", "friedman_metric_value": 10.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@12", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "10", "metric_symbol": "s", "metric_value": 10.0, "reference_text": "s = 10 | Trivial. |", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/12.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 10 | Trivial. |", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "b6e20361c3af5615", "improvement": -0.0005769586222079681, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.006918638920166e-07, "max_pair_overlap_depth": -2.082412255655086e-05, "metric_expression": "10.00058", "metric_symbol": "s", "metric_value": 10.000576958622208, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.0, "setup": "hexintri", "side": 10.000576958622208, "solution_path": "data/solutions/generated/hexintri_012_b6e20361c3af5615.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b6e20361c3af5615.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@13", "density": 0.7221828032992709, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "6\u221a3 = 10.3923+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.3923, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@13", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6\u221a3 = 10.3923+", "metric_symbol": "s", "metric_value": 10.3923, "reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "da55bd8ffef5f50e", "improvement": -0.00028846326752507423, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7733151165742505e-06, "max_pair_overlap_depth": -4.4289278687159594e-05, "metric_expression": "10.39259", "metric_symbol": "s", "metric_value": 10.392588463267526, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.3923, "setup": "hexintri", "side": 10.392588463267526, "solution_path": "data/solutions/generated/hexintri_013_da55bd8ffef5f50e.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/da55bd8ffef5f50e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@14", "density": 0.7777237680646841, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "6\u221a3 = 10.3923+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.3923, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@14", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6\u221a3 = 10.3923+", "metric_symbol": "s", "metric_value": 10.3923, "reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "272118e9284c579f", "improvement": -0.0003656905370288399, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.442529134834328e-06, "max_pair_overlap_depth": -4.275788154761706e-05, "metric_expression": "10.39267", "metric_symbol": "s", "metric_value": 10.39266569053703, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.3923, "setup": "hexintri", "side": 10.39266569053703, "solution_path": "data/solutions/generated/hexintri_014_272118e9284c579f.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/272118e9284c579f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@15", "density": 0.833304219754554, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "6\u221a3 = 10.3923+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.3923, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@15", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6\u221a3 = 10.3923+", "metric_symbol": "s", "metric_value": 10.3923, "reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "83460d360e93ff66", "improvement": -0.00018638448149665976, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.672744116747984e-07, "max_pair_overlap_depth": -2.8784199847526563e-05, "metric_expression": "10.39249", "metric_symbol": "s", "metric_value": 10.392486384481497, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.3923, "setup": "hexintri", "side": 10.392486384481497, "solution_path": "data/solutions/generated/hexintri_015_83460d360e93ff66.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/83460d360e93ff66.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@16", "density": 0.7429369875675369, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/16.gif", "friedman_metric_expression": "11.36599+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.36599, "friedman_reference": {"analytical_or_proved": false, "case": "hexintri@16", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "11.36599+", "metric_symbol": "s", "metric_value": 11.36599, "reference_text": "s = 11.36599+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/16.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 11.36599+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "89d5bef670451c83", "improvement": -0.0013703170956897992, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.657075146936789e-07, "max_pair_overlap_depth": -2.5616124350591463e-06, "metric_expression": "11.36736", "metric_symbol": "s", "metric_value": 11.36736031709569, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 11.36599, "setup": "hexintri", "side": 11.36736031709569, "solution_path": "data/solutions/generated/hexintri_16_89d5bef670451c83.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/89d5bef670451c83.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@17", "density": 0.7525303158572544, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/17.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "11.63457+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.63457, "friedman_reference": {"analytical_or_proved": false, "case": "hexintri@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "11.63457+", "metric_symbol": "s", "metric_value": 11.63457, "reference_text": "s = 11.63457+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/17.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 11.63457+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "8137d039fd339f48", "improvement": -0.007711226979866481, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.447216764354318e-05, "max_pair_overlap_depth": -1.3408176938511929e-05, "metric_expression": "11.64228", "metric_symbol": "s", "metric_value": 11.642281226979867, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 11.63457, "setup": "hexintri", "side": 11.642281226979867, "solution_path": "data/solutions/generated/hexintri_017_8137d039fd339f48.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T03:07:58+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/8137d039fd339f48.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@18", "density": 0.7507767554452202, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/19.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "12", "friedman_metric_symbol": "s", "friedman_metric_value": 12.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@18", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "12", "metric_symbol": "s", "metric_value": 12.0, "reference_text": "s = 12 | Trivial. |", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/19.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 12 | Trivial. |", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "8f10cd5e01be9029", "improvement": 0.0062092209316340075, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.734105056962567e-06, "max_pair_overlap_depth": -5.8920104462956147e-05, "metric_expression": "11.99379", "metric_symbol": "s", "metric_value": 11.993790779068366, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 12.0, "setup": "hexintri", "side": 11.993790779068366, "solution_path": "data/solutions/generated/hexintri_018_8f10cd5e01be9029.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/8f10cd5e01be9029.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@19", "density": 0.7866173060867477, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/19.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "12", "friedman_metric_symbol": "s", "friedman_metric_value": 12.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@19", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "12", "metric_symbol": "s", "metric_value": 12.0, "reference_text": "s = 12 | Trivial. |", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/19.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 12 | Trivial. |", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "18178ad9efaf7172", "improvement": -0.0384528795380632, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.242681221171444e-06, "max_pair_overlap_depth": -3.13088544121598e-05, "metric_expression": "12.03845", "metric_symbol": "s", "metric_value": 12.038452879538063, "n": 19, "notes": "Best available verified coordinate reconstruction; the Friedman reference remains the public metric until recovery improves.", "record_type": "verified", "reference_side": 12.0, "setup": "hexintri", "side": 12.038452879538063, "solution_path": "data/solutions/generated/hexintri_019_18178ad9efaf7172.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T03:07:58+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/18178ad9efaf7172.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@20", "density": 0.8162942117377381, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/21.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "7\u221a3 = 12.1243+", "friedman_metric_symbol": "s", "friedman_metric_value": 12.1243, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@20", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7\u221a3 = 12.1243+", "metric_symbol": "s", "metric_value": 12.1243, "reference_text": "s = 7\u221a3 = 12.1243+ | Trivial. |", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/21.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 7\u221a3 = 12.1243+ | Trivial. |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "f59179c5b6665dc4", "improvement": -0.00029566549525483765, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.116403950810266e-06, "max_pair_overlap_depth": -3.364591626842106e-05, "metric_expression": "12.12460", "metric_symbol": "s", "metric_value": 12.124595665495255, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 12.1243, "setup": "hexintri", "side": 12.124595665495255, "solution_path": "data/solutions/generated/hexintri_020_f59179c5b6665dc4.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/f59179c5b6665dc4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "hexintri@21", "density": 0.8571035425372456, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/21.gif", "friedman_metric_expression": "7\u221a3 = 12.1243+", "friedman_metric_symbol": "s", "friedman_metric_value": 12.1243, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@21", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7\u221a3 = 12.1243+", "metric_symbol": "s", "metric_value": 12.1243, "reference_text": "s = 7\u221a3 = 12.1243+ | Trivial. |", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/21.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 7\u221a3 = 12.1243+ | Trivial. |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "6cf726ac83ba7d7f", "improvement": -0.00033371669977810825, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7607980864287318e-06, "max_pair_overlap_depth": -2.977721456209892e-05, "metric_expression": "12.12463", "metric_symbol": "s", "metric_value": 12.124633716699778, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 12.1243, "setup": "hexintri", "side": 12.124633716699778, "solution_path": "data/solutions/generated/hexintri_021_6cf726ac83ba7d7f.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6cf726ac83ba7d7f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@1", "density": 0.9999940796878448, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/1.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial. |", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/1.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 1 | Trivial. |", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "13c63959e19cfeb5", "improvement": -2.96016922152198e-06, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1903479056983753e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00000", "metric_symbol": "s", "metric_value": 1.0000029601692215, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0, "setup": "octinoct", "side": 1.0000029601692215, "solution_path": "data/solutions/generated/octinoct_001_13c63959e19cfeb5.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/13c63959e19cfeb5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@2", "density": 0.4999889107582135, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/2.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial. |", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/2.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 2 | Trivial. |", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "937fb450c622d9b9", "improvement": -2.2178852493670576e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.909847599495863e-06, "max_pair_overlap_depth": -3.0343321494846753e-05, "metric_expression": "2.00002", "metric_symbol": "s", "metric_value": 2.0000221788524937, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "octinoct", "side": 2.0000221788524937, "solution_path": "data/solutions/generated/octinoct_002_937fb450c622d9b9.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/937fb450c622d9b9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@3", "density": 0.6361550822369931, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/3.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "5\u20132\u221a2 = 2.17157+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.17157, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@3", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5\u20132\u221a2 = 2.17157+", "metric_symbol": "s", "metric_value": 2.17157, "reference_text": "s = 5\u20132\u221a2 = 2.17157+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/3.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5\u20132\u221a2 = 2.17157+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "4cb37df94277ed6e", "improvement": -2.6468916854760494e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.359798521217012e-07, "max_pair_overlap_depth": -2.9093807346031175e-05, "metric_expression": "2.17160", "metric_symbol": "s", "metric_value": 2.1715964689168548, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.17157, "setup": "octinoct", "side": 2.1715964689168548, "solution_path": "data/solutions/generated/octinoct_003_4cb37df94277ed6e.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/4cb37df94277ed6e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@4", "density": 0.7008534259385343, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/4.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "\u221a(5+1/\u221a2) = 2.38895+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.38895, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@4", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "\u221a(5+1/\u221a2) = 2.38895+", "metric_symbol": "s", "metric_value": 2.38895, "reference_text": "s = \u221a(5+1/\u221a2) = 2.38895+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/4.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = \u221a(5+1/\u221a2) = 2.38895+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "8759832408431e84", "improvement": -5.1351046062464434e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.873179406841027e-06, "max_pair_overlap_depth": -4.8382172928693526e-05, "metric_expression": "2.38900", "metric_symbol": "s", "metric_value": 2.3890013510460624, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.38895, "setup": "octinoct", "side": 2.3890013510460624, "solution_path": "data/solutions/generated/octinoct_004_8759832408431e84.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/8759832408431e84.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@5", "density": 0.6822471038460284, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/5.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "2+1/\u221a2 = 2.70710+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.7071, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@5", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2+1/\u221a2 = 2.70710+", "metric_symbol": "s", "metric_value": 2.7071, "reference_text": "s = 2+1/\u221a2 = 2.70710+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/5.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 2+1/\u221a2 = 2.70710+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "d3468761b0fe9922", "improvement": -6.14172351678377e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3153662665326975e-06, "max_pair_overlap_depth": -4.586217379215496e-05, "metric_expression": "2.70716", "metric_symbol": "s", "metric_value": 2.707161417235168, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.7071, "setup": "octinoct", "side": 2.707161417235168, "solution_path": "data/solutions/generated/octinoct_005_d3468761b0fe9922.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/d3468761b0fe9922.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@6", "density": 0.6715848857993428, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/6.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "2.98915+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.98915, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@6", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.98915+", "metric_symbol": "s", "metric_value": 2.98915, "reference_text": "s = 2.98915+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/6.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 2.98915+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "3c4b2267adcca69f", "improvement": 0.0001551389501637246, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.973975728535522e-06, "max_pair_overlap_depth": -2.153911208435666e-05, "metric_expression": "2.98899", "metric_symbol": "s", "metric_value": 2.9889948610498362, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.98915, "setup": "octinoct", "side": 2.9889948610498362, "solution_path": "data/solutions/generated/octinoct_006_3c4b2267adcca69f.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/3c4b2267adcca69f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@7", "density": 0.7777522978920184, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/7.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial. |", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/7.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 3 | Trivial. |", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "a0053fa5c22778e9", "improvement": -4.914098707065051e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2079023857814054e-06, "max_pair_overlap_depth": -2.6395744560092638e-05, "metric_expression": "3.00005", "metric_symbol": "s", "metric_value": 3.0000491409870707, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "octinoct", "side": 3.0000491409870707, "solution_path": "data/solutions/generated/octinoct_007_a0053fa5c22778e9.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a0053fa5c22778e9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@8", "density": 0.7457363085593237, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/8.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "3.27526+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.27526, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@8", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3.27526+", "metric_symbol": "s", "metric_value": 3.27526, "reference_text": "s = 3.27526+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/8.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 3.27526+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "697fa4dfd3dbc6eb", "improvement": -4.953250458550329e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.188513199476399e-06, "max_pair_overlap_depth": -7.758148462100323e-06, "metric_expression": "3.27531", "metric_symbol": "s", "metric_value": 3.2753095325045853, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.27526, "setup": "octinoct", "side": 3.2753095325045853, "solution_path": "data/solutions/generated/octinoct_008_697fa4dfd3dbc6eb.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/697fa4dfd3dbc6eb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@9", "density": 0.7720692797565871, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/9.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "2+\u221a2 = 3.41421+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.41421, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@9", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2+\u221a2 = 3.41421+", "metric_symbol": "s", "metric_value": 3.41421, "reference_text": "s = 2+\u221a2 = 3.41421+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/9.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 2+\u221a2 = 3.41421+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "7face58a13302eec", "improvement": -2.270780627799951e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3803938489295472e-05, "max_pair_overlap_depth": -2.6349120965107886e-06, "metric_expression": "3.41423", "metric_symbol": "s", "metric_value": 3.414232707806278, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.41421, "setup": "octinoct", "side": 3.414232707806278, "solution_path": "data/solutions/generated/octinoct_009_7face58a13302eec.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/7face58a13302eec.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@10", "density": 0.7276473732898788, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/10.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "3+1/\u221a2 = 3.70710+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.7071, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@10", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3+1/\u221a2 = 3.70710+", "metric_symbol": "s", "metric_value": 3.7071, "reference_text": "s = 3+1/\u221a2 = 3.70710+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/10.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 3+1/\u221a2 = 3.70710+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "bcfc9e294bb15f21", "improvement": -4.452047150094174e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.744023326611903e-06, "max_pair_overlap_depth": -8.094414775450254e-06, "metric_expression": "3.70714", "metric_symbol": "s", "metric_value": 3.707144520471501, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.7071, "setup": "octinoct", "side": 3.707144520471501, "solution_path": "data/solutions/generated/octinoct_010_bcfc9e294bb15f21.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/bcfc9e294bb15f21.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@11", "density": 0.7111917967389362, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/11.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "3.93270+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.9327, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@11", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3.93270+", "metric_symbol": "s", "metric_value": 3.9327, "reference_text": "s = 3.93270+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/11.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 3.93270+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "5cff1aa2d330c49e", "improvement": -0.00011007046297972778, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.479089320941585e-06, "max_pair_overlap_depth": -8.811356971838435e-06, "metric_expression": "3.93281", "metric_symbol": "s", "metric_value": 3.93281007046298, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.9327, "setup": "octinoct", "side": 3.93281007046298, "solution_path": "data/solutions/generated/octinoct_011_5cff1aa2d330c49e.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/5cff1aa2d330c49e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@12", "density": 0.7499773041118397, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/12.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@12", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial. |", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/12.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 4 | Trivial. |", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "7bdd4832407bd3be", "improvement": -6.052374207055067e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.082066348113342e-06, "max_pair_overlap_depth": -1.2861422511445397e-05, "metric_expression": "4.00006", "metric_symbol": "s", "metric_value": 4.0000605237420706, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "octinoct", "side": 4.0000605237420706, "solution_path": "data/solutions/generated/octinoct_012_7bdd4832407bd3be.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7bdd4832407bd3be.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@13", "density": 0.7469920374756177, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/13.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "7\u20132\u221a2 = 4.17157+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.17157, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@13", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "7\u20132\u221a2 = 4.17157+", "metric_symbol": "s", "metric_value": 4.17157, "reference_text": "s = 7\u20132\u221a2 = 4.17157+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/13.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 7\u20132\u221a2 = 4.17157+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "867ad172957f0ddb", "improvement": -0.00013596034935225987, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.683174531654637e-06, "max_pair_overlap_depth": -2.596769201668181e-06, "metric_expression": "4.17171", "metric_symbol": "s", "metric_value": 4.171705960349352, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.17157, "setup": "octinoct", "side": 4.171705960349352, "solution_path": "data/solutions/generated/octinoct_013_867ad172957f0ddb.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/867ad172957f0ddb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@14", "density": 0.7637114154056861, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/14.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "4.28155+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.28155, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@14", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "4.28155+", "metric_symbol": "s", "metric_value": 4.28155, "reference_text": "s = 4.28155+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/14.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 4.28155+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "190d6c8f3f4e37d6", "improvement": 1.6200963535162316e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9636658184049817e-06, "max_pair_overlap_depth": -1.2788583332312697e-05, "metric_expression": "4.28153", "metric_symbol": "s", "metric_value": 4.281533799036465, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.28155, "setup": "octinoct", "side": 4.281533799036465, "solution_path": "data/solutions/generated/octinoct_014_190d6c8f3f4e37d6.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/190d6c8f3f4e37d6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@15", "density": 0.7554665958519668, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/15.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "4.45594+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.45594, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@15", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "4.45594+", "metric_symbol": "s", "metric_value": 4.45594, "reference_text": "s = 4.45594+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/15.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 4.45594+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "60843ec4ea56ad88", "improvement": 1.3726756218801484e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.050759161282626e-06, "max_pair_overlap_depth": -3.648956153723404e-05, "metric_expression": "4.45593", "metric_symbol": "s", "metric_value": 4.455926273243781, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.45594, "setup": "octinoct", "side": 4.455926273243781, "solution_path": "data/solutions/generated/octinoct_015_60843ec4ea56ad88.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/60843ec4ea56ad88.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@16", "density": 0.7566625545620705, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/16.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "4.59824+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.59824, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@16", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "4.59824+", "metric_symbol": "s", "metric_value": 4.59824, "reference_text": "s = 4.59824+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/16.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 4.59824+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "1178c6831eb46264", "improvement": -0.00018248651533525617, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.297640975948298e-06, "max_pair_overlap_depth": -9.126693151007004e-06, "metric_expression": "4.59842", "metric_symbol": "s", "metric_value": 4.598422486515335, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.59824, "setup": "octinoct", "side": 4.598422486515335, "solution_path": "data/solutions/generated/octinoct_016_1178c6831eb46264.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/1178c6831eb46264.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@17", "density": 0.7511087688524583, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/17.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "4.75741+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.75741, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@17", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "4.75741+", "metric_symbol": "s", "metric_value": 4.75741, "reference_text": "s = 4.75741+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/17.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 4.75741+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "ebcfc45448cb8c5f", "improvement": -2.6985802159096295e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.580862819791944e-06, "max_pair_overlap_depth": -2.6446396224955038e-05, "metric_expression": "4.75744", "metric_symbol": "s", "metric_value": 4.757436985802159, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.75741, "setup": "octinoct", "side": 4.757436985802159, "solution_path": "data/solutions/generated/octinoct_017_ebcfc45448cb8c5f.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/ebcfc45448cb8c5f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@18", "density": 0.7629999765077309, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/18.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "4.85717+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.85717, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@18", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "4.85717+", "metric_symbol": "s", "metric_value": 4.85717, "reference_text": "s = 4.85717+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/18.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 4.85717+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "46f5ac176d150ffa", "improvement": 0.00010416375427002578, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.225773906749903e-06, "max_pair_overlap_depth": -2.521618462036912e-05, "metric_expression": "4.85707", "metric_symbol": "s", "metric_value": 4.85706583624573, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.85717, "setup": "octinoct", "side": 4.85706583624573, "solution_path": "data/solutions/generated/octinoct_018_46f5ac176d150ffa.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/46f5ac176d150ffa.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@19", "density": 0.7795195170060211, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/19.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "4.93586+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.93586, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@19", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "4.93586+", "metric_symbol": "s", "metric_value": 4.93586, "reference_text": "s = 4.93586+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/19.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 4.93586+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "4a7169be4bda1825", "improvement": -0.0011420082689239663, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1928573301213418e-05, "max_pair_overlap_depth": -1.491417372267989e-06, "metric_expression": "4.93700", "metric_symbol": "s", "metric_value": 4.937002008268924, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.93586, "setup": "octinoct", "side": 4.937002008268924, "solution_path": "data/solutions/generated/octinoct_019_4a7169be4bda1825.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/4a7169be4bda1825.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@20", "density": 0.778654925537014, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/20.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "5.06781+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.06781, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@20", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.06781+", "metric_symbol": "s", "metric_value": 5.06781, "reference_text": "s = 5.06781+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "20.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/20.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.06781+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "ef9ffbe57f51321a", "improvement": -0.0002585475900360734, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.202696065204785e-06, "max_pair_overlap_depth": -3.692189855719619e-05, "metric_expression": "5.06807", "metric_symbol": "s", "metric_value": 5.068068547590036, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.06781, "setup": "octinoct", "side": 5.068068547590036, "solution_path": "data/solutions/generated/octinoct_020_ef9ffbe57f51321a.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/ef9ffbe57f51321a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@21", "density": 0.7774838310416734, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/21.gif", "friedman_metric_expression": "5.19715+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.19715, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@21", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.19715+", "metric_symbol": "s", "metric_value": 5.19715, "reference_text": "s = 5.19715+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/21.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.19715+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "92c4bcf9428983c2", "improvement": 1.540400287325383e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.1696825424876755e-06, "max_pair_overlap_depth": -2.317393259165801e-05, "metric_expression": "5.19713", "metric_symbol": "s", "metric_value": 5.1971345959971265, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.19715, "setup": "octinoct", "side": 5.1971345959971265, "solution_path": "data/solutions/generated/octinoct_021_92c4bcf9428983c2.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/92c4bcf9428983c2.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@22", "density": 0.7663269933124698, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/22.gif", "friedman_metric_expression": "5.35750+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.3575, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@22", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.35750+", "metric_symbol": "s", "metric_value": 5.3575, "reference_text": "s = 5.35750+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/22.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.35750+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "b8f2befaa2d4a75e", "improvement": -0.0005193617687213603, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.99203724676962e-06, "max_pair_overlap_depth": -1.57400449714018e-05, "metric_expression": "5.35802", "metric_symbol": "s", "metric_value": 5.358019361768721, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.3575, "setup": "octinoct", "side": 5.358019361768721, "solution_path": "data/solutions/generated/octinoct_022_b8f2befaa2d4a75e.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/b8f2befaa2d4a75e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@23", "density": 0.7845835529808874, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/23.gif", "friedman_metric_expression": "5.41440+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.4144, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@23", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.41440+", "metric_symbol": "s", "metric_value": 5.4144, "reference_text": "s = 5.41440+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/23.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.41440+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "d6528e8801e36ce0", "improvement": 7.503853147827755e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3646686331794102e-06, "max_pair_overlap_depth": -4.536567227386712e-05, "metric_expression": "5.41432", "metric_symbol": "s", "metric_value": 5.414324961468521, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.4144, "setup": "octinoct", "side": 5.414324961468521, "solution_path": "data/solutions/generated/octinoct_023_d6528e8801e36ce0.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/d6528e8801e36ce0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@24", "density": 0.777859618157, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/24.gif", "friedman_metric_expression": "5.55464+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.55464, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@24", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.55464+", "metric_symbol": "s", "metric_value": 5.55464, "reference_text": "s = 5.55464+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/24.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.55464+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "a34ac6a3c52f70df", "improvement": 1.1631965286973411e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.721783559078116e-06, "max_pair_overlap_depth": -2.4974900467711336e-05, "metric_expression": "5.55463", "metric_symbol": "s", "metric_value": 5.554628368034713, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.55464, "setup": "octinoct", "side": 5.554628368034713, "solution_path": "data/solutions/generated/octinoct_024_a34ac6a3c52f70df.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/a34ac6a3c52f70df.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@25", "density": 0.7805673327264805, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/25.gif", "friedman_metric_expression": "5.65593+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.65593, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@25", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.65593+", "metric_symbol": "s", "metric_value": 5.65593, "reference_text": "s = 5.65593+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/25.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.65593+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "113492960b7e7dce", "improvement": -0.003397389940226425, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.281212189125961e-06, "max_pair_overlap_depth": -2.6024797291412938e-05, "metric_expression": "5.65933", "metric_symbol": "s", "metric_value": 5.659327389940226, "n": 25, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.65593, "setup": "octinoct", "side": 5.659327389940226, "solution_path": "data/solutions/generated/octinoct_025_113492960b7e7dce.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/113492960b7e7dce.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@26", "density": 0.7881657239941322, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/26.gif", "friedman_metric_expression": "5.73940+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.7394, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@26", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.73940+", "metric_symbol": "s", "metric_value": 5.7394, "reference_text": "s = 5.73940+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/26.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.73940+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "581188c4895e4037", "improvement": -0.004116881359766289, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.43705423351296e-06, "max_pair_overlap_depth": -2.686615288460814e-05, "metric_expression": "5.74352", "metric_symbol": "s", "metric_value": 5.743516881359766, "n": 26, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.7394, "setup": "octinoct", "side": 5.743516881359766, "solution_path": "data/solutions/generated/octinoct_026_581188c4895e4037.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/581188c4895e4037.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@27", "density": 0.7989711846148311, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/27.gif", "friedman_metric_expression": "5.81238+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.81238, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@27", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.81238+", "metric_symbol": "s", "metric_value": 5.81238, "reference_text": "s = 5.81238+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/27.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.81238+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "25c7f85bc0b15209", "improvement": -0.0008341744667159645, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.851030556516347e-06, "max_pair_overlap_depth": -2.4661009629944886e-05, "metric_expression": "5.81321", "metric_symbol": "s", "metric_value": 5.813214174466716, "n": 27, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.81238, "setup": "octinoct", "side": 5.813214174466716, "solution_path": "data/solutions/generated/octinoct_027_25c7f85bc0b15209.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/25c7f85bc0b15209.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@28", "density": 0.8242204012359605, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/28.gif", "friedman_metric_expression": "3+2\u221a2 = 5.82842+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.82842, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@28", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3+2\u221a2 = 5.82842+", "metric_symbol": "s", "metric_value": 5.82842, "reference_text": "s = 3+2\u221a2 = 5.82842+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/28.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 3+2\u221a2 = 5.82842+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "f8d9c8a9f9a38d45", "improvement": -8.71778893589692e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.739062384519148e-06, "max_pair_overlap_depth": -9.673739088000843e-06, "metric_expression": "5.82851", "metric_symbol": "s", "metric_value": 5.828507177889359, "n": 28, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.82842, "setup": "octinoct", "side": 5.828507177889359, "solution_path": "data/solutions/generated/octinoct_028_f8d9c8a9f9a38d45.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/f8d9c8a9f9a38d45.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@29", "density": 0.7902111886471865, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/29.gif", "friedman_metric_expression": "6.05746+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.05746, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@29", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "6.05746+", "metric_symbol": "s", "metric_value": 6.05746, "reference_text": "s = 6.05746+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/29.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 6.05746+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "30789c6bb8fd99b7", "improvement": -0.0005140918720876186, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.805265150873157e-06, "max_pair_overlap_depth": -5.6301265520808386e-05, "metric_expression": "6.05797", "metric_symbol": "s", "metric_value": 6.0579740918720875, "n": 29, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.05746, "setup": "octinoct", "side": 6.0579740918720875, "solution_path": "data/solutions/generated/octinoct_029_30789c6bb8fd99b7.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/30789c6bb8fd99b7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "octinoct@30", "density": 0.7919434557655346, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/30.gif", "friedman_metric_expression": "6.15432+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.15432, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@30", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "6.15432+", "metric_symbol": "s", "metric_value": 6.15432, "reference_text": "s = 6.15432+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/30.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 6.15432+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "1a16dc8a4d741008", "improvement": -0.00047426130353134255, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1243912272535679e-05, "max_pair_overlap_depth": -2.8046648627544357e-05, "metric_expression": "6.15479", "metric_symbol": "s", "metric_value": 6.154794261303532, "n": 30, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.15432, "setup": "octinoct", "side": 6.154794261303532, "solution_path": "data/solutions/generated/octinoct_030_1a16dc8a4d741008.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/1a16dc8a4d741008.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@1", "density": 0.7568185573097284, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/1.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a((5+\u221a5)/10) = .86065+", "friedman_metric_symbol": "r", "friedman_metric_value": 0.86065, "friedman_reference": {"analytical_or_proved": true, "case": "penincir@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a((5+\u221a5)/10) = .86065+", "metric_symbol": "r", "metric_value": 0.86065, "reference_text": "r = \u221a((5+\u221a5)/10) = .86065+ | Trivial.", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/1.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = \u221a((5+\u221a5)/10) = .86065+ | Trivial.", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": false, "id": "ca75a5cd7cad481c", "improvement": 0.009994599443903796, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0926430426282394e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.85066", "metric_symbol": "r", "metric_value": 0.8506554005560962, "n": 1, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 0.86065, "setup": "penincir", "side": 1.7013108011121925, "solution_path": "data/solutions/generated/penincir_001_ca75a5cd7cad481c.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/ca75a5cd7cad481c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@2", "density": 0.462509898701161, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/2.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a(5+2\u221a5)/2 = 1.53884+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.53884, "friedman_reference": {"analytical_or_proved": true, "case": "penincir@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a(5+2\u221a5)/2 = 1.53884+", "metric_symbol": "r", "metric_value": 1.53884, "reference_text": "r = \u221a(5+2\u221a5)/2 = 1.53884+ | Trivial.", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/2.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = \u221a(5+2\u221a5)/2 = 1.53884+ | Trivial.", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "44da10d1d5897aae", "improvement": -3.743542237466535e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.684702690331392e-08, "max_pair_overlap_depth": -5.994543612077171e-05, "metric_expression": "1.53888", "metric_symbol": "r", "metric_value": 1.5388774354223747, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.53884, "setup": "penincir", "side": 3.0777548708447493, "solution_path": "data/solutions/generated/penincir_002_44da10d1d5897aae.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/44da10d1d5897aae.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@3", "density": 0.6275054691586099, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/3.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "(1+\u221a5)/2 = 1.61803+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.61803, "friedman_reference": {"analytical_or_proved": true, "case": "penincir@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(1+\u221a5)/2 = 1.61803+", "metric_symbol": "r", "metric_value": 1.61803, "reference_text": "r = (1+\u221a5)/2 = 1.61803+ | Trivial.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/3.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = (1+\u221a5)/2 = 1.61803+ | Trivial.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "350d35e2110a508b", "improvement": -5.532631868709892e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.919906156253262e-06, "max_pair_overlap_depth": -5.073652219536114e-05, "metric_expression": "1.61809", "metric_symbol": "r", "metric_value": 1.6180853263186872, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.61803, "setup": "penincir", "side": 3.2361706526373744, "solution_path": "data/solutions/generated/penincir_003_350d35e2110a508b.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/350d35e2110a508b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@4", "density": 0.6542873961171766, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/4.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "1.82971+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.82971, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@4", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.82971+", "metric_symbol": "r", "metric_value": 1.82971, "reference_text": "r = 1.82971+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/4.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 1.82971+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "d03dba2c6efc98dd", "improvement": -5.479863407176211e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.02075710251404e-06, "max_pair_overlap_depth": -3.1570603938990693e-05, "metric_expression": "1.82976", "metric_symbol": "r", "metric_value": 1.8297647986340717, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.82971, "setup": "penincir", "side": 3.6595295972681434, "solution_path": "data/solutions/generated/penincir_004_d03dba2c6efc98dd.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/d03dba2c6efc98dd.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@5", "density": 0.6626607161574516, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/5.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.03271+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.03271, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@5", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.03271+", "metric_symbol": "r", "metric_value": 2.03271, "reference_text": "r = 2.03271+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/5.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 2.03271+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "97027d267cb23b7f", "improvement": -6.326016348978669e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.209947377376011e-06, "max_pair_overlap_depth": -3.7991648266566536e-05, "metric_expression": "2.03277", "metric_symbol": "r", "metric_value": 2.0327732601634896, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.03271, "setup": "penincir", "side": 4.065546520326979, "solution_path": "data/solutions/generated/penincir_005_97027d267cb23b7f.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/97027d267cb23b7f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@6", "density": 0.6624931586206462, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/6.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a((25+11\u221a5)/10) = 2.22703+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.22703, "friedman_reference": {"analytical_or_proved": true, "case": "penincir@6", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "\u221a((25+11\u221a5)/10) = 2.22703+", "metric_symbol": "r", "metric_value": 2.22703, "reference_text": "r = \u221a((25+11\u221a5)/10) = 2.22703+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/6.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = \u221a((25+11\u221a5)/10) = 2.22703+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "4b5f96ff666b1595", "improvement": -4.3119632947785647e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3229349565289112e-06, "max_pair_overlap_depth": -2.759876285962637e-05, "metric_expression": "2.22707", "metric_symbol": "r", "metric_value": 2.227073119632948, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.22703, "setup": "penincir", "side": 4.454146239265896, "solution_path": "data/solutions/generated/penincir_006_4b5f96ff666b1595.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/4b5f96ff666b1595.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@7", "density": 0.7279626259462092, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/7.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.29474+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.29474, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@7", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.29474+", "metric_symbol": "r", "metric_value": 2.29474, "reference_text": "r = 2.29474+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/7.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 2.29474+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "ef28e611d9e22eb0", "improvement": -5.570482584049685e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.717800656082403e-06, "max_pair_overlap_depth": -4.1591260455220436e-05, "metric_expression": "2.29480", "metric_symbol": "r", "metric_value": 2.2947957048258405, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.29474, "setup": "penincir", "side": 4.589591409651681, "solution_path": "data/solutions/generated/penincir_007_ef28e611d9e22eb0.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/ef28e611d9e22eb0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@8", "density": 0.7115852708183573, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/8.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.48128+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.48128, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@8", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.48128+", "metric_symbol": "r", "metric_value": 2.48128, "reference_text": "r = 2.48128+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/8.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 2.48128+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "1a134d3903b2814e", "improvement": -3.0242757913079288e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.375187333127144e-07, "max_pair_overlap_depth": -8.34480700673268e-06, "metric_expression": "2.48131", "metric_symbol": "r", "metric_value": 2.481310242757913, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.48128, "setup": "penincir", "side": 4.962620485515826, "solution_path": "data/solutions/generated/penincir_008_1a134d3903b2814e.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/1a134d3903b2814e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@9", "density": 0.674675474238072, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/9.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.70171+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.70171, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@9", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.70171+", "metric_symbol": "r", "metric_value": 2.70171, "reference_text": "r = 2.70171+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/9.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 2.70171+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "1ca74640a9bafbff", "improvement": -0.0011487048015377255, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.541032137428402e-06, "max_pair_overlap_depth": -3.412963059545859e-05, "metric_expression": "2.70286", "metric_symbol": "r", "metric_value": 2.7028587048015376, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.70171, "setup": "penincir", "side": 5.405717409603075, "solution_path": "data/solutions/generated/penincir_009_1ca74640a9bafbff.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/1ca74640a9bafbff.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@10", "density": 0.6983515169709827, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/10.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.79768+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.79768, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@10", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.79768+", "metric_symbol": "r", "metric_value": 2.79768, "reference_text": "r = 2.79768+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/10.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 2.79768+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "b50c304b68607cc4", "improvement": -0.002671254854186955, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.377527520027513e-06, "max_pair_overlap_depth": -1.3481998292097508e-05, "metric_expression": "2.80035", "metric_symbol": "r", "metric_value": 2.800351254854187, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.79768, "setup": "penincir", "side": 5.600702509708374, "solution_path": "data/solutions/generated/penincir_010_b50c304b68607cc4.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b50c304b68607cc4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@11", "density": 0.7152777059461288, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/11.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.90180+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.9018, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.90180+", "metric_symbol": "r", "metric_value": 2.9018, "reference_text": "r = 2.90180+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/11.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 2.90180+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "37f9123e1b2153ff", "improvement": -0.0002744381126040807, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.569993058365583e-07, "max_pair_overlap_depth": -3.207817623795961e-05, "metric_expression": "2.90207", "metric_symbol": "r", "metric_value": 2.9020744381126042, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.9018, "setup": "penincir", "side": 5.8041488762252085, "solution_path": "data/solutions/generated/penincir_011_37f9123e1b2153ff.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/37f9123e1b2153ff.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@12", "density": 0.7213016172544706, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/12.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.01826+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.01826, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@12", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.01826+", "metric_symbol": "r", "metric_value": 3.01826, "reference_text": "r = 3.01826+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/12.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.01826+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "a564353850abf267", "improvement": -0.00017424806894128508, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.481723353017998e-06, "max_pair_overlap_depth": -2.751801632361328e-05, "metric_expression": "3.01843", "metric_symbol": "r", "metric_value": 3.0184342480689415, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.01826, "setup": "penincir", "side": 6.036868496137883, "solution_path": "data/solutions/generated/penincir_012_a564353850abf267.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/a564353850abf267.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@13", "density": 0.7515432650569983, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/13.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a(5+2\u221a5)/2 =3.07769+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.07769, "friedman_reference": {"analytical_or_proved": true, "case": "penincir@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "\u221a(5+2\u221a5)/2 =3.07769+", "metric_symbol": "r", "metric_value": 3.07769, "reference_text": "r = \u221a(5+2\u221a5)/2 =3.07769+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/13.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = \u221a(5+2\u221a5)/2 =3.07769+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "c39e9475d5bde4a6", "improvement": -0.0001371927275344298, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.94574749318366e-06, "max_pair_overlap_depth": -1.7086792766463077e-05, "metric_expression": "3.07783", "metric_symbol": "r", "metric_value": 3.0778271927275345, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.07769, "setup": "penincir", "side": 3.0778271927275345, "solution_path": "data/solutions/generated/penincir_013_c39e9475d5bde4a6.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/c39e9475d5bde4a6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@14", "density": 0.7320837719347534, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/14.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.23385+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.23385, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@14", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.23385+", "metric_symbol": "r", "metric_value": 3.23385, "reference_text": "r = 3.23385+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/14.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.23385+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "4798ec354812edfb", "improvement": -0.0023337860780814346, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.2470094596680497e-07, "max_pair_overlap_depth": -1.086583313658629e-05, "metric_expression": "3.23618", "metric_symbol": "r", "metric_value": 3.2361837860780813, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.23385, "setup": "penincir", "side": 6.472367572156163, "solution_path": "data/solutions/generated/penincir_014_4798ec354812edfb.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/4798ec354812edfb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@15", "density": 0.7364932523686155, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/15.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.33911+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.33911, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@15", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.33911+", "metric_symbol": "r", "metric_value": 3.33911, "reference_text": "r = 3.33911+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/15.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.33911+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "5fb8f24380ed14ba", "improvement": -0.0006156585665682002, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.531307800791922e-07, "max_pair_overlap_depth": -1.652091133119704e-05, "metric_expression": "3.33973", "metric_symbol": "r", "metric_value": 3.339725658566568, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.33911, "setup": "penincir", "side": 6.679451317133136, "solution_path": "data/solutions/generated/penincir_015_5fb8f24380ed14ba.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/5fb8f24380ed14ba.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@16", "density": 0.7415495303728211, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/16.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.43657+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.43657, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@16", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.43657+", "metric_symbol": "r", "metric_value": 3.43657, "reference_text": "r = 3.43657+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/16.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.43657+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "eb41133064f5f2da", "improvement": -0.0009042932667306935, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3274146969676224e-06, "max_pair_overlap_depth": -1.0162839398697443e-05, "metric_expression": "3.43747", "metric_symbol": "r", "metric_value": 3.437474293266731, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.43657, "setup": "penincir", "side": 6.874948586533462, "solution_path": "data/solutions/generated/penincir_016_eb41133064f5f2da.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/eb41133064f5f2da.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@17", "density": 0.7557819084683968, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/17.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.50958+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.50958, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.50958+", "metric_symbol": "r", "metric_value": 3.50958, "reference_text": "r = 3.50958+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/17.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.50958+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "9469ef74fc89a75b", "improvement": -0.00016662106909848262, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3042585254605683e-06, "max_pair_overlap_depth": -2.7118152433835974e-05, "metric_expression": "3.50975", "metric_symbol": "r", "metric_value": 3.5097466210690986, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.50958, "setup": "penincir", "side": 7.019493242138197, "solution_path": "data/solutions/generated/penincir_017_9469ef74fc89a75b.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/9469ef74fc89a75b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@18", "density": 0.7533573813986445, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/18.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.62080+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.6208, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@18", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.62080+", "metric_symbol": "r", "metric_value": 3.6208, "reference_text": "r = 3.62080+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/18.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.62080+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "767a627015948f32", "improvement": 0.00349375267338381, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7774333678797234e-06, "max_pair_overlap_depth": -3.109065453799076e-05, "metric_expression": "3.61731", "metric_symbol": "r", "metric_value": 3.617306247326616, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.6208, "setup": "penincir", "side": 7.234612494653232, "solution_path": "data/solutions/generated/penincir_018_767a627015948f32.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/767a627015948f32.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@19", "density": 0.771366675320165, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/19.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.67287+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.67287, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@19", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.67287+", "metric_symbol": "r", "metric_value": 3.67287, "reference_text": "r = 3.67287+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/19.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.67287+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "9d482c8012998879", "improvement": 8.154982387598153e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.817340790579493e-06, "max_pair_overlap_depth": -1.6517110137281854e-05, "metric_expression": "3.67279", "metric_symbol": "r", "metric_value": 3.672788450176124, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.67287, "setup": "penincir", "side": 7.345576900352248, "solution_path": "data/solutions/generated/penincir_019_9d482c8012998879.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/9d482c8012998879.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@20", "density": 0.7575056542556782, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/20.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.80415+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.80415, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@20", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.80415+", "metric_symbol": "r", "metric_value": 3.80415, "reference_text": "r = 3.80415+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/20.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.80415+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "5271226652403829", "improvement": 0.0016291138375779113, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.231424091092009e-06, "max_pair_overlap_depth": -3.071782635011289e-05, "metric_expression": "3.80252", "metric_symbol": "r", "metric_value": 3.802520886162422, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.80415, "setup": "penincir", "side": 7.605041772324844, "solution_path": "data/solutions/generated/penincir_020_5271226652403829.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/5271226652403829.svg", "tolerance": 1e-08, "verified": true}
-{"case": "penincir@21", "density": 0.7521173669587212, "friedman_analytical_or_proved": false, "friedman_credit": "Cyril Summerfield", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/21.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.90735+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.90735, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@21", "credit": "Cyril Summerfield", "date": "April 2026", "family_updated": "", "metric_expression": "3.90735+", "metric_symbol": "r", "metric_value": 3.90735, "reference_text": "r = 3.90735+ | Found by Cyril Summerfield | in April 2026.", "source_image": "21.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/21.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.90735+ | Found by Cyril Summerfield | in April 2026.", "friedman_source_image": "21.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "4625f732ad11a859", "improvement": -0.0030068209876064067, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9813425364011437e-06, "max_pair_overlap_depth": -1.4218243721247914e-05, "metric_expression": "3.91036", "metric_symbol": "r", "metric_value": 3.9103568209876065, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.90735, "setup": "penincir", "side": 3.9103568209876065, "solution_path": "data/solutions/generated/penincir_021_4625f732ad11a859.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Cyril Summerfield", "svg_path": "data/svg/4625f732ad11a859.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@1", "density": 0.9999780359352667, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/1.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "peninpen@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial.", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/1.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "b3173648630fcdeb", "improvement": -1.0982213277443975e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.256503348572871e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00001", "metric_symbol": "s", "metric_value": 1.0000109822132774, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0, "setup": "peninpen", "side": 1.0000109822132774, "solution_path": "data/solutions/generated/peninpen_001_b3173648630fcdeb.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b3173648630fcdeb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@2", "density": 0.4999891232484648, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/2.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "peninpen@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/2.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "edac094186a54129", "improvement": -2.175385798786067e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0537211348702708e-06, "max_pair_overlap_depth": -1.9731507571563167e-05, "metric_expression": "2.00002", "metric_symbol": "s", "metric_value": 2.000021753857988, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "peninpen", "side": 2.000021753857988, "solution_path": "data/solutions/generated/peninpen_002_edac094186a54129.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/edac094186a54129.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@3", "density": 0.599978840427638, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/3.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a5 = 2.23606+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.23606, "friedman_reference": {"analytical_or_proved": true, "case": "peninpen@3", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "\u221a5 = 2.23606+", "metric_symbol": "s", "metric_value": 2.23606, "reference_text": "s = \u221a5 = 2.23606+ | Found by Jonathan Viquerat in April 2026.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/3.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = \u221a5 = 2.23606+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "bf261a4dce178a90", "improvement": -4.7407077830463606e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7662857614262655e-06, "max_pair_overlap_depth": -2.7755970794667603e-05, "metric_expression": "2.23611", "metric_symbol": "s", "metric_value": 2.2361074070778306, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.23606, "setup": "peninpen", "side": 2.2361074070778306, "solution_path": "data/solutions/generated/peninpen_003_bf261a4dce178a90.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/bf261a4dce178a90.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@4", "density": 0.6678717661629443, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/4.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.44719+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.44719, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@4", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.44719+", "metric_symbol": "s", "metric_value": 2.44719, "reference_text": "s = 2.44719+ | Found by Jonathan Viquerat in April 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/4.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 2.44719+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "85435733af14070a", "improvement": -8.88306059834143e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0634336001709255e-06, "max_pair_overlap_depth": -3.575601486860425e-05, "metric_expression": "2.44728", "metric_symbol": "s", "metric_value": 2.4472788306059834, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.44719, "setup": "peninpen", "side": 2.4472788306059834, "solution_path": "data/solutions/generated/peninpen_004_85435733af14070a.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/85435733af14070a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@5", "density": 0.7294617516267693, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/6.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(3+\u221a5)/2 = 2.61803+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.61803, "friedman_reference": {"analytical_or_proved": true, "case": "peninpen@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(3+\u221a5)/2 = 2.61803+", "metric_symbol": "s", "metric_value": 2.61803, "reference_text": "s = (3+\u221a5)/2 = 2.61803+ | Trivial.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/6.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = (3+\u221a5)/2 = 2.61803+ | Trivial.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "28fa9f787595145b", "improvement": -5.4982701188421856e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.513169742800457e-06, "max_pair_overlap_depth": -2.4501936757292953e-06, "metric_expression": "2.61808", "metric_symbol": "s", "metric_value": 2.6180849827011885, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.61803, "setup": "peninpen", "side": 2.6180849827011885, "solution_path": "data/solutions/generated/peninpen_005_28fa9f787595145b.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/28fa9f787595145b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@6", "density": 0.8753627744848066, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/6.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(3+\u221a5)/2 = 2.61803+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.61803, "friedman_reference": {"analytical_or_proved": true, "case": "peninpen@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(3+\u221a5)/2 = 2.61803+", "metric_symbol": "s", "metric_value": 2.61803, "reference_text": "s = (3+\u221a5)/2 = 2.61803+ | Trivial.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/6.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = (3+\u221a5)/2 = 2.61803+ | Trivial.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "198bd12691483df1", "improvement": -4.201351602306502e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4414928963119693e-06, "max_pair_overlap_depth": -1.898562579671914e-05, "metric_expression": "2.61807", "metric_symbol": "s", "metric_value": 2.618072013516023, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.61803, "setup": "peninpen", "side": 2.618072013516023, "solution_path": "data/solutions/generated/peninpen_006_198bd12691483df1.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/198bd12691483df1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@7", "density": 0.7384663216832239, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/7.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.07866+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.07866, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@7", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.07866+", "metric_symbol": "s", "metric_value": 3.07866, "reference_text": "s = 3.07866+ | Found by Jonathan Viquerat in April 2026.", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/7.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 3.07866+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "5b6283ab3dc4c3b5", "improvement": -0.00015556249375014986, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.964189962454e-06, "max_pair_overlap_depth": -1.5079265303863387e-05, "metric_expression": "3.07882", "metric_symbol": "s", "metric_value": 3.0788155624937503, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.07866, "setup": "peninpen", "side": 3.0788155624937503, "solution_path": "data/solutions/generated/peninpen_007_5b6283ab3dc4c3b5.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/5b6283ab3dc4c3b5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@8", "density": 0.7191427449408644, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/8.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.33514+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.33514, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@8", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.33514+", "metric_symbol": "s", "metric_value": 3.33514, "reference_text": "s = 3.33514+ | Found by Jonathan Viquerat in April 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/8.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 3.33514+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "de5f4ff90a59b8bf", "improvement": -0.00017949381007964504, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.630985964589326e-06, "max_pair_overlap_depth": -3.1607294870367175e-05, "metric_expression": "3.33532", "metric_symbol": "s", "metric_value": 3.3353194938100796, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.33514, "setup": "peninpen", "side": 3.3353194938100796, "solution_path": "data/solutions/generated/peninpen_008_de5f4ff90a59b8bf.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/de5f4ff90a59b8bf.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@9", "density": 0.756208451025979, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/9.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.44956+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.44956, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@9", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.44956+", "metric_symbol": "s", "metric_value": 3.44956, "reference_text": "s = 3.44956+ | Found by Jonathan Viquerat in April 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/9.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 3.44956+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "291eb24a57cfcd61", "improvement": -0.00029221756483766015, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.66255661877786e-06, "max_pair_overlap_depth": -4.9483298464991776e-05, "metric_expression": "3.44985", "metric_symbol": "s", "metric_value": 3.4498522175648376, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.44956, "setup": "peninpen", "side": 3.4498522175648376, "solution_path": "data/solutions/generated/peninpen_009_291eb24a57cfcd61.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/291eb24a57cfcd61.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@10", "density": 0.7370589618267208, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/10.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.68324+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.68324, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@10", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.68324+", "metric_symbol": "s", "metric_value": 3.68324, "reference_text": "s = 3.68324+ | Found by Jonathan Viquerat in April 2026.", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/10.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 3.68324+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "90badee8fc213075", "improvement": -0.0001600058005193361, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0386948383243063e-06, "max_pair_overlap_depth": -3.744974033881299e-05, "metric_expression": "3.68340", "metric_symbol": "s", "metric_value": 3.6834000058005194, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.68324, "setup": "peninpen", "side": 3.6834000058005194, "solution_path": "data/solutions/generated/peninpen_010_90badee8fc213075.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/90badee8fc213075.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@11", "density": 0.7628515256681525, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/11.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.79704+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.79704, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.79704+", "metric_symbol": "s", "metric_value": 3.79704, "reference_text": "s = 3.79704+ | Found by Jonathan Viquerat in April 2026.", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/11.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 3.79704+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "e62e9f2d5556f0f5", "improvement": -0.0002724485549276956, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.930866570871943e-06, "max_pair_overlap_depth": -3.185052846443881e-05, "metric_expression": "3.79731", "metric_symbol": "s", "metric_value": 3.7973124485549277, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.79704, "setup": "peninpen", "side": 3.7973124485549277, "solution_path": "data/solutions/generated/peninpen_011_e62e9f2d5556f0f5.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/e62e9f2d5556f0f5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@12", "density": 0.7615744792401016, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/12.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.96926+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.96926, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@12", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.96926+", "metric_symbol": "s", "metric_value": 3.96926, "reference_text": "s = 3.96926+ | Found by Jonathan Viquerat in April 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/12.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 3.96926+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "6f961579a76d6be8", "improvement": -0.00022743831780980273, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.8385847579674985e-06, "max_pair_overlap_depth": -3.1875868486519465e-05, "metric_expression": "3.96949", "metric_symbol": "s", "metric_value": 3.9694874383178096, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.96926, "setup": "peninpen", "side": 3.9694874383178096, "solution_path": "data/solutions/generated/peninpen_012_6f961579a76d6be8.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/6f961579a76d6be8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@13", "density": 0.7865963698125578, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/13.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "4.06523+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.06523, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.06523+", "metric_symbol": "s", "metric_value": 4.06523, "reference_text": "s = 4.06523+ | Found by Jonathan Viquerat in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/13.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 4.06523+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "ce66c878ffee73fc", "improvement": -9.908623501608815e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.805365438846161e-06, "max_pair_overlap_depth": -6.344819103887289e-06, "metric_expression": "4.06533", "metric_symbol": "s", "metric_value": 4.065329086235016, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.06523, "setup": "peninpen", "side": 4.065329086235016, "solution_path": "data/solutions/generated/peninpen_013_ce66c878ffee73fc.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/ce66c878ffee73fc.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@14", "density": 0.7892057892472966, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/14.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "4.21120+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.2112, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@14", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.21120+", "metric_symbol": "s", "metric_value": 4.2112, "reference_text": "s = 4.21120+ | Found by Jonathan Viquerat in April 2026.", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/14.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 4.21120+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "367de42acefe1aa2", "improvement": -0.0006111178171677167, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.1042346422399235e-06, "max_pair_overlap_depth": -2.557915411749434e-05, "metric_expression": "4.21181", "metric_symbol": "s", "metric_value": 4.2118111178171675, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.2112, "setup": "peninpen", "side": 4.2118111178171675, "solution_path": "data/solutions/generated/peninpen_014_367de42acefe1aa2.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/367de42acefe1aa2.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@15", "density": 0.8358892308226911, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/15.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2+\u221a5 = 4.23606+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.23606, "friedman_reference": {"analytical_or_proved": true, "case": "peninpen@15", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2+\u221a5 = 4.23606+", "metric_symbol": "s", "metric_value": 4.23606, "reference_text": "s = 2+\u221a5 = 4.23606+ | Trivial.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/15.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 2+\u221a5 = 4.23606+ | Trivial.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "251c50a18682e896", "improvement": -8.936254454106063e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.6294493786472997e-06, "max_pair_overlap_depth": -5.5218241392651635e-06, "metric_expression": "4.23615", "metric_symbol": "s", "metric_value": 4.236149362544541, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.23606, "setup": "peninpen", "side": 4.236149362544541, "solution_path": "data/solutions/generated/peninpen_015_251c50a18682e896.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/251c50a18682e896.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@16", "density": 0.8202808577467057, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/16.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "4.41636+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.41636, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@16", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.41636+", "metric_symbol": "s", "metric_value": 4.41636, "reference_text": "s = 4.41636+ | Found by Jonathan Viquerat in April 2026.", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/16.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 4.41636+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "e2bb776d24739328", "improvement": -0.00014476044476463557, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7250867795500517e-06, "max_pair_overlap_depth": -1.097689597839846e-05, "metric_expression": "4.41650", "metric_symbol": "s", "metric_value": 4.416504760444765, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.41636, "setup": "peninpen", "side": 4.416504760444765, "solution_path": "data/solutions/generated/peninpen_016_e2bb776d24739328.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/e2bb776d24739328.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@17", "density": 0.7834157819221014, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/17.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "4.65561+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.65561, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.65561+", "metric_symbol": "s", "metric_value": 4.65561, "reference_text": "s = 4.65561+ | Found by Jonathan Viquerat in April 2026.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/17.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 4.65561+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "54856cb865a77d57", "improvement": -0.0026991003814726966, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9034050721792276e-06, "max_pair_overlap_depth": -2.8148707974851028e-05, "metric_expression": "4.65831", "metric_symbol": "s", "metric_value": 4.658309100381473, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.65561, "setup": "peninpen", "side": 4.658309100381473, "solution_path": "data/solutions/generated/peninpen_017_54856cb865a77d57.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/54856cb865a77d57.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@18", "density": 0.7892194761771266, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/18.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "4.77541+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.77541, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@18", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.77541+", "metric_symbol": "s", "metric_value": 4.77541, "reference_text": "s = 4.77541+ | Found by Jonathan Viquerat in April 2026.", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/18.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 4.77541+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "9c442356ca78c3de", "improvement": -0.0002934971525236918, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.1318004154280254e-06, "max_pair_overlap_depth": -1.258487199295999e-05, "metric_expression": "4.77570", "metric_symbol": "s", "metric_value": 4.775703497152524, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.77541, "setup": "peninpen", "side": 4.775703497152524, "solution_path": "data/solutions/generated/peninpen_018_9c442356ca78c3de.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/9c442356ca78c3de.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@19", "density": 0.7735353200149236, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/19.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "4.95269+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.95269, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@19", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "4.95269+", "metric_symbol": "s", "metric_value": 4.95269, "reference_text": "s = 4.95269+ | Found by Ian Watson in April 2026.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/19.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 4.95269+ | Found by Ian Watson in April 2026.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "b5440311f7b9b10b", "improvement": -0.0033719480383975764, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.8067945532274755e-06, "max_pair_overlap_depth": -5.844604614307913e-06, "metric_expression": "4.95606", "metric_symbol": "s", "metric_value": 4.956061948038397, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.95269, "setup": "peninpen", "side": 4.956061948038397, "solution_path": "data/solutions/generated/peninpen_019_b5440311f7b9b10b.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/b5440311f7b9b10b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@20", "density": 0.7890584950748962, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/20.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "5.04154+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.04154, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@20", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.04154+", "metric_symbol": "s", "metric_value": 5.04154, "reference_text": "s = 5.04154+ | Found by Jonathan Viquerat in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/20.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 5.04154+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "74f61b73dca76fef", "improvement": 0.0069930192193110585, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.063547145356236e-06, "max_pair_overlap_depth": -1.8870971417950244e-05, "metric_expression": "5.03455", "metric_symbol": "s", "metric_value": 5.034546980780689, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.04154, "setup": "peninpen", "side": 5.034546980780689, "solution_path": "data/solutions/generated/peninpen_020_74f61b73dca76fef.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/74f61b73dca76fef.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@21", "density": 0.7887631772097712, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/21.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "5.15948+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.15948, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@21", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "5.15948+", "metric_symbol": "s", "metric_value": 5.15948, "reference_text": "s = 5.15948+ | Found by Ignacio Vallejo in April 2026.", "source_image": "21.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/21.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 5.15948+ | Found by Ignacio Vallejo in April 2026.", "friedman_source_image": "21.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "b731ea055eb53ce1", "improvement": -0.00036117133928748046, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7271978645132776e-06, "max_pair_overlap_depth": -5.857494896677196e-06, "metric_expression": "5.15984", "metric_symbol": "s", "metric_value": 5.159841171339288, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.15948, "setup": "peninpen", "side": 5.159841171339288, "solution_path": "data/solutions/generated/peninpen_021_b731ea055eb53ce1.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/b731ea055eb53ce1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@22", "density": 0.7927891464448104, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/22.png", "friedman_metric_expression": "5.26740+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.2674, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@22", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "5.26740+", "metric_symbol": "s", "metric_value": 5.2674, "reference_text": "s = 5.26740+ | Found by Ignacio Vallejo in April 2026.", "source_image": "22.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/22.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 5.26740+ | Found by Ignacio Vallejo in April 2026.", "friedman_source_image": "22.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "07e217f200bb677e", "improvement": -0.0004389908125848052, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.647717325013132e-07, "max_pair_overlap_depth": -3.201222129578518e-05, "metric_expression": "5.26784", "metric_symbol": "s", "metric_value": 5.267838990812585, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.2674, "setup": "peninpen", "side": 5.267838990812585, "solution_path": "data/solutions/generated/peninpen_022_07e217f200bb677e.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/07e217f200bb677e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@23", "density": 0.8042579677659262, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/23.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "5.34794+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.34794, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@23", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "5.34794+", "metric_symbol": "s", "metric_value": 5.34794, "reference_text": "s = 5.34794+ | Found by Ignacio Vallejo in April 2026.", "source_image": "23.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/23.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 5.34794+ | Found by Ignacio Vallejo in April 2026.", "friedman_source_image": "23.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "4286cb7dfd8b00d7", "improvement": 0.00024989883471082663, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.435117490986131e-07, "max_pair_overlap_depth": -1.717758867814112e-05, "metric_expression": "5.34769", "metric_symbol": "s", "metric_value": 5.3476901011652895, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.34794, "setup": "peninpen", "side": 5.3476901011652895, "solution_path": "data/solutions/generated/peninpen_023_4286cb7dfd8b00d7.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/4286cb7dfd8b00d7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@24", "density": 0.7956104957168404, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/24.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "5.49241+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.49241, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@24", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "5.49241+", "metric_symbol": "s", "metric_value": 5.49241, "reference_text": "s = 5.49241+ | Found by Ignacio Vallejo in April 2026.", "source_image": "24.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/24.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 5.49241+ | Found by Ignacio Vallejo in April 2026.", "friedman_source_image": "24.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "8859e051020078b2", "improvement": 9.586393159466411e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.446346359580623e-06, "max_pair_overlap_depth": -9.458275242596459e-07, "metric_expression": "5.49231", "metric_symbol": "s", "metric_value": 5.492314136068405, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.49241, "setup": "peninpen", "side": 5.492314136068405, "solution_path": "data/solutions/generated/peninpen_024_8859e051020078b2.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/8859e051020078b2.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninpen@25", "density": 0.8019460079699491, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/25.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "5.58370+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.5837, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@25", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "5.58370+", "metric_symbol": "s", "metric_value": 5.5837, "reference_text": "s = 5.58370+ | Found by Ignacio Vallejo in April 2026.", "source_image": "25.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/25.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 5.58370+ | Found by Ignacio Vallejo in April 2026.", "friedman_source_image": "25.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "0f0645180a7be991", "improvement": 0.00031674927347591364, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.618192580437412e-06, "max_pair_overlap_depth": -2.8739793660470525e-06, "metric_expression": "5.58338", "metric_symbol": "s", "metric_value": 5.583383250726524, "n": 25, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.5837, "setup": "peninpen", "side": 5.583383250726524, "solution_path": "data/solutions/generated/peninpen_025_0f0645180a7be991.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/0f0645180a7be991.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@1", "density": 0.6736263863631768, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/1.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(3\u221a2+\u221a10+2\u221a(5+\u221a5))/8 = 1.59811+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.59811, "friedman_reference": {"analytical_or_proved": true, "case": "peninsqu@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(3\u221a2+\u221a10+2\u221a(5+\u221a5))/8 = 1.59811+", "metric_symbol": "s", "metric_value": 1.59811, "reference_text": "s = (3\u221a2+\u221a10+2\u221a(5+\u221a5))/8 = 1.59811+ | Trivial. |", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/1.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = (3\u221a2+\u221a10+2\u221a(5+\u221a5))/8 = 1.59811+ | Trivial. |", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "ec216f3e37e16785", "improvement": -3.043902908794749e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.733207856432108e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.59814", "metric_symbol": "s", "metric_value": 1.5981404390290879, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.59811, "setup": "peninsqu", "side": 1.5981404390290879, "solution_path": "data/solutions/generated/peninsqu_001_ec216f3e37e16785.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/ec216f3e37e16785.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@2", "density": 0.5669967061443146, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/2.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2.46345+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.46345, "friedman_reference": {"analytical_or_proved": true, "case": "peninsqu@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2.46345+", "metric_symbol": "s", "metric_value": 2.46345, "reference_text": "s = 2.46345+ | Trivial. |", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/2.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 2.46345+ | Trivial. |", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "bb5e10eb4917963f", "improvement": -3.0994037563569776e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.416866607183818e-07, "max_pair_overlap_depth": -1.473425980974552e-05, "metric_expression": "2.46348", "metric_symbol": "s", "metric_value": 2.4634809940375635, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.46345, "setup": "peninsqu", "side": 2.4634809940375635, "solution_path": "data/solutions/generated/peninsqu_002_bb5e10eb4917963f.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/bb5e10eb4917963f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@3", "density": 0.6102674136126431, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/3.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2.90812+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.90812, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@3", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.90812+", "metric_symbol": "s", "metric_value": 2.90812, "reference_text": "s = 2.90812+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/3.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 2.90812+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "52b96ec8c7f6b741", "improvement": -8.50430757179943e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.362613813108376e-06, "max_pair_overlap_depth": -5.5799797820554886e-05, "metric_expression": "2.90821", "metric_symbol": "s", "metric_value": 2.908205043075718, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.90812, "setup": "peninsqu", "side": 2.908205043075718, "solution_path": "data/solutions/generated/peninsqu_003_52b96ec8c7f6b741.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/52b96ec8c7f6b741.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@4", "density": 0.7148924907955395, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/4.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3.10252+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.10252, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@4", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.10252+", "metric_symbol": "s", "metric_value": 3.10252, "reference_text": "s = 3.10252+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/4.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 3.10252+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "a2fcd0b0beef7a6f", "improvement": -0.00013948165686938907, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2690386136380924e-06, "max_pair_overlap_depth": -3.0975557991252956e-05, "metric_expression": "3.10266", "metric_symbol": "s", "metric_value": 3.1026594816568696, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.10252, "setup": "peninsqu", "side": 3.1026594816568696, "solution_path": "data/solutions/generated/peninsqu_004_a2fcd0b0beef7a6f.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/a2fcd0b0beef7a6f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@5", "density": 0.6829538263136812, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/5.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3.54903+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.54903, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@5", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.54903+", "metric_symbol": "s", "metric_value": 3.54903, "reference_text": "s = 3.54903+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/5.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 3.54903+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "bda3fbac20c8a756", "improvement": -3.388580996466217e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.104606408908601e-07, "max_pair_overlap_depth": -5.251218256785428e-06, "metric_expression": "3.54906", "metric_symbol": "s", "metric_value": 3.549063885809965, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.54903, "setup": "peninsqu", "side": 3.549063885809965, "solution_path": "data/solutions/generated/peninsqu_005_bda3fbac20c8a756.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/bda3fbac20c8a756.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@6", "density": 0.6845927654826146, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/6.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3.88300+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.883, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@6", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.88300+", "metric_symbol": "s", "metric_value": 3.883, "reference_text": "s = 3.88300+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/6.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 3.88300+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "a267ec5809e79964", "improvement": -0.00014813728762552714, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0190677678703185e-06, "max_pair_overlap_depth": -5.678337215275775e-05, "metric_expression": "3.88315", "metric_symbol": "s", "metric_value": 3.8831481372876255, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.883, "setup": "peninsqu", "side": 3.8831481372876255, "solution_path": "data/solutions/generated/peninsqu_006_a267ec5809e79964.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/a267ec5809e79964.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@7", "density": 0.6891282499986281, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/7.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.18038+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.18038, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@7", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.18038+", "metric_symbol": "s", "metric_value": 4.18038, "reference_text": "s = 4.18038+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/7.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 4.18038+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "b5565b951637dc85", "improvement": -7.429190517971307e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4725627916323898e-06, "max_pair_overlap_depth": -1.919059115845778e-05, "metric_expression": "4.18045", "metric_symbol": "s", "metric_value": 4.18045429190518, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.18038, "setup": "peninsqu", "side": 4.18045429190518, "solution_path": "data/solutions/generated/peninsqu_007_b5565b951637dc85.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b5565b951637dc85.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@8", "density": 0.7167004559270932, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/8.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.38191+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.38191, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@8", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.38191+", "metric_symbol": "s", "metric_value": 4.38191, "reference_text": "s = 4.38191+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/8.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 4.38191+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "eac74e93d82a54dd", "improvement": -0.00037521167052823046, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2846178652549832e-06, "max_pair_overlap_depth": -2.0945039418324152e-05, "metric_expression": "4.38229", "metric_symbol": "s", "metric_value": 4.382285211670529, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.38191, "setup": "peninsqu", "side": 4.382285211670529, "solution_path": "data/solutions/generated/peninsqu_008_eac74e93d82a54dd.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/eac74e93d82a54dd.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@9", "density": 0.7314097601583741, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/9.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.60033+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.60033, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@9", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.60033+", "metric_symbol": "s", "metric_value": 4.60033, "reference_text": "s = 4.60033+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/9.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 4.60033+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "0270d827693b6f8d", "improvement": -0.0008091240419938472, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8644555136003204e-06, "max_pair_overlap_depth": -1.0559011279553943e-05, "metric_expression": "4.60114", "metric_symbol": "s", "metric_value": 4.601139124041993, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.60033, "setup": "peninsqu", "side": 4.601139124041993, "solution_path": "data/solutions/generated/peninsqu_009_0270d827693b6f8d.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/0270d827693b6f8d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@10", "density": 0.7142869465585481, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/10.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.90584+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.90584, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@10", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.90584+", "metric_symbol": "s", "metric_value": 4.90584, "reference_text": "s = 4.90584+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/10.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 4.90584+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "fba502b92276222e", "improvement": -0.0019743867107484547, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.710048632437797e-06, "max_pair_overlap_depth": -3.5387419318078095e-05, "metric_expression": "4.90781", "metric_symbol": "s", "metric_value": 4.907814386710749, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.90584, "setup": "peninsqu", "side": 4.907814386710749, "solution_path": "data/solutions/generated/peninsqu_010_fba502b92276222e.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/fba502b92276222e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@11", "density": 0.7230188560400136, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/11.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5.11557+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.11557, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.11557+", "metric_symbol": "s", "metric_value": 5.11557, "reference_text": "s = 5.11557+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/11.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 5.11557+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "3596998e61e47a27", "improvement": -0.0006123708648555493, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.520631561500943e-07, "max_pair_overlap_depth": -2.3531878005744744e-05, "metric_expression": "5.11618", "metric_symbol": "s", "metric_value": 5.1161823708648555, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.11557, "setup": "peninsqu", "side": 5.1161823708648555, "solution_path": "data/solutions/generated/peninsqu_011_3596998e61e47a27.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/3596998e61e47a27.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@12", "density": 0.7539256100231233, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/12.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5.23250+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.2325, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@12", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.23250+", "metric_symbol": "s", "metric_value": 5.2325, "reference_text": "s = 5.23250+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/12.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 5.23250+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "72fe010d0104fc76", "improvement": -0.0005015230417875216, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6940168948787563e-07, "max_pair_overlap_depth": -3.660774509528508e-05, "metric_expression": "5.23300", "metric_symbol": "s", "metric_value": 5.2330015230417875, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.2325, "setup": "peninsqu", "side": 5.2330015230417875, "solution_path": "data/solutions/generated/peninsqu_012_72fe010d0104fc76.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/72fe010d0104fc76.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@13", "density": 0.7325239130137876, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/13.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5.52183+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.52183, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.52183+", "metric_symbol": "s", "metric_value": 5.52183, "reference_text": "s = 5.52183+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/13.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 5.52183+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "b54e12045e01b08b", "improvement": -0.0038439980668076856, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7653030405329436e-07, "max_pair_overlap_depth": -2.666929117722816e-05, "metric_expression": "5.52567", "metric_symbol": "s", "metric_value": 5.525673998066807, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.52183, "setup": "peninsqu", "side": 5.525673998066807, "solution_path": "data/solutions/generated/peninsqu_013_b54e12045e01b08b.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b54e12045e01b08b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@14", "density": 0.7420924387932526, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/14.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5.69775+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.69775, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@14", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.69775+", "metric_symbol": "s", "metric_value": 5.69775, "reference_text": "s = 5.69775+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/14.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 5.69775+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "70191e649c0c34a4", "improvement": 0.0005757762276479284, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.376635005869759e-06, "max_pair_overlap_depth": -1.761308907194392e-05, "metric_expression": "5.69717", "metric_symbol": "s", "metric_value": 5.697174223772352, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.69775, "setup": "peninsqu", "side": 5.697174223772352, "solution_path": "data/solutions/generated/peninsqu_014_70191e649c0c34a4.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/70191e649c0c34a4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@15", "density": 0.737588729235034, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/15.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5.91106+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.91106, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@15", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.91106+", "metric_symbol": "s", "metric_value": 5.91106, "reference_text": "s = 5.91106+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/15.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 5.91106+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "f5d70026e15060e9", "improvement": -0.004052102776268995, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.031797802365455e-06, "max_pair_overlap_depth": -3.458627910934986e-05, "metric_expression": "5.91511", "metric_symbol": "s", "metric_value": 5.915112102776269, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.91106, "setup": "peninsqu", "side": 5.915112102776269, "solution_path": "data/solutions/generated/peninsqu_015_f5d70026e15060e9.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/f5d70026e15060e9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@16", "density": 0.7479800847585042, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/16.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.11021+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.11021, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@16", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.11021+", "metric_symbol": "s", "metric_value": 6.11021, "reference_text": "s = 6.11021+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/16.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 6.11021+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": false, "id": "4644b3e97f0bdea2", "improvement": 0.04369242522759276, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5325677563252472e-06, "max_pair_overlap_depth": -5.091262792966678e-06, "metric_expression": "6.06652", "metric_symbol": "s", "metric_value": 6.066517574772408, "n": 16, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 6.11021, "setup": "peninsqu", "side": 6.066517574772408, "solution_path": "data/solutions/generated/peninsqu_016_4644b3e97f0bdea2.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/4644b3e97f0bdea2.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@17", "density": 0.7514950307673967, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/17.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.26466+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.26466, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.26466+", "metric_symbol": "s", "metric_value": 6.26466, "reference_text": "s = 6.26466+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/17.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 6.26466+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": false, "id": "a6cd0104bc26d2a0", "improvement": 0.026077965234885703, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.35268853525978e-06, "max_pair_overlap_depth": -1.565351530530279e-05, "metric_expression": "6.23858", "metric_symbol": "s", "metric_value": 6.238582034765114, "n": 17, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 6.26466, "setup": "peninsqu", "side": 6.238582034765114, "solution_path": "data/solutions/generated/peninsqu_017_a6cd0104bc26d2a0.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/a6cd0104bc26d2a0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@18", "density": 0.7710926756280118, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/18.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.33632+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.33632, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@18", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.33632+", "metric_symbol": "s", "metric_value": 6.33632, "reference_text": "s = 6.33632+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/18.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 6.33632+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "9705d17615c32308", "improvement": -0.0010264494540361824, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.1888526227077803e-06, "max_pair_overlap_depth": -2.3019703547788595e-06, "metric_expression": "6.33735", "metric_symbol": "s", "metric_value": 6.337346449454036, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.33632, "setup": "peninsqu", "side": 6.337346449454036, "solution_path": "data/solutions/generated/peninsqu_018_9705d17615c32308.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/9705d17615c32308.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@19", "density": 0.759469213458349, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/19.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.55939+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.55939, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@19", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.55939+", "metric_symbol": "s", "metric_value": 6.55939, "reference_text": "s = 6.55939+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/19.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 6.55939+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "3b1113745987c1d6", "improvement": -0.0012498840705594816, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9916513391919655e-06, "max_pair_overlap_depth": -2.545361094719034e-05, "metric_expression": "6.56064", "metric_symbol": "s", "metric_value": 6.560639884070559, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.55939, "setup": "peninsqu", "side": 6.560639884070559, "solution_path": "data/solutions/generated/peninsqu_019_3b1113745987c1d6.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/3b1113745987c1d6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@20", "density": 0.7723902783602073, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/20.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.69588+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.69588, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@20", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.69588+", "metric_symbol": "s", "metric_value": 6.69588, "reference_text": "s = 6.69588+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/20.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 6.69588+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": false, "id": "eafd28d28fb8e254", "improvement": 0.021343927884186265, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.197045603091071e-06, "max_pair_overlap_depth": -2.9426570577584243e-05, "metric_expression": "6.67454", "metric_symbol": "s", "metric_value": 6.674536072115814, "n": 20, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 6.69588, "setup": "peninsqu", "side": 6.674536072115814, "solution_path": "data/solutions/generated/peninsqu_020_eafd28d28fb8e254.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/eafd28d28fb8e254.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@21", "density": 0.7688621226676746, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/21.png", "friedman_metric_expression": "6.85540+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.8554, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@21", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "6.85540+", "metric_symbol": "s", "metric_value": 6.8554, "reference_text": "s = 6.85540+ | Found by Emerson Connelly | in April 2026.", "source_image": "21.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/21.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 6.85540+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "21.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "9d320f91f05e2f0e", "improvement": 0.0003614666962281632, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5104768148309566e-06, "max_pair_overlap_depth": -4.3073173047414315e-05, "metric_expression": "6.85504", "metric_symbol": "s", "metric_value": 6.855038533303772, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.8554, "setup": "peninsqu", "side": 6.855038533303772, "solution_path": "data/solutions/generated/peninsqu_21_9d320f91f05e2f0e.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/9d320f91f05e2f0e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@22", "density": 0.7625061251104759, "friedman_analytical_or_proved": false, "friedman_credit": "Teodors Smith", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/22.png", "friedman_metric_expression": "7.04532+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.04532, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@22", "credit": "Teodors Smith", "date": "April 2026", "family_updated": "", "metric_expression": "7.04532+", "metric_symbol": "s", "metric_value": 7.04532, "reference_text": "s = 7.04532+ | Found by Teodors Smith | in April 2026.", "source_image": "22.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/22.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 7.04532+ | Found by Teodors Smith | in April 2026.", "friedman_source_image": "22.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "70964f50e662283c", "improvement": -0.0002179452221122702, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.550099486392989e-07, "max_pair_overlap_depth": -3.276547527986118e-05, "metric_expression": "7.04554", "metric_symbol": "s", "metric_value": 7.0455379452221125, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.04532, "setup": "peninsqu", "side": 7.0455379452221125, "solution_path": "data/solutions/generated/peninsqu_22_70964f50e662283c.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "Teodors Smith", "svg_path": "data/svg/70964f50e662283c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@23", "density": 0.7617923823536098, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/23.png", "friedman_metric_expression": "7.20731+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.20731, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@23", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "7.20731+", "metric_symbol": "s", "metric_value": 7.20731, "reference_text": "s = 7.20731+ | Found by Emerson Connelly | in April 2026.", "source_image": "23.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/23.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 7.20731+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "23.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "5b526f4bb0fda83c", "improvement": 5.162671385594564e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.635959726042516e-06, "max_pair_overlap_depth": -3.224520897293637e-05, "metric_expression": "7.20726", "metric_symbol": "s", "metric_value": 7.207258373286144, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.20731, "setup": "peninsqu", "side": 7.207258373286144, "solution_path": "data/solutions/generated/peninsqu_23_5b526f4bb0fda83c.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/5b526f4bb0fda83c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "peninsqu@24", "density": 0.766901141845129, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/24.png", "friedman_metric_expression": "7.33823+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.33823, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@24", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "7.33823+", "metric_symbol": "s", "metric_value": 7.33823, "reference_text": "s = 7.33823+ | Found by Emerson Connelly | in April 2026.", "source_image": "24.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/24.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 7.33823+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "24.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "3e42e9e085ed4c16", "improvement": 0.0005221765418985669, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.294217705264547e-06, "max_pair_overlap_depth": -1.6096651148655994e-05, "metric_expression": "7.33771", "metric_symbol": "s", "metric_value": 7.337707823458102, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.33823, "setup": "peninsqu", "side": 7.337707823458102, "solution_path": "data/solutions/generated/peninsqu_24_3e42e9e085ed4c16.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/3e42e9e085ed4c16.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@1", "density": 0.6366152339868586, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/1.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a2 / 2 = .707+", "friedman_metric_symbol": "r", "friedman_metric_value": 0.707, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a2 / 2 = .707+", "metric_symbol": "r", "metric_value": 0.707, "reference_text": "r = \u221a2 / 2 = .707+ | Trivial.", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/1.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a2 / 2 = .707+ | Trivial.", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "e1e16b7d18b1243e", "improvement": -0.00010930163680866745, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.331866965270656e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.70711", "metric_symbol": "r", "metric_value": 0.7071093016368086, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.707, "setup": "squincir", "side": 1.4142186032736173, "solution_path": "data/solutions/generated/squincir_001_e1e16b7d18b1243e.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/e1e16b7d18b1243e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@2", "density": 0.5092878238610891, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/2.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a5 / 2 = 1.118+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.118, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a5 / 2 = 1.118+", "metric_symbol": "r", "metric_value": 1.118, "reference_text": "r = \u221a5 / 2 = 1.118+ | Trivial.", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/2.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a5 / 2 = 1.118+ | Trivial.", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "e0e32383f27cd27d", "improvement": -4.2763322036343254e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.769902571588602e-06, "max_pair_overlap_depth": -3.18210820038356e-06, "metric_expression": "1.11804", "metric_symbol": "r", "metric_value": 1.1180427633220364, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.118, "setup": "squincir", "side": 2.236085526644073, "solution_path": "data/solutions/generated/squincir_002_e0e32383f27cd27d.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/e0e32383f27cd27d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@3", "density": 0.5751857401746373, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/3.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "5 \u221a17 / 16 = 1.288+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.288, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@3", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 \u221a17 / 16 = 1.288+", "metric_symbol": "r", "metric_value": 1.288, "reference_text": "r = 5 \u221a17 / 16 = 1.288+ | Found by Erich Friedman | in 1997.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/3.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 5 \u221a17 / 16 = 1.288+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "226172dcb69c3daf", "improvement": -0.0004917307904808155, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4552703365164916e-07, "max_pair_overlap_depth": -2.6483206490832134e-05, "metric_expression": "1.28849", "metric_symbol": "r", "metric_value": 1.2884917307904808, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.288, "setup": "squincir", "side": 2.5769834615809617, "solution_path": "data/solutions/generated/squincir_003_226172dcb69c3daf.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/226172dcb69c3daf.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@4", "density": 0.6365934661039774, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/4.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a2 = 1.414+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.414, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@4", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a2 = 1.414+", "metric_symbol": "r", "metric_value": 1.414, "reference_text": "r = \u221a2 = 1.414+ | Found by Erich Friedman | in 1997.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/4.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a2 = 1.414+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "59645297dac888b6", "improvement": -0.0002427821910928607, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.7659081458272254e-06, "max_pair_overlap_depth": -2.7879728913877792e-05, "metric_expression": "1.41424", "metric_symbol": "r", "metric_value": 1.4142427821910928, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.414, "setup": "squincir", "side": 2.8284855643821856, "solution_path": "data/solutions/generated/squincir_004_59645297dac888b6.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/59645297dac888b6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@5", "density": 0.6366006254695611, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/5.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a10 / 2 = 1.581+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.581, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@5", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a10 / 2 = 1.581+", "metric_symbol": "r", "metric_value": 1.581, "reference_text": "r = \u221a10 / 2 = 1.581+ | Found by Erich Friedman | in 1997.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/5.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a10 / 2 = 1.581+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "ee29f4bb5ac19bc4", "improvement": -0.00016260768909037715, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.383463508896625e-07, "max_pair_overlap_depth": -1.4376414261496606e-05, "metric_expression": "1.58116", "metric_symbol": "r", "metric_value": 1.5811626076890903, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.581, "setup": "squincir", "side": 3.1623252153781807, "solution_path": "data/solutions/generated/squincir_005_ee29f4bb5ac19bc4.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/ee29f4bb5ac19bc4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@6", "density": 0.6697921489003816, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/6.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "1.688+ *", "friedman_metric_symbol": "r", "friedman_metric_value": 1.688, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@6", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "1.688+ *", "metric_symbol": "r", "metric_value": 1.688, "reference_text": "r = 1.688+ * | Found by Erich Friedman | in 1997.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/6.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 1.688+ * | Found by Erich Friedman | in 1997.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "00816b044d90c514", "improvement": -0.0006150768345383195, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.526575301033731e-06, "max_pair_overlap_depth": -2.575413327610132e-05, "metric_expression": "1.68862", "metric_symbol": "r", "metric_value": 1.6886150768345383, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.688, "setup": "squincir", "side": 3.3772301536690765, "solution_path": "data/solutions/generated/squincir_006_00816b044d90c514.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/00816b044d90c514.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@7", "density": 0.685554128713642, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/7.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a13 / 2 = 1.802+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.802, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@7", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a13 / 2 = 1.802+", "metric_symbol": "r", "metric_value": 1.802, "reference_text": "r = \u221a13 / 2 = 1.802+ | Found by Erich Friedman | in 1997.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/7.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a13 / 2 = 1.802+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "f533c6ece1865b92", "improvement": -0.0008234908634709992, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3338190876808227e-06, "max_pair_overlap_depth": -2.5625121306882548e-05, "metric_expression": "1.80282", "metric_symbol": "r", "metric_value": 1.802823490863471, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.802, "setup": "squincir", "side": 3.605646981726942, "solution_path": "data/solutions/generated/squincir_007_f533c6ece1865b92.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f533c6ece1865b92.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@8", "density": 0.6501060351379704, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "March 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/8.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "1.978+ *", "friedman_metric_symbol": "r", "friedman_metric_value": 1.978, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@8", "credit": "David W. Cantrell", "date": "March 2002", "family_updated": "", "metric_expression": "1.978+ *", "metric_symbol": "r", "metric_value": 1.978, "reference_text": "r = 1.978+ * | Found by David W. Cantrell | in March 2002.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/8.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 1.978+ * | Found by David W. Cantrell | in March 2002.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "056dfca89502965a", "improvement": -0.0011465711618996455, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7255803199667241e-06, "max_pair_overlap_depth": -1.3772848531545812e-05, "metric_expression": "1.97915", "metric_symbol": "r", "metric_value": 1.9791465711618996, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.978, "setup": "squincir", "side": 3.9582931423237993, "solution_path": "data/solutions/generated/squincir_008_056dfca89502965a.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/056dfca89502965a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@9", "density": 0.6635107369929176, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/9.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a1105 / 16 = 2.077+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.077, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@9", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a1105 / 16 = 2.077+", "metric_symbol": "r", "metric_value": 2.077, "reference_text": "r = \u221a1105 / 16 = 2.077+ | Found by Erich Friedman | in 1997.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/9.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a1105 / 16 = 2.077+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "a0eb38ed275fd3c7", "improvement": -0.0008889848991340266, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.939578609861826e-06, "max_pair_overlap_depth": -1.5145046883935365e-05, "metric_expression": "2.07789", "metric_symbol": "r", "metric_value": 2.077888984899134, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.077, "setup": "squincir", "side": 4.155777969798268, "solution_path": "data/solutions/generated/squincir_009_a0eb38ed275fd3c7.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a0eb38ed275fd3c7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@10", "density": 0.7072738827790132, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/10.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3 \u221a2 / 2 = 2.121+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.121, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@10", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 \u221a2 / 2 = 2.121+", "metric_symbol": "r", "metric_value": 2.121, "reference_text": "r = 3 \u221a2 / 2 = 2.121+ | Found by Erich Friedman | in 1997.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/10.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3 \u221a2 / 2 = 2.121+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "eacd7cf43d9f9c99", "improvement": -0.00044244096873002903, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.093937328557672e-07, "max_pair_overlap_depth": -1.6684797704336063e-05, "metric_expression": "2.12144", "metric_symbol": "r", "metric_value": 2.12144244096873, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.121, "setup": "squincir", "side": 4.24288488193746, "solution_path": "data/solutions/generated/squincir_010_eacd7cf43d9f9c99.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/eacd7cf43d9f9c99.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@11", "density": 0.7141684249475255, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "March 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/11.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.214+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.214, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@11", "credit": "David W. Cantrell", "date": "March 2002", "family_updated": "", "metric_expression": "2.214+", "metric_symbol": "r", "metric_value": 2.214, "reference_text": "r = 2.214+ | Found by David W. Cantrell | in March 2002.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/11.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 2.214+ | Found by David W. Cantrell | in March 2002.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "ed2b520c292f3457", "improvement": -0.00022160321374320446, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.710372663145449e-07, "max_pair_overlap_depth": -1.7996528127950384e-05, "metric_expression": "2.21422", "metric_symbol": "r", "metric_value": 2.214221603213743, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.214, "setup": "squincir", "side": 4.428443206427486, "solution_path": "data/solutions/generated/squincir_011_ed2b520c292f3457.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/ed2b520c292f3457.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@12", "density": 0.7638630053365897, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/12.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a5 = 2.236+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.236, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@12", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a5 = 2.236+", "metric_symbol": "r", "metric_value": 2.236, "reference_text": "r = \u221a5 = 2.236+ | Found by Erich Friedman | in 1997.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/12.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a5 = 2.236+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "064666a02cb4bcc6", "improvement": -0.00018612303226639426, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.505810228203842e-06, "max_pair_overlap_depth": -1.897198109612752e-05, "metric_expression": "2.23619", "metric_symbol": "r", "metric_value": 2.2361861230322666, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.236, "setup": "squincir", "side": 4.472372246064533, "solution_path": "data/solutions/generated/squincir_012_064666a02cb4bcc6.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/064666a02cb4bcc6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@13", "density": 0.7419440056386433, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "November 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/13.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.3607+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.3607, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@13", "credit": "David W. Cantrell", "date": "November 2010", "family_updated": "", "metric_expression": "2.3607+", "metric_symbol": "r", "metric_value": 2.3607, "reference_text": "r = 2.3607+ | Found by David W. Cantrell | in November 2010.", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/13.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 2.3607+ | Found by David W. Cantrell | in November 2010.", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "1ff894504e55415a", "improvement": -0.0009262889169745492, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0049576295578788e-06, "max_pair_overlap_depth": -2.3549111853871096e-05, "metric_expression": "2.36163", "metric_symbol": "r", "metric_value": 2.3616262889169746, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.3607, "setup": "squincir", "side": 4.723252577833949, "solution_path": "data/solutions/generated/squincir_013_1ff894504e55415a.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/1ff894504e55415a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@14", "density": 0.7129258170438343, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/14.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.5", "friedman_metric_symbol": "r", "friedman_metric_value": 2.5, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@14", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "2.5", "metric_symbol": "r", "metric_value": 2.5, "reference_text": "r = 2.5 | Found by Erich Friedman | in 1997.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/14.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 2.5 | Found by Erich Friedman | in 1997.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "16de07d86777c846", "improvement": -0.0001548640652071498, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.318191377805647e-07, "max_pair_overlap_depth": -1.9980447390599565e-05, "metric_expression": "2.50015", "metric_symbol": "r", "metric_value": 2.500154864065207, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.5, "setup": "squincir", "side": 5.000309728130414, "solution_path": "data/solutions/generated/squincir_014_16de07d86777c846.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/16de07d86777c846.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@15", "density": 0.7434807535174716, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "June 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/15.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.533+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.533, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@15", "credit": "David W. Cantrell", "date": "June 2002", "family_updated": "", "metric_expression": "2.533+", "metric_symbol": "r", "metric_value": 2.533, "reference_text": "r = 2.533+ | Found by by David W. Cantrell | in June 2002.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/15.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 2.533+ | Found by by David W. Cantrell | in June 2002.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "4a2ad28ea2097ba3", "improvement": -0.001170482100864234, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5068887214985125e-06, "max_pair_overlap_depth": -8.103900655004326e-06, "metric_expression": "2.53417", "metric_symbol": "r", "metric_value": 2.534170482100864, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.533, "setup": "squincir", "side": 5.068340964201728, "solution_path": "data/solutions/generated/squincir_015_4a2ad28ea2097ba3.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/4a2ad28ea2097ba3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@16", "density": 0.7401412241351131, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/16.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a11009 / 40 = 2.623+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.623, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@16", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a11009 / 40 = 2.623+", "metric_symbol": "r", "metric_value": 2.623, "reference_text": "r = \u221a11009 / 40 = 2.623+ | Found by Erich Friedman | in 1997.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/16.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a11009 / 40 = 2.623+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "94d0a09f4c75dcc4", "improvement": -0.00017797945661435222, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.109474628753105e-06, "max_pair_overlap_depth": -5.84557896199378e-06, "metric_expression": "2.62318", "metric_symbol": "r", "metric_value": 2.6231779794566146, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.623, "setup": "squincir", "side": 5.246355958913229, "solution_path": "data/solutions/generated/squincir_016_94d0a09f4c75dcc4.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/94d0a09f4c75dcc4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@17", "density": 0.7544956917261334, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "December 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/17.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.677+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.677, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@17", "credit": "Maurizio Morandi", "date": "December 2010", "family_updated": "", "metric_expression": "2.677+", "metric_symbol": "r", "metric_value": 2.677, "reference_text": "r = 2.677+ | Found by Maurizio Morandi | in December 2010.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/17.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 2.677+ | Found by Maurizio Morandi | in December 2010.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "036910f6c0550e2f", "improvement": -0.0010651760341882088, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2297138575488873e-06, "max_pair_overlap_depth": -2.14353819976143e-05, "metric_expression": "2.67807", "metric_symbol": "r", "metric_value": 2.6780651760341883, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.677, "setup": "squincir", "side": 5.3561303520683765, "solution_path": "data/solutions/generated/squincir_017_036910f6c0550e2f.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/036910f6c0550e2f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@18", "density": 0.7617085472368383, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "March 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/18.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a481 / 8 = 2.741+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.741, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@18", "credit": "David W. Cantrell", "date": "March 2002", "family_updated": "", "metric_expression": "\u221a481 / 8 = 2.741+", "metric_symbol": "r", "metric_value": 2.741, "reference_text": "r = \u221a481 / 8 = 2.741+ | Found by David W. Cantrell | in March 2002.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/18.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a481 / 8 = 2.741+ | Found by David W. Cantrell | in March 2002.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "d726e310dc0e9eaa", "improvement": -0.001627976420446764, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.727763966399067e-06, "max_pair_overlap_depth": -2.573974201414142e-05, "metric_expression": "2.74263", "metric_symbol": "r", "metric_value": 2.742627976420447, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.741, "setup": "squincir", "side": 5.485255952840894, "solution_path": "data/solutions/generated/squincir_018_d726e310dc0e9eaa.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d726e310dc0e9eaa.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@19", "density": 0.76918194004105, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/19.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.802+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.802, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@19", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.802+", "metric_symbol": "r", "metric_value": 2.802, "reference_text": "r = 2.802+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/19.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 2.802+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "1966e8e14b94f055", "improvement": -0.002060145605657926, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.987997947874078e-06, "max_pair_overlap_depth": -1.7618196217927817e-05, "metric_expression": "2.80406", "metric_symbol": "r", "metric_value": 2.804060145605658, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.802, "setup": "squincir", "side": 2.804060145605658, "solution_path": "data/solutions/generated/squincir_019_1966e8e14b94f055.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/1966e8e14b94f055.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@20", "density": 0.7600583693203081, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "June 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/20.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.893+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.893, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@20", "credit": "David W. Cantrell", "date": "June 2002", "family_updated": "", "metric_expression": "2.893+", "metric_symbol": "r", "metric_value": 2.893, "reference_text": "r = 2.893+ | Found by David W. Cantrell | in June 2002.", "source_image": "20.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/20.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 2.893+ | Found by David W. Cantrell | in June 2002.", "friedman_source_image": "20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "d2a3e673398d1cd8", "improvement": -0.001120361357695998, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.0691232615918125e-06, "max_pair_overlap_depth": -2.4937171242911393e-05, "metric_expression": "2.89412", "metric_symbol": "r", "metric_value": 2.894120361357696, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.893, "setup": "squincir", "side": 2.894120361357696, "solution_path": "data/solutions/generated/squincir_020_d2a3e673398d1cd8.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d2a3e673398d1cd8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@21", "density": 0.7859297816487661, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/21.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a34 / 2 = 2.915+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.915, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@21", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a34 / 2 = 2.915+", "metric_symbol": "r", "metric_value": 2.915, "reference_text": "r = \u221a34 / 2 = 2.915+ | Found by Erich Friedman | in 1997.", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/21.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a34 / 2 = 2.915+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "af376e36d9f212b7", "improvement": -0.0013714498117400886, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.2933372016152873e-06, "max_pair_overlap_depth": -5.039957310160048e-06, "metric_expression": "2.91637", "metric_symbol": "r", "metric_value": 2.91637144981174, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.915, "setup": "squincir", "side": 2.91637144981174, "solution_path": "data/solutions/generated/squincir_021_af376e36d9f212b7.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/af376e36d9f212b7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@22", "density": 0.7599553171633651, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/22.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.033+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.033, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@22", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.033+", "metric_symbol": "r", "metric_value": 3.033, "reference_text": "r = 3.033+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/22.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.033+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "d08f4c9050539106", "improvement": -0.002584838809432277, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.8140931262198308e-06, "max_pair_overlap_depth": -1.4316626561150514e-05, "metric_expression": "3.03558", "metric_symbol": "r", "metric_value": 3.035584838809432, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.033, "setup": "squincir", "side": 3.035584838809432, "solution_path": "data/solutions/generated/squincir_022_d08f4c9050539106.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/d08f4c9050539106.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@23", "density": 0.7743799032674085, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "June 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/23.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.073+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.073, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@23", "credit": "David W. Cantrell", "date": "June 2002", "family_updated": "", "metric_expression": "3.073+", "metric_symbol": "r", "metric_value": 3.073, "reference_text": "r = 3.073+ | Found by David W. Cantrell | in June 2002.", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/23.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.073+ | Found by David W. Cantrell | in June 2002.", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "3cacab4fa2a4614f", "improvement": -0.0017651132247276102, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.255819924494972e-06, "max_pair_overlap_depth": -2.255301194975523e-05, "metric_expression": "3.07477", "metric_symbol": "r", "metric_value": 3.0747651132247276, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.073, "setup": "squincir", "side": 3.0747651132247276, "solution_path": "data/solutions/generated/squincir_023_3cacab4fa2a4614f.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/3cacab4fa2a4614f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@24", "density": 0.7887737589513796, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "October 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/24.gif", "friedman_metric_expression": "3.110+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.11, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@24", "credit": "Maurizio Morandi", "date": "October 2010", "family_updated": "", "metric_expression": "3.110+", "metric_symbol": "r", "metric_value": 3.11, "reference_text": "r = 3.110+ | Found by Maurizio Morandi | in October 2010.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/24.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.110+ | Found by Maurizio Morandi | in October 2010.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "35e22c7b35c718bd", "improvement": -0.0021065491772924716, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.225275177878075e-06, "max_pair_overlap_depth": -2.5024692080766187e-05, "metric_expression": "3.11211", "metric_symbol": "r", "metric_value": 3.1121065491772923, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.11, "setup": "squincir", "side": 6.224213098354585, "solution_path": "data/solutions/generated/squincir_024_35e22c7b35c718bd.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/35e22c7b35c718bd.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@25", "density": 0.7888590017487276, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "April 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/25.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a5809 / 24 = 3.175+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.175, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@25", "credit": "David W. Cantrell", "date": "April 2002", "family_updated": "", "metric_expression": "\u221a5809 / 24 = 3.175+", "metric_symbol": "r", "metric_value": 3.175, "reference_text": "r = \u221a5809 / 24 = 3.175+ | Found by David W. Cantrell | in April 2002.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/25.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a5809 / 24 = 3.175+ | Found by David W. Cantrell | in April 2002.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "0cabb86031dba8fd", "improvement": -0.0011088296688002153, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4103138832897173e-06, "max_pair_overlap_depth": -3.0578471952869e-05, "metric_expression": "3.17611", "metric_symbol": "r", "metric_value": 3.1761088296688, "n": 25, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.175, "setup": "squincir", "side": 3.1761088296688, "solution_path": "data/solutions/generated/squincir_025_0cabb86031dba8fd.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/0cabb86031dba8fd.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@26", "density": 0.806871444276765, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/26.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a41 / 2 = 3.201+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.201, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@26", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a41 / 2 = 3.201+", "metric_symbol": "r", "metric_value": 3.201, "reference_text": "r = \u221a41 / 2 = 3.201+ | Found by Erich Friedman | in 1997.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/26.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a41 / 2 = 3.201+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "d3c528742ca2afa0", "improvement": -0.0016506279557328973, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.80804369340504e-06, "max_pair_overlap_depth": -4.195014817909293e-06, "metric_expression": "3.20265", "metric_symbol": "r", "metric_value": 3.202650627955733, "n": 26, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.201, "setup": "squincir", "side": 3.202650627955733, "solution_path": "data/solutions/generated/squincir_026_d3c528742ca2afa0.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/d3c528742ca2afa0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@27", "density": 0.8080261870592086, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "September 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/27.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "13\u221a145/48 = 3.261+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.261, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@27", "credit": "Maurizio Morandi", "date": "September 2010", "family_updated": "", "metric_expression": "13\u221a145/48 = 3.261+", "metric_symbol": "r", "metric_value": 3.261, "reference_text": "r = 13\u221a145/48 = 3.261+ | Found by Maurizio Morandi | in September 2010.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/27.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 13\u221a145/48 = 3.261+ | Found by Maurizio Morandi | in September 2010.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "24101c6f5ef6d134", "improvement": -0.0003261127181617418, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.239249729032224e-06, "max_pair_overlap_depth": -1.3019517370316258e-05, "metric_expression": "3.26133", "metric_symbol": "r", "metric_value": 3.261326112718162, "n": 27, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.261, "setup": "squincir", "side": 3.261326112718162, "solution_path": "data/solutions/generated/squincir_027_24101c6f5ef6d134.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/24101c6f5ef6d134.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@28", "density": 0.8008141779745183, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/28.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.333+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.333, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@28", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.333+", "metric_symbol": "r", "metric_value": 3.333, "reference_text": "r = 3.333+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/28.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.333+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "33c1007b5e868482", "improvement": -0.0030934121555343808, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.8010138785924426e-06, "max_pair_overlap_depth": -3.2305275887267726e-05, "metric_expression": "3.33609", "metric_symbol": "r", "metric_value": 3.3360934121555346, "n": 28, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.333, "setup": "squincir", "side": 3.3360934121555346, "solution_path": "data/solutions/generated/squincir_028_33c1007b5e868482.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T03:19:00+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/33c1007b5e868482.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@29", "density": 0.7983676158390189, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/29.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.401+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.401, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@29", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.401+", "metric_symbol": "r", "metric_value": 3.401, "reference_text": "r = 3.401+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/29.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.401+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "ab533ab0ad8e3eeb", "improvement": 0.000657947854124874, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.686355668135576e-06, "max_pair_overlap_depth": -1.5593607360919748e-05, "metric_expression": "3.40034", "metric_symbol": "r", "metric_value": 3.400342052145875, "n": 29, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.401, "setup": "squincir", "side": 3.400342052145875, "solution_path": "data/solutions/generated/squincir_029_ab533ab0ad8e3eeb.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/ab533ab0ad8e3eeb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@30", "density": 0.7941259454921153, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/30.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.472+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.472, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@30", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.472+", "metric_symbol": "r", "metric_value": 3.472, "reference_text": "r = 3.472+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/30.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.472+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "d6d4b17471528bfa", "improvement": 0.004304153213534612, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2997799895563276e-06, "max_pair_overlap_depth": -1.7959424514213396e-05, "metric_expression": "3.46770", "metric_symbol": "r", "metric_value": 3.4676958467864654, "n": 30, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.472, "setup": "squincir", "side": 3.4676958467864654, "solution_path": "data/solutions/generated/squincir_030_d6d4b17471528bfa.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/d6d4b17471528bfa.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@31", "density": 0.7887915525098625, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/31.gif", "friedman_metric_expression": "5 \u221a2 / 2 = 3.535+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.535, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@31", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 \u221a2 / 2 = 3.535+", "metric_symbol": "r", "metric_value": 3.535, "reference_text": "r = 5 \u221a2 / 2 = 3.535+ | Found by Erich Friedman | in 1997.", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/31.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 5 \u221a2 / 2 = 3.535+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "fdf9170d30584f4f", "improvement": -0.0019163212872146396, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.62119976746078e-06, "max_pair_overlap_depth": -1.7231292489316985e-05, "metric_expression": "3.53692", "metric_symbol": "r", "metric_value": 3.536916321287215, "n": 31, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.535, "setup": "squincir", "side": 7.07383264257443, "solution_path": "data/solutions/generated/squincir_031_fdf9170d30584f4f.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/fdf9170d30584f4f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@32", "density": 0.7916012707303632, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/32.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.584+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.584, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@32", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.584+", "metric_symbol": "r", "metric_value": 3.584, "reference_text": "r = 3.584+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/32.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.584+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "b8bda371acb2b492", "improvement": -0.0031274839526354015, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.2509087862826505e-06, "max_pair_overlap_depth": -6.52310694349012e-09, "metric_expression": "3.58713", "metric_symbol": "r", "metric_value": 3.5871274839526355, "n": 32, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.584, "setup": "squincir", "side": 3.5871274839526355, "solution_path": "data/solutions/generated/squincir_032_b8bda371acb2b492.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b8bda371acb2b492.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@33", "density": 0.8076634866746639, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/33.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a13 = 3.605+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.605, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@33", "credit": "Maurizio Morandi", "date": "June 2010", "family_updated": "", "metric_expression": "\u221a13 = 3.605+", "metric_symbol": "r", "metric_value": 3.605, "reference_text": "r = \u221a13 = 3.605+ | Found by Maurizio Morandi | in June 2010.", "source_image": "33.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/33.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a13 = 3.605+ | Found by Maurizio Morandi | in June 2010.", "friedman_source_image": "33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "e5a9fc9a4b7235a0", "improvement": -0.0013411620456880158, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.493268955270025e-06, "max_pair_overlap_depth": -1.6481229587750335e-06, "metric_expression": "3.60634", "metric_symbol": "r", "metric_value": 3.606341162045688, "n": 33, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.605, "setup": "squincir", "side": 3.606341162045688, "solution_path": "data/solutions/generated/squincir_033_e5a9fc9a4b7235a0.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/e5a9fc9a4b7235a0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@34", "density": 0.819454861319337, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "December 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/34.gif", "friedman_metric_expression": "3.629+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.629, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@34", "credit": "Maurizio Morandi", "date": "December 2010", "family_updated": "", "metric_expression": "3.629+", "metric_symbol": "r", "metric_value": 3.629, "reference_text": "r = 3.629+ | Found by Maurizio Morandi | in December 2010.", "source_image": "34.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/34.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.629+ | Found by Maurizio Morandi | in December 2010.", "friedman_source_image": "34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "9214fdf4034b8f9a", "improvement": -0.005142931467513545, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0151594769247652e-06, "max_pair_overlap_depth": -1.71712946250846e-07, "metric_expression": "3.63414", "metric_symbol": "r", "metric_value": 3.6341429314675135, "n": 34, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.629, "setup": "squincir", "side": 7.268285862935027, "solution_path": "data/solutions/generated/squincir_34_9214fdf4034b8f9a.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T14:34:18+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/9214fdf4034b8f9a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squincir@35", "density": 0.8405414638481816, "friedman_analytical_or_proved": true, "friedman_credit": "David Eppstein", "friedman_date": "March 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/35.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a53 / 2 = 3.640+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.64, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@35", "credit": "David Eppstein", "date": "March 2002", "family_updated": "", "metric_expression": "\u221a53 / 2 = 3.640+", "metric_symbol": "r", "metric_value": 3.64, "reference_text": "r = \u221a53 / 2 = 3.640+ | Found by David Eppstein | in March 2002.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/35.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a53 / 2 = 3.640+ | Found by David Eppstein | in March 2002.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "f2d91d98b4c2ee39", "improvement": -0.0006549082553228835, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.121617296348745e-07, "max_pair_overlap_depth": -2.1344301219083928e-05, "metric_expression": "3.64065", "metric_symbol": "r", "metric_value": 3.640654908255323, "n": 35, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.64, "setup": "squincir", "side": 3.640654908255323, "solution_path": "data/solutions/generated/squincir_035_f2d91d98b4c2ee39.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David Eppstein", "svg_path": "data/svg/f2d91d98b4c2ee39.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@1", "density": 0.499990679660235, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/2.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "1-2. | | s = 1 | Trivial. |", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/2.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "1-2. | | s = 1 | Trivial. |", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "557f8de873a17a8e", "improvement": -9.320470070184328e-06, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7525252651882113e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00001", "metric_symbol": "s", "metric_value": 1.0000093204700702, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0, "setup": "squindom", "side": 2.0000186409401404, "solution_path": "data/solutions/generated/squindom_001_557f8de873a17a8e.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/557f8de873a17a8e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@2", "density": 0.9999882998008334, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/2.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "1-2. | | s = 1 | Trivial. |", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/2.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "1-2. | | s = 1 | Trivial. |", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "7684110dce2e5290", "improvement": -5.850150919473052e-06, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.463373909884382e-07, "max_pair_overlap_depth": -9.41881114351023e-06, "metric_expression": "1.00001", "metric_symbol": "s", "metric_value": 1.0000058501509195, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0, "setup": "squindom", "side": 2.000011700301839, "solution_path": "data/solutions/generated/squindom_002_7684110dce2e5290.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7684110dce2e5290.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@3", "density": 0.6666417193387711, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/3.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3/2 = 1.5", "friedman_metric_symbol": "s", "friedman_metric_value": 1.5, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3/2 = 1.5", "metric_symbol": "s", "metric_value": 1.5, "reference_text": "3. | | s = 3/2 = 1.5 | Trivial. |", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/3.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "3. | | s = 3/2 = 1.5 | Trivial. |", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "3047ca63de76a853", "improvement": -2.8066531593262667e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0617527398680835e-06, "max_pair_overlap_depth": -1.9554412996725112e-05, "metric_expression": "1.50003", "metric_symbol": "s", "metric_value": 1.5000280665315933, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.5, "setup": "squindom", "side": 3.0000561330631865, "solution_path": "data/solutions/generated/squindom_003_3047ca63de76a853.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3047ca63de76a853.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@4", "density": 0.5367260778952853, "friedman_analytical_or_proved": false, "friedman_credit": "Evert Stenlund", "friedman_date": "June 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/4.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "1.930+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.93, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@4", "credit": "Evert Stenlund", "date": "June 2010", "family_updated": "", "metric_expression": "1.930+", "metric_symbol": "s", "metric_value": 1.93, "reference_text": "4. | | s = 1.930+ | Found by Evert Stenlund | in June 2010.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/4.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "4. | | s = 1.930+ | Found by Evert Stenlund | in June 2010.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "8d067d60c7138964", "improvement": -0.00036150970269521743, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.561542731343124e-06, "max_pair_overlap_depth": -3.757775137364083e-05, "metric_expression": "1.93036", "metric_symbol": "s", "metric_value": 1.9303615097026952, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.93, "setup": "squindom", "side": 3.8607230194053903, "solution_path": "data/solutions/generated/squindom_004_8d067d60c7138964.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Evert Stenlund", "svg_path": "data/svg/8d067d60c7138964.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@5", "density": 0.6249853765160349, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "5-8. | | s = 2 | Trivial. |", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "5-8. | | s = 2 | Trivial. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "05f6000ecc64a077", "improvement": -2.3397984937112426e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3736510935966635e-06, "max_pair_overlap_depth": -1.2206465427119814e-05, "metric_expression": "2.00002", "metric_symbol": "s", "metric_value": 2.000023397984937, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "squindom", "side": 4.000046795969874, "solution_path": "data/solutions/generated/squindom_005_05f6000ecc64a077.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/05f6000ecc64a077.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@6", "density": 0.749983975114432, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "5-8. | | s = 2 | Trivial. |", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "5-8. | | s = 2 | Trivial. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "095d951ace7a918a", "improvement": -2.136685649301029e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.707507375492412e-07, "max_pair_overlap_depth": -1.170836956105551e-05, "metric_expression": "2.00002", "metric_symbol": "s", "metric_value": 2.000021366856493, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "squindom", "side": 4.000042733712986, "solution_path": "data/solutions/generated/squindom_006_095d951ace7a918a.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/095d951ace7a918a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@7", "density": 0.8749639011598754, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "5-8. | | s = 2 | Trivial. |", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "5-8. | | s = 2 | Trivial. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "50aa0817a0117c40", "improvement": -4.1257093861268856e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4800520663781356e-06, "max_pair_overlap_depth": -1.885899157594295e-05, "metric_expression": "2.00004", "metric_symbol": "s", "metric_value": 2.0000412570938613, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "squindom", "side": 4.0000825141877225, "solution_path": "data/solutions/generated/squindom_007_50aa0817a0117c40.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/50aa0817a0117c40.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@8", "density": 0.9999722279382355, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@8", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "5-8. | | s = 2 | Trivial. |", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "5-8. | | s = 2 | Trivial. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "0b83841ee591689c", "improvement": -2.7772640243561852e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0495464614356251e-07, "max_pair_overlap_depth": -1.777420024382249e-07, "metric_expression": "2.00003", "metric_symbol": "s", "metric_value": 2.0000277726402436, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "squindom", "side": 4.000055545280487, "solution_path": "data/solutions/generated/squindom_008_0b83841ee591689c.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/0b83841ee591689c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@9", "density": 0.7367411181657051, "friedman_analytical_or_proved": true, "friedman_credit": "Evert Stenlund", "friedman_date": "June 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/9.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2 + \u221a2/3 = 2.471+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.471, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@9", "credit": "Evert Stenlund", "date": "June 2010", "family_updated": "", "metric_expression": "2 + \u221a2/3 = 2.471+", "metric_symbol": "s", "metric_value": 2.471, "reference_text": "9. | | s = 2 + \u221a2/3 = 2.471+ | Found by Evert Stenlund | in June 2010.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/9.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "9. | | s = 2 + \u221a2/3 = 2.471+ | Found by Evert Stenlund | in June 2010.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "2963ae62200cf2c5", "improvement": -0.000432778719658522, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8492494266174617e-06, "max_pair_overlap_depth": -7.2416104319739105e-06, "metric_expression": "2.47143", "metric_symbol": "s", "metric_value": 2.4714327787196586, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.471, "setup": "squindom", "side": 4.942865557439317, "solution_path": "data/solutions/generated/squindom_009_2963ae62200cf2c5.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Evert Stenlund", "svg_path": "data/svg/2963ae62200cf2c5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@10", "density": 0.7999865137938963, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/10.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "5/2 = 2.5", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@10", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5/2 = 2.5", "metric_symbol": "s", "metric_value": 2.5, "reference_text": "10. | | s = 5/2 = 2.5 | Trivial. |", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/10.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "10. | | s = 5/2 = 2.5 | Trivial. |", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "7d3cb25d7577069f", "improvement": -2.1072463463500668e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5655128748903735e-06, "max_pair_overlap_depth": -9.554543500822632e-07, "metric_expression": "2.50002", "metric_symbol": "s", "metric_value": 2.5000210724634635, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.5, "setup": "squindom", "side": 5.000042144926927, "solution_path": "data/solutions/generated/squindom_010_7d3cb25d7577069f.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7d3cb25d7577069f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@11", "density": 0.7109112928838592, "friedman_analytical_or_proved": false, "friedman_credit": "Evert Stenlund", "friedman_date": "June 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/11.gif", "friedman_metric_expression": "2.779+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.779, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@11", "credit": "Evert Stenlund", "date": "June 2010", "family_updated": "", "metric_expression": "2.779+", "metric_symbol": "s", "metric_value": 2.779, "reference_text": "11. | | s = 2.779+ | Found by Evert Stenlund | in June 2010. |", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/11.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "11. | | s = 2.779+ | Found by Evert Stenlund | in June 2010. |", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "eadb11fb534aa6a1", "improvement": -0.0024652481713771657, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.886034921991467e-06, "max_pair_overlap_depth": -3.432035104045106e-05, "metric_expression": "2.78147", "metric_symbol": "s", "metric_value": 2.781465248171377, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.779, "setup": "squindom", "side": 5.562930496342754, "solution_path": "data/solutions/generated/squindom_11_eadb11fb534aa6a1.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "Evert Stenlund", "svg_path": "data/svg/eadb11fb534aa6a1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@12", "density": 0.7147860825866693, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/12.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "6 (1 + \u221a2) / 5 = 2.897+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.897, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@12", "credit": "Maurizio Morandi", "date": "June 2010", "family_updated": "", "metric_expression": "6 (1 + \u221a2) / 5 = 2.897+", "metric_symbol": "s", "metric_value": 2.897, "reference_text": "12. | | s = 6 (1 + \u221a2) / 5 = 2.897+ | Found by Maurizio Morandi | in June 2010. |", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/12.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "12. | | s = 6 (1 + \u221a2) / 5 = 2.897+ | Found by Maurizio Morandi | in June 2010. |", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "4c82fd3f736a139f", "improvement": -0.0002607386911455123, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.382389093311346e-06, "max_pair_overlap_depth": -2.837543626332817e-05, "metric_expression": "2.89726", "metric_symbol": "s", "metric_value": 2.8972607386911453, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.897, "setup": "squindom", "side": 2.8972607386911453, "solution_path": "data/solutions/generated/squindom_012_4c82fd3f736a139f.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/4c82fd3f736a139f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@13", "density": 0.7330751854461901, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/13.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2.976+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.976, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@13", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "2.976+", "metric_symbol": "s", "metric_value": 2.976, "reference_text": "13. | | s = 2.976+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in June 2010.)", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/13.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "13. | | s = 2.976+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in June 2010.)", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d738dfc349a8c940", "improvement": -0.0017101333567786092, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.7694174018241284e-06, "max_pair_overlap_depth": -2.048132261660285e-05, "metric_expression": "2.97771", "metric_symbol": "s", "metric_value": 2.9777101333567786, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.976, "setup": "squindom", "side": 2.9777101333567786, "solution_path": "data/solutions/generated/squindom_013_d738dfc349a8c940.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d738dfc349a8c940.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@14", "density": 0.7777474504876777, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@14", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "14-18. | | s = 3 | Trivial. |", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "167b47d5e39e076d", "improvement": -5.849005569213972e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0369016045785884e-06, "max_pair_overlap_depth": -1.5707310774493166e-05, "metric_expression": "3.00006", "metric_symbol": "s", "metric_value": 3.000058490055692, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.000116980111384, "solution_path": "data/solutions/generated/squindom_014_167b47d5e39e076d.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/167b47d5e39e076d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@15", "density": 0.8333077568349537, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@15", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "14-18. | | s = 3 | Trivial. |", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "6d686963e9ef0ece", "improvement": -4.603875684550118e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0382270816400307e-06, "max_pair_overlap_depth": -3.605764309855264e-06, "metric_expression": "3.00005", "metric_symbol": "s", "metric_value": 3.0000460387568455, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.000092077513691, "solution_path": "data/solutions/generated/squindom_015_6d686963e9ef0ece.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6d686963e9ef0ece.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@16", "density": 0.888839182217615, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@16", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "14-18. | | s = 3 | Trivial. |", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "0f551306e71d2ff0", "improvement": -8.388352586674586e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4593433725073055e-06, "max_pair_overlap_depth": -5.7555123290642385e-06, "metric_expression": "3.00008", "metric_symbol": "s", "metric_value": 3.0000838835258667, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.0001677670517335, "solution_path": "data/solutions/generated/squindom_016_0f551306e71d2ff0.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/0f551306e71d2ff0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@17", "density": 0.9444002665800334, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@17", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "14-18. | | s = 3 | Trivial. |", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "52c5d2070c49c79c", "improvement": -7.016730512532376e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.650126292496168e-06, "max_pair_overlap_depth": -3.2807804517531736e-07, "metric_expression": "3.00007", "metric_symbol": "s", "metric_value": 3.0000701673051253, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.000140334610251, "solution_path": "data/solutions/generated/squindom_017_52c5d2070c49c79c.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/52c5d2070c49c79c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@18", "density": 0.9999599238999353, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@18", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "14-18. | | s = 3 | Trivial. |", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "a6900e1fd3f925c6", "improvement": -6.011595701327366e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.408701424045262e-07, "max_pair_overlap_depth": -1.079909316636929e-06, "metric_expression": "3.00006", "metric_symbol": "s", "metric_value": 3.0000601159570133, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.0001202319140265, "solution_path": "data/solutions/generated/squindom_018_a6900e1fd3f925c6.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a6900e1fd3f925c6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@19", "density": 0.7988013748197259, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/19.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3 + 1/\u221a5 = 3.447+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.447, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@19", "credit": "Maurizio Morandi", "date": "June 2010", "family_updated": "", "metric_expression": "3 + 1/\u221a5 = 3.447+", "metric_symbol": "s", "metric_value": 3.447, "reference_text": "19. | | s = 3 + 1/\u221a5 = 3.447+ | Found by Maurizio Morandi | in June 2010.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/19.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "19. | | s = 3 + 1/\u221a5 = 3.447+ | Found by Maurizio Morandi | in June 2010.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "cddf920bff527727", "improvement": -0.0015966406761447338, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.98512899937964e-06, "max_pair_overlap_depth": -1.3782300862574814e-05, "metric_expression": "3.44860", "metric_symbol": "s", "metric_value": 3.448596640676145, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.447, "setup": "squindom", "side": 3.448596640676145, "solution_path": "data/solutions/generated/squindom_019_cddf920bff527727.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/cddf920bff527727.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@20", "density": 0.8162860196976518, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/21.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "7/2 = 3.5", "friedman_metric_symbol": "s", "friedman_metric_value": 3.5, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@20", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7/2 = 3.5", "metric_symbol": "s", "metric_value": 3.5, "reference_text": "20-21. | | s = 7/2 = 3.5 | Trivial. |", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/21.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "20-21. | | s = 7/2 = 3.5 | Trivial. |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "4a48c58d60a1f36b", "improvement": -8.6848505622239e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.863306484192975e-06, "max_pair_overlap_depth": -2.3696014737728532e-05, "metric_expression": "3.50009", "metric_symbol": "s", "metric_value": 3.5000868485056222, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.5, "setup": "squindom", "side": 7.0001736970112445, "solution_path": "data/solutions/generated/squindom_020_4a48c58d60a1f36b.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4a48c58d60a1f36b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@21", "density": 0.8571139620178546, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/21.gif", "friedman_metric_expression": "7/2 = 3.5", "friedman_metric_symbol": "s", "friedman_metric_value": 3.5, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@21", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7/2 = 3.5", "metric_symbol": "s", "metric_value": 3.5, "reference_text": "20-21. | | s = 7/2 = 3.5 | Trivial. |", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/21.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "20-21. | | s = 7/2 = 3.5 | Trivial. |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "f91a28177536f6d3", "improvement": -5.899570515355279e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8534056853702905e-06, "max_pair_overlap_depth": -1.3464050134803784e-06, "metric_expression": "3.50006", "metric_symbol": "s", "metric_value": 3.5000589957051536, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.5, "setup": "squindom", "side": 7.000117991410307, "solution_path": "data/solutions/generated/squindom_021_f91a28177536f6d3.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/f91a28177536f6d3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@22", "density": 0.783535190077589, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/22.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3.745+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.745, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@22", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "3.745+", "metric_symbol": "s", "metric_value": 3.745, "reference_text": "22. | | s = 3.745+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in June 2010.)", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/22.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "22. | | s = 3.745+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in June 2010.)", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d2d5380917cfe7ce", "improvement": -0.0018567500999222553, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.72695736733975e-06, "max_pair_overlap_depth": -9.08031145729904e-06, "metric_expression": "3.74686", "metric_symbol": "s", "metric_value": 3.7468567500999224, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.745, "setup": "squindom", "side": 3.7468567500999224, "solution_path": "data/solutions/generated/squindom_022_d2d5380917cfe7ce.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d2d5380917cfe7ce.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@23", "density": 0.7750857674093591, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/23.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3.849+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.849, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@23", "credit": "Maurizio Morandi", "date": "June 2010", "family_updated": "", "metric_expression": "3.849+", "metric_symbol": "s", "metric_value": 3.849, "reference_text": "23. | | s = 3.849+ | Found by Maurizio Morandi | in June 2010. |", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/23.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "23. | | s = 3.849+ | Found by Maurizio Morandi | in June 2010. |", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d12d3a9a87ba3b11", "improvement": -0.002891443742406352, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.707858897694649e-06, "max_pair_overlap_depth": -2.3445235071584136e-05, "metric_expression": "3.85189", "metric_symbol": "s", "metric_value": 3.8518914437424066, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.849, "setup": "squindom", "side": 3.8518914437424066, "solution_path": "data/solutions/generated/squindom_023_d12d3a9a87ba3b11.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/d12d3a9a87ba3b11.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@24", "density": 0.7961447064751783, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell after Maurizio Morandi", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/24.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3.882+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.882, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@24", "credit": "David W. Cantrell after Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "3.882+", "metric_symbol": "s", "metric_value": 3.882, "reference_text": "24. | | s = 3.882+ | Found by DC after MM | in August 2012. |", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/24.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "24. | | s = 3.882+ | Found by DC after MM | in August 2012. |", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "f954e5304808181f", "improvement": -0.0003493916461954072, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7445287290307476e-06, "max_pair_overlap_depth": -1.9183513864096113e-05, "metric_expression": "3.88235", "metric_symbol": "s", "metric_value": 3.8823493916461955, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.882, "setup": "squindom", "side": 3.8823493916461955, "solution_path": "data/solutions/generated/squindom_024_f954e5304808181f.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell after Maurizio Morandi", "svg_path": "data/svg/f954e5304808181f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@25", "density": 0.8043920132931294, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell after Maurizio Morandi", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/25.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3.939+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.939, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@25", "credit": "David W. Cantrell after Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "3.939+", "metric_symbol": "s", "metric_value": 3.939, "reference_text": "25. | | s = 3.939+ | Found by DC after MM | in August 2012. |", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/25.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "25. | | s = 3.939+ | Found by DC after MM | in August 2012. |", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "26273cb13b41bf72", "improvement": -0.003040951197706754, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.269696070480222e-06, "max_pair_overlap_depth": -2.5070959039568547e-05, "metric_expression": "3.94204", "metric_symbol": "s", "metric_value": 3.942040951197707, "n": 25, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.939, "setup": "squindom", "side": 3.942040951197707, "solution_path": "data/solutions/generated/squindom_025_26273cb13b41bf72.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell after Maurizio Morandi", "svg_path": "data/svg/26273cb13b41bf72.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@26", "density": 0.8201428647660397, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/26.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3.981+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.981, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@26", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "3.981+", "metric_symbol": "s", "metric_value": 3.981, "reference_text": "26. | | s = 3.981+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in July 2012.)", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/26.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "26. | | s = 3.981+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in July 2012.)", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "a46204eea3d7f9fe", "improvement": -0.00031848875831208545, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.947984543621445e-06, "max_pair_overlap_depth": -8.516034391559124e-06, "metric_expression": "3.98132", "metric_symbol": "s", "metric_value": 3.981318488758312, "n": 26, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.981, "setup": "squindom", "side": 3.981318488758312, "solution_path": "data/solutions/generated/squindom_026_a46204eea3d7f9fe.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/a46204eea3d7f9fe.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@27", "density": 0.8436994977206457, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@27", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "27-32. | | s = 4 | Trivial. | |", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "c68e03cc8f5005db", "improvement": -0.00011971448073744284, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.175631390130775e-07, "max_pair_overlap_depth": -1.4466911184563003e-06, "metric_expression": "4.00012", "metric_symbol": "s", "metric_value": 4.000119714480737, "n": 27, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.000239428961475, "solution_path": "data/solutions/generated/squindom_027_c68e03cc8f5005db.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/c68e03cc8f5005db.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@28", "density": 0.8749068629130922, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@28", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "27-32. | | s = 4 | Trivial. | |", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "2c8c79126d60cdac", "improvement": -0.00021290176655508475, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.243746303256302e-06, "max_pair_overlap_depth": -1.6386799395987463e-05, "metric_expression": "4.00021", "metric_symbol": "s", "metric_value": 4.000212901766555, "n": 28, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.00042580353311, "solution_path": "data/solutions/generated/squindom_028_2c8c79126d60cdac.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2c8c79126d60cdac.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@29", "density": 0.9062096331295707, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@29", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "27-32. | | s = 4 | Trivial. | |", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d2548471aff7d9f5", "improvement": -8.908848335043018e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8156140130519702e-06, "max_pair_overlap_depth": -2.026030487822794e-06, "metric_expression": "4.00009", "metric_symbol": "s", "metric_value": 4.00008908848335, "n": 29, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.0001781769667, "solution_path": "data/solutions/generated/squindom_029_d2548471aff7d9f5.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d2548471aff7d9f5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@30", "density": 0.9374597146255952, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@30", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "27-32. | | s = 4 | Trivial. | |", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d67a491f7b4498c0", "improvement": -8.594490193214455e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.188605266186585e-06, "max_pair_overlap_depth": -1.644068809536492e-06, "metric_expression": "4.00009", "metric_symbol": "s", "metric_value": 4.000085944901932, "n": 30, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.000171889803864, "solution_path": "data/solutions/generated/squindom_030_d67a491f7b4498c0.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d67a491f7b4498c0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@31", "density": 0.9683879031627185, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@31", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "27-32. | | s = 4 | Trivial. | |", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "86ec622f54cecf7c", "improvement": -0.000747764390431449, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.872732694063188e-07, "max_pair_overlap_depth": -2.9208615389531944e-06, "metric_expression": "4.00075", "metric_symbol": "s", "metric_value": 4.0007477643904314, "n": 31, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 4.0007477643904314, "solution_path": "data/solutions/generated/squindom_031_86ec622f54cecf7c.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/86ec622f54cecf7c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@32", "density": 0.9989694769831047, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@32", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "27-32. | | s = 4 | Trivial. | |", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "2873ac78de186b97", "improvement": -0.0020626403695489515, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0386280700511747e-06, "max_pair_overlap_depth": -2.476940408779882e-06, "metric_expression": "4.00206", "metric_symbol": "s", "metric_value": 4.002062640369549, "n": 32, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 4.002062640369549, "solution_path": "data/solutions/generated/squindom_032_2873ac78de186b97.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2873ac78de186b97.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@33", "density": 0.8455857799388354, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/33.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3 + \u221a2 = 4.414+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.414, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@33", "credit": "Maurizio Morandi", "date": "July 2012", "family_updated": "", "metric_expression": "3 + \u221a2 = 4.414+", "metric_symbol": "s", "metric_value": 4.414, "reference_text": "33. | | s = 3 + \u221a2 = 4.414+ | Found by Maurizio Morandi | in July 2012.", "source_image": "33.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/33.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "33. | | s = 3 + \u221a2 = 4.414+ | Found by Maurizio Morandi | in July 2012.", "friedman_source_image": "33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "167e5e2a52e9b5fe", "improvement": -0.00336348277204479, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.856211927670387e-06, "max_pair_overlap_depth": -6.591285420576298e-06, "metric_expression": "4.41736", "metric_symbol": "s", "metric_value": 4.4173634827720445, "n": 33, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.414, "setup": "squindom", "side": 4.4173634827720445, "solution_path": "data/solutions/generated/squindom_033_167e5e2a52e9b5fe.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/167e5e2a52e9b5fe.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@34", "density": 0.8499062715939519, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/34.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4 + \u221a2/3 = 4.471+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.471, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@34", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "4 + \u221a2/3 = 4.471+", "metric_symbol": "s", "metric_value": 4.471, "reference_text": "34. | | s = 4 + \u221a2/3 = 4.471+ | Found by David W. Cantrell | in July 2012.", "source_image": "34.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/34.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "34. | | s = 4 + \u221a2/3 = 4.471+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "b631e5a58c1517e1", "improvement": -0.0013825437310881483, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.603876101403557e-07, "max_pair_overlap_depth": -3.616786977911346e-06, "metric_expression": "4.47238", "metric_symbol": "s", "metric_value": 4.472382543731088, "n": 34, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.471, "setup": "squindom", "side": 4.472382543731088, "solution_path": "data/solutions/generated/squindom_034_b631e5a58c1517e1.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/b631e5a58c1517e1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@35", "density": 0.8625827503428114, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/36.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "9/2 = 4.5", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@35", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "9/2 = 4.5", "metric_symbol": "s", "metric_value": 4.5, "reference_text": "35-36. | | s = 9/2 = 4.5 | Trivial. |", "source_image": "36.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/36.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "35-36. | | s = 9/2 = 4.5 | Trivial. |", "friedman_source_image": "36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "dcd4cfae3ff101a2", "improvement": -0.004210097373901434, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9606355010637344e-06, "max_pair_overlap_depth": -1.1537592481902781e-05, "metric_expression": "4.50421", "metric_symbol": "s", "metric_value": 4.504210097373901, "n": 35, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.5, "setup": "squindom", "side": 4.504210097373901, "solution_path": "data/solutions/generated/squindom_035_dcd4cfae3ff101a2.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/dcd4cfae3ff101a2.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@36", "density": 0.8873457711540725, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/36.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "9/2 = 4.5", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@36", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "9/2 = 4.5", "metric_symbol": "s", "metric_value": 4.5, "reference_text": "35-36. | | s = 9/2 = 4.5 | Trivial. |", "source_image": "36.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/36.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "35-36. | | s = 9/2 = 4.5 | Trivial. |", "friedman_source_image": "36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "e0fc61bd01777ea3", "improvement": -0.003911109790387002, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.310647293148122e-07, "max_pair_overlap_depth": -3.943598063660403e-06, "metric_expression": "4.50391", "metric_symbol": "s", "metric_value": 4.503911109790387, "n": 36, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.5, "setup": "squindom", "side": 4.503911109790387, "solution_path": "data/solutions/generated/squindom_036_e0fc61bd01777ea3.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/e0fc61bd01777ea3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@37", "density": 0.8199742411033116, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/37.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "10 / 3 + \u221a2 = 4.747+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.747, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@37", "credit": "Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "10 / 3 + \u221a2 = 4.747+", "metric_symbol": "s", "metric_value": 4.747, "reference_text": "37. | | s = 10 / 3 + \u221a2 = 4.747+ | Found by Maurizio Morandi | in August 2012.", "source_image": "37.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/37.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "37. | | s = 10 / 3 + \u221a2 = 4.747+ | Found by Maurizio Morandi | in August 2012.", "friedman_source_image": "37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "09a846134d0a3602", "improvement": -0.0029141409565269782, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.702594714522945e-06, "max_pair_overlap_depth": -1.6080659807293785e-05, "metric_expression": "4.74991", "metric_symbol": "s", "metric_value": 4.749914140956527, "n": 37, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.747, "setup": "squindom", "side": 4.749914140956527, "solution_path": "data/solutions/generated/squindom_037_09a846134d0a3602.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/09a846134d0a3602.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@38", "density": 0.8220103806251009, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/38.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "(13 + \u221a2) / 3 = 4.804+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.804, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@38", "credit": "Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "(13 + \u221a2) / 3 = 4.804+", "metric_symbol": "s", "metric_value": 4.804, "reference_text": "38. | | s = (13 + \u221a2) / 3 = 4.804+ | Found by Maurizio Morandi | in August 2012.", "source_image": "38.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/38.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "38. | | s = (13 + \u221a2) / 3 = 4.804+ | Found by Maurizio Morandi | in August 2012.", "friedman_source_image": "38.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d4d1cf6ab169d8aa", "improvement": -0.0037087406838747228, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.518216488058613e-06, "max_pair_overlap_depth": -2.290026661011524e-05, "metric_expression": "4.80771", "metric_symbol": "s", "metric_value": 4.807708740683875, "n": 38, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.804, "setup": "squindom", "side": 4.807708740683875, "solution_path": "data/solutions/generated/squindom_038_d4d1cf6ab169d8aa.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/d4d1cf6ab169d8aa.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@39", "density": 0.8326335013079162, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/39.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4.839+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.839, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@39", "credit": "David W. Cantrell", "date": "August 2012", "family_updated": "", "metric_expression": "4.839+", "metric_symbol": "s", "metric_value": 4.839, "reference_text": "39. | | s = 4.839+ | Found by David W. Cantrell | in August 2012.", "source_image": "39.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/39.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "39. | | s = 4.839+ | Found by David W. Cantrell | in August 2012.", "friedman_source_image": "39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "a606e4f52500fa01", "improvement": -0.00038713064593576263, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.790990827618913e-06, "max_pair_overlap_depth": -1.236762303413741e-05, "metric_expression": "4.83939", "metric_symbol": "s", "metric_value": 4.839387130645936, "n": 39, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.839, "setup": "squindom", "side": 4.839387130645936, "solution_path": "data/solutions/generated/squindom_039_a606e4f52500fa01.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/a606e4f52500fa01.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@40", "density": 0.8345719865483486, "friedman_analytical_or_proved": false, "friedman_credit": "Sigvart Brendberg", "friedman_date": "June 2023", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/40.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4.895+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.895, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@40", "credit": "Sigvart Brendberg", "date": "June 2023", "family_updated": "", "metric_expression": "4.895+", "metric_symbol": "s", "metric_value": 4.895, "reference_text": "40. | | s = 4.895+ | Found by Sigvart Brendberg | in June 2023.", "source_image": "40.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/40.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "40. | | s = 4.895+ | Found by Sigvart Brendberg | in June 2023.", "friedman_source_image": "40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "bb2811be04efb5ee", "improvement": -0.0003426573425642232, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.37233266706005e-06, "max_pair_overlap_depth": -3.9677918853286975e-06, "metric_expression": "4.89534", "metric_symbol": "s", "metric_value": 4.895342657342564, "n": 40, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.895, "setup": "squindom", "side": 4.895342657342564, "solution_path": "data/solutions/generated/squindom_040_bb2811be04efb5ee.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Sigvart Brendberg", "svg_path": "data/svg/bb2811be04efb5ee.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@41", "density": 0.8199798328031639, "friedman_analytical_or_proved": false, "friedman_credit": "Sigvart Brendberg", "friedman_date": "June 2023", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/41.gif", "friedman_metric_expression": "4.927+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.927, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@41", "credit": "Sigvart Brendberg", "date": "June 2023", "family_updated": "", "metric_expression": "4.927+", "metric_symbol": "s", "metric_value": 4.927, "reference_text": "41. | | s = 4.927+ | Found by Sigvart Brendberg | in June 2023.", "source_image": "41.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/41.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "41. | | s = 4.927+ | Found by Sigvart Brendberg | in June 2023.", "friedman_source_image": "41.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "cb1462995bd3b2c0", "improvement": -0.07306148649036626, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.465020711057036e-07, "max_pair_overlap_depth": -6.131910444850952e-06, "metric_expression": "5.00006", "metric_symbol": "s", "metric_value": 5.000061486490366, "n": 41, "notes": "Best available verified coordinate reconstruction; the Friedman reference remains the public metric until recovery improves.", "record_type": "verified", "reference_side": 4.927, "setup": "squindom", "side": 10.000122972980732, "solution_path": "data/solutions/generated/squindom_41_cb1462995bd3b2c0.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "Sigvart Brendberg", "svg_path": "data/svg/cb1462995bd3b2c0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@42", "density": 0.8472590848618945, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/42.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4.989+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.989, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@42", "credit": "David W. Cantrell", "date": "August 2012", "family_updated": "", "metric_expression": "4.989+", "metric_symbol": "s", "metric_value": 4.989, "reference_text": "42. | | s = 4.989+ | Found by David W. Cantrell | in August 2012.", "source_image": "42.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/42.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "42. | | s = 4.989+ | Found by David W. Cantrell | in August 2012.", "friedman_source_image": "42.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": false, "id": "7faea8ae450b67a1", "improvement": 0.010465394774865189, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.54667148005916e-06, "max_pair_overlap_depth": -1.5216861767419765e-05, "metric_expression": "4.97853", "metric_symbol": "s", "metric_value": 4.978534605225135, "n": 42, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 4.989, "setup": "squindom", "side": 4.978534605225135, "solution_path": "data/solutions/generated/squindom_042_7faea8ae450b67a1.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/7faea8ae450b67a1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@43", "density": 0.859888117316887, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@43", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "7f9527ccddc9d648", "improvement": -0.00032527209571497906, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.1461540806485566e-08, "max_pair_overlap_depth": -2.005039219099558e-06, "metric_expression": "5.00033", "metric_symbol": "s", "metric_value": 5.000325272095715, "n": 43, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.00065054419143, "solution_path": "data/solutions/generated/squindom_043_7f9527ccddc9d648.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7f9527ccddc9d648.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@44", "density": 0.8799631966546829, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@44", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "1ffd758f1f058de5", "improvement": -0.00010455823792376862, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.245467354557775e-06, "max_pair_overlap_depth": -3.962242465904708e-06, "metric_expression": "5.00010", "metric_symbol": "s", "metric_value": 5.000104558237924, "n": 44, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000209116475848, "solution_path": "data/solutions/generated/squindom_044_1ffd758f1f058de5.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/1ffd758f1f058de5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@45", "density": 0.8999501217119913, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@45", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d085ed9682ca2968", "improvement": -0.00013855655918781906, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0485935568738114e-06, "max_pair_overlap_depth": -1.361329106774889e-06, "metric_expression": "5.00014", "metric_symbol": "s", "metric_value": 5.000138556559188, "n": 45, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000277113118376, "solution_path": "data/solutions/generated/squindom_045_d085ed9682ca2968.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d085ed9682ca2968.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@46", "density": 0.9198953665582859, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@46", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "3b408e84cd206068", "improvement": -0.00028435426002282327, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1757865436834436e-06, "max_pair_overlap_depth": -7.423155479191479e-07, "metric_expression": "5.00028", "metric_symbol": "s", "metric_value": 5.000284354260023, "n": 46, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000568708520046, "solution_path": "data/solutions/generated/squindom_046_3b408e84cd206068.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3b408e84cd206068.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@47", "density": 0.9398436614252309, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@47", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "2525e331a392c6b9", "improvement": -0.00041584595443833194, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3105818880431173e-06, "max_pair_overlap_depth": -3.7244704524930228e-06, "metric_expression": "5.00042", "metric_symbol": "s", "metric_value": 5.000415845954438, "n": 47, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 5.000415845954438, "solution_path": "data/solutions/generated/squindom_047_2525e331a392c6b9.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2525e331a392c6b9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@48", "density": 0.9599518570950978, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@48", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "0433c6294c4f3fe7", "improvement": -0.00012537686383318203, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0555756748686917e-06, "max_pair_overlap_depth": -4.3576353303542703e-07, "metric_expression": "5.00013", "metric_symbol": "s", "metric_value": 5.000125376863833, "n": 48, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000250753727666, "solution_path": "data/solutions/generated/squindom_048_0433c6294c4f3fe7.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/0433c6294c4f3fe7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@49", "density": 0.9798599360041962, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@49", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "c84a987ae840ebe3", "improvement": -0.0003573444166047324, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.316744310450815e-06, "max_pair_overlap_depth": -1.1644813818767474e-06, "metric_expression": "5.00036", "metric_symbol": "s", "metric_value": 5.000357344416605, "n": 49, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 5.000357344416605, "solution_path": "data/solutions/generated/squindom_049_c84a987ae840ebe3.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/c84a987ae840ebe3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squindom@50", "density": 0.9999308512263965, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@50", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "df0259815cc5320a", "improvement": -0.00017288089993794387, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.912590304139712e-09, "max_pair_overlap_depth": -7.234911549747203e-06, "metric_expression": "5.00017", "metric_symbol": "s", "metric_value": 5.000172880899938, "n": 50, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 5.000172880899938, "solution_path": "data/solutions/generated/squindom_050_df0259815cc5320a.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/df0259815cc5320a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@1", "density": 0.6187919623349352, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/1.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(3+\u221a3)/6 = 0.778+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.778, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(3+\u221a3)/6 = 0.778+", "metric_symbol": "s", "metric_value": 0.778, "reference_text": "s = (3+\u221a3)/6 = 0.778+ | Trivial. |", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/1.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = (3+\u221a3)/6 = 0.778+ | Trivial. |", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "7a4ae27d520be7d9", "improvement": -0.01068162910289272, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.500249687614274e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.78868", "metric_symbol": "s", "metric_value": 0.7886816291028927, "n": 1, "notes": "Best available verified coordinate reconstruction; the Friedman reference remains the public metric until recovery improves.", "record_type": "verified", "reference_side": 0.778, "setup": "squinhex", "side": 0.7886816291028927, "solution_path": "data/solutions/generated/squinhex_001_7a4ae27d520be7d9.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T03:07:58+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7a4ae27d520be7d9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@2", "density": 0.5773349997096187, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/2.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2/\u221a3 = 1.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.154, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2/\u221a3 = 1.154+", "metric_symbol": "s", "metric_value": 1.154, "reference_text": "s = 2/\u221a3 = 1.154+ | Trivial. |", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/2.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2/\u221a3 = 1.154+ | Trivial. |", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "21fb35b9b51e59c5", "improvement": -0.0007158081621452261, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.964730208767781e-07, "max_pair_overlap_depth": -4.154668371614267e-06, "metric_expression": "1.15472", "metric_symbol": "s", "metric_value": 1.1547158081621451, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.154, "setup": "squinhex", "side": 1.1547158081621451, "solution_path": "data/solutions/generated/squinhex_002_21fb35b9b51e59c5.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/21fb35b9b51e59c5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@3", "density": 0.6187647598988664, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/3.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(1+\u221a3)/2 = 1.366+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.366, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@3", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "(1+\u221a3)/2 = 1.366+", "metric_symbol": "s", "metric_value": 1.366, "reference_text": "s = (1+\u221a3)/2 = 1.366+ | Found by Maurizio Morandi | in March 2026.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/3.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = (1+\u221a3)/2 = 1.366+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "a5c3dc7083e49634", "improvement": -6.667945486871218e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5592677317674368e-06, "max_pair_overlap_depth": -3.548280330942233e-05, "metric_expression": "1.36607", "metric_symbol": "s", "metric_value": 1.3660666794548688, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.366, "setup": "squinhex", "side": 1.3660666794548688, "solution_path": "data/solutions/generated/squinhex_003_a5c3dc7083e49634.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/a5c3dc7083e49634.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@4", "density": 0.6666229789487077, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/4.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2/3 1/4 = 1.519+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.519, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@4", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "2/3 1/4 = 1.519+", "metric_symbol": "s", "metric_value": 1.519, "reference_text": "s = 2/3 1/4 = 1.519+ | Found by Maurizio Morandi | in March 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/4.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2/3 1/4 = 1.519+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "49c10fcc6144044b", "improvement": -0.0007211669812854371, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.944520361455586e-06, "max_pair_overlap_depth": -3.91225282254104e-05, "metric_expression": "1.51972", "metric_symbol": "s", "metric_value": 1.5197211669812853, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.519, "setup": "squinhex", "side": 1.5197211669812853, "solution_path": "data/solutions/generated/squinhex_004_49c10fcc6144044b.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/49c10fcc6144044b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@5", "density": 0.6414868196410884, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/5.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a3 = 1.732+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.732, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3 = 1.732+", "metric_symbol": "s", "metric_value": 1.732, "reference_text": "s = \u221a3 = 1.732+ | Trivial. |", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/5.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = \u221a3 = 1.732+ | Trivial. |", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "1d1f44767245ecd6", "improvement": -6.900512462770791e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1790980134561805e-06, "max_pair_overlap_depth": -6.972508607594463e-06, "metric_expression": "1.73207", "metric_symbol": "s", "metric_value": 1.7320690051246277, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.732, "setup": "squinhex", "side": 1.7320690051246277, "solution_path": "data/solutions/generated/squinhex_005_1d1f44767245ecd6.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/1d1f44767245ecd6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@6", "density": 0.6953530967749227, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/6.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.821+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.821, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@6", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "1.821+", "metric_symbol": "s", "metric_value": 1.821, "reference_text": "s = 1.821+ | Found by Maurizio Morandi | in March 2026.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/6.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 1.821+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "974baf9dad527270", "improvement": -0.001413758154348077, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6930873179275063e-06, "max_pair_overlap_depth": -8.425702388847878e-06, "metric_expression": "1.82241", "metric_symbol": "s", "metric_value": 1.822413758154348, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.821, "setup": "squinhex", "side": 1.822413758154348, "solution_path": "data/solutions/generated/squinhex_006_974baf9dad527270.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/974baf9dad527270.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@7", "density": 0.777727236784812, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/7.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(\u221a2)3 1/4 = 1.861+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.861, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@7", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "(\u221a2)3 1/4 = 1.861+", "metric_symbol": "s", "metric_value": 1.861, "reference_text": "s = (\u221a2)3 1/4 = 1.861+ | Found by Maurizio Morandi | in March 2026.", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/7.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = (\u221a2)3 1/4 = 1.861+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "3877067eb2c9921c", "improvement": -0.0002701930433344213, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.58302978193184e-06, "max_pair_overlap_depth": -6.096494394869367e-07, "metric_expression": "1.86127", "metric_symbol": "s", "metric_value": 1.8612701930433344, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.861, "setup": "squinhex", "side": 1.8612701930433344, "solution_path": "data/solutions/generated/squinhex_007_3877067eb2c9921c.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/3877067eb2c9921c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@8", "density": 0.6631734535885272, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/8.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.15378+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.15378, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@8", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.15378+", "metric_symbol": "s", "metric_value": 2.15378, "reference_text": "s = 2.15378+ | Found by Ian Watson | in April 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/8.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.15378+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "15b61ced9a55fd6e", "improvement": -0.0010126406302717328, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9874417895385996e-06, "max_pair_overlap_depth": -2.4862858205326743e-05, "metric_expression": "2.15479", "metric_symbol": "s", "metric_value": 2.1547926406302715, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.15378, "setup": "squinhex", "side": 2.1547926406302715, "solution_path": "data/solutions/generated/squinhex_008_15b61ced9a55fd6e.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/15b61ced9a55fd6e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@9", "density": 0.6952132004138262, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/9.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1/2+\u221a3 = 2.23205+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.23205, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@9", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1/2+\u221a3 = 2.23205+", "metric_symbol": "s", "metric_value": 2.23205, "reference_text": "s = 1/2+\u221a3 = 2.23205+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/9.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 1/2+\u221a3 = 2.23205+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "55005bb44a4d74d4", "improvement": -0.00016646218885707498, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6345465208580115e-06, "max_pair_overlap_depth": -3.4334606535568746e-05, "metric_expression": "2.23222", "metric_symbol": "s", "metric_value": 2.232216462188857, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.23205, "setup": "squinhex", "side": 2.232216462188857, "solution_path": "data/solutions/generated/squinhex_009_55005bb44a4d74d4.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/55005bb44a4d74d4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@10", "density": 0.721657360923786, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/10.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "4/\u221a3 = 2.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.309, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@10", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4/\u221a3 = 2.309+", "metric_symbol": "s", "metric_value": 2.309, "reference_text": "s = 4/\u221a3 = 2.309+ | Trivial. |", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/10.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 4/\u221a3 = 2.309+ | Trivial. |", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "1dc9bb5dff8e4212", "improvement": -0.0004498392040641619, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.82433796662779e-07, "max_pair_overlap_depth": -1.056245311698012e-05, "metric_expression": "2.30945", "metric_symbol": "s", "metric_value": 2.3094498392040643, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.309, "setup": "squinhex", "side": 2.3094498392040643, "solution_path": "data/solutions/generated/squinhex_010_1dc9bb5dff8e4212.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/1dc9bb5dff8e4212.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@11", "density": 0.7291333157228544, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/11.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.40944+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.40944, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.40944+", "metric_symbol": "s", "metric_value": 2.40944, "reference_text": "s = 2.40944+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/11.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.40944+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "705689a83f76dc66", "improvement": -0.00028191921915876605, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.56887695052102e-06, "max_pair_overlap_depth": -2.8090377436340574e-05, "metric_expression": "2.40972", "metric_symbol": "s", "metric_value": 2.409721919219159, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.40944, "setup": "squinhex", "side": 2.409721919219159, "solution_path": "data/solutions/generated/squinhex_011_705689a83f76dc66.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/705689a83f76dc66.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@12", "density": 0.7736310958293992, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/12.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1+5\u221a3/6 = 2.443+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.443, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@12", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "1+5\u221a3/6 = 2.443+", "metric_symbol": "s", "metric_value": 2.443, "reference_text": "s = 1+5\u221a3/6 = 2.443+ | Found by Maurizio Morandi | in March 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/12.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 1+5\u221a3/6 = 2.443+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "2605d29267e364f4", "improvement": -0.00041773030028169345, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4756288282540256e-06, "max_pair_overlap_depth": -2.0709990555811686e-06, "metric_expression": "2.44342", "metric_symbol": "s", "metric_value": 2.4434177303002818, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.443, "setup": "squinhex", "side": 2.4434177303002818, "solution_path": "data/solutions/generated/squinhex_012_2605d29267e364f4.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/2605d29267e364f4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@13", "density": 0.7555572001975633, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/13.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.572+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.572, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@13", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "2.572+", "metric_symbol": "s", "metric_value": 2.572, "reference_text": "s = 2.572+ | Found by Maurizio Morandi | in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/13.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.572+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "051502d315bb277c", "improvement": -0.001428217365099016, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2709109889167893e-06, "max_pair_overlap_depth": -2.7796124429491087e-05, "metric_expression": "2.57343", "metric_symbol": "s", "metric_value": 2.573428217365099, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.572, "setup": "squinhex", "side": 2.573428217365099, "solution_path": "data/solutions/generated/squinhex_013_051502d315bb277c.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/051502d315bb277c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@14", "density": 0.7839123498532168, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/14.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.62126+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.62126, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@14", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.62126+", "metric_symbol": "s", "metric_value": 2.62126, "reference_text": "s = 2.62126+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/14.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.62126+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "525e7cc585bd027c", "improvement": -0.000568749699766613, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5893267782285044e-06, "max_pair_overlap_depth": -1.4910539989809735e-05, "metric_expression": "2.62183", "metric_symbol": "s", "metric_value": 2.6218287496997665, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.62126, "setup": "squinhex", "side": 2.6218287496997665, "solution_path": "data/solutions/generated/squinhex_014_525e7cc585bd027c.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/525e7cc585bd027c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@15", "density": 0.7612787773180276, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/15.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.75365+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.75365, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@15", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.75365+", "metric_symbol": "s", "metric_value": 2.75365, "reference_text": "s = 2.75365+ | Found by Ian Watson | in April 2026.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/15.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.75365+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "b68c71af3475ffed", "improvement": -0.00024783676108830477, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.513954837939443e-06, "max_pair_overlap_depth": -3.0296502803062175e-05, "metric_expression": "2.75390", "metric_symbol": "s", "metric_value": 2.7538978367610882, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.75365, "setup": "squinhex", "side": 2.7538978367610882, "solution_path": "data/solutions/generated/squinhex_015_b68c71af3475ffed.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/b68c71af3475ffed.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@16", "density": 0.7742958402345232, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/16.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.82004+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.82004, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@16", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.82004+", "metric_symbol": "s", "metric_value": 2.82004, "reference_text": "s = 2.82004+ | Found by Ian Watson | in April 2026.", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/16.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.82004+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "89a68dd125601a79", "improvement": -0.0001643950752789891, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.5133109066385543e-06, "max_pair_overlap_depth": -7.772490123292286e-06, "metric_expression": "2.82020", "metric_symbol": "s", "metric_value": 2.820204395075279, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.82004, "setup": "squinhex", "side": 2.820204395075279, "solution_path": "data/solutions/generated/squinhex_016_89a68dd125601a79.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/89a68dd125601a79.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@17", "density": 0.7850485918652831, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/17.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "5/\u221a3 = 2.88675+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.88675, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@17", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5/\u221a3 = 2.88675+", "metric_symbol": "s", "metric_value": 2.88675, "reference_text": "s = 5/\u221a3 = 2.88675+ | Trivial. |", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/17.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 5/\u221a3 = 2.88675+ | Trivial. |", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "68ba30b81c3844a7", "improvement": -0.0002730281066796181, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9250177449939015e-07, "max_pair_overlap_depth": -5.409854801341485e-05, "metric_expression": "2.88702", "metric_symbol": "s", "metric_value": 2.8870230281066798, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.88675, "setup": "squinhex", "side": 2.8870230281066798, "solution_path": "data/solutions/generated/squinhex_017_68ba30b81c3844a7.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/68ba30b81c3844a7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@18", "density": 0.8107158050212577, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/18.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.92111+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.92111, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@18", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.92111+", "metric_symbol": "s", "metric_value": 2.92111, "reference_text": "s = 2.92111+ | Found by Ian Watson | in April 2026.", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/18.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.92111+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "e136eeec3de765c3", "improvement": -0.0022075205933282938, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.264526989622803e-06, "max_pair_overlap_depth": -7.64695502597501e-06, "metric_expression": "2.92332", "metric_symbol": "s", "metric_value": 2.9233175205933284, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.92111, "setup": "squinhex", "side": 2.9233175205933284, "solution_path": "data/solutions/generated/squinhex_018_e136eeec3de765c3.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/e136eeec3de765c3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@19", "density": 0.8472470384096148, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/19.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.93511+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.93511, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@19", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.93511+", "metric_symbol": "s", "metric_value": 2.93511, "reference_text": "s = 2.93511+ | Found by Ian Watson | in April 2026.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/19.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.93511+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "ae3850bdb0ac1880", "improvement": -0.0028496545666087947, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.965076151420116e-06, "max_pair_overlap_depth": -7.201359447650546e-06, "metric_expression": "2.93796", "metric_symbol": "s", "metric_value": 2.9379596545666087, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.93511, "setup": "squinhex", "side": 2.9379596545666087, "solution_path": "data/solutions/generated/squinhex_019_ae3850bdb0ac1880.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/ae3850bdb0ac1880.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@20", "density": 0.7690325218237841, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/20.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.16123+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.16123, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@20", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.16123+", "metric_symbol": "s", "metric_value": 3.16123, "reference_text": "s = 3.16123+ | Found by Ian Watson | in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/20.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.16123+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "79028917ac70723d", "improvement": -0.0026259473298471114, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.697008230249216e-06, "max_pair_overlap_depth": -2.836652900617409e-05, "metric_expression": "3.16386", "metric_symbol": "s", "metric_value": 3.1638559473298473, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.16123, "setup": "squinhex", "side": 3.1638559473298473, "solution_path": "data/solutions/generated/squinhex_020_79028917ac70723d.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/79028917ac70723d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@21", "density": 0.7707944086673769, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/21.png", "friedman_metric_expression": "3.23496+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.23496, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@21", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.23496+", "metric_symbol": "s", "metric_value": 3.23496, "reference_text": "s = 3.23496+ | Found by Ian Watson | in April 2026.", "source_image": "21.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/21.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.23496+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "21.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "1d2ec517035658cd", "improvement": -0.0033202143530659534, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4830527974218215e-06, "max_pair_overlap_depth": -2.662964935497389e-05, "metric_expression": "3.23828", "metric_symbol": "s", "metric_value": 3.238280214353066, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.23496, "setup": "squinhex", "side": 3.238280214353066, "solution_path": "data/solutions/generated/squinhex_21_1d2ec517035658cd.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/1d2ec517035658cd.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@22", "density": 0.770525380181529, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/22.png", "friedman_metric_expression": "3.31573+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.31573, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@22", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.31573+", "metric_symbol": "s", "metric_value": 3.31573, "reference_text": "s = 3.31573+ | Found by Ian Watson | in April 2026.", "source_image": "22.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/22.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.31573+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "22.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "a1e7eaf12003fa2e", "improvement": 0.0006659560348909466, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.169977234601665e-06, "max_pair_overlap_depth": -1.9814540655582746e-05, "metric_expression": "3.31506", "metric_symbol": "s", "metric_value": 3.315064043965109, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.31573, "setup": "squinhex", "side": 3.315064043965109, "solution_path": "data/solutions/generated/squinhex_022_a1e7eaf12003fa2e.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/a1e7eaf12003fa2e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@23", "density": 0.7808954728618008, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/23.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.36437+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.36437, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@23", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.36437+", "metric_symbol": "s", "metric_value": 3.36437, "reference_text": "s = 3.36437+ | Found by Ian Watson | in April 2026.", "source_image": "23.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/23.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.36437+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "23.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "de78b89055181fd4", "improvement": -0.00261764110148599, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.240294795076849e-06, "max_pair_overlap_depth": -2.4390771091731267e-07, "metric_expression": "3.36699", "metric_symbol": "s", "metric_value": 3.366987641101486, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.36437, "setup": "squinhex", "side": 3.366987641101486, "solution_path": "data/solutions/generated/squinhex_023_de78b89055181fd4.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/de78b89055181fd4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@24", "density": 0.7904120022571139, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/24.png", "friedman_metric_expression": "3.41524+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.41524, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@24", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.41524+", "metric_symbol": "s", "metric_value": 3.41524, "reference_text": "s = 3.41524+ | Found by Ian Watson | in April 2026.", "source_image": "24.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/24.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.41524+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "24.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "73f1ceeaee97d32f", "improvement": -0.0033964158861361504, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0115328209217065e-06, "max_pair_overlap_depth": -2.5222031625055097e-05, "metric_expression": "3.41864", "metric_symbol": "s", "metric_value": 3.418636415886136, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.41524, "setup": "squinhex", "side": 3.418636415886136, "solution_path": "data/solutions/generated/squinhex_24_73f1ceeaee97d32f.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T14:34:18+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/73f1ceeaee97d32f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@25", "density": 0.8017157641998289, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/25.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2\u221a3 = 3.46410+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.4641, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@25", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2\u221a3 = 3.46410+", "metric_symbol": "s", "metric_value": 3.4641, "reference_text": "s = 2\u221a3 = 3.46410+ | Trivial. |", "source_image": "25.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/25.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2\u221a3 = 3.46410+ | Trivial. |", "friedman_source_image": "25.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "803edc96e272ee7a", "improvement": -0.00034642351026814566, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4684956309553432e-06, "max_pair_overlap_depth": -2.379294205911986e-05, "metric_expression": "3.46445", "metric_symbol": "s", "metric_value": 3.4644464235102683, "n": 25, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.4641, "setup": "squinhex", "side": 3.4644464235102683, "solution_path": "data/solutions/generated/squinhex_025_803edc96e272ee7a.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/803edc96e272ee7a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@26", "density": 0.8183272598135501, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/26.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.4939927+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.4939927, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@26", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.4939927+", "metric_symbol": "s", "metric_value": 3.4939927, "reference_text": "s = 3.4939927+ | Found by Ian Watson | in April 2026.", "source_image": "26.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/26.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.4939927+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "26.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "5fb7804364b5caaf", "improvement": -0.003020096739570377, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.392756549893556e-06, "max_pair_overlap_depth": -2.620566831379456e-05, "metric_expression": "3.49701", "metric_symbol": "s", "metric_value": 3.4970127967395706, "n": 26, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.4939927, "setup": "squinhex", "side": 3.4970127967395706, "solution_path": "data/solutions/generated/squinhex_026_5fb7804364b5caaf.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/5fb7804364b5caaf.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@27", "density": 0.8382703933092833, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/27.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(9+7\u221a3)/6 = 3.52072+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.52072, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@27", "credit": "Erich Friedman", "date": "April 2026", "family_updated": "", "metric_expression": "(9+7\u221a3)/6 = 3.52072+", "metric_symbol": "s", "metric_value": 3.52072, "reference_text": "s = (9+7\u221a3)/6 = 3.52072+ | Found by Erich Friedman | in April 2026.", "source_image": "27.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/27.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = (9+7\u221a3)/6 = 3.52072+ | Found by Erich Friedman | in April 2026.", "friedman_source_image": "27.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "9cebbb5918ba8e81", "improvement": -0.00026256883024800004, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.315485793993667e-06, "max_pair_overlap_depth": -2.3466684225592616e-05, "metric_expression": "3.52098", "metric_symbol": "s", "metric_value": 3.520982568830248, "n": 27, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.52072, "setup": "squinhex", "side": 3.520982568830248, "solution_path": "data/solutions/generated/squinhex_027_9cebbb5918ba8e81.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/9cebbb5918ba8e81.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@28", "density": 0.8183465665432688, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/28.png", "friedman_metric_expression": "3.62565+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.62565, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@28", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.62565+", "metric_symbol": "s", "metric_value": 3.62565, "reference_text": "s = 3.62565+ | Found by Ian Watson | in April 2026.", "source_image": "28.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/28.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.62565+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "28.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "184baa4d16ce7f9e", "improvement": -0.00332887546169669, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.700161000518847e-07, "max_pair_overlap_depth": -3.0973746406592895e-06, "metric_expression": "3.62898", "metric_symbol": "s", "metric_value": 3.6289788754616965, "n": 28, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.62565, "setup": "squinhex", "side": 3.6289788754616965, "solution_path": "data/solutions/generated/squinhex_28_184baa4d16ce7f9e.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T14:34:18+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/184baa4d16ce7f9e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@29", "density": 0.829741145219895, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/29.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.66437+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.66437, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@29", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.66437+", "metric_symbol": "s", "metric_value": 3.66437, "reference_text": "s = 3.66437+ | Found by Ian Watson | in April 2026.", "source_image": "29.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/29.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.66437+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "29.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "3b329ff6f7babb15", "improvement": -0.0033970322125265184, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.61123003800209e-07, "max_pair_overlap_depth": -5.962934573305878e-06, "metric_expression": "3.66777", "metric_symbol": "s", "metric_value": 3.6677670322125264, "n": 29, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.66437, "setup": "squinhex", "side": 3.6677670322125264, "solution_path": "data/solutions/generated/squinhex_029_3b329ff6f7babb15.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/3b329ff6f7babb15.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@30", "density": 0.8440944212011869, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/30.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.69554+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.69554, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@30", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.69554+", "metric_symbol": "s", "metric_value": 3.69554, "reference_text": "s = 3.69554+ | Found by Ian Watson | in April 2026.", "source_image": "30.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/30.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.69554+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "30.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "0b645f27b41cbed7", "improvement": -0.0030753622873653796, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7116676711139033e-06, "max_pair_overlap_depth": -4.452216716410717e-06, "metric_expression": "3.69862", "metric_symbol": "s", "metric_value": 3.698615362287365, "n": 30, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.69554, "setup": "squinhex", "side": 3.698615362287365, "solution_path": "data/solutions/generated/squinhex_030_0b645f27b41cbed7.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/0b645f27b41cbed7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@31", "density": 0.83334567619728, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/31.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.78410+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.7841, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.78410+", "metric_symbol": "s", "metric_value": 3.7841, "reference_text": "s = 3.78410+ | Found by Emerson Connelly | in April 2026.", "source_image": "31.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/31.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.78410+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "31.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "83b47696242e2320", "improvement": 0.00017682363310500548, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0307084382837186e-06, "max_pair_overlap_depth": -2.9858800666679453e-05, "metric_expression": "3.78392", "metric_symbol": "s", "metric_value": 3.783923176366895, "n": 31, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.7841, "setup": "squinhex", "side": 3.783923176366895, "solution_path": "data/solutions/generated/squinhex_031_83b47696242e2320.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/83b47696242e2320.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@32", "density": 0.8302114212681493, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/32.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.84795+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.84795, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@32", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.84795+", "metric_symbol": "s", "metric_value": 3.84795, "reference_text": "s = 3.84795+ | Found by Emerson Connelly | in April 2026.", "source_image": "32.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/32.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.84795+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "32.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "b233eac9bd8cc0d8", "improvement": -0.003769870972095468, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1351534734416697e-07, "max_pair_overlap_depth": -1.2013295066060081e-05, "metric_expression": "3.85172", "metric_symbol": "s", "metric_value": 3.8517198709720954, "n": 32, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.84795, "setup": "squinhex", "side": 3.8517198709720954, "solution_path": "data/solutions/generated/squinhex_032_b233eac9bd8cc0d8.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/b233eac9bd8cc0d8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@33", "density": 0.8381951703957873, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/33.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.88897+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.88897, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@33", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.88897+", "metric_symbol": "s", "metric_value": 3.88897, "reference_text": "s = 3.88897+ | Found by Emerson Connelly | in April 2026.", "source_image": "33.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/33.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.88897+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "33.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "b14228e93eb45276", "improvement": -0.003797355726243623, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.797125554432215e-06, "max_pair_overlap_depth": -8.18730119578337e-06, "metric_expression": "3.89277", "metric_symbol": "s", "metric_value": 3.8927673557262437, "n": 33, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.88897, "setup": "squinhex", "side": 3.8927673557262437, "solution_path": "data/solutions/generated/squinhex_033_b14228e93eb45276.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/b14228e93eb45276.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@34", "density": 0.8486832973249016, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/34.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.92704+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.92704, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@34", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.92704+", "metric_symbol": "s", "metric_value": 3.92704, "reference_text": "s = 3.92704+ | Found by Emerson Connelly | in April 2026.", "source_image": "34.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/34.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.92704+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "34.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "fc2f6f3b6cabd5ed", "improvement": 0.0002227711135822652, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.30656742709778e-06, "max_pair_overlap_depth": -1.5057714820798651e-05, "metric_expression": "3.92682", "metric_symbol": "s", "metric_value": 3.9268172288864176, "n": 34, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.92704, "setup": "squinhex", "side": 3.9268172288864176, "solution_path": "data/solutions/generated/squinhex_034_fc2f6f3b6cabd5ed.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/fc2f6f3b6cabd5ed.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@35", "density": 0.8429987640989463, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/35.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.99833+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.99833, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@35", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.99833+", "metric_symbol": "s", "metric_value": 3.99833, "reference_text": "s = 3.99833+ | Found by Emerson Connelly | in April 2026.", "source_image": "35.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/35.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.99833+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "35.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "1ea058e93fb408a4", "improvement": 0.0007735056288105113, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.597269939847931e-06, "max_pair_overlap_depth": -1.4652234902046146e-05, "metric_expression": "3.99756", "metric_symbol": "s", "metric_value": 3.9975564943711897, "n": 35, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.99833, "setup": "squinhex", "side": 3.9975564943711897, "solution_path": "data/solutions/generated/squinhex_035_1ea058e93fb408a4.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/1ea058e93fb408a4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@36", "density": 0.8531956067904081, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/36.png", "friedman_metric_expression": "2+7\u221a3/6 = 4.02072+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.02072, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@36", "credit": "Maurizio Morandi", "date": "May 2026", "family_updated": "", "metric_expression": "2+7\u221a3/6 = 4.02072+", "metric_symbol": "s", "metric_value": 4.02072, "reference_text": "s = 2+7\u221a3/6 = 4.02072+ | Found by Maurizio Morandi | in May 2026.", "source_image": "36.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/36.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2+7\u221a3/6 = 4.02072+ | Found by Maurizio Morandi | in May 2026.", "friedman_source_image": "36.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "b7083884dfb200c3", "improvement": -0.009242471992498835, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.100230231840266e-07, "max_pair_overlap_depth": -2.2076071326537772e-07, "metric_expression": "4.02996", "metric_symbol": "s", "metric_value": 4.029962471992499, "n": 36, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.02072, "setup": "squinhex", "side": 4.029962471992499, "solution_path": "data/solutions/record_hunt/squinhex_36_b7083884dfb200c3.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/b7083884dfb200c3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@37", "density": 0.8718839158050435, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/37.png", "friedman_metric_expression": "7/\u221a3 = 4.04145+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.04145, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@37", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7/\u221a3 = 4.04145+", "metric_symbol": "s", "metric_value": 4.04145, "reference_text": "s = 7/\u221a3 = 4.04145+ | Trivial. |", "source_image": "37.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/37.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 7/\u221a3 = 4.04145+ | Trivial. |", "friedman_source_image": "37.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "a455ecb05be301ad", "improvement": -7.79426845127773e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1637259667017477e-06, "max_pair_overlap_depth": -1.0162747484221413e-05, "metric_expression": "4.04153", "metric_symbol": "s", "metric_value": 4.041527942684513, "n": 37, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.04145, "setup": "squinhex", "side": 4.041527942684513, "solution_path": "data/solutions/generated/squinhex_037_a455ecb05be301ad.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a455ecb05be301ad.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@38", "density": 0.8354467372021385, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/38.png", "friedman_metric_expression": "4.18476+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.18476, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@38", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.18476+", "metric_symbol": "s", "metric_value": 4.18476, "reference_text": "s = 4.18476+ | Found by Emerson Connelly | in April 2026.", "source_image": "38.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/38.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 4.18476+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "38.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "657f5bc54656101d", "improvement": 0.0006174563357532747, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.3358638004937594e-07, "max_pair_overlap_depth": -2.4459335521420655e-06, "metric_expression": "4.18414", "metric_symbol": "s", "metric_value": 4.1841425436642465, "n": 38, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.18476, "setup": "squinhex", "side": 4.184142543664246, "solution_path": "data/solutions/record_hunt/squinhex_38_657f5bc54656101d.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/657f5bc54656101d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@39", "density": 0.8275155837189719, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/39.png", "friedman_metric_expression": "4.25744+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.25744, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@39", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.25744+", "metric_symbol": "s", "metric_value": 4.25744, "reference_text": "s = 4.25744+ | Found by Emerson Connelly | in April 2026.", "source_image": "39.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/39.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 4.25744+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "39.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "85e7e9eb3b3b7f5f", "improvement": -0.001664243334674076, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.204496689819836e-06, "max_pair_overlap_depth": -1.602842749781619e-05, "metric_expression": "4.25910", "metric_symbol": "s", "metric_value": 4.259104243334674, "n": 39, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.25744, "setup": "squinhex", "side": 4.259104243334674, "solution_path": "data/solutions/generated/squinhex_039_85e7e9eb3b3b7f5f.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/85e7e9eb3b3b7f5f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinhex@40", "density": 0.8288203735668404, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/40.png", "friedman_metric_expression": "4.31157+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.31157, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@40", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.31157+", "metric_symbol": "s", "metric_value": 4.31157, "reference_text": "s = 4.31157+ | Found by Emerson Connelly | in April 2026.", "source_image": "40.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/40.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 4.31157+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "40.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "9859865bba74b499", "improvement": 0.0016040074615322197, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.921136271031391e-07, "max_pair_overlap_depth": -1.8467799072219293e-06, "metric_expression": "4.30997", "metric_symbol": "s", "metric_value": 4.3099659925384675, "n": 40, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.31157, "setup": "squinhex", "side": 4.309965992538467, "solution_path": "data/solutions/record_hunt/squinhex_40_9859865bba74b499.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/9859865bba74b499.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@1", "density": 0.7071036297921394, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/1.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "1/\u221a(2+\u221a2) = .54119+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.54119, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1/\u221a(2+\u221a2) = .54119+", "metric_symbol": "s", "metric_value": 0.54119, "reference_text": "s = 1/\u221a(2+\u221a2) = .54119+ | Trivial. |", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/1.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 1/\u221a(2+\u221a2) = .54119+ | Trivial. |", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "b63c90f60717ba08", "improvement": -7.30613665711477e-06, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1705346114441018e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.54120", "metric_symbol": "s", "metric_value": 0.5411973061366571, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.54119, "setup": "squinoct", "side": 0.5411973061366571, "solution_path": "data/solutions/generated/squinoct_001_b63c90f60717ba08.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b63c90f60717ba08.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@2", "density": 0.5389047922692529, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "December 2013", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/2.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": ".87667+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.87667, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@2", "credit": "Maurizio Morandi", "date": "December 2013", "family_updated": "", "metric_expression": ".87667+", "metric_symbol": "s", "metric_value": 0.87667, "reference_text": "s = .87667+ | Found by Maurizio Morandi | in December 2013.", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/2.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = .87667+ | Found by Maurizio Morandi | in December 2013.", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "7194ac58cba12fa7", "improvement": -4.0351762315649076e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.607981571160174e-06, "max_pair_overlap_depth": -1.1896285956325414e-06, "metric_expression": "0.87671", "metric_symbol": "s", "metric_value": 0.8767103517623156, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.87667, "setup": "squinoct", "side": 0.8767103517623156, "solution_path": "data/solutions/generated/squinoct_002_7194ac58cba12fa7.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/7194ac58cba12fa7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@3", "density": 0.5862650280603752, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "December 2013", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/3.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "18 \u2013 12\u221a2 = 1.02940+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0294, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@3", "credit": "Erich Friedman", "date": "December 2013", "family_updated": "", "metric_expression": "18 \u2013 12\u221a2 = 1.02940+", "metric_symbol": "s", "metric_value": 1.0294, "reference_text": "s = 18 \u2013 12\u221a2 = 1.02940+ | Found by Erich Friedman | in December 2013.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/3.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 18 \u2013 12\u221a2 = 1.02940+ | Found by Erich Friedman | in December 2013.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "3d85991afac60053", "improvement": -6.311956912941064e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.835100684478856e-06, "max_pair_overlap_depth": -3.4409461626638826e-05, "metric_expression": "1.02946", "metric_symbol": "s", "metric_value": 1.0294631195691295, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0294, "setup": "squinoct", "side": 1.0294631195691295, "solution_path": "data/solutions/generated/squinoct_003_3d85991afac60053.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3d85991afac60053.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@4", "density": 0.7070860385519216, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/4.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "\u221a(4\u20132\u221a2) = 1.08239+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.08239, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a(4\u20132\u221a2) = 1.08239+", "metric_symbol": "s", "metric_value": 1.08239, "reference_text": "s = \u221a(4\u20132\u221a2) = 1.08239+ | Trivial. |", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/4.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = \u221a(4\u20132\u221a2) = 1.08239+ | Trivial. |", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "30bfa7c46cf6dd87", "improvement": -1.8076366912644914e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.1007450963540665e-06, "max_pair_overlap_depth": -1.5610612916431253e-05, "metric_expression": "1.08241", "metric_symbol": "s", "metric_value": 1.0824080763669126, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.08239, "setup": "squinoct", "side": 1.0824080763669126, "solution_path": "data/solutions/generated/squinoct_004_30bfa7c46cf6dd87.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/30bfa7c46cf6dd87.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@5", "density": 0.6705984373456575, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/5.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "3\u221a2 \u2013 3 = 1.24261+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.24261, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3\u221a2 \u2013 3 = 1.24261+", "metric_symbol": "s", "metric_value": 1.24261, "reference_text": "s = 3\u221a2 \u2013 3 = 1.24261+ | Trivial. |", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/5.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 3\u221a2 \u2013 3 = 1.24261+ | Trivial. |", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "3bf97518bf42702c", "improvement": -4.591996404545995e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2899385621933342e-06, "max_pair_overlap_depth": -9.75715015771872e-06, "metric_expression": "1.24266", "metric_symbol": "s", "metric_value": 1.2426559199640455, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.24261, "setup": "squinoct", "side": 1.2426559199640455, "solution_path": "data/solutions/generated/squinoct_005_3bf97518bf42702c.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3bf97518bf42702c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@6", "density": 0.6887710971991018, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "December 2013", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/6.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "7 \u2013 4\u221a2 = 1.34314+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.34314, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@6", "credit": "Maurizio Morandi", "date": "December 2013", "family_updated": "", "metric_expression": "7 \u2013 4\u221a2 = 1.34314+", "metric_symbol": "s", "metric_value": 1.34314, "reference_text": "s = 7 \u2013 4\u221a2 = 1.34314+ | Found by Maurizio Morandi | in December 2013.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/6.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 7 \u2013 4\u221a2 = 1.34314+ | Found by Maurizio Morandi | in December 2013.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "6368fdc8880da7e0", "improvement": -4.34357115446371e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.645757686457117e-06, "max_pair_overlap_depth": -2.457675057909725e-05, "metric_expression": "1.34318", "metric_symbol": "s", "metric_value": 1.3431834357115446, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.34314, "setup": "squinoct", "side": 1.3431834357115446, "solution_path": "data/solutions/generated/squinoct_006_6368fdc8880da7e0.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/6368fdc8880da7e0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@7", "density": 0.702115846733366, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "December 2013", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/7.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "1.43691+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.43691, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@7", "credit": "Maurizio Morandi", "date": "December 2013", "family_updated": "", "metric_expression": "1.43691+", "metric_symbol": "s", "metric_value": 1.43691, "reference_text": "s = 1.43691+ | Found by Maurizio Morandi | in December 2013.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/7.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 1.43691+ | Found by Maurizio Morandi | in December 2013.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "35d0c06f5c122826", "improvement": -4.0447358323758564e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.691262438170272e-07, "max_pair_overlap_depth": -1.2495412028745712e-05, "metric_expression": "1.43695", "metric_symbol": "s", "metric_value": 1.4369504473583237, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.43691, "setup": "squinoct", "side": 1.4369504473583237, "solution_path": "data/solutions/generated/squinoct_007_35d0c06f5c122826.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/35d0c06f5c122826.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@8", "density": 0.6722160358662614, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/8.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "1.56967+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.56967, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@8", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.56967+", "metric_symbol": "s", "metric_value": 1.56967, "reference_text": "s = 1.56967+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/8.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 1.56967+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "ecaef16600354c9f", "improvement": -0.00028685978676801227, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.463017743294653e-07, "max_pair_overlap_depth": -1.4183990756477804e-05, "metric_expression": "1.56996", "metric_symbol": "s", "metric_value": 1.569956859786768, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.56967, "setup": "squinoct", "side": 1.569956859786768, "solution_path": "data/solutions/generated/squinoct_008_ecaef16600354c9f.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/ecaef16600354c9f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@9", "density": 0.707064805820287, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/9.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "3/\u221a(2+\u221a2) = 1.62358+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.62358, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@9", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3/\u221a(2+\u221a2) = 1.62358+", "metric_symbol": "s", "metric_value": 1.62358, "reference_text": "s = 3/\u221a(2+\u221a2) = 1.62358+ | Trivial. |", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/9.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 3/\u221a(2+\u221a2) = 1.62358+ | Trivial. |", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "703059104a2b70f8", "improvement": -5.649241589167353e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6691071523245226e-06, "max_pair_overlap_depth": -7.928509179855325e-06, "metric_expression": "1.62364", "metric_symbol": "s", "metric_value": 1.6236364924158917, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.62358, "setup": "squinoct", "side": 1.6236364924158917, "solution_path": "data/solutions/generated/squinoct_009_703059104a2b70f8.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/703059104a2b70f8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@10", "density": 0.732083417533705, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "December 2013", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/10.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2^(3/4) = 1.68179+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.68179, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@10", "credit": "Maurizio Morandi", "date": "December 2013", "family_updated": "", "metric_expression": "2^(3/4) = 1.68179+", "metric_symbol": "s", "metric_value": 1.68179, "reference_text": "s = 2^(3/4) = 1.68179+ | Found by Maurizio Morandi | in December 2013.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/10.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2^(3/4) = 1.68179+ | Found by Maurizio Morandi | in December 2013.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "03facc71923f1148", "improvement": -0.00017469136174708133, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.2978224497762506e-06, "max_pair_overlap_depth": -2.3560442145575955e-05, "metric_expression": "1.68196", "metric_symbol": "s", "metric_value": 1.681964691361747, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.68179, "setup": "squinoct", "side": 1.681964691361747, "solution_path": "data/solutions/generated/squinoct_010_03facc71923f1148.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/03facc71923f1148.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@11", "density": 0.7597329550134904, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/11.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "1.73180+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.7318, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@11", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.73180+", "metric_symbol": "s", "metric_value": 1.7318, "reference_text": "s = 1.73180+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/11.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 1.73180+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "78c32afdb2263b2d", "improvement": 0.00013842799014240903, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.409684401929808e-09, "max_pair_overlap_depth": -1.1021961062729346e-05, "metric_expression": "1.73166", "metric_symbol": "s", "metric_value": 1.7316615720098576, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.7318, "setup": "squinoct", "side": 1.7316615720098576, "solution_path": "data/solutions/generated/squinoct_011_78c32afdb2263b2d.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/78c32afdb2263b2d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@12", "density": 0.8083084838512736, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "December 2013", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/12.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "1.7533+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.7533, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@12", "credit": "Maurizio Morandi", "date": "December 2013", "family_updated": "", "metric_expression": "1.7533+", "metric_symbol": "s", "metric_value": 1.7533, "reference_text": "s = 1.7533+ | Found by Maurizio Morandi | in December 2013.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/12.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 1.7533+ | Found by Maurizio Morandi | in December 2013.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "7a9451cae59dbb40", "improvement": -0.00017352882969978722, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.088823188597466e-06, "max_pair_overlap_depth": -2.0797652260817756e-05, "metric_expression": "1.75347", "metric_symbol": "s", "metric_value": 1.7534735288296999, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.7533, "setup": "squinoct", "side": 1.7534735288296999, "solution_path": "data/solutions/generated/squinoct_012_7a9451cae59dbb40.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/7a9451cae59dbb40.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@13", "density": 0.7475846505942733, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/13.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "1.89760+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.8976, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@13", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.89760+", "metric_symbol": "s", "metric_value": 1.8976, "reference_text": "s = 1.89760+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/13.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 1.89760+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "c5bdcacf094e428e", "improvement": -0.00014846020367564883, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.439964464558682e-06, "max_pair_overlap_depth": -2.7883136850093138e-05, "metric_expression": "1.89775", "metric_symbol": "s", "metric_value": 1.8977484602036756, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.8976, "setup": "squinoct", "side": 1.8977484602036756, "solution_path": "data/solutions/generated/squinoct_013_c5bdcacf094e428e.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/c5bdcacf094e428e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@14", "density": 0.7369251630765782, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/14.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "1.98350+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.9835, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@14", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.98350+", "metric_symbol": "s", "metric_value": 1.9835, "reference_text": "s = 1.98350+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/14.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 1.98350+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "994e0dd57a80d4a3", "improvement": -7.891975572338339e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.49261632156563e-06, "max_pair_overlap_depth": -2.894591689805992e-05, "metric_expression": "1.98358", "metric_symbol": "s", "metric_value": 1.9835789197557234, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.9835, "setup": "squinoct", "side": 1.9835789197557234, "solution_path": "data/solutions/generated/squinoct_014_994e0dd57a80d4a3.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/994e0dd57a80d4a3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@15", "density": 0.7515649396279209, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/15.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.03259+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.03259, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@15", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.03259+", "metric_symbol": "s", "metric_value": 2.03259, "reference_text": "s = 2.03259+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/15.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.03259+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "c898d8b3cd2b55db", "improvement": -0.0005137104569299389, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7983161874182644e-07, "max_pair_overlap_depth": -1.9704810269915995e-05, "metric_expression": "2.03310", "metric_symbol": "s", "metric_value": 2.03310371045693, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.03259, "setup": "squinoct", "side": 2.03310371045693, "solution_path": "data/solutions/generated/squinoct_015_c898d8b3cd2b55db.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/c898d8b3cd2b55db.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@16", "density": 0.7725175260502497, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/16.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "5\u221a2 \u2013 5 = 2.07106+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.07106, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@16", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5\u221a2 \u2013 5 = 2.07106+", "metric_symbol": "s", "metric_value": 2.07106, "reference_text": "s = 5\u221a2 \u2013 5 = 2.07106+ | Trivial. |", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/16.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 5\u221a2 \u2013 5 = 2.07106+ | Trivial. |", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "3a2e6d99239355fd", "improvement": -4.911646819039106e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5220970331887e-06, "max_pair_overlap_depth": -8.060079695892952e-07, "metric_expression": "2.07111", "metric_symbol": "s", "metric_value": 2.0711091164681905, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.07106, "setup": "squinoct", "side": 2.0711091164681905, "solution_path": "data/solutions/generated/squinoct_016_3a2e6d99239355fd.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3a2e6d99239355fd.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@17", "density": 0.8208079603816107, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/17.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "5\u221a2 \u2013 5 = 2.07106+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.07106, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@17", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5\u221a2 \u2013 5 = 2.07106+", "metric_symbol": "s", "metric_value": 2.07106, "reference_text": "s = 5\u221a2 \u2013 5 = 2.07106+ | Trivial. |", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/17.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 5\u221a2 \u2013 5 = 2.07106+ | Trivial. |", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "ab660f0510a3dd1c", "improvement": -3.891119068244464e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.368661750365744e-09, "max_pair_overlap_depth": -1.0480566122961221e-05, "metric_expression": "2.07110", "metric_symbol": "s", "metric_value": 2.0710989111906826, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.07106, "setup": "squinoct", "side": 2.0710989111906826, "solution_path": "data/solutions/generated/squinoct_017_ab660f0510a3dd1c.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/ab660f0510a3dd1c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@18", "density": 0.7934296897658486, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/18.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.16778+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.16778, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@18", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.16778+", "metric_symbol": "s", "metric_value": 2.16778, "reference_text": "s = 2.16778+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/18.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.16778+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "a7275e2ab7024187", "improvement": 0.00017977070139485463, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9711774532196686e-06, "max_pair_overlap_depth": -1.4522556675777842e-05, "metric_expression": "2.16760", "metric_symbol": "s", "metric_value": 2.167600229298605, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.16778, "setup": "squinoct", "side": 2.167600229298605, "solution_path": "data/solutions/generated/squinoct_018_a7275e2ab7024187.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/a7275e2ab7024187.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@19", "density": 0.793469906795261, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/19.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.22823+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.22823, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@19", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.22823+", "metric_symbol": "s", "metric_value": 2.22823, "reference_text": "s = 2.22823+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/19.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.22823+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "330906f1db929cdb", "improvement": 0.0012889040224068893, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0543459428412958e-06, "max_pair_overlap_depth": -1.4850274468458657e-05, "metric_expression": "2.22694", "metric_symbol": "s", "metric_value": 2.226941095977593, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.22823, "setup": "squinoct", "side": 2.226941095977593, "solution_path": "data/solutions/generated/squinoct_019_330906f1db929cdb.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/330906f1db929cdb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@20", "density": 0.7912106699597153, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/20.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.28730+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.2873, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@20", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.28730+", "metric_symbol": "s", "metric_value": 2.2873, "reference_text": "s = 2.28730+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "20.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/20.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.28730+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "d1597d2fdf3beee1", "improvement": -0.0007530500807275686, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.367182536359887e-05, "max_pair_overlap_depth": -8.661317408398261e-06, "metric_expression": "2.28805", "metric_symbol": "s", "metric_value": 2.2880530500807277, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.2873, "setup": "squinoct", "side": 2.2880530500807277, "solution_path": "data/solutions/generated/squinoct_020_d1597d2fdf3beee1.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/d1597d2fdf3beee1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@21", "density": 0.8091484006744551, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/21.gif", "friedman_metric_expression": "2.32167+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.32167, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@21", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.32167+", "metric_symbol": "s", "metric_value": 2.32167, "reference_text": "s = 2.32167+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/21.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.32167+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "a3759fa0325bb184", "improvement": 0.0032467851108277657, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.288437924896875e-06, "max_pair_overlap_depth": -7.589979277911052e-06, "metric_expression": "2.31842", "metric_symbol": "s", "metric_value": 2.3184232148891724, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.32167, "setup": "squinoct", "side": 2.3184232148891724, "solution_path": "data/solutions/generated/squinoct_021_a3759fa0325bb184.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/a3759fa0325bb184.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@22", "density": 0.779088370401765, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/22.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.41811+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.41811, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@22", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.41811+", "metric_symbol": "s", "metric_value": 2.41811, "reference_text": "s = 2.41811+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/22.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.41811+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "c01cc63fb6a654c8", "improvement": -0.00021763715445111131, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4586625716053672e-06, "max_pair_overlap_depth": -2.804598842454098e-05, "metric_expression": "2.41833", "metric_symbol": "s", "metric_value": 2.418327637154451, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.41811, "setup": "squinoct", "side": 2.418327637154451, "solution_path": "data/solutions/generated/squinoct_022_c01cc63fb6a654c8.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/c01cc63fb6a654c8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@23", "density": 0.7777209980912639, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/23.gif", "friedman_metric_expression": "2.47329+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.47329, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@23", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.47329+", "metric_symbol": "s", "metric_value": 2.47329, "reference_text": "s = 2.47329+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/23.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.47329+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "004d1c5d4d7bca47", "improvement": -0.0015616154620987643, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.985478608046375e-06, "max_pair_overlap_depth": -1.4996046389881279e-05, "metric_expression": "2.47485", "metric_symbol": "s", "metric_value": 2.4748516154620988, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.47329, "setup": "squinoct", "side": 2.4748516154620988, "solution_path": "data/solutions/generated/squinoct_023_004d1c5d4d7bca47.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/004d1c5d4d7bca47.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@24", "density": 0.804709078453695, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/24.gif", "friedman_metric_expression": "6\u221a2 \u2013 6 = 2.48528+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.48528, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@24", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6\u221a2 \u2013 6 = 2.48528+", "metric_symbol": "s", "metric_value": 2.48528, "reference_text": "s = 6\u221a2 \u2013 6 = 2.48528+ | Trivial.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/24.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 6\u221a2 \u2013 6 = 2.48528+ | Trivial.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "4f255616aca5e547", "improvement": -4.580955124477981e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.67146332394231e-06, "max_pair_overlap_depth": -9.95881463161652e-06, "metric_expression": "2.48533", "metric_symbol": "s", "metric_value": 2.4853258095512447, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.48528, "setup": "squinoct", "side": 2.4853258095512447, "solution_path": "data/solutions/generated/squinoct_024_4f255616aca5e547.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4f255616aca5e547.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@25", "density": 0.8352526418920933, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/25.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "11/(3+\u221a2) = 2.49195+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.49195, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@25", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "11/(3+\u221a2) = 2.49195+", "metric_symbol": "s", "metric_value": 2.49195, "reference_text": "s = 11/(3+\u221a2) = 2.49195+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/25.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 11/(3+\u221a2) = 2.49195+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "0d3ca23da8f19da7", "improvement": 0.0021857033362460143, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.3164106628379786e-06, "max_pair_overlap_depth": -1.12459370678053e-05, "metric_expression": "2.48976", "metric_symbol": "s", "metric_value": 2.489764296663754, "n": 25, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.49195, "setup": "squinoct", "side": 2.489764296663754, "solution_path": "data/solutions/generated/squinoct_025_0d3ca23da8f19da7.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/0d3ca23da8f19da7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@26", "density": 0.8421766228224283, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/26.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.52841+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.52841, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@26", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.52841+", "metric_symbol": "s", "metric_value": 2.52841, "reference_text": "s = 2.52841+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/26.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.52841+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "6c0cffa2ebc39455", "improvement": -0.00020227836257324938, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3125873649857306e-07, "max_pair_overlap_depth": -6.684300461579085e-06, "metric_expression": "2.52861", "metric_symbol": "s", "metric_value": 2.5286122783625733, "n": 26, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.52841, "setup": "squinoct", "side": 2.5286122783625733, "solution_path": "data/solutions/generated/squinoct_026_6c0cffa2ebc39455.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/6c0cffa2ebc39455.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@27", "density": 0.8354347334144809, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/27.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.58695+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.58695, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@27", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.58695+", "metric_symbol": "s", "metric_value": 2.58695, "reference_text": "s = 2.58695+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/27.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.58695+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "9243c0031663b10e", "improvement": -0.0002069616913620287, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1228507689492215e-06, "max_pair_overlap_depth": -1.654834709174935e-05, "metric_expression": "2.58716", "metric_symbol": "s", "metric_value": 2.587156961691362, "n": 27, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.58695, "setup": "squinoct", "side": 2.587156961691362, "solution_path": "data/solutions/generated/squinoct_027_9243c0031663b10e.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/9243c0031663b10e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@28", "density": 0.8375235613860185, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/28.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.63116+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.63116, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@28", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.63116+", "metric_symbol": "s", "metric_value": 2.63116, "reference_text": "s = 2.63116+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/28.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.63116+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "f74e0430c683aaa0", "improvement": -0.0001841851467743183, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.686366302485504e-06, "max_pair_overlap_depth": -8.053108887406957e-06, "metric_expression": "2.63134", "metric_symbol": "s", "metric_value": 2.6313441851467743, "n": 28, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.63116, "setup": "squinoct", "side": 2.6313441851467743, "solution_path": "data/solutions/generated/squinoct_028_f74e0430c683aaa0.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/f74e0430c683aaa0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@29", "density": 0.8192713469622411, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/29.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.70811+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.70811, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@29", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.70811+", "metric_symbol": "s", "metric_value": 2.70811, "reference_text": "s = 2.70811+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/29.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.70811+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "7d4fefbdad04f318", "improvement": 0.0005239033117416803, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.8746413323634285e-06, "max_pair_overlap_depth": -4.299633667326219e-06, "metric_expression": "2.70759", "metric_symbol": "s", "metric_value": 2.7075860966882583, "n": 29, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.70811, "setup": "squinoct", "side": 2.7075860966882583, "solution_path": "data/solutions/generated/squinoct_029_7d4fefbdad04f318.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/7d4fefbdad04f318.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinoct@30", "density": 0.8256466842229233, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/30.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.74220+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.7422, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@30", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.74220+", "metric_symbol": "s", "metric_value": 2.7422, "reference_text": "s = 2.74220+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/30.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.74220+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "e6809975fe4af696", "improvement": -0.0010201760226595802, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.275653162265926e-06, "max_pair_overlap_depth": -1.2928136431611392e-05, "metric_expression": "2.74322", "metric_symbol": "s", "metric_value": 2.7432201760226596, "n": 30, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.7422, "setup": "squinoct", "side": 2.7432201760226596, "solution_path": "data/solutions/generated/squinoct_030_e6809975fe4af696.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/e6809975fe4af696.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@1", "density": 0.6621920830904189, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/1.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": ".936+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.936, "friedman_reference": {"analytical_or_proved": true, "case": "squinpen@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": ".936+", "metric_symbol": "s", "metric_value": 0.936, "reference_text": "s = cos(27\u00b0) / cos(18\u00b0) | s = .936+ | Trivial. |", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/1.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = cos(27\u00b0) / cos(18\u00b0) | s = .936+ | Trivial. |", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "7c4deaf46828e45f", "improvement": -0.0008790374068264883, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1116494686738636e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.93688", "metric_symbol": "s", "metric_value": 0.9368790374068265, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.936, "setup": "squinpen", "side": 0.9368790374068265, "solution_path": "data/solutions/generated/squinpen_001_7c4deaf46828e45f.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7c4deaf46828e45f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@2", "density": 0.49287502021209156, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "March 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/2.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "1.535+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.535, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@2", "credit": "Erich Friedman", "date": "March 2012", "family_updated": "", "metric_expression": "1.535+", "metric_symbol": "s", "metric_value": 1.535, "reference_text": "s = 4\u221a(1\u20132/\u221a5)+\u221a5\u20132 | s = 1.535+ | Found by Erich Friedman | in March 2012.", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/2.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4\u221a(1\u20132/\u221a5)+\u221a5\u20132 | s = 1.535+ | Found by Erich Friedman | in March 2012.", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "e0d70422bfa035f2", "improvement": -0.0007556004426736429, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.603512178169518e-07, "max_pair_overlap_depth": -4.1992473865937185e-06, "metric_expression": "1.53576", "metric_symbol": "s", "metric_value": 1.5357556004426736, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.535, "setup": "squinpen", "side": 1.5357556004426736, "solution_path": "data/solutions/generated/squinpen_002_e0d70422bfa035f2.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/e0d70422bfa035f2.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@3", "density": 0.6064545372240481, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/3.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "1.695+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.695, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@3", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "1.695+", "metric_symbol": "s", "metric_value": 1.695, "reference_text": "s = 1.695+ | Found by David W. Cantrell | in July 2012. |", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/3.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 1.695+ | Found by David W. Cantrell | in July 2012. |", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "20345bd1acea07ee", "improvement": -0.0006531320538840646, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.096669049020619e-07, "max_pair_overlap_depth": -2.3507012493845236e-05, "metric_expression": "1.69565", "metric_symbol": "s", "metric_value": 1.6956531320538841, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.695, "setup": "squinpen", "side": 1.6956531320538841, "solution_path": "data/solutions/generated/squinpen_003_20345bd1acea07ee.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/20345bd1acea07ee.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@4", "density": 0.6707731008924028, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/4.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "1.861+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.861, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@4", "credit": "Maurizio Morandi", "date": "July 2012", "family_updated": "", "metric_expression": "1.861+", "metric_symbol": "s", "metric_value": 1.861, "reference_text": "s = 1.861+ | Found by Maurizio Morandi | in July 2012. |", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/4.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 1.861+ | Found by Maurizio Morandi | in July 2012. |", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "3b3728a72d1e463a", "improvement": -0.0007342580262259535, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8617284829398191e-06, "max_pair_overlap_depth": -1.61527471932571e-05, "metric_expression": "1.86173", "metric_symbol": "s", "metric_value": 1.861734258026226, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.861, "setup": "squinpen", "side": 1.861734258026226, "solution_path": "data/solutions/generated/squinpen_004_3b3728a72d1e463a.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/3b3728a72d1e463a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@5", "density": 0.7633756933560932, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "March 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/5.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "1.951+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.951, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@5", "credit": "Erich Friedman", "date": "March 2012", "family_updated": "", "metric_expression": "1.951+", "metric_symbol": "s", "metric_value": 1.951, "reference_text": "s = 1 + cos(18\u00b0) | s = 1.951+ | Found by Erich Friedman | in March 2012.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/5.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 1 + cos(18\u00b0) | s = 1.951+ | Found by Erich Friedman | in March 2012.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "dfdcf10f7cfe5f16", "improvement": -0.00015311806809270578, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.72748390891526e-06, "max_pair_overlap_depth": -2.251588894827683e-05, "metric_expression": "1.95115", "metric_symbol": "s", "metric_value": 1.9511531180680928, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.951, "setup": "squinpen", "side": 1.9511531180680928, "solution_path": "data/solutions/generated/squinpen_005_dfdcf10f7cfe5f16.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/dfdcf10f7cfe5f16.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@6", "density": 0.7300382566069571, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "March 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/6.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "2.185+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.185, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@6", "credit": "Erich Friedman", "date": "March 2012", "family_updated": "", "metric_expression": "2.185+", "metric_symbol": "s", "metric_value": 2.185, "reference_text": "s = 6\u221a(1\u20132/\u221a5)+\u221a5\u20132 | s = 2.185+ | Found by Erich Friedman | in March 2012.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/6.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 6\u221a(1\u20132/\u221a5)+\u221a5\u20132 | s = 2.185+ | Found by Erich Friedman | in March 2012.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "fbc8fa063fa8154f", "improvement": -0.0006384860733987807, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.046305607470139e-06, "max_pair_overlap_depth": -1.535481943848449e-05, "metric_expression": "2.18564", "metric_symbol": "s", "metric_value": 2.185638486073399, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.185, "setup": "squinpen", "side": 2.185638486073399, "solution_path": "data/solutions/generated/squinpen_006_fbc8fa063fa8154f.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/fbc8fa063fa8154f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@7", "density": 0.6937337333431398, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/7.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "2.421+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.421, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@7", "credit": "Erich Friedman", "date": "July 2012", "family_updated": "", "metric_expression": "2.421+", "metric_symbol": "s", "metric_value": 2.421, "reference_text": "s = 6\u221a(1\u20132/\u221a5)+2\u221a5\u20134 | s = 2.421+ | Found by Erich Friedman | in July 2012.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/7.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 6\u221a(1\u20132/\u221a5)+2\u221a5\u20134 | s = 2.421+ | Found by Erich Friedman | in July 2012.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "a9079c85f43f1608", "improvement": -0.0007434143519042813, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.2537276094194668e-06, "max_pair_overlap_depth": -4.1098373515402464e-05, "metric_expression": "2.42174", "metric_symbol": "s", "metric_value": 2.421743414351904, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.421, "setup": "squinpen", "side": 2.421743414351904, "solution_path": "data/solutions/generated/squinpen_007_a9079c85f43f1608.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a9079c85f43f1608.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@8", "density": 0.7064703560745587, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/8.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "2.565+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.565, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@8", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "2.565+", "metric_symbol": "s", "metric_value": 2.565, "reference_text": "s = 2.565+ | Found by David W. Cantrell | in July 2012. |", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/8.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 2.565+ | Found by David W. Cantrell | in July 2012. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "eedfd38786ce3d06", "improvement": -0.0005089596009386632, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2839977085297321e-06, "max_pair_overlap_depth": -2.052513031336467e-05, "metric_expression": "2.56551", "metric_symbol": "s", "metric_value": 2.5655089596009386, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.565, "setup": "squinpen", "side": 2.5655089596009386, "solution_path": "data/solutions/generated/squinpen_008_eedfd38786ce3d06.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/eedfd38786ce3d06.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@9", "density": 0.7204851744660768, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/9.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "2.693+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.693, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@9", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "2.693+", "metric_symbol": "s", "metric_value": 2.693, "reference_text": "s = 2.693+ | Found by Timo Berthold et al | in January 2026. |", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/9.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 2.693+ | Found by Timo Berthold et al | in January 2026. |", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "b050939c544c8091", "improvement": -0.0015375753912136858, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.502705162025631e-06, "max_pair_overlap_depth": -2.4857642315079254e-05, "metric_expression": "2.69454", "metric_symbol": "s", "metric_value": 2.6945375753912137, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.693, "setup": "squinpen", "side": 2.6945375753912137, "solution_path": "data/solutions/generated/squinpen_009_b050939c544c8091.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/b050939c544c8091.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@10", "density": 0.7436124164686567, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/10.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "2.793+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.793, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@10", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "2.793+", "metric_symbol": "s", "metric_value": 2.793, "reference_text": "s = 2.793+ | Found by Timo Berthold et al | in January 2026.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/10.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 2.793+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "5e3ff0fc92953694", "improvement": -0.0027748736770187854, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.695299675143831e-07, "max_pair_overlap_depth": -2.06179882833446e-05, "metric_expression": "2.79577", "metric_symbol": "s", "metric_value": 2.795774873677019, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.793, "setup": "squinpen", "side": 2.795774873677019, "solution_path": "data/solutions/generated/squinpen_010_5e3ff0fc92953694.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/5e3ff0fc92953694.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@11", "density": 0.7644165493663143, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/11.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "2.890+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.89, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@11", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "2.890+", "metric_symbol": "s", "metric_value": 2.89, "reference_text": "s = 2.890+ | Found by Timo Berthold et al | in January 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/11.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 2.890+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "eed6c273a2c34445", "improvement": -0.002056785153238039, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.157725543754771e-07, "max_pair_overlap_depth": -2.6396452143861993e-05, "metric_expression": "2.89206", "metric_symbol": "s", "metric_value": 2.892056785153238, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.89, "setup": "squinpen", "side": 2.892056785153238, "solution_path": "data/solutions/generated/squinpen_011_eed6c273a2c34445.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/eed6c273a2c34445.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@12", "density": 0.7899142461050866, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/12.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "2.969+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.969, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@12", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "2.969+", "metric_symbol": "s", "metric_value": 2.969, "reference_text": "s = 2.969+ | Found by Timo Berthold et al | in January 2026.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/12.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 2.969+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "0dc0d2750d6b1986", "improvement": -0.0025029358297610216, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6525891073399466e-06, "max_pair_overlap_depth": -2.2708088333933674e-05, "metric_expression": "2.97150", "metric_symbol": "s", "metric_value": 2.971502935829761, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.969, "setup": "squinpen", "side": 2.971502935829761, "solution_path": "data/solutions/generated/squinpen_012_0dc0d2750d6b1986.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/0dc0d2750d6b1986.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@13", "density": 0.7748259330672266, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/13.gif", "friedman_metric_expression": "3.121+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.121, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@13", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "3.121+", "metric_symbol": "s", "metric_value": 3.121, "reference_text": "s = 2\u221a(1\u20132/\u221a5)+2\u221a5\u20132 | s = 3.121+ | Found by David W. Cantrell | in July 2012.", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/13.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 2\u221a(1\u20132/\u221a5)+2\u221a5\u20132 | s = 3.121+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "13449338c659bacc", "improvement": -0.0018067975436526496, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.882913072523962e-06, "max_pair_overlap_depth": -3.6524180263897144e-05, "metric_expression": "3.12281", "metric_symbol": "s", "metric_value": 3.1228067975436526, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.121, "setup": "squinpen", "side": 3.1228067975436526, "solution_path": "data/solutions/generated/squinpen_13_13449338c659bacc.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/13449338c659bacc.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@14", "density": 0.8005863693145643, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/14.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.187+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.187, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@14", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "3.187+", "metric_symbol": "s", "metric_value": 3.187, "reference_text": "s = 3.187+ | Found by David W. Cantrell | in July 2012. |", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/14.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.187+ | Found by David W. Cantrell | in July 2012. |", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "c910e2e69c5e952c", "improvement": -0.0011256975399334124, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.345127969056754e-07, "max_pair_overlap_depth": -6.672013015962541e-06, "metric_expression": "3.18813", "metric_symbol": "s", "metric_value": 3.1881256975399332, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.187, "setup": "squinpen", "side": 3.1881256975399332, "solution_path": "data/solutions/generated/squinpen_014_c910e2e69c5e952c.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/c910e2e69c5e952c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@15", "density": 0.8079241717093747, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/15.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.284+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.284, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@15", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "3.284+", "metric_symbol": "s", "metric_value": 3.284, "reference_text": "s = (\u221a(125\u201338\u221a5)+8\u20132\u221a5)/3 | s = 3.284+ | Found by David W. Cantrell | in July 2012.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/15.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = (\u221a(125\u201338\u221a5)+8\u20132\u221a5)/3 | s = 3.284+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "d5cf3c3cc0980b70", "improvement": -0.001003544834322856, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.349360852435288e-06, "max_pair_overlap_depth": -2.4505079251779804e-05, "metric_expression": "3.28500", "metric_symbol": "s", "metric_value": 3.2850035448343227, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.284, "setup": "squinpen", "side": 3.2850035448343227, "solution_path": "data/solutions/generated/squinpen_015_d5cf3c3cc0980b70.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d5cf3c3cc0980b70.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@16", "density": 0.7674551673762713, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/16.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.480+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.48, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@16", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3.480+", "metric_symbol": "s", "metric_value": 3.48, "reference_text": "s = 3.480+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/16.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.480+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "9ca0c9eddf5257a6", "improvement": -0.0010398730905230025, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.610061847707955e-09, "max_pair_overlap_depth": -2.0020049716018562e-05, "metric_expression": "3.48104", "metric_symbol": "s", "metric_value": 3.481039873090523, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.48, "setup": "squinpen", "side": 3.481039873090523, "solution_path": "data/solutions/generated/squinpen_016_9ca0c9eddf5257a6.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/9ca0c9eddf5257a6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@17", "density": 0.7871473019367167, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/17.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.541+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.541, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@17", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3.541+", "metric_symbol": "s", "metric_value": 3.541, "reference_text": "s = 3.541+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/17.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.541+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "c1324401d6fef4ff", "improvement": -0.0020066620107224153, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3581595293550302e-06, "max_pair_overlap_depth": -1.9942519651539037e-05, "metric_expression": "3.54301", "metric_symbol": "s", "metric_value": 3.5430066620107223, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.541, "setup": "squinpen", "side": 3.5430066620107223, "solution_path": "data/solutions/generated/squinpen_017_c1324401d6fef4ff.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/c1324401d6fef4ff.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@18", "density": 0.7949963635829483, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/18.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.626+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.626, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@18", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3.626+", "metric_symbol": "s", "metric_value": 3.626, "reference_text": "s = 3.626+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/18.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.626+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "521e6ed1020d53e6", "improvement": -0.001681876520355896, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.132718950610418e-06, "max_pair_overlap_depth": -5.952134619235938e-06, "metric_expression": "3.62768", "metric_symbol": "s", "metric_value": 3.627681876520356, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.626, "setup": "squinpen", "side": 3.627681876520356, "solution_path": "data/solutions/generated/squinpen_018_521e6ed1020d53e6.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/521e6ed1020d53e6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@19", "density": 0.8016091392666894, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/19.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.710+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.71, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@19", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3.710+", "metric_symbol": "s", "metric_value": 3.71, "reference_text": "s = 3.710+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/19.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.710+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "66aeca1af6b9bf65", "improvement": -0.001683910599032501, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.240508232322902e-06, "max_pair_overlap_depth": -2.033383687716217e-05, "metric_expression": "3.71168", "metric_symbol": "s", "metric_value": 3.7116839105990325, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.71, "setup": "squinpen", "side": 3.7116839105990325, "solution_path": "data/solutions/generated/squinpen_019_66aeca1af6b9bf65.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/66aeca1af6b9bf65.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@20", "density": 0.8030009751844291, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/20.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.802+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.802, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@20", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3.802+", "metric_symbol": "s", "metric_value": 3.802, "reference_text": "s = 3.802+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "20.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/20.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.802+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "8424f972564f51a8", "improvement": -0.00280562795003636, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.488842199203112e-06, "max_pair_overlap_depth": -3.2279665458823104e-05, "metric_expression": "3.80481", "metric_symbol": "s", "metric_value": 3.8048056279500364, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.802, "setup": "squinpen", "side": 3.8048056279500364, "solution_path": "data/solutions/generated/squinpen_020_8424f972564f51a8.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/8424f972564f51a8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@21", "density": 0.8168349282229689, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/21.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.864+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.864, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@21", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "3.864+", "metric_symbol": "s", "metric_value": 3.864, "reference_text": "s = 3.864+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/21.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.864+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "7f1bf6bbe100ca6c", "improvement": -0.0016097781764203845, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0975793470973527e-06, "max_pair_overlap_depth": -2.933142426009283e-05, "metric_expression": "3.86561", "metric_symbol": "s", "metric_value": 3.8656097781764203, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.864, "setup": "squinpen", "side": 3.8656097781764203, "solution_path": "data/solutions/generated/squinpen_021_7f1bf6bbe100ca6c.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/7f1bf6bbe100ca6c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@22", "density": 0.8236901899623935, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/22.gif", "friedman_metric_expression": "3.937+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.937, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@22", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "3.937+", "metric_symbol": "s", "metric_value": 3.937, "reference_text": "s = 3.937+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/22.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.937+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "33b95c59890c6cd7", "improvement": -0.0030787983666398233, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.1912141987053604e-06, "max_pair_overlap_depth": -2.2837560550437175e-06, "metric_expression": "3.94008", "metric_symbol": "s", "metric_value": 3.9400787983666397, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.937, "setup": "squinpen", "side": 3.9400787983666397, "solution_path": "data/solutions/generated/squinpen_022_33b95c59890c6cd7.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/33b95c59890c6cd7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@23", "density": 0.8342850114016098, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/23.gif", "friedman_metric_expression": "4.002+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.002, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@23", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.002+", "metric_symbol": "s", "metric_value": 4.002, "reference_text": "s = 4.002+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/23.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.002+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "125d15e1f42d4cc7", "improvement": -0.000968856434470311, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.142206802972531e-06, "max_pair_overlap_depth": -1.3381668518996914e-05, "metric_expression": "4.00297", "metric_symbol": "s", "metric_value": 4.00296885643447, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.002, "setup": "squinpen", "side": 4.00296885643447, "solution_path": "data/solutions/generated/squinpen_023_125d15e1f42d4cc7.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/125d15e1f42d4cc7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@24", "density": 0.8254992867774645, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/24.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.107+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.107, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@24", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.107+", "metric_symbol": "s", "metric_value": 4.107, "reference_text": "s = 4.107+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/24.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.107+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "7b6bed1452ec8f75", "improvement": -0.003766262034448964, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.187233974203672e-07, "max_pair_overlap_depth": -1.4055675707780324e-06, "metric_expression": "4.11077", "metric_symbol": "s", "metric_value": 4.110766262034449, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.107, "setup": "squinpen", "side": 4.110766262034449, "solution_path": "data/solutions/generated/squinpen_024_7b6bed1452ec8f75.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/7b6bed1452ec8f75.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@25", "density": 0.8346264431391957, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/25.gif", "friedman_metric_expression": "4.169+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.169, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@25", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.169+", "metric_symbol": "s", "metric_value": 4.169, "reference_text": "s = 4.169+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/25.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.169+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "965f543dd3a1a5b2", "improvement": -0.003529809873594303, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.8760801669609005e-06, "max_pair_overlap_depth": -9.22783607931743e-06, "metric_expression": "4.17253", "metric_symbol": "s", "metric_value": 4.172529809873594, "n": 25, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.169, "setup": "squinpen", "side": 4.172529809873594, "solution_path": "data/solutions/generated/squinpen_025_965f543dd3a1a5b2.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/965f543dd3a1a5b2.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@26", "density": 0.846720772454187, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/26.gif", "friedman_metric_expression": "4.223+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.223, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@26", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.223+", "metric_symbol": "s", "metric_value": 4.223, "reference_text": "s = 4.223+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/26.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.223+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "12ea954c92ca25d9", "improvement": -0.0016630925283189768, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.329734777233707e-07, "max_pair_overlap_depth": -5.080948892621784e-06, "metric_expression": "4.22466", "metric_symbol": "s", "metric_value": 4.224663092528319, "n": 26, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.223, "setup": "squinpen", "side": 4.224663092528319, "solution_path": "data/solutions/generated/squinpen_026_12ea954c92ca25d9.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/12ea954c92ca25d9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@27", "density": 0.8275684076653715, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/27.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.350+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.35, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@27", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.350+", "metric_symbol": "s", "metric_value": 4.35, "reference_text": "s = 4.350+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/27.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.350+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "fc7fa66fe1928440", "improvement": -0.0046719542874926745, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1174449550978842e-06, "max_pair_overlap_depth": -1.176344423625686e-05, "metric_expression": "4.35467", "metric_symbol": "s", "metric_value": 4.354671954287492, "n": 27, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.35, "setup": "squinpen", "side": 4.354671954287492, "solution_path": "data/solutions/generated/squinpen_027_fc7fa66fe1928440.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/fc7fa66fe1928440.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@28", "density": 0.8327551046015365, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/28.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.421+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.421, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@28", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.421+", "metric_symbol": "s", "metric_value": 4.421, "reference_text": "s = 4.421+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/28.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.421+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "81b95a9d0915b9b0", "improvement": 0.0002507922234444848, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3068707244667255e-06, "max_pair_overlap_depth": -1.2232567722092469e-05, "metric_expression": "4.42075", "metric_symbol": "s", "metric_value": 4.420749207776556, "n": 28, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.421, "setup": "squinpen", "side": 4.420749207776556, "solution_path": "data/solutions/generated/squinpen_028_81b95a9d0915b9b0.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/81b95a9d0915b9b0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@29", "density": 0.8326213695314967, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/29.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.499+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.499, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@29", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.499+", "metric_symbol": "s", "metric_value": 4.499, "reference_text": "s = 4.499+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/29.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.499+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "7cc5c6e51420a8c4", "improvement": -0.0003599298979271026, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.900416840645271e-06, "max_pair_overlap_depth": -1.3857736228028994e-05, "metric_expression": "4.49936", "metric_symbol": "s", "metric_value": 4.499359929897927, "n": 29, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.499, "setup": "squinpen", "side": 4.499359929897927, "solution_path": "data/solutions/generated/squinpen_029_7cc5c6e51420a8c4.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/7cc5c6e51420a8c4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@30", "density": 0.8380230132738162, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/30.gif", "friedman_metric_expression": "4.550+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.55, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@30", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.550+", "metric_symbol": "s", "metric_value": 4.55, "reference_text": "s = 4.550+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/30.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.550+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "45dff6d0a270c6c5", "improvement": -0.011505140806147729, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.226407328203014e-06, "max_pair_overlap_depth": -5.655700288098586e-07, "metric_expression": "4.56151", "metric_symbol": "s", "metric_value": 4.5615051408061476, "n": 30, "notes": "Best available verified coordinate reconstruction; the Friedman reference remains the public metric until recovery improves.", "record_type": "verified", "reference_side": 4.55, "setup": "squinpen", "side": 4.5615051408061476, "solution_path": "data/solutions/generated/squinpen_30_45dff6d0a270c6c5.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T14:34:18+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/45dff6d0a270c6c5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@31", "density": 0.8407016075258085, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/31.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.63041+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.63041, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.63041+", "metric_symbol": "s", "metric_value": 4.63041, "reference_text": "s = 4.63041+ | Found by Emerson Connelly | in April 2026.", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/31.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.63041+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "8b17df7ad9586809", "improvement": 0.0008957860403464579, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9662880539345523e-06, "max_pair_overlap_depth": -1.825266620247845e-05, "metric_expression": "4.62951", "metric_symbol": "s", "metric_value": 4.629514213959654, "n": 31, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.63041, "setup": "squinpen", "side": 4.629514213959654, "solution_path": "data/solutions/generated/squinpen_031_8b17df7ad9586809.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/8b17df7ad9586809.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@32", "density": 0.839043760254319, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/32.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.70774+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.70774, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@32", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.70774+", "metric_symbol": "s", "metric_value": 4.70774, "reference_text": "s = 4.70774+ | Found by Emerson Connelly | in April 2026.", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/32.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.70774+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "ded24abc59c82646", "improvement": -0.000495710626569057, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.910425347385825e-07, "max_pair_overlap_depth": -1.0586072088480947e-05, "metric_expression": "4.70824", "metric_symbol": "s", "metric_value": 4.708235710626569, "n": 32, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.70774, "setup": "squinpen", "side": 4.708235710626569, "solution_path": "data/solutions/generated/squinpen_032_ded24abc59c82646.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/ded24abc59c82646.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@33", "density": 0.8454321018371573, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/33.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.75774+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.75774, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@33", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.75774+", "metric_symbol": "s", "metric_value": 4.75774, "reference_text": "s = 4.75774+ | Found by Emerson Connelly | in April 2026.", "source_image": "33.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/33.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.75774+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "a9622b0aeea26f74", "improvement": -0.00539747702354898, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5906514265382727e-06, "max_pair_overlap_depth": -5.445984948337923e-06, "metric_expression": "4.76314", "metric_symbol": "s", "metric_value": 4.763137477023549, "n": 33, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.75774, "setup": "squinpen", "side": 4.763137477023549, "solution_path": "data/solutions/generated/squinpen_033_a9622b0aeea26f74.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T03:19:00+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/a9622b0aeea26f74.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@34", "density": 0.8467800838831615, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/34.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.81817+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.81817, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@34", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.81817+", "metric_symbol": "s", "metric_value": 4.81817, "reference_text": "s = 4.81817+ | Found by Emerson Connelly | in April 2026.", "source_image": "34.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/34.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.81817+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "d7af1c3ed90603fd", "improvement": -0.012747878392191048, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.414710391441419e-07, "max_pair_overlap_depth": -6.11976491438071e-06, "metric_expression": "4.83092", "metric_symbol": "s", "metric_value": 4.830917878392191, "n": 34, "notes": "Best available verified coordinate reconstruction; the Friedman reference remains the public metric until recovery improves.", "record_type": "verified", "reference_side": 4.81817, "setup": "squinpen", "side": 4.830917878392191, "solution_path": "data/solutions/generated/squinpen_034_d7af1c3ed90603fd.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/d7af1c3ed90603fd.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@35", "density": 0.8578692979479161, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/35.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.86994+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.86994, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@35", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.86994+", "metric_symbol": "s", "metric_value": 4.86994, "reference_text": "s = 4.86994+ | Found by Emerson Connelly | in April 2026.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/35.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.86994+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "d0212c88c498f951", "improvement": 0.0002762594618035408, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.74898699366526e-06, "max_pair_overlap_depth": -8.959642868955342e-06, "metric_expression": "4.86966", "metric_symbol": "s", "metric_value": 4.869663740538196, "n": 35, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.86994, "setup": "squinpen", "side": 4.869663740538196, "solution_path": "data/solutions/generated/squinpen_035_d0212c88c498f951.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/d0212c88c498f951.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@36", "density": 0.8582852742974874, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/36.gif", "friedman_metric_expression": "4.93902+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.93902, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@36", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.93902+", "metric_symbol": "s", "metric_value": 4.93902, "reference_text": "s = 4.93902+ | Found by Emerson Connelly | in April 2026.", "source_image": "36.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/36.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.93902+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "e43d96c031faf38c", "improvement": 0.0014765141690533667, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.258589189166597e-07, "max_pair_overlap_depth": -1.4614402910062552e-07, "metric_expression": "4.93754", "metric_symbol": "s", "metric_value": 4.937543485830947, "n": 36, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.93902, "setup": "squinpen", "side": 4.937543485830948, "solution_path": "data/solutions/record_hunt/squinpen_36_e43d96c031faf38c.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/e43d96c031faf38c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@37", "density": 0.8661967376016183, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/37.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.98234+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.98234, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@37", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.98234+", "metric_symbol": "s", "metric_value": 4.98234, "reference_text": "s = 4.98234+ | Found by Emerson Connelly | in April 2026.", "source_image": "37.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/37.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.98234+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "dba281681536eecb", "improvement": -0.00039859816128817727, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.161377353190805e-07, "max_pair_overlap_depth": -4.181534561720213e-06, "metric_expression": "4.98274", "metric_symbol": "s", "metric_value": 4.982738598161288, "n": 37, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.98234, "setup": "squinpen", "side": 4.982738598161288, "solution_path": "data/solutions/generated/squinpen_037_dba281681536eecb.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/dba281681536eecb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@38", "density": 0.8656682257376033, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/38.gif", "friedman_metric_expression": "5.05233+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.05233, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@38", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "5.05233+", "metric_symbol": "s", "metric_value": 5.05233, "reference_text": "s = 5.05233+ | Found by Emerson Connelly | in April 2026.", "source_image": "38.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/38.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 5.05233+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "38.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "00496638a2011285", "improvement": 0.0011647868098014769, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.776103493630765e-07, "max_pair_overlap_depth": -5.692587072836375e-07, "metric_expression": "5.05117", "metric_symbol": "s", "metric_value": 5.051165213190199, "n": 38, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.05233, "setup": "squinpen", "side": 5.051165213190198, "solution_path": "data/solutions/record_hunt/squinpen_38_00496638a2011285.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/00496638a2011285.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@39", "density": 0.8585952949364365, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/39.gif", "friedman_metric_expression": "5.13846+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.13846, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@39", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "5.13846+", "metric_symbol": "s", "metric_value": 5.13846, "reference_text": "s = 5.13846+ | Found by Nathan Roll | in April 2026.", "source_image": "39.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/39.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 5.13846+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "b01313a4744f0eda", "improvement": 0.0002297068768006838, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2716746597467209e-07, "max_pair_overlap_depth": -7.26967224418118e-07, "metric_expression": "5.13823", "metric_symbol": "s", "metric_value": 5.1382302931232, "n": 39, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.13846, "setup": "squinpen", "side": 5.1382302931232005, "solution_path": "data/solutions/record_hunt/squinpen_39_b01313a4744f0eda.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/b01313a4744f0eda.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squinpen@40", "density": 0.8691683896670805, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/40.gif", "friedman_metric_expression": "5.17250+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.1725, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@40", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "5.17250+", "metric_symbol": "s", "metric_value": 5.1725, "reference_text": "s = 5.17250+ | Found by Emerson Connelly | in April 2026.", "source_image": "40.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/40.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 5.17250+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "153dbd88227e7083", "improvement": 0.0005591675539955432, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.137577186347585e-07, "max_pair_overlap_depth": -4.818340240131391e-07, "metric_expression": "5.17194", "metric_symbol": "s", "metric_value": 5.171940832446005, "n": 40, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.1725, "setup": "squinpen", "side": 5.171940832446005, "solution_path": "data/solutions/record_hunt/squinpen_40_153dbd88227e7083.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/153dbd88227e7083.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@1", "density": 0.49741240626661626, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts1.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "1 + 2 / \u221a3 = 2.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1 + 2 / \u221a3 = 2.154+", "metric_symbol": "s", "metric_value": 2.154, "reference_text": "s = 1 + 2 / \u221a3 = 2.154+ | Trivial.", "source_image": "ts1.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts1.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 1 + 2 / \u221a3 = 2.154+ | Trivial.", "friedman_source_image": "ts1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "bc8215883b0d29fa", "improvement": -0.0007226428063047408, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.189638991319711e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "2.15472", "metric_symbol": "s", "metric_value": 2.1547226428063047, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.154, "setup": "squintri", "side": 2.1547226428063047, "solution_path": "data/solutions/generated/squintri_001_bc8215883b0d29fa.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/bc8215883b0d29fa.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@2", "density": 0.4640775964340414, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts2.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2 + 2 / \u221a3 = 3.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 2 / \u221a3 = 3.154+", "metric_symbol": "s", "metric_value": 3.154, "reference_text": "s = 2 + 2 / \u221a3 = 3.154+ | Trivial.", "source_image": "ts2.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts2.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 2 + 2 / \u221a3 = 3.154+ | Trivial.", "friedman_source_image": "ts2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "9fcb7fc91bdb6f8d", "improvement": -0.000782174335970165, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.2952441585539525e-06, "max_pair_overlap_depth": -3.4919698730116744e-05, "metric_expression": "3.15478", "metric_symbol": "s", "metric_value": 3.15478217433597, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.154, "setup": "squintri", "side": 3.15478217433597, "solution_path": "data/solutions/generated/squintri_002_9fcb7fc91bdb6f8d.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9fcb7fc91bdb6f8d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@3", "density": 0.6632023535454387, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts3.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3 / 2 + \u221a3 = 3.232+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.232, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@3", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 / 2 + \u221a3 = 3.232+", "metric_symbol": "s", "metric_value": 3.232, "reference_text": "s = 3 / 2 + \u221a3 = 3.232+ | Found by Erich Friedman | in 1997.", "source_image": "ts3.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts3.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 / 2 + \u221a3 = 3.232+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "6a0fd09e2fb037d5", "improvement": -0.00011853663309757678, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0862631378148764e-06, "max_pair_overlap_depth": -4.0896047962125925e-05, "metric_expression": "3.23212", "metric_symbol": "s", "metric_value": 3.232118536633098, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.232, "setup": "squintri", "side": 3.232118536633098, "solution_path": "data/solutions/generated/squintri_003_6a0fd09e2fb037d5.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/6a0fd09e2fb037d5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@4", "density": 0.5351297741951115, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts4.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3 + 2 / \u221a3 = 4.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@4", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 2 / \u221a3 = 4.154+", "metric_symbol": "s", "metric_value": 4.154, "reference_text": "s = 3 + 2 / \u221a3 = 4.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts4.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts4.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 2 / \u221a3 = 4.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "a14dd3a45e182f53", "improvement": -0.000799971071848482, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3541096500101422e-06, "max_pair_overlap_depth": -2.3979848717270436e-05, "metric_expression": "4.15480", "metric_symbol": "s", "metric_value": 4.154799971071848, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.154, "setup": "squintri", "side": 4.154799971071848, "solution_path": "data/solutions/generated/squintri_004_a14dd3a45e182f53.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a14dd3a45e182f53.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@5", "density": 0.6217402820027407, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts6.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2 + 4 / \u221a3 = 4.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.309, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@5", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "2 + 4 / \u221a3 = 4.309+", "metric_symbol": "s", "metric_value": 4.309, "reference_text": "s = 2 + 4 / \u221a3 = 4.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts6.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts6.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 2 + 4 / \u221a3 = 4.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "b7ec125fb5317da3", "improvement": -0.0005327081561983249, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.818148398440643e-06, "max_pair_overlap_depth": -2.4242157156018074e-05, "metric_expression": "4.30953", "metric_symbol": "s", "metric_value": 4.3095327081561985, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.309, "setup": "squintri", "side": 4.3095327081561985, "solution_path": "data/solutions/generated/squintri_005_b7ec125fb5317da3.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/b7ec125fb5317da3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@6", "density": 0.7460914321932061, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts6.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2 + 4 / \u221a3 = 4.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.309, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@6", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "2 + 4 / \u221a3 = 4.309+", "metric_symbol": "s", "metric_value": 4.309, "reference_text": "s = 2 + 4 / \u221a3 = 4.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts6.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts6.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 2 + 4 / \u221a3 = 4.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "cafd65072dcc4c6f", "improvement": -0.0005237730564999765, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6885485560193324e-06, "max_pair_overlap_depth": -3.2625800104213454e-05, "metric_expression": "4.30952", "metric_symbol": "s", "metric_value": 4.3095237730565, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.309, "setup": "squintri", "side": 4.3095237730565, "solution_path": "data/solutions/generated/squintri_006_cafd65072dcc4c6f.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/cafd65072dcc4c6f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@7", "density": 0.6083690864087284, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts7.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 2 / \u221a3 = 5.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@7", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 2 / \u221a3 = 5.154+", "metric_symbol": "s", "metric_value": 5.154, "reference_text": "s = 4 + 2 / \u221a3 = 5.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts7.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts7.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 2 / \u221a3 = 5.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "b90420d34e1bd6e3", "improvement": -0.0008393265287383045, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.7967605582714725e-06, "max_pair_overlap_depth": -3.132565940933141e-05, "metric_expression": "5.15484", "metric_symbol": "s", "metric_value": 5.154839326528738, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.154, "setup": "squintri", "side": 5.154839326528738, "solution_path": "data/solutions/generated/squintri_007_b90420d34e1bd6e3.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/b90420d34e1bd6e3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@8", "density": 0.6568042680930309, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts8.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5.301+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.301, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@8", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "5.301+", "metric_symbol": "s", "metric_value": 5.301, "reference_text": "s = 5.301+ | Found by David W. Cantrell | in July 2002.", "source_image": "ts8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts8.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 5.301+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "ts8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "9523798d0eb30de3", "improvement": -0.0026724224763796656, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4466353188141736e-06, "max_pair_overlap_depth": -2.591508822799571e-05, "metric_expression": "5.30367", "metric_symbol": "s", "metric_value": 5.30367242247638, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.301, "setup": "squintri", "side": 5.30367242247638, "solution_path": "data/solutions/generated/squintri_008_9523798d0eb30de3.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/9523798d0eb30de3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@9", "density": 0.7370953908179042, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts9.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3 + 4 / \u221a3 = 5.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.309, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@9", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 4 / \u221a3 = 5.309+", "metric_symbol": "s", "metric_value": 5.309, "reference_text": "s = 3 + 4 / \u221a3 = 5.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts9.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts9.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 4 / \u221a3 = 5.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "221ec438e9e78bec", "improvement": -0.0011781204346883456, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.476651564783165e-07, "max_pair_overlap_depth": -2.048243915964143e-05, "metric_expression": "5.31018", "metric_symbol": "s", "metric_value": 5.3101781204346885, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.309, "setup": "squintri", "side": 5.3101781204346885, "solution_path": "data/solutions/generated/squintri_009_221ec438e9e78bec.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/221ec438e9e78bec.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@10", "density": 0.773471405329893, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts10.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2 + 2 \u221a3 = 5.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.464, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@10", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "2 + 2 \u221a3 = 5.464+", "metric_symbol": "s", "metric_value": 5.464, "reference_text": "s = 2 + 2 \u221a3 = 5.464+ | Found by Erich Friedman | in 1997.", "source_image": "ts10.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts10.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 2 + 2 \u221a3 = 5.464+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "4b92d4f5e6704540", "improvement": -0.00021212423215644094, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1415169103720102e-06, "max_pair_overlap_depth": -1.0706995126841257e-05, "metric_expression": "5.46421", "metric_symbol": "s", "metric_value": 5.464212124232157, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.464, "setup": "squintri", "side": 5.464212124232157, "solution_path": "data/solutions/generated/squintri_010_4b92d4f5e6704540.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/4b92d4f5e6704540.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@11", "density": 0.6705894339586221, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts11.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5 + 2 / \u221a3 = 6.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@11", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 + 2 / \u221a3 = 6.154+", "metric_symbol": "s", "metric_value": 6.154, "reference_text": "s = 5 + 2 / \u221a3 = 6.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts11.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts11.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 5 + 2 / \u221a3 = 6.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "c18f0f7b3b1e0227", "improvement": -0.0008528095099347155, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.062945956811758e-06, "max_pair_overlap_depth": -2.0039414804218225e-05, "metric_expression": "6.15485", "metric_symbol": "s", "metric_value": 6.154852809509935, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.154, "setup": "squintri", "side": 6.154852809509935, "solution_path": "data/solutions/generated/squintri_011_c18f0f7b3b1e0227.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/c18f0f7b3b1e0227.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@12", "density": 0.6970880090801083, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts12.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.301+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.301, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@12", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "6.301+", "metric_symbol": "s", "metric_value": 6.301, "reference_text": "s = 6.301+ | Found by David W. Cantrell | in July 2002.", "source_image": "ts12.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts12.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 6.301+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "ts12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "d248ead86c97d2e6", "improvement": -0.004165620418586258, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6132876791233732e-06, "max_pair_overlap_depth": -5.9825433180948906e-05, "metric_expression": "6.30517", "metric_symbol": "s", "metric_value": 6.305165620418586, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.301, "setup": "squintri", "side": 6.305165620418586, "solution_path": "data/solutions/generated/squintri_012_d248ead86c97d2e6.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d248ead86c97d2e6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@13", "density": 0.7541442288903037, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts13.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 4 / \u221a3 = 6.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.309, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@13", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 4 / \u221a3 = 6.309+", "metric_symbol": "s", "metric_value": 6.309, "reference_text": "s = 4 + 4 / \u221a3 = 6.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts13.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts13.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 4 / \u221a3 = 6.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "3b7ab78f832490ed", "improvement": -0.0004884860946248182, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1403459653225352e-06, "max_pair_overlap_depth": -1.2003638542146255e-05, "metric_expression": "6.30949", "metric_symbol": "s", "metric_value": 6.309488486094625, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.309, "setup": "squintri", "side": 6.309488486094625, "solution_path": "data/solutions/generated/squintri_013_3b7ab78f832490ed.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3b7ab78f832490ed.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@14", "density": 0.7737221641470288, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts15.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3 + 2 \u221a3 = 6.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.464, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@14", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 2 \u221a3 = 6.464+", "metric_symbol": "s", "metric_value": 6.464, "reference_text": "s = 3 + 2 \u221a3 = 6.464+ | Found by Erich Friedman | in 1997.", "source_image": "ts15.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts15.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 2 \u221a3 = 6.464+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "ffc152ae79167d33", "improvement": -0.00029520096988377986, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.958378324017974e-07, "max_pair_overlap_depth": -1.6732970481436737e-05, "metric_expression": "6.46430", "metric_symbol": "s", "metric_value": 6.464295200969884, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.464, "setup": "squintri", "side": 6.464295200969884, "solution_path": "data/solutions/generated/squintri_014_ffc152ae79167d33.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/ffc152ae79167d33.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@15", "density": 0.8289675963035161, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts15.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3 + 2 \u221a3 = 6.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.464, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@15", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 2 \u221a3 = 6.464+", "metric_symbol": "s", "metric_value": 6.464, "reference_text": "s = 3 + 2 \u221a3 = 6.464+ | Found by Erich Friedman | in 1997.", "source_image": "ts15.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts15.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 2 \u221a3 = 6.464+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "37ace9570dd5c15b", "improvement": -0.0003748832881260711, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.039916977622108e-06, "max_pair_overlap_depth": -1.0702944550522453e-05, "metric_expression": "6.46437", "metric_symbol": "s", "metric_value": 6.4643748832881265, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.464, "setup": "squintri", "side": 6.4643748832881265, "solution_path": "data/solutions/generated/squintri_015_37ace9570dd5c15b.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/37ace9570dd5c15b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@16", "density": 0.7217958800018128, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts16.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6 + 2 / \u221a3 = 7.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@16", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "6 + 2 / \u221a3 = 7.154+", "metric_symbol": "s", "metric_value": 7.154, "reference_text": "s = 6 + 2 / \u221a3 = 7.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts16.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts16.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 6 + 2 / \u221a3 = 7.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "36afe756f11f62d6", "improvement": -0.0008819711932614993, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.5571215058068575e-06, "max_pair_overlap_depth": -2.4408532427300234e-05, "metric_expression": "7.15488", "metric_symbol": "s", "metric_value": 7.154881971193261, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.154, "setup": "squintri", "side": 7.154881971193261, "solution_path": "data/solutions/generated/squintri_016_36afe756f11f62d6.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/36afe756f11f62d6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@17", "density": 0.7357616143895036, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts17.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "7.301+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.301, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@17", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "7.301+", "metric_symbol": "s", "metric_value": 7.301, "reference_text": "s = 7.301+ | Found by David W. Cantrell | in July 2002.", "source_image": "ts17.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts17.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 7.301+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "ts17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "3a2c987432da5f1d", "improvement": -0.0037537403553287163, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.4850519352755694e-10, "max_pair_overlap_depth": -2.8059876787889237e-05, "metric_expression": "7.30475", "metric_symbol": "s", "metric_value": 7.304753740355329, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.301, "setup": "squintri", "side": 7.304753740355329, "solution_path": "data/solutions/generated/squintri_017_3a2c987432da5f1d.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/3a2c987432da5f1d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@18", "density": 0.7780278532013658, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts18.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5 + 4 / \u221a3 = 7.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.309, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@18", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 + 4 / \u221a3 = 7.309+", "metric_symbol": "s", "metric_value": 7.309, "reference_text": "s = 5 + 4 / \u221a3 = 7.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts18.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 5 + 4 / \u221a3 = 7.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "e729e13f2fcc5856", "improvement": -0.0005116409635963493, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1021248752761939e-06, "max_pair_overlap_depth": -8.579145527987109e-06, "metric_expression": "7.30951", "metric_symbol": "s", "metric_value": 7.3095116409635965, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.309, "setup": "squintri", "side": 7.3095116409635965, "solution_path": "data/solutions/generated/squintri_018_e729e13f2fcc5856.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/e729e13f2fcc5856.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@19", "density": 0.787537700321716, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts19.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 2 \u221a3 = 7.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.464, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@19", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 2 \u221a3 = 7.464+", "metric_symbol": "s", "metric_value": 7.464, "reference_text": "s = 4 + 2 \u221a3 = 7.464+ | Found by Erich Friedman | in 1997.", "source_image": "ts19.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts19.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 2 \u221a3 = 7.464+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "805d1e50c1db2842", "improvement": -0.0003295610354463463, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0609380559736765e-06, "max_pair_overlap_depth": -1.836020690809681e-05, "metric_expression": "7.46433", "metric_symbol": "s", "metric_value": 7.464329561035447, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.464, "setup": "squintri", "side": 7.464329561035447, "solution_path": "data/solutions/generated/squintri_019_805d1e50c1db2842.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/805d1e50c1db2842.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@20", "density": 0.795657127614163, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts21.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3 + 8 / \u221a3 = 7.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.618, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@20", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 8 / \u221a3 = 7.618+", "metric_symbol": "s", "metric_value": 7.618, "reference_text": "s = 3 + 8 / \u221a3 = 7.618+ | Found by Erich Friedman | in 1997.", "source_image": "ts21.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts21.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 8 / \u221a3 = 7.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "f4d21a73470ce742", "improvement": -0.0010653922379448844, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.382631857088383e-07, "max_pair_overlap_depth": -1.1470843414018717e-05, "metric_expression": "7.61907", "metric_symbol": "s", "metric_value": 7.619065392237945, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.618, "setup": "squintri", "side": 7.619065392237945, "solution_path": "data/solutions/generated/squintri_020_f4d21a73470ce742.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f4d21a73470ce742.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@21", "density": 0.8354483980307021, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts21.gif", "friedman_metric_expression": "3 + 8 / \u221a3 = 7.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.618, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@21", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 8 / \u221a3 = 7.618+", "metric_symbol": "s", "metric_value": 7.618, "reference_text": "s = 3 + 8 / \u221a3 = 7.618+ | Found by Erich Friedman | in 1997.", "source_image": "ts21.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts21.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 8 / \u221a3 = 7.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "a2e24601c2557198", "improvement": -0.0010270252659259071, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8035391721895233e-06, "max_pair_overlap_depth": -9.7266150509423e-06, "metric_expression": "7.61903", "metric_symbol": "s", "metric_value": 7.619027025265926, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.618, "setup": "squintri", "side": 7.619027025265926, "solution_path": "data/solutions/generated/squintri_021_a2e24601c2557198.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a2e24601c2557198.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@22", "density": 0.7639711289147475, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts22.gif", "friedman_metric_expression": "7 + 2 / \u221a3 = 8.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@22", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "7 + 2 / \u221a3 = 8.154+", "metric_symbol": "s", "metric_value": 8.154, "reference_text": "s = 7 + 2 / \u221a3 = 8.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts22.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts22.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 7 + 2 / \u221a3 = 8.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "bbb028380d997350", "improvement": -0.00097340620692421, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4334644744109685e-07, "max_pair_overlap_depth": -2.16343036094635e-05, "metric_expression": "8.15497", "metric_symbol": "s", "metric_value": 8.154973406206924, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.154, "setup": "squintri", "side": 8.154973406206924, "solution_path": "data/solutions/generated/squintri_022_bbb028380d997350.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/bbb028380d997350.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@23", "density": 0.7697946105392801, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts23.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "8.301+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.301, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@23", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "8.301+", "metric_symbol": "s", "metric_value": 8.301, "reference_text": "s = 8.301+ | Found by David W. Cantrell | in July 2002.", "source_image": "ts23.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts23.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 8.301+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "ts23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "b1534ea73e00f283", "improvement": -0.005654877387726742, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7355116411507935e-05, "max_pair_overlap_depth": 9.193993899536679e-09, "metric_expression": "8.30665", "metric_symbol": "s", "metric_value": 8.306654877387727, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.301, "setup": "squintri", "side": 8.306654877387727, "solution_path": "data/solutions/generated/squintri_023_b1534ea73e00f283.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/b1534ea73e00f283.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@24", "density": 0.8026821430235043, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts24.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6 + 4 / \u221a3 = 8.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.309, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@24", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "6 + 4 / \u221a3 = 8.309+", "metric_symbol": "s", "metric_value": 8.309, "reference_text": "s = 6 + 4 / \u221a3 = 8.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts24.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts24.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 6 + 4 / \u221a3 = 8.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "648f7ef8ee4a197f", "improvement": -0.0006647382800331059, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.518509956206752e-06, "max_pair_overlap_depth": -1.764131142900638e-05, "metric_expression": "8.30966", "metric_symbol": "s", "metric_value": 8.309664738280032, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.309, "setup": "squintri", "side": 8.309664738280032, "solution_path": "data/solutions/generated/squintri_024_648f7ef8ee4a197f.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/648f7ef8ee4a197f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@25", "density": 0.8057999996349686, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts25.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5 + 2 \u221a3 = 8.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.464, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@25", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 + 2 \u221a3 = 8.464+", "metric_symbol": "s", "metric_value": 8.464, "reference_text": "s = 5 + 2 \u221a3 = 8.464+ | Found by Erich Friedman | in 1997.", "source_image": "ts25.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts25.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 5 + 2 \u221a3 = 8.464+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "a868d8af97a256ad", "improvement": -0.0005924926961089483, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7884356498379287e-06, "max_pair_overlap_depth": -2.4212693654490636e-06, "metric_expression": "8.46459", "metric_symbol": "s", "metric_value": 8.46459249269611, "n": 25, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.464, "setup": "squintri", "side": 8.46459249269611, "solution_path": "data/solutions/generated/squintri_025_a868d8af97a256ad.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a868d8af97a256ad.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@26", "density": 0.8076243715506172, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts27.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 8 / \u221a3 = 8.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.618, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@26", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 8 / \u221a3 = 8.618+", "metric_symbol": "s", "metric_value": 8.618, "reference_text": "s = 4 + 8 / \u221a3 = 8.618+ | Found by Erich Friedman | in 1997.", "source_image": "ts27.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts27.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 8 / \u221a3 = 8.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "249af47eefa3a74b", "improvement": -0.004469124601865104, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.4769017068652995e-06, "max_pair_overlap_depth": -2.5879119370686432e-05, "metric_expression": "8.62247", "metric_symbol": "s", "metric_value": 8.622469124601865, "n": 26, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.618, "setup": "squintri", "side": 8.622469124601865, "solution_path": "data/solutions/generated/squintri_026_249af47eefa3a74b.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/249af47eefa3a74b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@27", "density": 0.8380226358442442, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts27.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 8 / \u221a3 = 8.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.618, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@27", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 8 / \u221a3 = 8.618+", "metric_symbol": "s", "metric_value": 8.618, "reference_text": "s = 4 + 8 / \u221a3 = 8.618+ | Found by Erich Friedman | in 1997.", "source_image": "ts27.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts27.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 8 / \u221a3 = 8.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "3c62cd2126467cec", "improvement": -0.007885505517393199, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.235025861662866e-06, "max_pair_overlap_depth": -2.0721793894562524e-05, "metric_expression": "8.62589", "metric_symbol": "s", "metric_value": 8.625885505517394, "n": 27, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.618, "setup": "squintri", "side": 8.625885505517394, "solution_path": "data/solutions/generated/squintri_027_3c62cd2126467cec.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3c62cd2126467cec.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@28", "density": 0.8400113660191094, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts28.gif", "friedman_metric_expression": "3 + 10 / \u221a3 = 8.773+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.773, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@28", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 10 / \u221a3 = 8.773+", "metric_symbol": "s", "metric_value": 8.773, "reference_text": "s = 3 + 10 / \u221a3 = 8.773+ | Found by Erich Friedman | in 1997.", "source_image": "ts28.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts28.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 10 / \u221a3 = 8.773+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "0d20b359435ed501", "improvement": -0.0007673944032049462, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0145694937175628e-06, "max_pair_overlap_depth": -1.3941973455278145e-05, "metric_expression": "8.77377", "metric_symbol": "s", "metric_value": 8.773767394403205, "n": 28, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.773, "setup": "squintri", "side": 8.773767394403205, "solution_path": "data/solutions/generated/squintri_028_0d20b359435ed501.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/0d20b359435ed501.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@29", "density": 0.7990846545133872, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts29.gif", "friedman_metric_expression": "8 + 2 / \u221a3 = 9.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@29", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "8 + 2 / \u221a3 = 9.154+", "metric_symbol": "s", "metric_value": 9.154, "reference_text": "s = 8 + 2 / \u221a3 = 9.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts29.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts29.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 8 + 2 / \u221a3 = 9.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "3947aab39c0a1a92", "improvement": -0.000872191450978832, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2628787428294703e-06, "max_pair_overlap_depth": -1.875431036468811e-05, "metric_expression": "9.15487", "metric_symbol": "s", "metric_value": 9.154872191450979, "n": 29, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.154, "setup": "squintri", "side": 9.154872191450979, "solution_path": "data/solutions/generated/squintri_029_3947aab39c0a1a92.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3947aab39c0a1a92.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@30", "density": 0.7992784153759376, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts30.gif", "friedman_metric_expression": "9.301+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.301, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@30", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "9.301+", "metric_symbol": "s", "metric_value": 9.301, "reference_text": "s = 9.301+ | Found by David W. Cantrell | in July 2002.", "source_image": "ts30.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts30.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 9.301+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "ts30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "e0b7505d8b6b1d46", "improvement": -0.009248370808917272, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5690858985090003e-06, "max_pair_overlap_depth": -2.654303096472077e-05, "metric_expression": "9.31025", "metric_symbol": "s", "metric_value": 9.310248370808917, "n": 30, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.301, "setup": "squintri", "side": 9.310248370808917, "solution_path": "data/solutions/generated/squintri_030_e0b7505d8b6b1d46.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/e0b7505d8b6b1d46.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@31", "density": 0.825993871568094, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts31.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "7 + 4 / \u221a3 = 9.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.309, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@31", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "7 + 4 / \u221a3 = 9.309+", "metric_symbol": "s", "metric_value": 9.309, "reference_text": "s = 7 + 4 / \u221a3 = 9.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts31.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts31.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 7 + 4 / \u221a3 = 9.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "22b5f5f5707858bb", "improvement": -0.0008378379274667225, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6472234336534766e-06, "max_pair_overlap_depth": -3.090204228461957e-05, "metric_expression": "9.30984", "metric_symbol": "s", "metric_value": 9.309837837927466, "n": 31, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.309, "setup": "squintri", "side": 9.309837837927466, "solution_path": "data/solutions/generated/squintri_031_22b5f5f5707858bb.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/22b5f5f5707858bb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@32", "density": 0.8249717531277396, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts32.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6 + 2 \u221a3 = 9.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.464, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@32", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "6 + 2 \u221a3 = 9.464+", "metric_symbol": "s", "metric_value": 9.464, "reference_text": "s = 6 + 2 \u221a3 = 9.464+ | Found by Erich Friedman | in 1997.", "source_image": "ts32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts32.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 6 + 2 \u221a3 = 9.464+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "a44fbd99145b1ab3", "improvement": -0.0006624941961259623, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.1832282131171085e-07, "max_pair_overlap_depth": -1.699280632205813e-06, "metric_expression": "9.46466", "metric_symbol": "s", "metric_value": 9.464662494196126, "n": 32, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.464, "setup": "squintri", "side": 9.464662494196126, "solution_path": "data/solutions/generated/squintri_032_a44fbd99145b1ab3.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a44fbd99145b1ab3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@33", "density": 0.8228028312642216, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts34.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5 + 8 / \u221a3 = 9.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.618, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@33", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 + 8 / \u221a3 = 9.618+", "metric_symbol": "s", "metric_value": 9.618, "reference_text": "s = 5 + 8 / \u221a3 = 9.618+ | Found by Erich Friedman | in 1997.", "source_image": "ts34.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts34.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 5 + 8 / \u221a3 = 9.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "67de1e08be2d150c", "improvement": -0.00606977957198751, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.226435414550563e-06, "max_pair_overlap_depth": -3.0944328952564604e-06, "metric_expression": "9.62407", "metric_symbol": "s", "metric_value": 9.624069779571988, "n": 33, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.618, "setup": "squintri", "side": 9.624069779571988, "solution_path": "data/solutions/generated/squintri_033_67de1e08be2d150c.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/67de1e08be2d150c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@34", "density": 0.8485900582279403, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts34.gif", "friedman_metric_expression": "5 + 8 / \u221a3 = 9.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.618, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@34", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 + 8 / \u221a3 = 9.618+", "metric_symbol": "s", "metric_value": 9.618, "reference_text": "s = 5 + 8 / \u221a3 = 9.618+ | Found by Erich Friedman | in 1997.", "source_image": "ts34.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts34.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 5 + 8 / \u221a3 = 9.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "3661302cdd9d0eb0", "improvement": -0.0012269381751668362, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.159334309452278e-06, "max_pair_overlap_depth": -1.741303470170763e-05, "metric_expression": "9.61923", "metric_symbol": "s", "metric_value": 9.619226938175167, "n": 34, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.618, "setup": "squintri", "side": 9.619226938175167, "solution_path": "data/solutions/generated/squintri_034_3661302cdd9d0eb0.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3661302cdd9d0eb0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@35", "density": 0.8451911560111607, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts36.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 10 / \u221a3 = 9.773+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.773, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@35", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 10 / \u221a3 = 9.773+", "metric_symbol": "s", "metric_value": 9.773, "reference_text": "s = 4 + 10 / \u221a3 = 9.773+ | Found by Erich Friedman | in 1997.", "source_image": "ts36.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts36.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 10 / \u221a3 = 9.773+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "39d52a823b632afe", "improvement": -0.006265389488104844, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.195518233524095e-06, "max_pair_overlap_depth": -2.612964804571405e-05, "metric_expression": "9.77927", "metric_symbol": "s", "metric_value": 9.779265389488105, "n": 35, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.773, "setup": "squintri", "side": 9.779265389488105, "solution_path": "data/solutions/generated/squintri_035_39d52a823b632afe.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/39d52a823b632afe.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@36", "density": 0.8698156108347095, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts36.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 10 / \u221a3 = 9.773+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.773, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@36", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 10 / \u221a3 = 9.773+", "metric_symbol": "s", "metric_value": 9.773, "reference_text": "s = 4 + 10 / \u221a3 = 9.773+ | Found by Erich Friedman | in 1997.", "source_image": "ts36.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts36.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 10 / \u221a3 = 9.773+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "f0473aaa767ebe98", "improvement": -0.0035884436987085877, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9270562690497914e-06, "max_pair_overlap_depth": -7.45737725771356e-06, "metric_expression": "9.77659", "metric_symbol": "s", "metric_value": 9.776588443698708, "n": 36, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 9.773, "setup": "squintri", "side": 9.776588443698708, "solution_path": "data/solutions/generated/squintri_036_f0473aaa767ebe98.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f0473aaa767ebe98.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@37", "density": 0.8285699034870297, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts37.gif", "friedman_metric_expression": "10.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.154, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@37", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "10.154+", "metric_symbol": "s", "metric_value": 10.154, "reference_text": "s = 10.154+ | Found by Emerson Connelly | in April 2026.", "source_image": "ts37.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts37.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 10.154+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "ts37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "8530be000ac2e9ed", "improvement": -0.0011412138177444575, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3566887422690367e-06, "max_pair_overlap_depth": -1.2239987045248313e-05, "metric_expression": "10.15514", "metric_symbol": "s", "metric_value": 10.155141213817744, "n": 37, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.154, "setup": "squintri", "side": 10.155141213817744, "solution_path": "data/solutions/generated/squintri_037_8530be000ac2e9ed.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/8530be000ac2e9ed.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@38", "density": 0.8256293316995508, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts38.gif", "friedman_metric_expression": "10.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.309, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@38", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "10.309+", "metric_symbol": "s", "metric_value": 10.309, "reference_text": "s = 10.309+ | Found by Emerson Connelly | in April 2026.", "source_image": "ts38.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts38.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 10.309+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "ts38.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "896b636f9d899bf5", "improvement": -0.0007687369401132571, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.231911901047403e-06, "max_pair_overlap_depth": -1.6045122417374102e-05, "metric_expression": "10.30977", "metric_symbol": "s", "metric_value": 10.309768736940113, "n": 38, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.309, "setup": "squintri", "side": 10.309768736940113, "solution_path": "data/solutions/generated/squintri_038_896b636f9d899bf5.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/896b636f9d899bf5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@39", "density": 0.8473534790778084, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts39.gif", "friedman_metric_expression": "10.310+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.31, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@39", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "10.310+", "metric_symbol": "s", "metric_value": 10.31, "reference_text": "s = 10.310+ | Found by Emerson Connelly | in April 2026.", "source_image": "ts39.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts39.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 10.310+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "ts39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "c0988e1f3bf189d9", "improvement": 0.00021337573934943066, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.79079015949438e-07, "max_pair_overlap_depth": -3.1188368032619707e-06, "metric_expression": "10.30979", "metric_symbol": "s", "metric_value": 10.309786624260651, "n": 39, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.31, "setup": "squintri", "side": 10.30978662426065, "solution_path": "data/solutions/record_hunt/squintri_39_c0988e1f3bf189d9.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/c0988e1f3bf189d9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "squintri@40", "density": 0.8436138407048179, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts40.gif", "friedman_metric_expression": "10.46426+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.46426, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@40", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "10.46426+", "metric_symbol": "s", "metric_value": 10.46426, "reference_text": "s = 10.46426+ | Found by Nathan Roll | in April 2026.", "source_image": "ts40.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts40.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 10.46426+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "ts40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "13b03f25af76ce87", "improvement": 1.6681613434599285e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.839061064144293e-07, "max_pair_overlap_depth": -1.4626365829606414e-06, "metric_expression": "10.46424", "metric_symbol": "s", "metric_value": 10.464243318386565, "n": 40, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.46426, "setup": "squintri", "side": 10.464243318386565, "solution_path": "data/solutions/record_hunt/squintri_40_13b03f25af76ce87.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/13b03f25af76ce87.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@1", "density": 0.41349293224745, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc1.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1/\u221a3 = .577+", "friedman_metric_symbol": "r", "friedman_metric_value": 0.577, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1/\u221a3 = .577+", "metric_symbol": "r", "metric_value": 0.577, "reference_text": "r = 1/\u221a3 = .577+ | Trivial.", "source_image": "tc1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc1.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1/\u221a3 = .577+ | Trivial.", "friedman_source_image": "tc1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "49954bfbb560620d", "improvement": -0.00035287974439102143, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.775275519510515e-07, "max_pair_overlap_depth": 0.0, "metric_expression": "0.57735", "metric_symbol": "r", "metric_value": 0.577352879744391, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.577, "setup": "triincir", "side": 1.154705759488782, "solution_path": "data/solutions/generated/triincir_001_49954bfbb560620d.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/49954bfbb560620d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@2", "density": 0.3675414068402789, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc2.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "\u221a3/2 = .866+", "friedman_metric_symbol": "r", "friedman_metric_value": 0.866, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3/2 = .866+", "metric_symbol": "r", "metric_value": 0.866, "reference_text": "r = \u221a3/2 = .866+ | Trivial.", "source_image": "tc2.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc2.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = \u221a3/2 = .866+ | Trivial.", "friedman_source_image": "tc2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "92f5511b689fe041", "improvement": -3.858712037896339e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.072458078845486e-06, "max_pair_overlap_depth": -1.0890566688626091e-05, "metric_expression": "0.86604", "metric_symbol": "r", "metric_value": 0.866038587120379, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.866, "setup": "triincir", "side": 1.732077174240758, "solution_path": "data/solutions/generated/triincir_002_92f5511b689fe041.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/92f5511b689fe041.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@3", "density": 0.45476420302551357, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc3.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "3\u221a3 - 3\u221a2 = .953+", "friedman_metric_symbol": "r", "friedman_metric_value": 0.953, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@3", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3\u221a3 - 3\u221a2 = .953+", "metric_symbol": "r", "metric_value": 0.953, "reference_text": "r = 3\u221a3 - 3\u221a2 = .953+ | Found by Erich Friedman | in 1997.", "source_image": "tc3.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc3.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 3\u221a3 - 3\u221a2 = .953+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "6ede8225b5939550", "improvement": -0.0005486885675038966, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4231654198736408e-06, "max_pair_overlap_depth": -2.0408752807413166e-05, "metric_expression": "0.95355", "metric_symbol": "r", "metric_value": 0.9535486885675039, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.953, "setup": "triincir", "side": 1.9070973771350077, "solution_path": "data/solutions/generated/triincir_003_6ede8225b5939550.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/6ede8225b5939550.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@4", "density": 0.5732690550709103, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc4.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": ".980+", "friedman_metric_symbol": "r", "friedman_metric_value": 0.98, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@4", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": ".980+", "metric_symbol": "r", "metric_value": 0.98, "reference_text": "r = .980+ | Found by Erich Friedman | in 1997.", "source_image": "tc4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc4.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = .980+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "7f9481f33de5ecda", "improvement": -0.0006773112916812485, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.642177417024662e-06, "max_pair_overlap_depth": -2.677155749965807e-05, "metric_expression": "0.98068", "metric_symbol": "r", "metric_value": 0.9806773112916812, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.98, "setup": "triincir", "side": 1.9613546225833625, "solution_path": "data/solutions/generated/triincir_004_7f9481f33de5ecda.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/7f9481f33de5ecda.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@5", "density": 0.6891272508887964, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc5.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1", "friedman_metric_symbol": "r", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "r", "metric_value": 1.0, "reference_text": "r = 1 | Trivial.", "source_image": "tc5.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc5.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1 | Trivial.", "friedman_source_image": "tc5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "cf17d061ebfe392b", "improvement": -2.4573090296353683e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.800898255106432e-06, "max_pair_overlap_depth": -9.00554050570603e-06, "metric_expression": "1.00002", "metric_symbol": "r", "metric_value": 1.0000245730902964, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0, "setup": "triincir", "side": 2.0000491461805927, "solution_path": "data/solutions/generated/triincir_005_cf17d061ebfe392b.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/cf17d061ebfe392b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@6", "density": 0.826963635452614, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc6.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1", "friedman_metric_symbol": "r", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "r", "metric_value": 1.0, "reference_text": "r = 1 | Trivial.", "source_image": "tc6.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc6.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1 | Trivial.", "friedman_source_image": "tc6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "fe93ba87f23c6c01", "improvement": -1.7961741001881748e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.861140867618019e-08, "max_pair_overlap_depth": -9.037179045745058e-06, "metric_expression": "1.00002", "metric_symbol": "r", "metric_value": 1.0000179617410019, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0, "setup": "triincir", "side": 2.0000359234820038, "solution_path": "data/solutions/generated/triincir_006_fe93ba87f23c6c01.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fe93ba87f23c6c01.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@7", "density": 0.7265087780749782, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc7.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.152+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.152, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@7", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "1.152+", "metric_symbol": "r", "metric_value": 1.152, "reference_text": "r = 1.152+ | Found by Erich Friedman | in 1997.", "source_image": "tc7.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc7.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.152+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "4d32cb6092e127b9", "improvement": -0.0004019079826915739, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.469968968942808e-06, "max_pair_overlap_depth": -4.662019983989185e-06, "metric_expression": "1.15240", "metric_symbol": "r", "metric_value": 1.1524019079826915, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.152, "setup": "triincir", "side": 2.304803815965383, "solution_path": "data/solutions/generated/triincir_007_4d32cb6092e127b9.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/4d32cb6092e127b9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@8", "density": 0.6908236123299982, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc8.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.263+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.263, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@8", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "", "metric_expression": "1.263+", "metric_symbol": "r", "metric_value": 1.263, "reference_text": "r = 1.263+ | Found by Maurizio Morandi | in April 2008.", "source_image": "tc8.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc8.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.263+ | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "tc8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "33389e3c966a2171", "improvement": -0.0003881191911174753, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.08502399998045e-06, "max_pair_overlap_depth": -2.667573266001977e-06, "metric_expression": "1.26339", "metric_symbol": "r", "metric_value": 1.2633881191911174, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.263, "setup": "triincir", "side": 2.5267762383822348, "solution_path": "data/solutions/generated/triincir_008_33389e3c966a2171.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/33389e3c966a2171.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@9", "density": 0.7175995300521633, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc9.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.314+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.314, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@9", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "1.314+", "metric_symbol": "r", "metric_value": 1.314, "reference_text": "r = 1.314+ | Found by David W. Cantrell | in July 2002.", "source_image": "tc9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc9.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.314+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "tc9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "68692b6dc093b182", "improvement": -0.0007874911911363025, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.28479338054899e-06, "max_pair_overlap_depth": -4.021549232235966e-06, "metric_expression": "1.31479", "metric_symbol": "r", "metric_value": 1.3147874911911364, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.314, "setup": "triincir", "side": 2.6295749823822727, "solution_path": "data/solutions/generated/triincir_009_68692b6dc093b182.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/68692b6dc093b182.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@10", "density": 0.7187883776258066, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc10.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.385+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.385, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@10", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "1.385+", "metric_symbol": "r", "metric_value": 1.385, "reference_text": "r = 1.385+ | Found by David W. Cantrell | in July 2002.", "source_image": "tc10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc10.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.385+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "tc10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "676585e5c31a1d2f", "improvement": 0.0002388888356295027, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.405400112705848e-06, "max_pair_overlap_depth": -1.0902511275390214e-05, "metric_expression": "1.38476", "metric_symbol": "r", "metric_value": 1.3847611111643705, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.385, "setup": "triincir", "side": 2.769522222328741, "solution_path": "data/solutions/generated/triincir_010_676585e5c31a1d2f.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/676585e5c31a1d2f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@11", "density": 0.7044397586610418, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell after Maurizio Morandi", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc11.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.466+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.466, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@11", "credit": "David W. Cantrell after Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "1.466+", "metric_symbol": "r", "metric_value": 1.466, "reference_text": "r = 1.466+ | Found by DC after MM | in August 2012.", "source_image": "tc11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc11.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.466+ | Found by DC after MM | in August 2012.", "friedman_source_image": "tc11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "27341d9b331a210b", "improvement": -0.0010664804404059147, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5710045151440255e-06, "max_pair_overlap_depth": -1.3965597729533735e-05, "metric_expression": "1.46707", "metric_symbol": "r", "metric_value": 1.4670664804404059, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.466, "setup": "triincir", "side": 2.9341329608808118, "solution_path": "data/solutions/generated/triincir_011_27341d9b331a210b.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell after Maurizio Morandi", "svg_path": "data/svg/27341d9b331a210b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@12", "density": 0.7276318026083056, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "October 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc12.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.507+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.507, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@12", "credit": "David W. Cantrell", "date": "October 2005", "family_updated": "", "metric_expression": "1.507+", "metric_symbol": "r", "metric_value": 1.507, "reference_text": "r = 1.507+ | Found by David W. Cantrell | in October 2005.", "source_image": "tc12.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc12.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.507+ | Found by David W. Cantrell | in October 2005.", "friedman_source_image": "tc12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "0430950ce5a6ad65", "improvement": -0.0006834530254535842, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.988712507651542e-07, "max_pair_overlap_depth": -1.4096456774082e-06, "metric_expression": "1.50768", "metric_symbol": "r", "metric_value": 1.5076834530254535, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.507, "setup": "triincir", "side": 3.015366906050907, "solution_path": "data/solutions/generated/triincir_012_0430950ce5a6ad65.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/0430950ce5a6ad65.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@13", "density": 0.7678670311765523, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc13.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "\u221a(7/3) = 1.527+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.527, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@13", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a(7/3) = 1.527+", "metric_symbol": "r", "metric_value": 1.527, "reference_text": "r = \u221a(7/3) = 1.527+ | Found by Erich Friedman | in 1997.", "source_image": "tc13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc13.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = \u221a(7/3) = 1.527+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "2b014eeead965100", "improvement": -0.0005802936296193106, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.995535488687807e-06, "max_pair_overlap_depth": -6.278738689302621e-06, "metric_expression": "1.52758", "metric_symbol": "r", "metric_value": 1.5275802936296192, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.527, "setup": "triincir", "side": 3.0551605872592384, "solution_path": "data/solutions/generated/triincir_013_2b014eeead965100.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/2b014eeead965100.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@14", "density": 0.7492543391037492, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell after Maurizio Morandi", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc14.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.609+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.609, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@14", "credit": "David W. Cantrell after Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "1.609+", "metric_symbol": "r", "metric_value": 1.609, "reference_text": "r = 1.609+ | Found by DC after MM | in August 2012.", "source_image": "tc14.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc14.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.609+ | Found by DC after MM | in August 2012.", "friedman_source_image": "tc14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": false, "id": "da4114b8f6b7e383", "improvement": 0.0041857717254343285, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4204418656849072e-06, "max_pair_overlap_depth": -6.9567849071972e-06, "metric_expression": "1.60481", "metric_symbol": "r", "metric_value": 1.6048142282745657, "n": 14, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 1.609, "setup": "triincir", "side": 3.2096284565491313, "solution_path": "data/solutions/generated/triincir_014_da4114b8f6b7e383.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/da4114b8f6b7e383.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@15", "density": 0.7710002838935242, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc15.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.636+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.636, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@15", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "1.636+", "metric_symbol": "r", "metric_value": 1.636, "reference_text": "r = 1.636+ | Found by David W. Cantrell | in July 2002.", "source_image": "tc15.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc15.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.636+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "tc15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "e8f573cc9f431ed8", "improvement": -0.001546881163444036, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.392157777674768e-06, "max_pair_overlap_depth": -3.2056821285264547e-06, "metric_expression": "1.63755", "metric_symbol": "r", "metric_value": 1.637546881163444, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.636, "setup": "triincir", "side": 1.637546881163444, "solution_path": "data/solutions/generated/triincir_015_e8f573cc9f431ed8.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/e8f573cc9f431ed8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@16", "density": 0.7739404139048575, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc16.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.687+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.687, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@16", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "", "metric_expression": "1.687+", "metric_symbol": "r", "metric_value": 1.687, "reference_text": "r = 1.687+ | Found by Maurizio Morandi | in April 2008.", "source_image": "tc16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc16.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.687+ | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "tc16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "8bb5531a7df45934", "improvement": -0.0010356342518775818, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.068784430077415e-07, "max_pair_overlap_depth": -9.622687440202782e-07, "metric_expression": "1.68804", "metric_symbol": "r", "metric_value": 1.6880356342518776, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.687, "setup": "triincir", "side": 3.3760712685037553, "solution_path": "data/solutions/generated/triincir_16_8bb5531a7df45934.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/8bb5531a7df45934.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@17", "density": 0.7824875965148136, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc17.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.730+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.73, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@17", "credit": "David W. Cantrell", "date": "July 2005", "family_updated": "", "metric_expression": "1.730+", "metric_symbol": "r", "metric_value": 1.73, "reference_text": "r = 1.730+ | Found by David W. Cantrell | in July 2005.", "source_image": "tc17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc17.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.730+ | Found by David W. Cantrell | in July 2005.", "friedman_source_image": "tc17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "47c836632f7377eb", "improvement": -0.00045819251380652126, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3405989713065622e-06, "max_pair_overlap_depth": -3.2207400054717894e-06, "metric_expression": "1.73046", "metric_symbol": "r", "metric_value": 1.7304581925138065, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.73, "setup": "triincir", "side": 3.460916385027613, "solution_path": "data/solutions/generated/triincir_017_47c836632f7377eb.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/47c836632f7377eb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@18", "density": 0.763334042584198, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc18.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "\u221a13/2 = 1.802+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.802, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@18", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a13/2 = 1.802+", "metric_symbol": "r", "metric_value": 1.802, "reference_text": "r = \u221a13/2 = 1.802+ | Found by Erich Friedman | in 1997.", "source_image": "tc18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc18.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = \u221a13/2 = 1.802+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "8d495d33c78cce38", "improvement": -0.000828100052461922, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0688499290090192e-06, "max_pair_overlap_depth": -2.7547144944017887e-06, "metric_expression": "1.80283", "metric_symbol": "r", "metric_value": 1.802828100052462, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.802, "setup": "triincir", "side": 3.605656200104924, "solution_path": "data/solutions/generated/triincir_018_8d495d33c78cce38.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/8d495d33c78cce38.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@19", "density": 0.7847061820967525, "friedman_analytical_or_proved": true, "friedman_credit": "Haym Hirsh", "friedman_date": "August 2019", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc19.gif", "friedman_metric_expression": "\u221a(10/3) = 1.825+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.825, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@19", "credit": "Haym Hirsh", "date": "August 2019", "family_updated": "", "metric_expression": "\u221a(10/3) = 1.825+", "metric_symbol": "r", "metric_value": 1.825, "reference_text": "r = \u221a(10/3) = 1.825+ | Found by Haym Hirsh | in August 2019.", "source_image": "tc19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc19.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = \u221a(10/3) = 1.825+ | Found by Haym Hirsh | in August 2019.", "friedman_source_image": "tc19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "975121af758119a3", "improvement": -0.001832145989409506, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1422789436554837e-06, "max_pair_overlap_depth": -1.3515939734642934e-06, "metric_expression": "1.82683", "metric_symbol": "r", "metric_value": 1.8268321459894095, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.825, "setup": "triincir", "side": 3.653664291978819, "solution_path": "data/solutions/generated/triincir_19_975121af758119a3.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T14:34:18+00:00", "submitted_by": "Haym Hirsh", "svg_path": "data/svg/975121af758119a3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@20", "density": 0.7838286390350678, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc20.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.874+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.874, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@20", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "1.874+", "metric_symbol": "r", "metric_value": 1.874, "reference_text": "r = 1.874+ | Found by Erich Friedman | in 1997.", "source_image": "tc20.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc20.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.874+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "1217493f9010e655", "improvement": -0.0013391312177313264, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.256942872007727e-07, "max_pair_overlap_depth": -1.3401476308239069e-05, "metric_expression": "1.87534", "metric_symbol": "r", "metric_value": 1.8753391312177314, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.874, "setup": "triincir", "side": 3.750678262435463, "solution_path": "data/solutions/generated/triincir_020_1217493f9010e655.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/1217493f9010e655.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@21", "density": 0.7923995336773865, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc21.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.910+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.91, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@21", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "1.910+", "metric_symbol": "r", "metric_value": 1.91, "reference_text": "r = 1.910+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tc21.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc21.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.910+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tc21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "46b1371ff82573dd", "improvement": -0.001229866251050593, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3922552147160872e-05, "max_pair_overlap_depth": -4.454706840822942e-06, "metric_expression": "1.91123", "metric_symbol": "r", "metric_value": 1.9112298662510505, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.91, "setup": "triincir", "side": 1.9112298662510505, "solution_path": "data/solutions/generated/triincir_021_46b1371ff82573dd.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/46b1371ff82573dd.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@22", "density": 0.8017796736685735, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc22.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.943+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.943, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@22", "credit": "Maurizio Morandi", "date": "June 2008", "family_updated": "", "metric_expression": "1.943+", "metric_symbol": "r", "metric_value": 1.943, "reference_text": "r = 1.943+ | Found by Maurizio Morandi | in June 2008.", "source_image": "tc22.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc22.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.943+ | Found by Maurizio Morandi | in June 2008.", "friedman_source_image": "tc22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "69faaab68dcae114", "improvement": -0.0017294951342150533, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.28347034731047e-06, "max_pair_overlap_depth": -4.770520652064292e-06, "metric_expression": "1.94473", "metric_symbol": "r", "metric_value": 1.9447294951342151, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.943, "setup": "triincir", "side": 1.9447294951342151, "solution_path": "data/solutions/generated/triincir_022_69faaab68dcae114.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/69faaab68dcae114.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@23", "density": 0.8020036651008373, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc23.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.988+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.988, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@23", "credit": "Maurizio Morandi", "date": "June 2008", "family_updated": "", "metric_expression": "1.988+", "metric_symbol": "r", "metric_value": 1.988, "reference_text": "r = 1.988+ | Found by Maurizio Morandi | in June 2008.", "source_image": "tc23.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc23.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.988+ | Found by Maurizio Morandi | in June 2008.", "friedman_source_image": "tc23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "bc03a535bc1be813", "improvement": -0.0001590443736558811, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0403434641002036e-06, "max_pair_overlap_depth": -2.2042403398092603e-06, "metric_expression": "1.98816", "metric_symbol": "r", "metric_value": 1.9881590443736559, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.988, "setup": "triincir", "side": 1.9881590443736559, "solution_path": "data/solutions/generated/triincir_023_bc03a535bc1be813.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/bc03a535bc1be813.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@24", "density": 0.82672890671807, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc24.gif", "friedman_metric_expression": "2", "friedman_metric_symbol": "r", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@24", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "r", "metric_value": 2.0, "reference_text": "r = 2 | Trivial.", "source_image": "tc24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc24.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2 | Trivial.", "friedman_source_image": "tc24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "e1de703df3a339d6", "improvement": -0.00031983310403260745, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.886123831735148e-06, "max_pair_overlap_depth": -6.330279525013083e-07, "metric_expression": "2.00032", "metric_symbol": "r", "metric_value": 2.0003198331040326, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "triincir", "side": 4.000639666208065, "solution_path": "data/solutions/generated/triincir_24_e1de703df3a339d6.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/e1de703df3a339d6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@25", "density": 0.814798554480224, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc25.gif", "friedman_metric_expression": "2.055+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.055, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@25", "credit": "David W. Cantrell", "date": "July 2005", "family_updated": "", "metric_expression": "2.055+", "metric_symbol": "r", "metric_value": 2.055, "reference_text": "r = 2.055+ | Found by David W. Cantrell | in July 2005.", "source_image": "tc25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc25.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.055+ | Found by David W. Cantrell | in July 2005.", "friedman_source_image": "tc25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "460bfc63493e0a99", "improvement": -0.0014599742843675045, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.499842621325968e-07, "max_pair_overlap_depth": -5.533591762807433e-06, "metric_expression": "2.05646", "metric_symbol": "r", "metric_value": 2.0564599742843677, "n": 25, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.055, "setup": "triincir", "side": 4.112919948568735, "solution_path": "data/solutions/generated/triincir_025_460bfc63493e0a99.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/460bfc63493e0a99.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@26", "density": 0.8055666262709128, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc26.gif", "friedman_metric_expression": "2.109+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.109, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@26", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "", "metric_expression": "2.109+", "metric_symbol": "r", "metric_value": 2.109, "reference_text": "r = 2.109+ | Found by Maurizio Morandi | in April 2008.", "source_image": "tc26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc26.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.109+ | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "tc26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "8245fa6b2a472124", "improvement": -0.00016872463985961872, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.221408735321063e-06, "max_pair_overlap_depth": -3.3631357487795555e-07, "metric_expression": "2.10917", "metric_symbol": "r", "metric_value": 2.1091687246398596, "n": 26, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.109, "setup": "triincir", "side": 4.218337449279719, "solution_path": "data/solutions/generated/triincir_26_8245fa6b2a472124.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/8245fa6b2a472124.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@27", "density": 0.8132267625810148, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc27.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "2.138+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.138, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@27", "credit": "David W. Cantrell", "date": "July 2005", "family_updated": "", "metric_expression": "2.138+", "metric_symbol": "r", "metric_value": 2.138, "reference_text": "r = 2.138+ | Found by David W. Cantrell | in July 2005.", "source_image": "tc27.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc27.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.138+ | Found by David W. Cantrell | in July 2005.", "friedman_source_image": "tc27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "6af16151fd4089a6", "improvement": -0.0012002098551513463, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.142316272914286e-07, "max_pair_overlap_depth": -2.7141719629586447e-06, "metric_expression": "2.13920", "metric_symbol": "r", "metric_value": 2.1392002098551512, "n": 27, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.138, "setup": "triincir", "side": 2.1392002098551512, "solution_path": "data/solutions/generated/triincir_027_6af16151fd4089a6.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/6af16151fd4089a6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@28", "density": 0.8029867070046219, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc28.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "2.191+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.191, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@28", "credit": "David W. Cantrell", "date": "July 2005", "family_updated": "", "metric_expression": "2.191+", "metric_symbol": "r", "metric_value": 2.191, "reference_text": "r = 2.191+ | Found by David W. Cantrell | in July 2005.", "source_image": "tc28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc28.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.191+ | Found by David W. Cantrell | in July 2005.", "friedman_source_image": "tc28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "b8a030f48c690095", "improvement": -0.0013011891131600173, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3556564240069235e-06, "max_pair_overlap_depth": -5.7499508869796045e-06, "metric_expression": "2.19230", "metric_symbol": "r", "metric_value": 2.19230118911316, "n": 28, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.191, "setup": "triincir", "side": 2.19230118911316, "solution_path": "data/solutions/generated/triincir_028_b8a030f48c690095.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/b8a030f48c690095.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@29", "density": 0.8182348688742829, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc29.gif", "friedman_metric_expression": "2.215+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.215, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@29", "credit": "Maurizio Morandi", "date": "June 2008", "family_updated": "", "metric_expression": "2.215+", "metric_symbol": "r", "metric_value": 2.215, "reference_text": "r = 2.215+ | Found by Maurizio Morandi | in June 2008.", "source_image": "tc29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc29.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.215+ | Found by Maurizio Morandi | in June 2008.", "friedman_source_image": "tc29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": false, "id": "73bcaf90b9fd2226", "improvement": 0.004780586805306797, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.1271053040349273e-06, "max_pair_overlap_depth": -1.2019034158372577e-05, "metric_expression": "2.21022", "metric_symbol": "r", "metric_value": 2.210219413194693, "n": 29, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 2.215, "setup": "triincir", "side": 4.420438826389386, "solution_path": "data/solutions/generated/triincir_029_73bcaf90b9fd2226.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/73bcaf90b9fd2226.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@30", "density": 0.8110819400631206, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc30.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "2.258+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.258, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@30", "credit": "Maurizio Morandi", "date": "July 2008", "family_updated": "", "metric_expression": "2.258+", "metric_symbol": "r", "metric_value": 2.258, "reference_text": "r = 2.258+ | Found by Maurizio Morandi | in July 2008.", "source_image": "tc30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc30.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.258+ | Found by Maurizio Morandi | in July 2008.", "friedman_source_image": "tc30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "f720ec98817b9983", "improvement": 0.00010551194829711008, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9289733554094255e-06, "max_pair_overlap_depth": -4.657725350298847e-06, "metric_expression": "2.25789", "metric_symbol": "r", "metric_value": 2.257894488051703, "n": 30, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.258, "setup": "triincir", "side": 2.257894488051703, "solution_path": "data/solutions/generated/triincir_030_f720ec98817b9983.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/f720ec98817b9983.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triincir@31", "density": 0.8263348764897189, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc31.gif", "friedman_metric_expression": "2.272+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.272, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@31", "credit": "Maurizio Morandi", "date": "July 2008", "family_updated": "", "metric_expression": "2.272+", "metric_symbol": "r", "metric_value": 2.272, "reference_text": "r = 2.272+ | Found by Maurizio Morandi | in July 2008.", "source_image": "tc31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc31.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.272+ | Found by Maurizio Morandi | in July 2008.", "friedman_source_image": "tc31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "ed4a5072cc8d0adf", "improvement": -0.0019357369126002943, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3803713964376527e-06, "max_pair_overlap_depth": -5.497010580973738e-06, "metric_expression": "2.27394", "metric_symbol": "r", "metric_value": 2.2739357369126, "n": 31, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.272, "setup": "triincir", "side": 4.5478714738252, "solution_path": "data/solutions/generated/triincir_31_ed4a5072cc8d0adf.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/ed4a5072cc8d0adf.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@1", "density": 0.2886702903701934, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/2.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "\u221a3/2", "friedman_metric_symbol": "s", "friedman_metric_value": 0.8660254037844386, "friedman_reference": {"analytical_or_proved": true, "case": "triindom@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3/2", "metric_symbol": "s", "metric_value": 0.8660254037844386, "reference_text": "1-2. | | s = \u221a3/2 | Trivial. |", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/2.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "1-2. | | s = \u221a3/2 | Trivial. |", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "37b1dca676a779e3", "improvement": -7.266428382335377e-06, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4019537994690587e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.86603", "metric_symbol": "s", "metric_value": 0.8660326702128209, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.8660254037844386, "setup": "triindom", "side": 1.7320653404256419, "solution_path": "data/solutions/generated/triindom_001_37b1dca676a779e3.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/37b1dca676a779e3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@2", "density": 0.5773351362696808, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/2.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "\u221a3/2", "friedman_metric_symbol": "s", "friedman_metric_value": 0.8660254037844386, "friedman_reference": {"analytical_or_proved": true, "case": "triindom@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3/2", "metric_symbol": "s", "metric_value": 0.8660254037844386, "reference_text": "1-2. | | s = \u221a3/2 | Trivial. |", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/2.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "1-2. | | s = \u221a3/2 | Trivial. |", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "31518429031d2b51", "improvement": -1.1349913078717577e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.727531273176268e-07, "max_pair_overlap_depth": -0.13184728443338464, "metric_expression": "0.86604", "metric_symbol": "s", "metric_value": 0.8660367536975173, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.8660254037844386, "setup": "triindom", "side": 1.7320735073950346, "solution_path": "data/solutions/generated/triindom_002_31518429031d2b51.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/31518429031d2b51.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@3", "density": 0.6931800833949334, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/3.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": ".96796+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.96796, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@3", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": ".96796+", "metric_symbol": "s", "metric_value": 0.96796, "reference_text": "3. | | s = .96796+ | Found by Jonathan Viquerat | in April 2026", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/3.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "3. | | s = .96796+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "b10402bd10926cd1", "improvement": -3.454421484250236e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.601436348197538e-06, "max_pair_overlap_depth": -2.0206571078401625e-05, "metric_expression": "0.96799", "metric_symbol": "s", "metric_value": 0.9679945442148425, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.96796, "setup": "triindom", "side": 1.935989088429685, "solution_path": "data/solutions/generated/triindom_003_b10402bd10926cd1.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b10402bd10926cd1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@4", "density": 0.6899032945238053, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/4.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.12036+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.12036, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@4", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.12036+", "metric_symbol": "s", "metric_value": 1.12036, "reference_text": "4. | | s = 1.12036+ | Found by Jonathan Viquerat | in April 2026", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/4.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "4. | | s = 1.12036+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "daed4fc0474960d9", "improvement": -3.511453092652772e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.5154853256512695e-07, "max_pair_overlap_depth": -1.4193770285886309e-05, "metric_expression": "1.12040", "metric_symbol": "s", "metric_value": 1.1203951145309266, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.12036, "setup": "triindom", "side": 2.240790229061853, "solution_path": "data/solutions/generated/triindom_004_daed4fc0474960d9.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/daed4fc0474960d9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@5", "density": 0.6784100035927487, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/5.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.26315+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.26315, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@5", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.26315+", "metric_symbol": "s", "metric_value": 1.26315, "reference_text": "5. | | s = 1.26315+ | Found by Jonathan Viquerat | in April 2026", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/5.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "5. | | s = 1.26315+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "5f00d1fbe39c88e1", "improvement": -5.606079820497101e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.863453948036664e-06, "max_pair_overlap_depth": -2.090220821476807e-05, "metric_expression": "1.26321", "metric_symbol": "s", "metric_value": 1.263206060798205, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.26315, "setup": "triindom", "side": 2.52641212159641, "solution_path": "data/solutions/generated/triindom_005_5f00d1fbe39c88e1.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/5f00d1fbe39c88e1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@6", "density": 0.6857340379834939, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/6.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.37629+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.37629, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@6", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.37629+", "metric_symbol": "s", "metric_value": 1.37629, "reference_text": "6. | | s = 1.37629+ | Found by Jonathan Viquerat | in April 2026", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/6.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "6. | | s = 1.37629+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "cd22df1373fe40ae", "improvement": -7.332504683676966e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4791971315062398e-06, "max_pair_overlap_depth": -2.311431619983484e-05, "metric_expression": "1.37636", "metric_symbol": "s", "metric_value": 1.3763633250468368, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.37629, "setup": "triindom", "side": 2.7527266500936736, "solution_path": "data/solutions/generated/triindom_006_cd22df1373fe40ae.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/cd22df1373fe40ae.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@7", "density": 0.6816978937448389, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/7.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.49076+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.49076, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@7", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.49076+", "metric_symbol": "s", "metric_value": 1.49076, "reference_text": "7. | | s = 1.49076+ | Found by Jonathan Viquerat | in April 2026", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/7.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "7. | | s = 1.49076+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "070c4bf393378882", "improvement": -0.0002768061413715994, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6376140599128062e-06, "max_pair_overlap_depth": -1.7596834968304498e-05, "metric_expression": "1.49104", "metric_symbol": "s", "metric_value": 1.4910368061413717, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.49076, "setup": "triindom", "side": 2.9820736122827434, "solution_path": "data/solutions/generated/triindom_007_070c4bf393378882.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/070c4bf393378882.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@8", "density": 0.7089538292098625, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/8.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.56293+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.56293, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@8", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.56293+", "metric_symbol": "s", "metric_value": 1.56293, "reference_text": "8. | | s = 1.56293+ | Found by Jonathan Viquerat | in April 2026", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/8.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "8. | | s = 1.56293+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "dc89b8abdea24aaa", "improvement": -0.00011448128988833517, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.119934312653896e-07, "max_pair_overlap_depth": -2.4977335194534267e-05, "metric_expression": "1.56304", "metric_symbol": "s", "metric_value": 1.5630444812898883, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.56293, "setup": "triindom", "side": 3.1260889625797765, "solution_path": "data/solutions/generated/triindom_008_dc89b8abdea24aaa.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/dc89b8abdea24aaa.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@9", "density": 0.7166974833845204, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/9.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.64908+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.64908, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@9", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.64908+", "metric_symbol": "s", "metric_value": 1.64908, "reference_text": "9. | | s = 1.64908+ | Found by Jonathan Viquerat | in April 2026", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/9.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "9. | | s = 1.64908+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "9817bdf32d5c0df3", "improvement": 0.00020157610904458778, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.318788145301177e-06, "max_pair_overlap_depth": -1.0913014285005018e-05, "metric_expression": "1.64888", "metric_symbol": "s", "metric_value": 1.6488784238909555, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.64908, "setup": "triindom", "side": 1.6488784238909555, "solution_path": "data/solutions/generated/triindom_009_9817bdf32d5c0df3.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/9817bdf32d5c0df3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@10", "density": 0.7634583385020529, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/10.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.68303+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.68303, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@10", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.68303+", "metric_symbol": "s", "metric_value": 1.68303, "reference_text": "10. | | s = 1.68303+ | Found by Jonathan Viquerat | in April 2026", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/10.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "10. | | s = 1.68303+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "d60b37ee7b5fe8bb", "improvement": -0.0009721776557043427, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.2406370054083306e-07, "max_pair_overlap_depth": -6.360676094274709e-06, "metric_expression": "1.68400", "metric_symbol": "s", "metric_value": 1.6840021776557044, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.68303, "setup": "triindom", "side": 1.6840021776557044, "solution_path": "data/solutions/generated/triindom_010_d60b37ee7b5fe8bb.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/d60b37ee7b5fe8bb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@11", "density": 0.7938130122561631, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/11.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.73127+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.73127, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.73127+", "metric_symbol": "s", "metric_value": 1.73127, "reference_text": "11. | | s = 1.73127+ | Found by Jonathan Viquerat | in April 2026", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/11.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "11. | | s = 1.73127+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "92f51b49b50f6469", "improvement": -0.0008283817611449695, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.752164488808219e-06, "max_pair_overlap_depth": -9.417608768792381e-06, "metric_expression": "1.73210", "metric_symbol": "s", "metric_value": 1.732098381761145, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.73127, "setup": "triindom", "side": 3.46419676352229, "solution_path": "data/solutions/generated/triindom_011_92f51b49b50f6469.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/92f51b49b50f6469.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@12", "density": 0.8578074620402373, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/12.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.74040+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.7404, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@12", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.74040+", "metric_symbol": "s", "metric_value": 1.7404, "reference_text": "12. | | s = 1.74040+ | Found by Jonathan Viquerat | in April 2026", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/12.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "12. | | s = 1.74040+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "d4d97a50d19ba7f9", "improvement": 7.229766750849187e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6179955143756928e-06, "max_pair_overlap_depth": -3.432318619178698e-05, "metric_expression": "1.74033", "metric_symbol": "s", "metric_value": 1.7403277023324915, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.7404, "setup": "triindom", "side": 3.480655404664983, "solution_path": "data/solutions/generated/triindom_012_d4d97a50d19ba7f9.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/d4d97a50d19ba7f9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@13", "density": 0.819430202315155, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/13.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.85319+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.85319, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.85319+", "metric_symbol": "s", "metric_value": 1.85319, "reference_text": "13. | | s = 1.85319+ | Found by Jonathan Viquerat | in April 2026", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/13.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "13. | | s = 1.85319+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "734b8b487f97f149", "improvement": -0.00013256217488510913, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3279304722789718e-06, "max_pair_overlap_depth": -3.056740874492897e-05, "metric_expression": "1.85332", "metric_symbol": "s", "metric_value": 1.853322562174885, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.85319, "setup": "triindom", "side": 3.70664512434977, "solution_path": "data/solutions/generated/triindom_013_734b8b487f97f149.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/734b8b487f97f149.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@14", "density": 0.837880420446461, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/14.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "3(3\u2013\u221a3)/2 = 1.90192+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.90192, "friedman_reference": {"analytical_or_proved": true, "case": "triindom@14", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3(3\u2013\u221a3)/2 = 1.90192+", "metric_symbol": "s", "metric_value": 1.90192, "reference_text": "14. | | s = 3(3\u2013\u221a3)/2 = 1.90192+ | Found by Jonathan Viquerat | in April 2026", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/14.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "14. | | s = 3(3\u2013\u221a3)/2 = 1.90192+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "050249738efebc30", "improvement": -7.038452971785958e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8171942911825454e-06, "max_pair_overlap_depth": -1.6337035744662387e-05, "metric_expression": "1.90199", "metric_symbol": "s", "metric_value": 1.901990384529718, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.90192, "setup": "triindom", "side": 3.803980769059436, "solution_path": "data/solutions/generated/triindom_014_050249738efebc30.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/050249738efebc30.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@15", "density": 0.827720714535801, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/15.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.97950+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.9795, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@15", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.97950+", "metric_symbol": "s", "metric_value": 1.9795, "reference_text": "15. | | s = 1.97950+ | Found by Jonathan Viquerat | in April 2026", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/15.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "15. | | s = 1.97950+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "a012f18849dc81a9", "improvement": -0.001292749811030447, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8417244084533166e-06, "max_pair_overlap_depth": -2.1063364326057155e-05, "metric_expression": "1.98079", "metric_symbol": "s", "metric_value": 1.9807927498110305, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.9795, "setup": "triindom", "side": 3.961585499622061, "solution_path": "data/solutions/generated/triindom_015_a012f18849dc81a9.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/a012f18849dc81a9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@16", "density": 0.8658669941172707, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/16.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@16", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "16. | | s = 2 | Found by Jonathan Viquerat | in April 2026", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/16.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "16. | | s = 2 | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "97ac1b8d90281e30", "improvement": -0.0001829408254119258, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4402032156322093e-07, "max_pair_overlap_depth": -8.654468217628164e-06, "metric_expression": "2.00018", "metric_symbol": "s", "metric_value": 2.000182940825412, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "triindom", "side": 2.000182940825412, "solution_path": "data/solutions/generated/triindom_016_97ac1b8d90281e30.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/97ac1b8d90281e30.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@17", "density": 0.8312856464398565, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/17.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "2.10306+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.10306, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.10306+", "metric_symbol": "s", "metric_value": 2.10306, "reference_text": "17. | | s = 2.10306+ | Found by Jonathan Viquerat | in April 2026", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/17.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "17. | | s = 2.10306+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "3b35e95b5dec0d61", "improvement": -0.001128486819858221, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9321384165115205e-07, "max_pair_overlap_depth": -4.175356206781444e-05, "metric_expression": "2.10419", "metric_symbol": "s", "metric_value": 2.1041884868198584, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.10306, "setup": "triindom", "side": 2.1041884868198584, "solution_path": "data/solutions/generated/triindom_017_3b35e95b5dec0d61.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/3b35e95b5dec0d61.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@18", "density": 0.8436836711045761, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/18.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "2.14913.+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.14913, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@18", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.14913.+", "metric_symbol": "s", "metric_value": 2.14913, "reference_text": "18. | | s = 2.14913.+ | Found by Jonathan Viquerat | in April 2026", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/18.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "18. | | s = 2.14913.+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "c7d745720c03d379", "improvement": -9.434274541320065e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.205887803834571e-06, "max_pair_overlap_depth": -1.5204184911299379e-05, "metric_expression": "2.14922", "metric_symbol": "s", "metric_value": 2.149224342745413, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.14913, "setup": "triindom", "side": 4.298448685490826, "solution_path": "data/solutions/generated/triindom_018_c7d745720c03d379.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/c7d745720c03d379.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@19", "density": 0.815605476746989, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/19.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "2.24848+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.24848, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@19", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.24848+", "metric_symbol": "s", "metric_value": 2.24848, "reference_text": "19. | | s = 2.24848+ | Found by Jonathan Viquerat | in April 2026", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/19.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "19. | | s = 2.24848+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "ccdcfac9f05e2d80", "improvement": 0.002674950513982033, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.889707633764061e-07, "max_pair_overlap_depth": -1.2330676740557323e-05, "metric_expression": "2.24581", "metric_symbol": "s", "metric_value": 2.245805049486018, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.24848, "setup": "triindom", "side": 4.491610098972036, "solution_path": "data/solutions/generated/triindom_019_ccdcfac9f05e2d80.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/ccdcfac9f05e2d80.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triindom@20", "density": 0.820310655560819, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/20.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "2.29730+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.2973, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@20", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.29730+", "metric_symbol": "s", "metric_value": 2.2973, "reference_text": "20. | | s = 2.29730+ | Found by Jonathan Viquerat | in April 2026", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/20.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "20. | | s = 2.29730+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "b651b20201d5daf5", "improvement": -0.00022973895965483493, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6504306361575516e-06, "max_pair_overlap_depth": -2.6692790362048413e-05, "metric_expression": "2.29753", "metric_symbol": "s", "metric_value": 2.2975297389596547, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.2973, "setup": "triindom", "side": 2.2975297389596547, "solution_path": "data/solutions/generated/triindom_020_b651b20201d5daf5.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b651b20201d5daf5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@1", "density": 0.4999941032613389, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/1.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1/\u221a3 = .577+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.577, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1/\u221a3 = .577+", "metric_symbol": "s", "metric_value": 0.577, "reference_text": "s = 1/\u221a3 = .577+ | Trivial.", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/1.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1/\u221a3 = .577+ | Trivial.", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "e9bcdcd079c1c3b7", "improvement": -0.0003536737033924542, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.663290249487904e-07, "max_pair_overlap_depth": 0.0, "metric_expression": "0.57735", "metric_symbol": "s", "metric_value": 0.5773536737033924, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.577, "setup": "triinhex", "side": 0.5773536737033924, "solution_path": "data/solutions/generated/triinhex_001_e9bcdcd079c1c3b7.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/e9bcdcd079c1c3b7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@2", "density": 0.4444266358884555, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/2.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a3/2 = .866+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.866, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3/2 = .866+", "metric_symbol": "s", "metric_value": 0.866, "reference_text": "s = \u221a3/2 = .866+ | Trivial.", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/2.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = \u221a3/2 = .866+ | Trivial.", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "132c3ed1fad59f87", "improvement": -4.275480049964475e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.75428637420658e-06, "max_pair_overlap_depth": -4.36783093215443e-06, "metric_expression": "0.86604", "metric_symbol": "s", "metric_value": 0.8660427548004996, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.866, "setup": "triinhex", "side": 0.8660427548004996, "solution_path": "data/solutions/generated/triinhex_002_132c3ed1fad59f87.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/132c3ed1fad59f87.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@3", "density": 0.5108536175549566, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/3.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a((6+\u221a33)/12) = .989+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.989, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@3", "credit": "Erich Friedman", "date": "April 2005", "family_updated": "", "metric_expression": "\u221a((6+\u221a33)/12) = .989+", "metric_symbol": "s", "metric_value": 0.989, "reference_text": "s = \u221a((6+\u221a33)/12) = .989+ | Found by Erich Friedman | in April 2005.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/3.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = \u221a((6+\u221a33)/12) = .989+ | Found by Erich Friedman | in April 2005.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "3f19c46d9d07a54d", "improvement": -0.00031994709745319927, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.001855317125196e-07, "max_pair_overlap_depth": -5.956334362088034e-06, "metric_expression": "0.98932", "metric_symbol": "s", "metric_value": 0.9893199470974532, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.989, "setup": "triinhex", "side": 0.9893199470974532, "solution_path": "data/solutions/generated/triinhex_003_3f19c46d9d07a54d.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3f19c46d9d07a54d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@4", "density": 0.6666347131299533, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/4.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/4.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "9a8f72cbad0da23e", "improvement": -2.396601406218224e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.5130077190296347e-08, "max_pair_overlap_depth": -1.1324383645797624e-05, "metric_expression": "1.00002", "metric_symbol": "s", "metric_value": 1.0000239660140622, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0, "setup": "triinhex", "side": 1.0000239660140622, "solution_path": "data/solutions/generated/triinhex_004_9a8f72cbad0da23e.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9a8f72cbad0da23e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@5", "density": 0.8333022730508762, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/5.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/5.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "0d61e48787b7d5d1", "improvement": -1.863669045065741e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0710555572355552e-06, "max_pair_overlap_depth": -2.545819687190834e-06, "metric_expression": "1.00002", "metric_symbol": "s", "metric_value": 1.0000186366904507, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0, "setup": "triinhex", "side": 1.0000186366904507, "solution_path": "data/solutions/generated/triinhex_005_0d61e48787b7d5d1.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/0d61e48787b7d5d1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@6", "density": 0.9999558832137269, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/6.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/6.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "163de15143701be8", "improvement": -2.2059123022444282e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.751347439371281e-07, "max_pair_overlap_depth": -4.053173147537176e-06, "metric_expression": "1.00002", "metric_symbol": "s", "metric_value": 1.0000220591230224, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0, "setup": "triinhex", "side": 1.0000220591230224, "solution_path": "data/solutions/generated/triinhex_006_163de15143701be8.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/163de15143701be8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@7", "density": 0.7149966699512036, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "May 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/7.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.277+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.277, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@7", "credit": "David W. Cantrell", "date": "May 2005", "family_updated": "", "metric_expression": "1.277+", "metric_symbol": "s", "metric_value": 1.277, "reference_text": "s = 1.277+ | Found by David W. Cantrell | in May 2005.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/7.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.277+ | Found by David W. Cantrell | in May 2005.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "af96c8dfd69688ff", "improvement": -0.0003837447071250999, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8980834118002716e-06, "max_pair_overlap_depth": -1.0312174508100846e-05, "metric_expression": "1.27738", "metric_symbol": "s", "metric_value": 1.277383744707125, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.277, "setup": "triinhex", "side": 1.277383744707125, "solution_path": "data/solutions/generated/triinhex_007_af96c8dfd69688ff.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/af96c8dfd69688ff.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@8", "density": 0.724469693124155, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/8.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.356+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.356, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@8", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "1.356+", "metric_symbol": "s", "metric_value": 1.356, "reference_text": "s = 1.356+ | Found by David W. Cantrell | in July 2012.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/8.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.356+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "d557994d1935d6fb", "improvement": -0.0006232543073974028, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6958384610035216e-07, "max_pair_overlap_depth": -6.193690244168937e-06, "metric_expression": "1.35662", "metric_symbol": "s", "metric_value": 1.3566232543073975, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.356, "setup": "triinhex", "side": 1.3566232543073975, "solution_path": "data/solutions/generated/triinhex_008_d557994d1935d6fb.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d557994d1935d6fb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@9", "density": 0.7288897241647109, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "April 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/9.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.434+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.434, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@9", "credit": "Erich Friedman", "date": "April 2005", "family_updated": "", "metric_expression": "1.434+", "metric_symbol": "s", "metric_value": 1.434, "reference_text": "s = 1.434+ | Found by Erich Friedman | in April 2005.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/9.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.434+ | Found by Erich Friedman | in April 2005.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "e1592cc41ab92d4d", "improvement": -0.0005467824277936018, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.288341076546003e-06, "max_pair_overlap_depth": -3.69502244107478e-06, "metric_expression": "1.43455", "metric_symbol": "s", "metric_value": 1.4345467824277935, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.434, "setup": "triinhex", "side": 1.4345467824277935, "solution_path": "data/solutions/generated/triinhex_009_e1592cc41ab92d4d.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/e1592cc41ab92d4d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@10", "density": 0.7432082873431878, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/10.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.497+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.497, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@10", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "1.497+", "metric_symbol": "s", "metric_value": 1.497, "reference_text": "s = 1.497+ | Found by Timo Berthold et al | in January 2026.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/10.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.497+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "27f77a1826f41a7d", "improvement": -0.0005078337450310499, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4736979206377467e-06, "max_pair_overlap_depth": -5.4498855950879665e-06, "metric_expression": "1.49751", "metric_symbol": "s", "metric_value": 1.4975078337450312, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.497, "setup": "triinhex", "side": 1.4975078337450312, "solution_path": "data/solutions/generated/triinhex_010_27f77a1826f41a7d.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/27f77a1826f41a7d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@11", "density": 0.8147753094192889, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/11.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3/2 = 1.500", "friedman_metric_symbol": "s", "friedman_metric_value": 1.5, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@11", "credit": "Erich Friedman", "date": "April 2005", "family_updated": "", "metric_expression": "3/2 = 1.500", "metric_symbol": "s", "metric_value": 1.5, "reference_text": "s = 3/2 = 1.500 | Found by Erich Friedman | in April 2005.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/11.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 3/2 = 1.500 | Found by Erich Friedman | in April 2005.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "a4d7af61e2935a14", "improvement": -3.636424319730658e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0384668058064728e-06, "max_pair_overlap_depth": -9.97710385775008e-06, "metric_expression": "1.50004", "metric_symbol": "s", "metric_value": 1.5000363642431973, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.5, "setup": "triinhex", "side": 1.5000363642431973, "solution_path": "data/solutions/generated/triinhex_011_a4d7af61e2935a14.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a4d7af61e2935a14.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@12", "density": 0.7583806241339298, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/12.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.623+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.623, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@12", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "1.623+", "metric_symbol": "s", "metric_value": 1.623, "reference_text": "s = 1.623+ | Found by Timo Berthold et al | in January 2026.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/12.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.623+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "9de540041e59931a", "improvement": -0.0009452508446496921, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.928415721340796e-06, "max_pair_overlap_depth": -1.3315156502324843e-05, "metric_expression": "1.62395", "metric_symbol": "s", "metric_value": 1.6239452508446497, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.623, "setup": "triinhex", "side": 1.6239452508446497, "solution_path": "data/solutions/generated/triinhex_012_9de540041e59931a.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/9de540041e59931a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@13", "density": 0.7821155301010267, "friedman_analytical_or_proved": true, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/13.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a(587+51\u221a17)/12\u221a2 = 1.663829+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.663829, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@13", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "\u221a(587+51\u221a17)/12\u221a2 = 1.663829+", "metric_symbol": "s", "metric_value": 1.663829, "reference_text": "s = \u221a(587+51\u221a17)/12\u221a2 = 1.663829+ | Found by Ian Watson | in April 2026.", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/13.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = \u221a(587+51\u221a17)/12\u221a2 = 1.663829+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "efc43a6cbf81ae24", "improvement": -0.0005820721473515089, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.9742574580490386e-06, "max_pair_overlap_depth": -2.1993638058326326e-05, "metric_expression": "1.66441", "metric_symbol": "s", "metric_value": 1.6644110721473515, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.663829, "setup": "triinhex", "side": 1.6644110721473515, "solution_path": "data/solutions/generated/triinhex_013_efc43a6cbf81ae24.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/efc43a6cbf81ae24.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@14", "density": 0.7828804179520853, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/14.gif", "friedman_metric_expression": "1.725+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.725, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@14", "credit": "David W. Cantrell", "date": "August 2012", "family_updated": "", "metric_expression": "1.725+", "metric_symbol": "s", "metric_value": 1.725, "reference_text": "s = 1.725+ | Found by David W. Cantrell | in August 2012.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/14.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.725+ | Found by David W. Cantrell | in August 2012.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "dbfe389f1c3777d5", "improvement": -0.0013970193042536216, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.5600740703080906e-07, "max_pair_overlap_depth": -1.248505498985253e-07, "metric_expression": "1.72640", "metric_symbol": "s", "metric_value": 1.7263970193042537, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.725, "setup": "triinhex", "side": 1.7263970193042537, "solution_path": "data/solutions/generated/triinhex_14_dbfe389f1c3777d5.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/dbfe389f1c3777d5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@15", "density": 0.8162509918871499, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/15.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "7/4 = 1.750", "friedman_metric_symbol": "s", "friedman_metric_value": 1.75, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@15", "credit": "Erich Friedman", "date": "April 2005", "family_updated": "", "metric_expression": "7/4 = 1.750", "metric_symbol": "s", "metric_value": 1.75, "reference_text": "s = 7/4 = 1.750 | Found by Erich Friedman | in April 2005.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/15.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 7/4 = 1.750 | Found by Erich Friedman | in April 2005.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "20b1a0e905211ce9", "improvement": -8.097369067638915e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.758472784251539e-06, "max_pair_overlap_depth": -9.75502017092067e-06, "metric_expression": "1.75008", "metric_symbol": "s", "metric_value": 1.7500809736906764, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.75, "setup": "triinhex", "side": 1.7500809736906764, "solution_path": "data/solutions/generated/triinhex_015_20b1a0e905211ce9.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/20b1a0e905211ce9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@16", "density": 0.8229668363309378, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/16.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "9/5 = 1.800", "friedman_metric_symbol": "s", "friedman_metric_value": 1.8, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@16", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "", "metric_expression": "9/5 = 1.800", "metric_symbol": "s", "metric_value": 1.8, "reference_text": "s = 9/5 = 1.800 | Found by Maurizio Morandi | in April 2008.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/16.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 9/5 = 1.800 | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "a0b20d24de5f00ac", "improvement": -8.577060222680899e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.740972449159294e-06, "max_pair_overlap_depth": -2.3139042383268915e-06, "metric_expression": "1.80009", "metric_symbol": "s", "metric_value": 1.8000857706022269, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.8, "setup": "triinhex", "side": 1.8000857706022269, "solution_path": "data/solutions/generated/triinhex_016_a0b20d24de5f00ac.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/a0b20d24de5f00ac.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@17", "density": 0.842931104392013, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/17.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "11/6 = 1.833+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.833, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@17", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "", "metric_expression": "11/6 = 1.833+", "metric_symbol": "s", "metric_value": 1.833, "reference_text": "s = 11/6 = 1.833+ | Found by Maurizio Morandi | in April 2008.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/17.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 11/6 = 1.833+ | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "b8b9e4b4d61ec96c", "improvement": -0.00038129277582044097, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4990722513823532e-07, "max_pair_overlap_depth": -1.1854387088505725e-05, "metric_expression": "1.83338", "metric_symbol": "s", "metric_value": 1.8333812927758204, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.833, "setup": "triinhex", "side": 1.8333812927758204, "solution_path": "data/solutions/generated/triinhex_017_b8b9e4b4d61ec96c.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/b8b9e4b4d61ec96c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@18", "density": 0.8400311467465807, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/18.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "17/9 = 1.888+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.888, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@18", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "", "metric_expression": "17/9 = 1.888+", "metric_symbol": "s", "metric_value": 1.888, "reference_text": "s = 17/9 = 1.888+ | Found by Maurizio Morandi | in April 2008.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/18.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 17/9 = 1.888+ | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "7671cfd644a7af24", "improvement": -0.001787329223866152, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.484924045151217e-06, "max_pair_overlap_depth": -8.796062168636709e-06, "metric_expression": "1.88979", "metric_symbol": "s", "metric_value": 1.889787329223866, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.888, "setup": "triinhex", "side": 1.889787329223866, "solution_path": "data/solutions/generated/triinhex_018_7671cfd644a7af24.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/7671cfd644a7af24.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@19", "density": 0.8283361930438612, "friedman_analytical_or_proved": false, "friedman_credit": "Alex Adri\u00e1n Nieto", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/19.gif", "friedman_metric_expression": "1.953+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.953, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@19", "credit": "Alex Adri\u00e1n Nieto", "date": "April 2026", "family_updated": "", "metric_expression": "1.953+", "metric_symbol": "s", "metric_value": 1.953, "reference_text": "s = 1.953+ | Found by Alex Adri\u00e1n Nieto | in April 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/19.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.953+ | Found by Alex Adri\u00e1n Nieto | in April 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "451018f1291a5791", "improvement": -0.0022300191069650754, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7079807782201595e-06, "max_pair_overlap_depth": -1.3478523960663846e-05, "metric_expression": "1.95523", "metric_symbol": "s", "metric_value": 1.9552300191069651, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.953, "setup": "triinhex", "side": 1.9552300191069651, "solution_path": "data/solutions/generated/triinhex_19_451018f1291a5791.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "Alex Adri\u00e1n Nieto", "svg_path": "data/svg/451018f1291a5791.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@20", "density": 0.8524037354438128, "friedman_analytical_or_proved": false, "friedman_credit": "Aaden Abraham", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/20.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.97591+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.97591, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@20", "credit": "Aaden Abraham", "date": "April 2026", "family_updated": "", "metric_expression": "1.97591+", "metric_symbol": "s", "metric_value": 1.97591, "reference_text": "s = 1.97591+ | Found by Aaden Abraham | in April 2026.", "source_image": "20.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/20.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.97591+ | Found by Aaden Abraham | in April 2026.", "friedman_source_image": "20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "388fc73ef451e99b", "improvement": -0.0015909487133152211, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7558573966506123e-06, "max_pair_overlap_depth": -5.2993150605473716e-06, "metric_expression": "1.97750", "metric_symbol": "s", "metric_value": 1.9775009487133153, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.97591, "setup": "triinhex", "side": 1.9775009487133153, "solution_path": "data/solutions/generated/triinhex_020_388fc73ef451e99b.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Aaden Abraham", "svg_path": "data/svg/388fc73ef451e99b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@21", "density": 0.8749527320468723, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/21.gif", "friedman_metric_expression": "1.99878+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.99878, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@21", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "1.99878+", "metric_symbol": "s", "metric_value": 1.99878, "reference_text": "s = 1.99878+ | Found by Ian Watson | in April 2026.", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/21.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.99878+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "96f74df50573bb81", "improvement": -0.0012740227066210252, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.545179480294009e-07, "max_pair_overlap_depth": -6.765642482919887e-06, "metric_expression": "2.00005", "metric_symbol": "s", "metric_value": 2.000054022706621, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.99878, "setup": "triinhex", "side": 2.000054022706621, "solution_path": "data/solutions/generated/triinhex_021_96f74df50573bb81.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/96f74df50573bb81.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@22", "density": 0.9166150664792877, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/24.gif", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@22", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial. |", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/24.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2 | Trivial. |", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "38182a59461da74e", "improvement": -5.629349013291929e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.91184500961419e-07, "max_pair_overlap_depth": -9.942352799940934e-07, "metric_expression": "2.00006", "metric_symbol": "s", "metric_value": 2.000056293490133, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "triinhex", "side": 2.000056293490133, "solution_path": "data/solutions/generated/triinhex_022_38182a59461da74e.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/38182a59461da74e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@23", "density": 0.958278890368788, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/24.gif", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@23", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial. |", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/24.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2 | Trivial. |", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "dcd331994270a997", "improvement": -5.681247061106731e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0734329507489061e-06, "max_pair_overlap_depth": -9.41664729126046e-06, "metric_expression": "2.00006", "metric_symbol": "s", "metric_value": 2.000056812470611, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "triinhex", "side": 2.000056812470611, "solution_path": "data/solutions/generated/triinhex_023_dcd331994270a997.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/dcd331994270a997.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@24", "density": 0.9999420291406096, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/24.gif", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@24", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial. |", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/24.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2 | Trivial. |", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "13936a08bfd4d152", "improvement": -5.7973379977482864e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3635152509087512e-06, "max_pair_overlap_depth": -3.8859579855957235e-06, "metric_expression": "2.00006", "metric_symbol": "s", "metric_value": 2.0000579733799775, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "triinhex", "side": 2.0000579733799775, "solution_path": "data/solutions/generated/triinhex_024_13936a08bfd4d152.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/13936a08bfd4d152.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@25", "density": 0.8871759236838516, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/25.gif", "friedman_metric_expression": "13/6 = 2.166+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.166, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@25", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "13/6 = 2.166+", "metric_symbol": "s", "metric_value": 2.166, "reference_text": "s = 13/6 = 2.166+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/25.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 13/6 = 2.166+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "71a7801cfb4a8d31", "improvement": -0.0011526610601886311, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.780494307380179e-07, "max_pair_overlap_depth": -1.3308976067882838e-06, "metric_expression": "2.16715", "metric_symbol": "s", "metric_value": 2.1671526610601886, "n": 25, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.166, "setup": "triinhex", "side": 2.1671526610601886, "solution_path": "data/solutions/generated/triinhex_025_71a7801cfb4a8d31.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/71a7801cfb4a8d31.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@26", "density": 0.878430458154361, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/26.gif", "friedman_metric_expression": "2.21991+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.21991, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@26", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.21991+", "metric_symbol": "s", "metric_value": 2.21991, "reference_text": "s = 2.21991+ | Found by Ian Watson | in April 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/26.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.21991+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "658d7f93b604b663", "improvement": -0.0011349903788469007, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.149641376243232e-07, "max_pair_overlap_depth": -5.41804289733161e-06, "metric_expression": "2.22104", "metric_symbol": "s", "metric_value": 2.221044990378847, "n": 26, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.21991, "setup": "triinhex", "side": 2.221044990378847, "solution_path": "data/solutions/generated/triinhex_026_658d7f93b604b663.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/658d7f93b604b663.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@27", "density": 0.8525233088554102, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/27.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.29541+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.29541, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@27", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.29541+", "metric_symbol": "s", "metric_value": 2.29541, "reference_text": "s = 2.29541+ | Found by Ian Watson | in April 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/27.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.29541+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "ace1d8ac458c5bc4", "improvement": -0.0020773347367342154, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.449964623966915e-06, "max_pair_overlap_depth": -5.244956223338093e-06, "metric_expression": "2.29749", "metric_symbol": "s", "metric_value": 2.297487334736734, "n": 27, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.29541, "setup": "triinhex", "side": 2.297487334736734, "solution_path": "data/solutions/generated/triinhex_027_ace1d8ac458c5bc4.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/ace1d8ac458c5bc4.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@28", "density": 0.8636324031387004, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/28.gif", "friedman_metric_expression": "2.32289+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.32289, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@28", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.32289+", "metric_symbol": "s", "metric_value": 2.32289, "reference_text": "s = 2.32289+ | Found by Ian Watson | in April 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/28.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.32289+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "3435500023858121", "improvement": -0.0016601826518787277, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.60736131059532e-06, "max_pair_overlap_depth": -1.0038160656700157e-05, "metric_expression": "2.32455", "metric_symbol": "s", "metric_value": 2.324550182651879, "n": 28, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.32289, "setup": "triinhex", "side": 2.324550182651879, "solution_path": "data/solutions/generated/triinhex_28_3435500023858121.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/3435500023858121.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@29", "density": 0.8876189549602443, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/29.gif", "friedman_metric_expression": "7/3 = 2.333+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.333, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@29", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "7/3 = 2.333+", "metric_symbol": "s", "metric_value": 2.333, "reference_text": "s = 7/3 = 2.333+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/29.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 7/3 = 2.333+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "4339d1ebd3160f94", "improvement": -0.0005122751747741461, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.371335525980428e-06, "max_pair_overlap_depth": -4.303492490143057e-07, "metric_expression": "2.33351", "metric_symbol": "s", "metric_value": 2.3335122751747743, "n": 29, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.333, "setup": "triinhex", "side": 2.3335122751747743, "solution_path": "data/solutions/generated/triinhex_29_4339d1ebd3160f94.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/4339d1ebd3160f94.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@30", "density": 0.8862493271562808, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/30.gif", "friedman_metric_expression": "19/8 = 2.375", "friedman_metric_symbol": "s", "friedman_metric_value": 2.375, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@30", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "19/8 = 2.375", "metric_symbol": "s", "metric_value": 2.375, "reference_text": "s = 19/8 = 2.375 | Found by Ignacio Vallejo | in April 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/30.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 19/8 = 2.375 | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "7274d354e73d8050", "improvement": -0.00023750926269094919, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.946881910542413e-07, "max_pair_overlap_depth": -3.1129912760885503e-06, "metric_expression": "2.37524", "metric_symbol": "s", "metric_value": 2.375237509262691, "n": 30, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.375, "setup": "triinhex", "side": 2.375237509262691, "solution_path": "data/solutions/generated/triinhex_30_7274d354e73d8050.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/7274d354e73d8050.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@31", "density": 0.896119877899338, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/31.gif", "friedman_metric_expression": "12/5 = 2.400", "friedman_metric_symbol": "s", "friedman_metric_value": 2.4, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "12/5 = 2.400", "metric_symbol": "s", "metric_value": 2.4, "reference_text": "s = 12/5 = 2.400 | Found by Emerson Connelly | in April 2026.", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/31.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 12/5 = 2.400 | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "cae7087451040120", "improvement": -0.0011658949471424762, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9409739826535097e-07, "max_pair_overlap_depth": -2.8541100203227643e-06, "metric_expression": "2.40117", "metric_symbol": "s", "metric_value": 2.4011658949471424, "n": 31, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.4, "setup": "triinhex", "side": 2.4011658949471424, "solution_path": "data/solutions/generated/triinhex_31_cae7087451040120.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/cae7087451040120.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@32", "density": 0.8767870241121234, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/32.gif", "friedman_metric_expression": "2.465+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.465, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@32", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "2.465+", "metric_symbol": "s", "metric_value": 2.465, "reference_text": "s = 2.465+ | Found by Emerson Connelly | in April 2026.", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/32.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.465+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "8674581436e5e74d", "improvement": -0.0013363682337788063, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.925122041572899e-07, "max_pair_overlap_depth": -1.2807310418028184e-06, "metric_expression": "2.46634", "metric_symbol": "s", "metric_value": 2.4663363682337787, "n": 32, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.465, "setup": "triinhex", "side": 2.4663363682337787, "solution_path": "data/solutions/generated/triinhex_032_8674581436e5e74d.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/8674581436e5e74d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@33", "density": 0.8799443949567897, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/33.gif", "friedman_metric_expression": "2.498+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.498, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@33", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "2.498+", "metric_symbol": "s", "metric_value": 2.498, "reference_text": "s = 2.498+ | Found by Emerson Connelly | in April 2026.", "source_image": "33.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/33.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.498+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "5b836a27d2bb958c", "improvement": -0.00207898817969987, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8634496562140136e-06, "max_pair_overlap_depth": -5.829272115620565e-06, "metric_expression": "2.50008", "metric_symbol": "s", "metric_value": 2.5000789881797, "n": 33, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.498, "setup": "triinhex", "side": 2.5000789881797, "solution_path": "data/solutions/generated/triinhex_033_5b836a27d2bb958c.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/5b836a27d2bb958c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@34", "density": 0.9066102013779274, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "friedman_metric_expression": "5/2 = 2.500", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@34", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "5/2 = 2.500", "metric_symbol": "s", "metric_value": 2.5, "reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "a712744ff3b5bcc9", "improvement": -7.7851001305973e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2753913921730298e-06, "max_pair_overlap_depth": -7.960301433351802e-06, "metric_expression": "2.50008", "metric_symbol": "s", "metric_value": 2.500077851001306, "n": 34, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.5, "setup": "triinhex", "side": 2.500077851001306, "solution_path": "data/solutions/generated/triinhex_034_a712744ff3b5bcc9.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/a712744ff3b5bcc9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@35", "density": 0.93316535041203, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "friedman_metric_expression": "5/2 = 2.500", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@35", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "5/2 = 2.500", "metric_symbol": "s", "metric_value": 2.5, "reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "ec49778cf450c37b", "improvement": -0.00022500750012577342, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6207066397040256e-06, "max_pair_overlap_depth": -1.5837731757728513e-06, "metric_expression": "2.50023", "metric_symbol": "s", "metric_value": 2.5002250075001258, "n": 35, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.5, "setup": "triinhex", "side": 2.5002250075001258, "solution_path": "data/solutions/generated/triinhex_35_ec49778cf450c37b.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/ec49778cf450c37b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@36", "density": 0.8942711196026824, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/36.gif", "friedman_metric_expression": "2.59123+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.59123, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@36", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "2.59123+", "metric_symbol": "s", "metric_value": 2.59123, "reference_text": "s = 2.59123+ | Found by Nathan Roll | in April 2026.", "source_image": "36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/36.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.59123+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "d30ec5c19b8369f3", "improvement": 0.00098393596194013, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.744185746811731e-07, "max_pair_overlap_depth": -1.519675368655271e-06, "metric_expression": "2.59025", "metric_symbol": "s", "metric_value": 2.59024606403806, "n": 36, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.59123, "setup": "triinhex", "side": 2.5902460640380602, "solution_path": "data/solutions/record_hunt/triinhex_36_d30ec5c19b8369f3.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/d30ec5c19b8369f3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@37", "density": 0.899950790089805, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/37.gif", "friedman_metric_expression": "2.61943+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.61943, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@37", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "2.61943+", "metric_symbol": "s", "metric_value": 2.61943, "reference_text": "s = 2.61943+ | Found by Nathan Roll | in April 2026.", "source_image": "37.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/37.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.61943+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "c791d76ac960cbca", "improvement": 0.0017542149405089802, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.903248309191345e-07, "max_pair_overlap_depth": -8.609976616114068e-07, "metric_expression": "2.61768", "metric_symbol": "s", "metric_value": 2.617675785059491, "n": 37, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.61943, "setup": "triinhex", "side": 2.617675785059491, "solution_path": "data/solutions/record_hunt/triinhex_37_c791d76ac960cbca.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/c791d76ac960cbca.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@38", "density": 0.8907392561961515, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", "friedman_metric_expression": "8/3 = 2.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.666, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@38", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "8/3 = 2.666+", "metric_symbol": "s", "metric_value": 2.666, "reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", "source_image": "39.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "76fd196e8c79a5c9", "improvement": -0.0004956329108489754, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7850003642140776e-06, "max_pair_overlap_depth": -3.863933217385096e-06, "metric_expression": "2.66650", "metric_symbol": "s", "metric_value": 2.666495632910849, "n": 38, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.666, "setup": "triinhex", "side": 2.666495632910849, "solution_path": "data/solutions/generated/triinhex_038_76fd196e8c79a5c9.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/76fd196e8c79a5c9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@39", "density": 0.9136318214810123, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", "friedman_metric_expression": "8/3 = 2.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.666, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@39", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "8/3 = 2.666+", "metric_symbol": "s", "metric_value": 2.666, "reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", "source_image": "39.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "a314b4bb8467e5e8", "improvement": -0.0012951149704507614, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.891404577705932e-07, "max_pair_overlap_depth": -6.19203720853001e-07, "metric_expression": "2.66730", "metric_symbol": "s", "metric_value": 2.6672951149704507, "n": 39, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.666, "setup": "triinhex", "side": 2.6672951149704507, "solution_path": "data/solutions/generated/triinhex_39_a314b4bb8467e5e8.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/a314b4bb8467e5e8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinhex@40", "density": 0.9051419512423183, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/40.gif", "friedman_metric_expression": "2.71622+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.71622, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@40", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "2.71622+", "metric_symbol": "s", "metric_value": 2.71622, "reference_text": "s = 2.71622+ | Found by Nathan Roll | in April 2026.", "source_image": "40.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/40.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.71622+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "8b6d682b66356562", "improvement": 0.0023063615780789526, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2777984478773679e-07, "max_pair_overlap_depth": -4.470141119483628e-07, "metric_expression": "2.71391", "metric_symbol": "s", "metric_value": 2.713913638421921, "n": 40, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.71622, "setup": "triinhex", "side": 2.713913638421921, "solution_path": "data/solutions/record_hunt/triinhex_40_8b6d682b66356562.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/8b6d682b66356562.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@1", "density": 0.4152122478191623, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/1.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": ".46470+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.4647, "friedman_reference": {"analytical_or_proved": true, "case": "triinoct@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": ".46470+", "metric_symbol": "s", "metric_value": 0.4647, "reference_text": "s = .46470+ | Trivial. |", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/1.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = .46470+ | Trivial. |", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "92c66f395ff03ab9", "improvement": -4.251046545072912e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3282842257895898e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.46474", "metric_symbol": "s", "metric_value": 0.46474251046545073, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.4647, "setup": "triinoct", "side": 0.46474251046545073, "solution_path": "data/solutions/generated/triinoct_001_92c66f395ff03ab9.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/92c66f395ff03ab9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@2", "density": 0.40823825072318354, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/2.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "\u221a3/\u221a(4+2\u221a2) = .66282+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.66282, "friedman_reference": {"analytical_or_proved": true, "case": "triinoct@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3/\u221a(4+2\u221a2) = .66282+", "metric_symbol": "s", "metric_value": 0.66282, "reference_text": "s = \u221a3/\u221a(4+2\u221a2) = .66282+ | Trivial. |", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/2.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = \u221a3/\u221a(4+2\u221a2) = .66282+ | Trivial. |", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "5706480ca6648c20", "improvement": -1.5298424264065247e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.197393779037103e-06, "max_pair_overlap_depth": -5.033792755404498e-07, "metric_expression": "0.66284", "metric_symbol": "s", "metric_value": 0.662835298424264, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.66282, "setup": "triinoct", "side": 0.662835298424264, "solution_path": "data/solutions/generated/triinoct_002_5706480ca6648c20.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/5706480ca6648c20.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@3", "density": 0.49592364137420614, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/3.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": ".73651+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.73651, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@3", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": ".73651+", "metric_symbol": "s", "metric_value": 0.73651, "reference_text": "s = .73651+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/3.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = .73651+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "f3eee07e4d3df15b", "improvement": -3.739655668555386e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.662035887956301e-06, "max_pair_overlap_depth": -2.2250265259982083e-06, "metric_expression": "0.73655", "metric_symbol": "s", "metric_value": 0.7365473965566856, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.73651, "setup": "triinoct", "side": 0.7365473965566856, "solution_path": "data/solutions/generated/triinoct_003_f3eee07e4d3df15b.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/f3eee07e4d3df15b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@4", "density": 0.593540115018531, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/4.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": ".77733+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.77733, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@4", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": ".77733+", "metric_symbol": "s", "metric_value": 0.77733, "reference_text": "s = .77733+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/4.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = .77733+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "daeed392d4fde104", "improvement": -8.413889807601382e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.025085624522854e-06, "max_pair_overlap_depth": -2.0675467336764286e-05, "metric_expression": "0.77741", "metric_symbol": "s", "metric_value": 0.777414138898076, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.77733, "setup": "triinoct", "side": 0.777414138898076, "solution_path": "data/solutions/generated/triinoct_004_daeed392d4fde104.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/daeed392d4fde104.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@5", "density": 0.6941546982548611, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/5.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": ".80367+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.80367, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@5", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": ".80367+", "metric_symbol": "s", "metric_value": 0.80367, "reference_text": "s = .80367+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/5.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = .80367+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "a0addde9772ff582", "improvement": -4.920494762683525e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8458025218911135e-06, "max_pair_overlap_depth": -2.636296310584707e-05, "metric_expression": "0.80372", "metric_symbol": "s", "metric_value": 0.8037192049476268, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.80367, "setup": "triinoct", "side": 0.8037192049476268, "solution_path": "data/solutions/generated/triinoct_005_a0addde9772ff582.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/a0addde9772ff582.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@6", "density": 0.8085136476900721, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/6.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": ".81574+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.81574, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@6", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": ".81574+", "metric_symbol": "s", "metric_value": 0.81574, "reference_text": "s = .81574+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/6.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = .81574+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "e7ac2b490b8118b6", "improvement": -5.196804299267832e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.999019070763348e-06, "max_pair_overlap_depth": -1.523375553563397e-05, "metric_expression": "0.81579", "metric_symbol": "s", "metric_value": 0.8157919680429927, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.81574, "setup": "triinoct", "side": 0.8157919680429927, "solution_path": "data/solutions/generated/triinoct_006_e7ac2b490b8118b6.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/e7ac2b490b8118b6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@7", "density": 0.7344053269175495, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/7.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": ".92453+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.92453, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@7", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": ".92453+", "metric_symbol": "s", "metric_value": 0.92453, "reference_text": "s = .92453+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/7.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = .92453+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "f1951fc893682894", "improvement": -1.6137063615029312e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.877767128601322e-06, "max_pair_overlap_depth": -4.487804835917597e-06, "metric_expression": "0.92455", "metric_symbol": "s", "metric_value": 0.924546137063615, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.92453, "setup": "triinoct", "side": 0.924546137063615, "solution_path": "data/solutions/generated/triinoct_007_f1951fc893682894.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/f1951fc893682894.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@8", "density": 0.7351940708693757, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/8.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": ".98782+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.98782, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@8", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": ".98782+", "metric_symbol": "s", "metric_value": 0.98782, "reference_text": "s = .98782+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/8.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = .98782+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "cdfdb28bbd7c958a", "improvement": -3.106647506945226e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6027742606714177e-06, "max_pair_overlap_depth": -8.534452089259403e-06, "metric_expression": "0.98785", "metric_symbol": "s", "metric_value": 0.9878510664750695, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.98782, "setup": "triinoct", "side": 0.9878510664750695, "solution_path": "data/solutions/generated/triinoct_008_cdfdb28bbd7c958a.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/cdfdb28bbd7c958a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@9", "density": 0.7931382286148597, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/9.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.00875+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.00875, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@9", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.00875+", "metric_symbol": "s", "metric_value": 1.00875, "reference_text": "s = 1.00875+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/9.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.00875+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "0a4766beda41f84b", "improvement": -2.4953411784212065e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.992353374519455e-07, "max_pair_overlap_depth": -5.523000253249144e-06, "metric_expression": "1.00877", "metric_symbol": "s", "metric_value": 1.0087749534117842, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.00875, "setup": "triinoct", "side": 1.0087749534117842, "solution_path": "data/solutions/generated/triinoct_009_0a4766beda41f84b.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/0a4766beda41f84b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@10", "density": 0.7747683390429897, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/10.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.07565+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.07565, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@10", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.07565+", "metric_symbol": "s", "metric_value": 1.07565, "reference_text": "s = 1.07565+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/10.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.07565+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "df6976316812d041", "improvement": -0.00022432883231693346, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3385179497460716e-06, "max_pair_overlap_depth": -1.8668924055276204e-05, "metric_expression": "1.07587", "metric_symbol": "s", "metric_value": 1.075874328832317, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.07565, "setup": "triinoct", "side": 1.075874328832317, "solution_path": "data/solutions/generated/triinoct_010_df6976316812d041.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/df6976316812d041.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@11", "density": 0.7515390994133405, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/11.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.14571+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.14571, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@11", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.14571+", "metric_symbol": "s", "metric_value": 1.14571, "reference_text": "s = 1.14571+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/11.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.14571+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "11036479a300844a", "improvement": 1.760556295749005e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.2318282708152993e-06, "max_pair_overlap_depth": -7.937162014970411e-06, "metric_expression": "1.14569", "metric_symbol": "s", "metric_value": 1.1456923944370425, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.14571, "setup": "triinoct", "side": 1.1456923944370425, "solution_path": "data/solutions/generated/triinoct_011_11036479a300844a.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/11036479a300844a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@12", "density": 0.74788706506276, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/12.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.19929+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.19929, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@12", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.19929+", "metric_symbol": "s", "metric_value": 1.19929, "reference_text": "s = 1.19929+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/12.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.19929+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "1fa5264e0f579a94", "improvement": -0.0002647906752053508, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2204761718366964e-06, "max_pair_overlap_depth": -1.4073169272155184e-06, "metric_expression": "1.19955", "metric_symbol": "s", "metric_value": 1.1995547906752053, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.19929, "setup": "triinoct", "side": 1.1995547906752053, "solution_path": "data/solutions/generated/triinoct_012_1fa5264e0f579a94.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/1fa5264e0f579a94.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@13", "density": 0.7604201292225432, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/13.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.23759+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.23759, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@13", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.23759+", "metric_symbol": "s", "metric_value": 1.23759, "reference_text": "s = 1.23759+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/13.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.23759+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "ebb5825b80b24749", "improvement": -0.0006144239243295324, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.763415979262496e-06, "max_pair_overlap_depth": -3.350137919300167e-06, "metric_expression": "1.23820", "metric_symbol": "s", "metric_value": 1.2382044239243295, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.23759, "setup": "triinoct", "side": 1.2382044239243295, "solution_path": "data/solutions/generated/triinoct_013_ebb5825b80b24749.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/ebb5825b80b24749.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@14", "density": 0.7704458772851622, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/14.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.27677+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.27677, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@14", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.27677+", "metric_symbol": "s", "metric_value": 1.27677, "reference_text": "s = 1.27677+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/14.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.27677+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "e00a18106d2fcfee", "improvement": 0.00021236461691676034, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.593194459024069e-06, "max_pair_overlap_depth": -5.874056281496287e-06, "metric_expression": "1.27656", "metric_symbol": "s", "metric_value": 1.2765576353830832, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.27677, "setup": "triinoct", "side": 1.2765576353830832, "solution_path": "data/solutions/generated/triinoct_014_e00a18106d2fcfee.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/e00a18106d2fcfee.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@15", "density": 0.7924237027261557, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/15.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.30275+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.30275, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@15", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.30275+", "metric_symbol": "s", "metric_value": 1.30275, "reference_text": "s = 1.30275+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/15.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.30275+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "e14eda962017d06d", "improvement": -0.0001598827367932376, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3032093708899356e-06, "max_pair_overlap_depth": -6.895788391969582e-06, "metric_expression": "1.30291", "metric_symbol": "s", "metric_value": 1.3029098827367933, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.30275, "setup": "triinoct", "side": 1.3029098827367933, "solution_path": "data/solutions/generated/triinoct_015_e14eda962017d06d.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/e14eda962017d06d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@16", "density": 0.7920450821484247, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/16.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.34584+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.34584, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@16", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.34584+", "metric_symbol": "s", "metric_value": 1.34584, "reference_text": "s = 1.34584+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/16.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.34584+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "640352ab60961c58", "improvement": -0.00012112963758759143, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.464975954309231e-06, "max_pair_overlap_depth": -7.38266966116008e-07, "metric_expression": "1.34596", "metric_symbol": "s", "metric_value": 1.3459611296375875, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.34584, "setup": "triinoct", "side": 1.3459611296375875, "solution_path": "data/solutions/generated/triinoct_16_640352ab60961c58.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/640352ab60961c58.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@17", "density": 0.7893502976721624, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/17.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.38964+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.38964, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@17", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.38964+", "metric_symbol": "s", "metric_value": 1.38964, "reference_text": "s = 1.38964+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/17.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.38964+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "6f00b933407c7c80", "improvement": -0.0001111742572472707, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.693112749768053e-07, "max_pair_overlap_depth": -2.384134229149737e-06, "metric_expression": "1.38975", "metric_symbol": "s", "metric_value": 1.3897511742572473, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.38964, "setup": "triinoct", "side": 1.3897511742572473, "solution_path": "data/solutions/generated/triinoct_017_6f00b933407c7c80.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/6f00b933407c7c80.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@18", "density": 0.8027011318687396, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/18.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.41836+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.41836, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@18", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.41836+", "metric_symbol": "s", "metric_value": 1.41836, "reference_text": "s = 1.41836+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/18.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.41836+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "91a694d1aa76789c", "improvement": 0.0002602110634655208, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.835525154559248e-06, "max_pair_overlap_depth": -9.960192069113916e-06, "metric_expression": "1.41810", "metric_symbol": "s", "metric_value": 1.4180997889365345, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.41836, "setup": "triinoct", "side": 1.4180997889365345, "solution_path": "data/solutions/generated/triinoct_018_91a694d1aa76789c.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/91a694d1aa76789c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@19", "density": 0.7911354610751659, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/19.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.46749+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.46749, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@19", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.46749+", "metric_symbol": "s", "metric_value": 1.46749, "reference_text": "s = 1.46749+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/19.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.46749+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "f92a5a5a3b54909a", "improvement": -8.00911599283527e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.983793930902579e-06, "max_pair_overlap_depth": -8.286875174401409e-06, "metric_expression": "1.46757", "metric_symbol": "s", "metric_value": 1.4675700911599283, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.46749, "setup": "triinoct", "side": 1.4675700911599283, "solution_path": "data/solutions/generated/triinoct_019_f92a5a5a3b54909a.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/f92a5a5a3b54909a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@20", "density": 0.7990175319982477, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/20.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.49832+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.49832, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@20", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.49832+", "metric_symbol": "s", "metric_value": 1.49832, "reference_text": "s = 1.49832+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "20.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/20.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.49832+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "ead92a925f9f00a7", "improvement": 6.988050923584233e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3276003687482804e-06, "max_pair_overlap_depth": -2.2158170135400113e-06, "metric_expression": "1.49825", "metric_symbol": "s", "metric_value": 1.4982501194907643, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.49832, "setup": "triinoct", "side": 1.4982501194907643, "solution_path": "data/solutions/generated/triinoct_020_ead92a925f9f00a7.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/ead92a925f9f00a7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@21", "density": 0.8078137663314585, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/21.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.52615+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.52615, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@21", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.52615+", "metric_symbol": "s", "metric_value": 1.52615, "reference_text": "s = 1.52615+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/21.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.52615+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "e350f46688d32a68", "improvement": -0.0007180233722772211, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.407426670083604e-06, "max_pair_overlap_depth": -1.1097264517423433e-05, "metric_expression": "1.52687", "metric_symbol": "s", "metric_value": 1.5268680233722771, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.52615, "setup": "triinoct", "side": 1.5268680233722771, "solution_path": "data/solutions/generated/triinoct_021_e350f46688d32a68.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/e350f46688d32a68.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@22", "density": 0.8155579012833859, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/22.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.55545+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.55545, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@22", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.55545+", "metric_symbol": "s", "metric_value": 1.55545, "reference_text": "s = 1.55545+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/22.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.55545+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "c1314ed0099ebdb6", "improvement": 8.823677085589843e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5323101194107617e-06, "max_pair_overlap_depth": -6.490133103720375e-06, "metric_expression": "1.55536", "metric_symbol": "s", "metric_value": 1.555361763229144, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.55545, "setup": "triinoct", "side": 1.555361763229144, "solution_path": "data/solutions/generated/triinoct_022_c1314ed0099ebdb6.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/c1314ed0099ebdb6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@23", "density": 0.8210902170609379, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/23.gif", "friedman_metric_expression": "1.58458+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.58458, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@23", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.58458+", "metric_symbol": "s", "metric_value": 1.58458, "reference_text": "s = 1.58458+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/23.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.58458+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "4ca1a825a22b1f04", "improvement": -0.0003714249913302492, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.321393677649287e-06, "max_pair_overlap_depth": -1.5960535921788477e-06, "metric_expression": "1.58495", "metric_symbol": "s", "metric_value": 1.5849514249913303, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.58458, "setup": "triinoct", "side": 1.5849514249913303, "solution_path": "data/solutions/generated/triinoct_023_4ca1a825a22b1f04.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/4ca1a825a22b1f04.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@24", "density": 0.8373304168984339, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/24.gif", "friedman_metric_expression": "1.60261+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.60261, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@24", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.60261+", "metric_symbol": "s", "metric_value": 1.60261, "reference_text": "s = 1.60261+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/24.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.60261+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "e197b92ecac28558", "improvement": -0.0006526103809141315, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.085338835265612e-06, "max_pair_overlap_depth": -9.98840296584369e-06, "metric_expression": "1.60326", "metric_symbol": "s", "metric_value": 1.6032626103809142, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.60261, "setup": "triinoct", "side": 1.6032626103809142, "solution_path": "data/solutions/generated/triinoct_024_e197b92ecac28558.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/e197b92ecac28558.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@25", "density": 0.8354970198268096, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/25.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.63797+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.63797, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@25", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.63797+", "metric_symbol": "s", "metric_value": 1.63797, "reference_text": "s = 1.63797+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/25.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.63797+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "e322401fc5bafe4d", "improvement": -0.00014742221399233202, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.350806641875647e-06, "max_pair_overlap_depth": -2.3762008580785476e-06, "metric_expression": "1.63812", "metric_symbol": "s", "metric_value": 1.6381174222139923, "n": 25, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.63797, "setup": "triinoct", "side": 1.6381174222139923, "solution_path": "data/solutions/generated/triinoct_025_e322401fc5bafe4d.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/e322401fc5bafe4d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@26", "density": 0.8292613760828756, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/26.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.67666+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.67666, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@26", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.67666+", "metric_symbol": "s", "metric_value": 1.67666, "reference_text": "s = 1.67666+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/26.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.67666+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "7cb3797cd10b21d8", "improvement": -0.0001676725391090983, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6314473196666756e-05, "max_pair_overlap_depth": -5.591511314406716e-06, "metric_expression": "1.67683", "metric_symbol": "s", "metric_value": 1.6768276725391091, "n": 26, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.67666, "setup": "triinoct", "side": 1.6768276725391091, "solution_path": "data/solutions/generated/triinoct_026_7cb3797cd10b21d8.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/7cb3797cd10b21d8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@27", "density": 0.8297818493323376, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/27.gif", "friedman_metric_expression": "1.70882+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.70882, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@27", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.70882+", "metric_symbol": "s", "metric_value": 1.70882, "reference_text": "s = 1.70882+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/27.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.70882+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "c7748eb19b34ff2e", "improvement": 0.0005858720037632548, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.055699391789948e-06, "max_pair_overlap_depth": -3.877030372301249e-07, "metric_expression": "1.70823", "metric_symbol": "s", "metric_value": 1.7082341279962368, "n": 27, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.70882, "setup": "triinoct", "side": 1.7082341279962368, "solution_path": "data/solutions/generated/triinoct_27_c7748eb19b34ff2e.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/c7748eb19b34ff2e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@28", "density": 0.8299146316504411, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/28.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.73954+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.73954, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@28", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.73954+", "metric_symbol": "s", "metric_value": 1.73954, "reference_text": "s = 1.73954+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/28.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.73954+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "26e8795d82a701d8", "improvement": 9.867973407984287e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2195571711171888e-06, "max_pair_overlap_depth": -8.650951147992636e-06, "metric_expression": "1.73944", "metric_symbol": "s", "metric_value": 1.7394413202659202, "n": 28, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.73954, "setup": "triinoct", "side": 1.7394413202659202, "solution_path": "data/solutions/generated/triinoct_028_26e8795d82a701d8.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/26e8795d82a701d8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@29", "density": 0.8439313801960897, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/29.gif", "friedman_metric_expression": "1.75576+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.75576, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@29", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.75576+", "metric_symbol": "s", "metric_value": 1.75576, "reference_text": "s = 1.75576+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/29.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.75576+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "0575780ea85d68dd", "improvement": 0.00029203482208850495, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.530531415212805e-06, "max_pair_overlap_depth": -1.1528245769332202e-05, "metric_expression": "1.75547", "metric_symbol": "s", "metric_value": 1.7554679651779115, "n": 29, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.75576, "setup": "triinoct", "side": 1.7554679651779115, "solution_path": "data/solutions/generated/triinoct_29_0575780ea85d68dd.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/0575780ea85d68dd.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinoct@30", "density": 0.8466211131481989, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/30.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.78101+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.78101, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@30", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.78101+", "metric_symbol": "s", "metric_value": 1.78101, "reference_text": "s = 1.78101+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/30.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.78101+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "4c08385c68f67e77", "improvement": -0.0016296314900829945, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.472767326523421e-06, "max_pair_overlap_depth": -3.1822885246945987e-06, "metric_expression": "1.78264", "metric_symbol": "s", "metric_value": 1.782639631490083, "n": 30, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.78101, "setup": "triinoct", "side": 1.782639631490083, "solution_path": "data/solutions/generated/triinoct_030_4c08385c68f67e77.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/4c08385c68f67e77.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@1", "density": 0.4121951299103008, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/1.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": ".781+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.781, "friedman_reference": {"analytical_or_proved": true, "case": "triinpen@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": ".781+", "metric_symbol": "s", "metric_value": 0.781, "reference_text": "s = cos(42\u00b0) / cos(18\u00b0) | s = .781+ | Trivial. |", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/1.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = cos(42\u00b0) / cos(18\u00b0) | s = .781+ | Trivial. |", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "6d3af7533c544064", "improvement": -0.0004017935524769367, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.010333690326338e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.78140", "metric_symbol": "s", "metric_value": 0.781401793552477, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.781, "setup": "triinpen", "side": 0.781401793552477, "solution_path": "data/solutions/generated/triinpen_001_6d3af7533c544064.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6d3af7533c544064.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@2", "density": 0.4392461428307515, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/2.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.070+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.07, "friedman_reference": {"analytical_or_proved": true, "case": "triinpen@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1.070+", "metric_symbol": "s", "metric_value": 1.07, "reference_text": "s = \u221a(6/(3+\u221a5)) | s = 1.070+ | Trivial. |", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/2.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = \u221a(6/(3+\u221a5)) | s = 1.070+ | Trivial. |", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "eebb278fed1959e5", "improvement": -0.0005004432401414238, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.540244142797655e-06, "max_pair_overlap_depth": -4.677606255020384e-06, "metric_expression": "1.07050", "metric_symbol": "s", "metric_value": 1.0705004432401415, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.07, "setup": "triinpen", "side": 1.0705004432401415, "solution_path": "data/solutions/generated/triinpen_002_eebb278fed1959e5.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/eebb278fed1959e5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@3", "density": 0.5773621092377123, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/3.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.143+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.143, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@3", "credit": "Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "1.143+", "metric_symbol": "s", "metric_value": 1.143, "reference_text": "s = 1.143+ | Found by Maurizio Morandi | in August 2012. |", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/3.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.143+ | Found by Maurizio Morandi | in August 2012. |", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "6ec3c588fb678646", "improvement": -0.0005688574140261249, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4403665141977852e-06, "max_pair_overlap_depth": -4.738354987265936e-06, "metric_expression": "1.14357", "metric_symbol": "s", "metric_value": 1.1435688574140261, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.143, "setup": "triinpen", "side": 1.1435688574140261, "solution_path": "data/solutions/generated/triinpen_003_6ec3c588fb678646.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/6ec3c588fb678646.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@4", "density": 0.6999227205319379, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/4.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.199+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.199, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@4", "credit": "Maurizio Morandi", "date": "July 2012", "family_updated": "", "metric_expression": "1.199+", "metric_symbol": "s", "metric_value": 1.199, "reference_text": "s = 1.199+ | Found by Maurizio Morandi | in July 2012. |", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/4.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.199+ | Found by Maurizio Morandi | in July 2012. |", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "c45dd63fd5a8806b", "improvement": -0.0003081410566185827, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9487802360351315e-06, "max_pair_overlap_depth": -2.0674441409240796e-05, "metric_expression": "1.19931", "metric_symbol": "s", "metric_value": 1.1993081410566186, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.199, "setup": "triinpen", "side": 1.1993081410566186, "solution_path": "data/solutions/generated/triinpen_004_c45dd63fd5a8806b.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/c45dd63fd5a8806b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@5", "density": 0.818292233169356, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "March 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/5.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.240+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.24, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@5", "credit": "Erich Friedman", "date": "March 2012", "family_updated": "", "metric_expression": "1.240+", "metric_symbol": "s", "metric_value": 1.24, "reference_text": "s = 1 + 2sin(12\u00b0)/\u221a3 | s = 1.240+ | Found by Erich Friedman | in March 2012.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/5.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1 + 2sin(12\u00b0)/\u221a3 | s = 1.240+ | Found by Erich Friedman | in March 2012.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "a99f063702c9d4e8", "improvement": -9.974870359874544e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.477405190997757e-06, "max_pair_overlap_depth": -1.386320747689962e-05, "metric_expression": "1.24010", "metric_symbol": "s", "metric_value": 1.2400997487035987, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.24, "setup": "triinpen", "side": 1.2400997487035987, "solution_path": "data/solutions/generated/triinpen_005_a99f063702c9d4e8.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a99f063702c9d4e8.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@6", "density": 0.7771309707429922, "friedman_analytical_or_proved": false, "friedman_credit": "Dominik Kamp", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/6.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.393+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.393, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@6", "credit": "Dominik Kamp", "date": "April 2026", "family_updated": "", "metric_expression": "1.393+", "metric_symbol": "s", "metric_value": 1.393, "reference_text": "s = 1.393+ | Found by Dominik Kamp | in April 2026.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/6.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.393+ | Found by Dominik Kamp | in April 2026.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "8d12939187f41554", "improvement": -0.0009729568537997224, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4992545061493154e-06, "max_pair_overlap_depth": -6.033095825291185e-06, "metric_expression": "1.39397", "metric_symbol": "s", "metric_value": 1.3939729568537997, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.393, "setup": "triinpen", "side": 1.3939729568537997, "solution_path": "data/solutions/generated/triinpen_006_8d12939187f41554.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Dominik Kamp", "svg_path": "data/svg/8d12939187f41554.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@7", "density": 0.7037922543518549, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/7.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.581+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.581, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@7", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "1.581+", "metric_symbol": "s", "metric_value": 1.581, "reference_text": "s = 1.581+ | Found by David W. Cantrell | in July 2012.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/7.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.581+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "d149692a95cde76c", "improvement": -0.0011680508685436575, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5825119043122948e-06, "max_pair_overlap_depth": -1.774591125650038e-05, "metric_expression": "1.58217", "metric_symbol": "s", "metric_value": 1.5821680508685436, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.581, "setup": "triinpen", "side": 1.5821680508685436, "solution_path": "data/solutions/generated/triinpen_007_d149692a95cde76c.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d149692a95cde76c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@8", "density": 0.7311911118691055, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/8.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.659+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.659, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@8", "credit": "Maurizio Morandi", "date": "July 2012", "family_updated": "", "metric_expression": "1.659+", "metric_symbol": "s", "metric_value": 1.659, "reference_text": "s = 1.659+ | Found by Maurizio Morandi | in July 2012.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/8.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.659+ | Found by Maurizio Morandi | in July 2012.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "2ac4629603f9ce00", "improvement": -0.0004163833859911481, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2708160845523109e-06, "max_pair_overlap_depth": -2.147963686768839e-05, "metric_expression": "1.65942", "metric_symbol": "s", "metric_value": 1.6594163833859912, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.659, "setup": "triinpen", "side": 1.6594163833859912, "solution_path": "data/solutions/generated/triinpen_008_2ac4629603f9ce00.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/2ac4629603f9ce00.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@9", "density": 0.7240666005522262, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/9.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.767+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.767, "friedman_reference": {"analytical_or_proved": true, "case": "triinpen@9", "credit": "Maurizio Morandi", "date": "July 2012", "family_updated": "", "metric_expression": "1.767+", "metric_symbol": "s", "metric_value": 1.767, "reference_text": "s = 1.767+ | Found by Maurizio Morandi in July 2012. | Improved slightly by David W. Cantrell in August 2012.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/9.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.767+ | Found by Maurizio Morandi in July 2012. | Improved slightly by David W. Cantrell in August 2012.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "72279a3b92ae69ca", "improvement": -0.0017148783247660226, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.918216101924126e-06, "max_pair_overlap_depth": -2.2857021414468548e-08, "metric_expression": "1.76871", "metric_symbol": "s", "metric_value": 1.768714878324766, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.767, "setup": "triinpen", "side": 1.768714878324766, "solution_path": "data/solutions/generated/triinpen_009_72279a3b92ae69ca.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/72279a3b92ae69ca.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@10", "density": 0.7626299188806616, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/10.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.815+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.815, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@10", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "1.815+", "metric_symbol": "s", "metric_value": 1.815, "reference_text": "s = 1.815+ | Found by Timo Berthold et al | in January 2026.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/10.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.815+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "f05f442bda1c650b", "improvement": -0.0016401662315850274, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.176875820916777e-06, "max_pair_overlap_depth": -3.546498963225009e-06, "metric_expression": "1.81664", "metric_symbol": "s", "metric_value": 1.816640166231585, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.815, "setup": "triinpen", "side": 1.816640166231585, "solution_path": "data/solutions/generated/triinpen_010_f05f442bda1c650b.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/f05f442bda1c650b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@11", "density": 0.7898687600512876, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/11.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.871+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.871, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@11", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "1.871+", "metric_symbol": "s", "metric_value": 1.871, "reference_text": "s = 1.871+ | Found by Timo Berthold et al | in January 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/11.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.871+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "ca41afc6c9b25b21", "improvement": -0.0011675168118052692, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.191218876790302e-06, "max_pair_overlap_depth": -1.6407344776342425e-05, "metric_expression": "1.87217", "metric_symbol": "s", "metric_value": 1.8721675168118053, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.871, "setup": "triinpen", "side": 1.8721675168118053, "solution_path": "data/solutions/generated/triinpen_011_ca41afc6c9b25b21.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/ca41afc6c9b25b21.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@12", "density": 0.774567014021955, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/12.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.973+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.973, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@12", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.973+", "metric_symbol": "s", "metric_value": 1.973, "reference_text": "s = 1.973+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/12.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.973+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "bf7ddcc735398246", "improvement": -0.0016355699317964767, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.087403974087953e-06, "max_pair_overlap_depth": -1.3435552785878713e-05, "metric_expression": "1.97464", "metric_symbol": "s", "metric_value": 1.9746355699317966, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.973, "setup": "triinpen", "side": 1.9746355699317966, "solution_path": "data/solutions/generated/triinpen_012_bf7ddcc735398246.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/bf7ddcc735398246.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@13", "density": 0.8029051605923488, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/13.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.017+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.017, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@13", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "2.017+", "metric_symbol": "s", "metric_value": 2.017, "reference_text": "s = 2.017+ | Found by Timo Berthold et al | in January 2026.", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/13.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.017+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "ff07ea08219353cb", "improvement": -0.0016702531518792973, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.615779077054135e-07, "max_pair_overlap_depth": -4.3930654851198625e-05, "metric_expression": "2.01867", "metric_symbol": "s", "metric_value": 2.018670253151879, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.017, "setup": "triinpen", "side": 2.018670253151879, "solution_path": "data/solutions/generated/triinpen_013_ff07ea08219353cb.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/ff07ea08219353cb.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@14", "density": 0.8070990172681682, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/14.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.088+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.088, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@14", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "2.088+", "metric_symbol": "s", "metric_value": 2.088, "reference_text": "s = 2.088+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/14.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.088+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "bc900c323229ee0c", "improvement": -0.0014233339778941811, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2035527022044616e-06, "max_pair_overlap_depth": -3.9869300561434784e-05, "metric_expression": "2.08942", "metric_symbol": "s", "metric_value": 2.0894233339778943, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.088, "setup": "triinpen", "side": 2.0894233339778943, "solution_path": "data/solutions/generated/triinpen_014_bc900c323229ee0c.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/bc900c323229ee0c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@15", "density": 0.8370246386712876, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/15.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.123+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.123, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@15", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.123+", "metric_symbol": "s", "metric_value": 2.123, "reference_text": "s = 2.123+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/15.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.123+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "39b13eec51bf2ea6", "improvement": -0.000744857935933041, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.959192129393714e-06, "max_pair_overlap_depth": -3.112528873211451e-05, "metric_expression": "2.12374", "metric_symbol": "s", "metric_value": 2.1237448579359333, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.123, "setup": "triinpen", "side": 2.1237448579359333, "solution_path": "data/solutions/generated/triinpen_015_39b13eec51bf2ea6.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/39b13eec51bf2ea6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@16", "density": 0.8497807683719915, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/16.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.17598+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.17598, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@16", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.17598+", "metric_symbol": "s", "metric_value": 2.17598, "reference_text": "s = 2.17598+ | Found by Ian Watson | in April 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/16.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.17598+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "852225bc33edeebe", "improvement": -0.0008893975034314217, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.337598971955443e-06, "max_pair_overlap_depth": -3.396060024770525e-05, "metric_expression": "2.17687", "metric_symbol": "s", "metric_value": 2.1768693975034314, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.17598, "setup": "triinpen", "side": 2.1768693975034314, "solution_path": "data/solutions/generated/triinpen_016_852225bc33edeebe.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/852225bc33edeebe.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@17", "density": 0.8546812093159708, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/17.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.23658+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.23658, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@17", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.23658+", "metric_symbol": "s", "metric_value": 2.23658, "reference_text": "s = 2.23658+ | Found by Ian Watson | in April 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/17.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.23658+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "a8618eee7fdaba2a", "improvement": -0.0008435999825509022, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.39992551681681e-07, "max_pair_overlap_depth": -1.7412872881594765e-05, "metric_expression": "2.23742", "metric_symbol": "s", "metric_value": 2.237423599982551, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.23658, "setup": "triinpen", "side": 2.237423599982551, "solution_path": "data/solutions/generated/triinpen_017_a8618eee7fdaba2a.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/a8618eee7fdaba2a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@18", "density": 0.8594655455888001, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/18.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.29382+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.29382, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@18", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.29382+", "metric_symbol": "s", "metric_value": 2.29382, "reference_text": "s = 2.29382+ | Found by Ian Watson | in April 2026.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/18.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.29382+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "6bf4f9b0915d39ee", "improvement": -0.0020529370652000445, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4774387061990001e-05, "max_pair_overlap_depth": -2.9541451285197695e-08, "metric_expression": "2.29587", "metric_symbol": "s", "metric_value": 2.2958729370652002, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.29382, "setup": "triinpen", "side": 2.2958729370652002, "solution_path": "data/solutions/generated/triinpen_018_6bf4f9b0915d39ee.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/6bf4f9b0915d39ee.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@19", "density": 0.868021512388566, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/19.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.344+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.344, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@19", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.344+", "metric_symbol": "s", "metric_value": 2.344, "reference_text": "s = 2.344+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/19.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.344+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "eaa1430599d74148", "improvement": -0.003131316702724707, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9139572150349214e-06, "max_pair_overlap_depth": -1.0705368373109536e-05, "metric_expression": "2.34713", "metric_symbol": "s", "metric_value": 2.3471313167027246, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.344, "setup": "triinpen", "side": 2.3471313167027246, "solution_path": "data/solutions/generated/triinpen_19_eaa1430599d74148.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/eaa1430599d74148.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@20", "density": 0.8487102106795418, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/20.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.43350+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.4335, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@20", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.43350+", "metric_symbol": "s", "metric_value": 2.4335, "reference_text": "s = 2.43350+ | Found by Ian Watson | in April 2026.", "source_image": "20.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/20.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.43350+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "760faebc4d0b131b", "improvement": -0.001848490502489586, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.176183787084704e-07, "max_pair_overlap_depth": -8.807499014995557e-06, "metric_expression": "2.43535", "metric_symbol": "s", "metric_value": 2.4353484905024896, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.4335, "setup": "triinpen", "side": 2.4353484905024896, "solution_path": "data/solutions/generated/triinpen_020_760faebc4d0b131b.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/760faebc4d0b131b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@21", "density": 0.8589998748063701, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/21.gif", "friedman_metric_expression": "2.479+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.479, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@21", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.479+", "metric_symbol": "s", "metric_value": 2.479, "reference_text": "s = 2.479+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/21.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.479+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "115c20fad199fe9d", "improvement": -0.0014982727861765355, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.811024693767028e-06, "max_pair_overlap_depth": -2.8209410938284663e-06, "metric_expression": "2.48050", "metric_symbol": "s", "metric_value": 2.4804982727861766, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.479, "setup": "triinpen", "side": 2.4804982727861766, "solution_path": "data/solutions/generated/triinpen_021_115c20fad199fe9d.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/115c20fad199fe9d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@22", "density": 0.8389716031782235, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/22.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.56740+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5674, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@22", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.56740+", "metric_symbol": "s", "metric_value": 2.5674, "reference_text": "s = 2.56740+ | Found by Ian Watson | in April 2026.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/22.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.56740+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "929a1aedb3cfabb6", "improvement": -0.0015966714037367602, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.137059676377362e-07, "max_pair_overlap_depth": -8.172541590134585e-06, "metric_expression": "2.56900", "metric_symbol": "s", "metric_value": 2.568996671403737, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.5674, "setup": "triinpen", "side": 2.568996671403737, "solution_path": "data/solutions/generated/triinpen_022_929a1aedb3cfabb6.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/929a1aedb3cfabb6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@23", "density": 0.8432972889244729, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/23.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.61831+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.61831, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@23", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.61831+", "metric_symbol": "s", "metric_value": 2.61831, "reference_text": "s = 2.61831+ | Found by Ian Watson | in April 2026.", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/23.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.61831+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "c0d3905a26f7ef33", "improvement": -0.001678576890473682, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.992582537021139e-06, "max_pair_overlap_depth": -4.362218652298466e-06, "metric_expression": "2.61999", "metric_symbol": "s", "metric_value": 2.619988576890474, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.61831, "setup": "triinpen", "side": 2.619988576890474, "solution_path": "data/solutions/generated/triinpen_023_c0d3905a26f7ef33.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/c0d3905a26f7ef33.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@24", "density": 0.8488309418600493, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/24.gif", "friedman_metric_expression": "2.66641+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.66641, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@24", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.66641+", "metric_symbol": "s", "metric_value": 2.66641, "reference_text": "s = 2.66641+ | Found by Ian Watson | in April 2026.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/24.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.66641+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "7bc3466b163ec849", "improvement": -0.0011908775198370414, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.421732660736982e-06, "max_pair_overlap_depth": -3.3708625798434255e-05, "metric_expression": "2.66760", "metric_symbol": "s", "metric_value": 2.667600877519837, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.66641, "setup": "triinpen", "side": 2.667600877519837, "solution_path": "data/solutions/generated/triinpen_24_7bc3466b163ec849.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/7bc3466b163ec849.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@25", "density": 0.8491607934363294, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/25.gif", "friedman_metric_expression": "2.720+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.72, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@25", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.720+", "metric_symbol": "s", "metric_value": 2.72, "reference_text": "s = 2.720+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/25.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.720+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "0f20f89eba855e21", "improvement": -0.0020799025832594076, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.997044244500036e-06, "max_pair_overlap_depth": -8.121085018242269e-06, "metric_expression": "2.72208", "metric_symbol": "s", "metric_value": 2.7220799025832596, "n": 25, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.72, "setup": "triinpen", "side": 2.7220799025832596, "solution_path": "data/solutions/generated/triinpen_25_0f20f89eba855e21.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/0f20f89eba855e21.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@26", "density": 0.8520841245254238, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/26.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.771+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.771, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@26", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.771+", "metric_symbol": "s", "metric_value": 2.771, "reference_text": "s = 2.771+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/26.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.771+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "f991467944e6e634", "improvement": -0.00022168609627781777, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2986540650513047e-06, "max_pair_overlap_depth": -8.90414151222263e-06, "metric_expression": "2.77122", "metric_symbol": "s", "metric_value": 2.7712216860962777, "n": 26, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.771, "setup": "triinpen", "side": 2.7712216860962777, "solution_path": "data/solutions/generated/triinpen_026_f991467944e6e634.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/f991467944e6e634.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@27", "density": 0.8551322064876743, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/27.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.81761+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.81761, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@27", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.81761+", "metric_symbol": "s", "metric_value": 2.81761, "reference_text": "s = 2.81761+ | Found by Ian Watson | in April 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/27.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.81761+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "d6f1d3073209f163", "improvement": -0.0013640775641627911, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.642891580892597e-06, "max_pair_overlap_depth": -2.2253597504739986e-06, "metric_expression": "2.81897", "metric_symbol": "s", "metric_value": 2.818974077564163, "n": 27, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.81761, "setup": "triinpen", "side": 2.818974077564163, "solution_path": "data/solutions/generated/triinpen_027_d6f1d3073209f163.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/d6f1d3073209f163.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@28", "density": 0.8558072548880046, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/28.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.868+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.868, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@28", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.868+", "metric_symbol": "s", "metric_value": 2.868, "reference_text": "s = 2.868+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/28.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.868+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "2caa877f5aa08139", "improvement": -0.0015702794780372287, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.305293369328324e-06, "max_pair_overlap_depth": -1.9072024115107666e-05, "metric_expression": "2.86957", "metric_symbol": "s", "metric_value": 2.869570279478037, "n": 28, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.868, "setup": "triinpen", "side": 2.869570279478037, "solution_path": "data/solutions/generated/triinpen_028_2caa877f5aa08139.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T03:19:00+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/2caa877f5aa08139.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@29", "density": 0.8642558401713053, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/29.gif", "friedman_metric_expression": "2.903+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.903, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@29", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "2.903+", "metric_symbol": "s", "metric_value": 2.903, "reference_text": "s = 2.903+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/29.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.903+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "9730db96941e5b52", "improvement": -0.003053928319848076, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.800701997238751e-06, "max_pair_overlap_depth": -5.0014115865471e-06, "metric_expression": "2.90605", "metric_symbol": "s", "metric_value": 2.906053928319848, "n": 29, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.903, "setup": "triinpen", "side": 2.906053928319848, "solution_path": "data/solutions/generated/triinpen_29_9730db96941e5b52.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/9730db96941e5b52.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@30", "density": 0.8662355090431414, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/30.gif", "friedman_metric_expression": "2.952+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.952, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@30", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "2.952+", "metric_symbol": "s", "metric_value": 2.952, "reference_text": "s = 2.952+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/30.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.952+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "29e4e0e684891d8a", "improvement": -0.00035425771249730786, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.990812967166742e-06, "max_pair_overlap_depth": -2.800943013125945e-05, "metric_expression": "2.95235", "metric_symbol": "s", "metric_value": 2.9523542577124973, "n": 30, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.952, "setup": "triinpen", "side": 2.9523542577124973, "solution_path": "data/solutions/generated/triinpen_30_29e4e0e684891d8a.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/29e4e0e684891d8a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@31", "density": 0.8713888671331285, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/31.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.992+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.992, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "2.992+", "metric_symbol": "s", "metric_value": 2.992, "reference_text": "s = 2.992+ | Found by Emerson Connelly | in April 2026.", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/31.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.992+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "44a0bb3f581e92e2", "improvement": -0.0002692889761504169, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7795814276238894e-06, "max_pair_overlap_depth": -1.3596177029562284e-05, "metric_expression": "2.99227", "metric_symbol": "s", "metric_value": 2.9922692889761504, "n": 31, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.992, "setup": "triinpen", "side": 2.9922692889761504, "solution_path": "data/solutions/generated/triinpen_031_44a0bb3f581e92e2.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/44a0bb3f581e92e2.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@32", "density": 0.8781198099192385, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/32.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.029+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.029, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@32", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.029+", "metric_symbol": "s", "metric_value": 3.029, "reference_text": "s = 3.029+ | Found by Emerson Connelly | in April 2026.", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/32.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.029+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "310badf403ba4fce", "improvement": 0.0005254132016050406, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.877563686504985e-07, "max_pair_overlap_depth": -2.5172545809359903e-05, "metric_expression": "3.02847", "metric_symbol": "s", "metric_value": 3.028474586798395, "n": 32, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.029, "setup": "triinpen", "side": 3.028474586798395, "solution_path": "data/solutions/generated/triinpen_032_310badf403ba4fce.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/310badf403ba4fce.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@33", "density": 0.8767362450469284, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/33.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.078+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.078, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@33", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.078+", "metric_symbol": "s", "metric_value": 3.078, "reference_text": "s = 3.078+ | Found by Emerson Connelly | in April 2026.", "source_image": "33.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/33.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.078+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "2cf1b1f38810795d", "improvement": 0.00014382895343612034, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9579559668821387e-06, "max_pair_overlap_depth": -6.152141676318834e-06, "metric_expression": "3.07786", "metric_symbol": "s", "metric_value": 3.0778561710465637, "n": 33, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.078, "setup": "triinpen", "side": 3.0778561710465637, "solution_path": "data/solutions/generated/triinpen_033_2cf1b1f38810795d.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/2cf1b1f38810795d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@34", "density": 0.8805067280135348, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/34.gif", "friedman_metric_expression": "3.114+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.114, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@34", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.114+", "metric_symbol": "s", "metric_value": 3.114, "reference_text": "s = 3.114+ | Found by Emerson Connelly | in April 2026.", "source_image": "34.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/34.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.114+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "faff9872314d8b1a", "improvement": -0.003446083050953952, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4825467113644516e-07, "max_pair_overlap_depth": -8.773672566686486e-06, "metric_expression": "3.11745", "metric_symbol": "s", "metric_value": 3.117446083050954, "n": 34, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.114, "setup": "triinpen", "side": 3.117446083050954, "solution_path": "data/solutions/generated/triinpen_34_faff9872314d8b1a.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T14:34:18+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/faff9872314d8b1a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@35", "density": 0.8875886550862264, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/35.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.150+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.15, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@35", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.150+", "metric_symbol": "s", "metric_value": 3.15, "reference_text": "s = 3.150+ | Found by Emerson Connelly | in April 2026.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/35.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.150+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "8fe5da1df972c0b3", "improvement": -0.0003150122852528803, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4152162741179605e-06, "max_pair_overlap_depth": -9.46789992128938e-06, "metric_expression": "3.15032", "metric_symbol": "s", "metric_value": 3.150315012285253, "n": 35, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.15, "setup": "triinpen", "side": 3.150315012285253, "solution_path": "data/solutions/generated/triinpen_035_8fe5da1df972c0b3.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/8fe5da1df972c0b3.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@36", "density": 0.8985572699958073, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/36.gif", "friedman_metric_expression": "3.177+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.177, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@36", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.177+", "metric_symbol": "s", "metric_value": 3.177, "reference_text": "s = 3.177+ | Found by Emerson Connelly | in April 2026.", "source_image": "36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/36.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.177+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "5183a03ffc16bcf9", "improvement": 0.0015578799748001693, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.498547309348074e-07, "max_pair_overlap_depth": -2.3029919912209706e-06, "metric_expression": "3.17544", "metric_symbol": "s", "metric_value": 3.1754421200252, "n": 36, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.177, "setup": "triinpen", "side": 3.1754421200252, "solution_path": "data/solutions/record_hunt/triinpen_36_5183a03ffc16bcf9.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/5183a03ffc16bcf9.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@37", "density": 0.9016389565813171, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/37.gif", "friedman_metric_expression": "3.21529+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.21529, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@37", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "3.21529+", "metric_symbol": "s", "metric_value": 3.21529, "reference_text": "s = 3.21529+ | Found by Nathan Roll | in April 2026.", "source_image": "37.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/37.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.21529+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "08996b45ff04652c", "improvement": 0.0015527990845796147, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.641732962649716e-07, "max_pair_overlap_depth": -1.845774424413804e-07, "metric_expression": "3.21374", "metric_symbol": "s", "metric_value": 3.2137372009154204, "n": 37, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.21529, "setup": "triinpen", "side": 3.213737200915421, "solution_path": "data/solutions/record_hunt/triinpen_37_08996b45ff04652c.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/08996b45ff04652c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@38", "density": 0.9010687984988149, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/38.gif", "friedman_metric_expression": "3.25877+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.25877, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@38", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "3.25877+", "metric_symbol": "s", "metric_value": 3.25877, "reference_text": "s = 3.25877+ | Found by Nathan Roll | in April 2026.", "source_image": "38.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/38.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.25877+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "38.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "cca381f54ee9b299", "improvement": 0.0008632130758914336, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.177049878621715e-07, "max_pair_overlap_depth": -8.114145322313249e-07, "metric_expression": "3.25791", "metric_symbol": "s", "metric_value": 3.2579067869241087, "n": 38, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.25877, "setup": "triinpen", "side": 3.2579067869241087, "solution_path": "data/solutions/record_hunt/triinpen_38_cca381f54ee9b299.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/cca381f54ee9b299.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@39", "density": 0.9025101556269666, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/39.gif", "friedman_metric_expression": "3.295+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.295, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@39", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.295+", "metric_symbol": "s", "metric_value": 3.295, "reference_text": "s = 3.295+ | Found by Emerson Connelly | in April 2026.", "source_image": "39.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/39.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.295+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "67a6bbf24108966a", "improvement": -0.002859023967915597, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9740347935670834e-06, "max_pair_overlap_depth": -1.0007850107629501e-05, "metric_expression": "3.29786", "metric_symbol": "s", "metric_value": 3.2978590239679155, "n": 39, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.295, "setup": "triinpen", "side": 3.2978590239679155, "solution_path": "data/solutions/generated/triinpen_039_67a6bbf24108966a.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/67a6bbf24108966a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinpen@40", "density": 0.9091859235070038, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/40.gif", "friedman_metric_expression": "3.32786+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.32786, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@40", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "3.32786+", "metric_symbol": "s", "metric_value": 3.32786, "reference_text": "s = 3.32786+ | Found by Nathan Roll | in April 2026.", "source_image": "40.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/40.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.32786+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "032ceee4e789eff6", "improvement": 0.0002725631872597667, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4037127820689358e-07, "max_pair_overlap_depth": -1.074379339049969e-06, "metric_expression": "3.32759", "metric_symbol": "s", "metric_value": 3.32758743681274, "n": 40, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.32786, "setup": "triinpen", "side": 3.3275874368127405, "solution_path": "data/solutions/record_hunt/triinpen_40_032ceee4e789eff6.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/032ceee4e789eff6.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@1", "density": 0.4640965865243202, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t1.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(\u221a2+\u221a6)/4 = .965+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.965, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(\u221a2+\u221a6)/4 = .965+", "metric_symbol": "s", "metric_value": 0.965, "reference_text": "s = (\u221a2+\u221a6)/4 = .965+ | Trivial.", "source_image": "t1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t1.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = (\u221a2+\u221a6)/4 = .965+ | Trivial.", "friedman_source_image": "t1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "d0da9c9aae58c37e", "improvement": -0.0009310593101359288, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.031269735994051e-07, "max_pair_overlap_depth": 0.0, "metric_expression": "0.96593", "metric_symbol": "s", "metric_value": 0.9659310593101359, "n": 1, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 0.965, "setup": "triinsqu", "side": 0.9659310593101359, "solution_path": "data/solutions/generated/triinsqu_001_d0da9c9aae58c37e.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d0da9c9aae58c37e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@2", "density": 0.5773345801938596, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t2.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a6/2 = 1.224+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.224, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a6/2 = 1.224+", "metric_symbol": "s", "metric_value": 1.224, "reference_text": "s = \u221a6/2 = 1.224+ | Trivial.", "source_image": "t2.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t2.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = \u221a6/2 = 1.224+ | Trivial.", "friedman_source_image": "t2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "536a8bec14614050", "improvement": -0.0007615124236886039, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.5296142902332335e-06, "max_pair_overlap_depth": -8.602274652714007e-06, "metric_expression": "1.22476", "metric_symbol": "s", "metric_value": 1.2247615124236886, "n": 2, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.224, "setup": "triinsqu", "side": 1.2247615124236886, "solution_path": "data/solutions/generated/triinsqu_002_536a8bec14614050.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/536a8bec14614050.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@3", "density": 0.594304450868366, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1996", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t3.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a3/2+\u221a6/4 = 1.478+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.478, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@3", "credit": "Erich Friedman", "date": "1996", "family_updated": "", "metric_expression": "\u221a3/2+\u221a6/4 = 1.478+", "metric_symbol": "s", "metric_value": 1.478, "reference_text": "s = \u221a3/2+\u221a6/4 = 1.478+ | Found by Erich Friedman | in 1996.", "source_image": "t3.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t3.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = \u221a3/2+\u221a6/4 = 1.478+ | Found by Erich Friedman | in 1996.", "friedman_source_image": "t3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "85d51e7a9d3c3790", "improvement": -0.00044936123470140643, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.5025974997417175e-07, "max_pair_overlap_depth": -2.9708917167997617e-05, "metric_expression": "1.47845", "metric_symbol": "s", "metric_value": 1.4784493612347014, "n": 3, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.478, "setup": "triinsqu", "side": 1.4784493612347014, "solution_path": "data/solutions/generated/triinsqu_003_85d51e7a9d3c3790.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/85d51e7a9d3c3790.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@4", "density": 0.6961073829345906, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1996", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t4.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1+1/\u221a3 = 1.577+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.577, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@4", "credit": "Erich Friedman", "date": "1996", "family_updated": "", "metric_expression": "1+1/\u221a3 = 1.577+", "metric_symbol": "s", "metric_value": 1.577, "reference_text": "s = 1+1/\u221a3 = 1.577+ | Found by Erich Friedman | in 1996.", "source_image": "t4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t4.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 1+1/\u221a3 = 1.577+ | Found by Erich Friedman | in 1996.", "friedman_source_image": "t4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "10004832454b0f3c", "improvement": -0.00040129748580719493, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.301219061208287e-07, "max_pair_overlap_depth": -2.378752457826705e-05, "metric_expression": "1.57740", "metric_symbol": "s", "metric_value": 1.5774012974858072, "n": 4, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.577, "setup": "triinsqu", "side": 1.5774012974858072, "solution_path": "data/solutions/generated/triinsqu_004_10004832454b0f3c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/10004832454b0f3c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@5", "density": 0.6658433228069607, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1996", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t5.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.803+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.803, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@5", "credit": "Erich Friedman", "date": "1996", "family_updated": "", "metric_expression": "1.803+", "metric_symbol": "s", "metric_value": 1.803, "reference_text": "s = 1.803+ | Found by Erich Friedman | in 1996.", "source_image": "t5.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t5.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 1.803+ | Found by Erich Friedman | in 1996.", "friedman_source_image": "t5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "f41eeaee150f1c15", "improvement": -0.0002224086857132157, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6751531743821273e-06, "max_pair_overlap_depth": -2.226142234541817e-05, "metric_expression": "1.80322", "metric_symbol": "s", "metric_value": 1.8032224086857132, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.803, "setup": "triinsqu", "side": 1.8032224086857132, "solution_path": "data/solutions/generated/triinsqu_005_f41eeaee150f1c15.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f41eeaee150f1c15.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@6", "density": 0.7181966221398873, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1996", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t6.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "9/2\u20133\u221a3/2 = 1.901+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.901, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@6", "credit": "Erich Friedman", "date": "1996", "family_updated": "", "metric_expression": "9/2\u20133\u221a3/2 = 1.901+", "metric_symbol": "s", "metric_value": 1.901, "reference_text": "s = 9/2\u20133\u221a3/2 = 1.901+ | Found by Erich Friedman | in 1996.", "source_image": "t6.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t6.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 9/2\u20133\u221a3/2 = 1.901+ | Found by Erich Friedman | in 1996.", "friedman_source_image": "t6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "47de8ba8f0c9310c", "improvement": -0.0009726348176062327, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.131503356462894e-07, "max_pair_overlap_depth": -1.0226066655058541e-06, "metric_expression": "1.90197", "metric_symbol": "s", "metric_value": 1.9019726348176063, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.901, "setup": "triinsqu", "side": 1.9019726348176063, "solution_path": "data/solutions/generated/triinsqu_006_47de8ba8f0c9310c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/47de8ba8f0c9310c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@7", "density": 0.7576992844833931, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1996", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t7.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@7", "credit": "Erich Friedman", "date": "1996", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Found by Erich Friedman | in 1996.", "source_image": "t7.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t7.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2 | Found by Erich Friedman | in 1996.", "friedman_source_image": "t7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "d4eecc11473f7bf7", "improvement": -9.626783869354227e-05, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.867700488247924e-07, "max_pair_overlap_depth": -1.9950288099523666e-05, "metric_expression": "2.00010", "metric_symbol": "s", "metric_value": 2.0000962678386935, "n": 7, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "triinsqu", "side": 2.0000962678386935, "solution_path": "data/solutions/generated/triinsqu_007_d4eecc11473f7bf7.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/d4eecc11473f7bf7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@8", "density": 0.7868902975296777, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1996", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t8.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3\u221a3/2\u20131/2 = 2.098+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.098, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@8", "credit": "Erich Friedman", "date": "1996", "family_updated": "", "metric_expression": "3\u221a3/2\u20131/2 = 2.098+", "metric_symbol": "s", "metric_value": 2.098, "reference_text": "s = 3\u221a3/2\u20131/2 = 2.098+ | Found by Erich Friedman | in 1996.", "source_image": "t8.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t8.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 3\u221a3/2\u20131/2 = 2.098+ | Found by Erich Friedman | in 1996.", "friedman_source_image": "t8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "bd28b88a6ba9b99d", "improvement": -0.00015813059780489525, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0169734543243664e-06, "max_pair_overlap_depth": -2.636538231393093e-05, "metric_expression": "2.09816", "metric_symbol": "s", "metric_value": 2.0981581305978048, "n": 8, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.098, "setup": "triinsqu", "side": 2.0981581305978048, "solution_path": "data/solutions/generated/triinsqu_008_bd28b88a6ba9b99d.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/bd28b88a6ba9b99d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@9", "density": 0.7446998211864622, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t9.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.287+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.287, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@9", "credit": "Maurizio Morandi", "date": "July 2008", "family_updated": "", "metric_expression": "2.287+", "metric_symbol": "s", "metric_value": 2.287, "reference_text": "s = 2.287+ | Found by Maurizio Morandi | in July 2008.", "source_image": "t9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t9.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.287+ | Found by Maurizio Morandi | in July 2008.", "friedman_source_image": "t9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "458e49fec5e5573c", "improvement": -0.0006045303821253789, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.745466993473377e-07, "max_pair_overlap_depth": -5.124657140087585e-06, "metric_expression": "2.28760", "metric_symbol": "s", "metric_value": 2.2876045303821253, "n": 9, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.287, "setup": "triinsqu", "side": 2.2876045303821253, "solution_path": "data/solutions/generated/triinsqu_009_458e49fec5e5573c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/458e49fec5e5573c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@10", "density": 0.7662234065167978, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t10.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.377+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.377, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@10", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "2.377+", "metric_symbol": "s", "metric_value": 2.377, "reference_text": "s = 2.377+ | Found by David W. Cantrell | in July 2002.", "source_image": "t10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t10.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.377+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "t10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "5b0eaf84f54fe48c", "improvement": -0.00023770927049060475, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.324926068512563e-07, "max_pair_overlap_depth": -1.1896205976635876e-05, "metric_expression": "2.37724", "metric_symbol": "s", "metric_value": 2.3772377092704904, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.377, "setup": "triinsqu", "side": 2.3772377092704904, "solution_path": "data/solutions/generated/triinsqu_10_5b0eaf84f54fe48c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/5b0eaf84f54fe48c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@11", "density": 0.7671858882860172, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t11.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.490+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.49, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@11", "credit": "Maurizio Morandi", "date": "June 2008", "family_updated": "", "metric_expression": "2.490+", "metric_symbol": "s", "metric_value": 2.49, "reference_text": "s = 2.490+ | Found by Maurizio Morandi | in June 2008.", "source_image": "t11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t11.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.490+ | Found by Maurizio Morandi | in June 2008.", "friedman_source_image": "t11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "b95fbc7e7009bb63", "improvement": -0.0017034740442234408, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.171896607131735e-07, "max_pair_overlap_depth": -3.917476745979309e-05, "metric_expression": "2.49170", "metric_symbol": "s", "metric_value": 2.4917034740442237, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.49, "setup": "triinsqu", "side": 2.4917034740442237, "solution_path": "data/solutions/generated/triinsqu_011_b95fbc7e7009bb63.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/b95fbc7e7009bb63.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@12", "density": 0.792908624011719, "friedman_analytical_or_proved": false, "friedman_credit": "Dominik Kamp", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t12.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.558+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.558, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@12", "credit": "Dominik Kamp", "date": "April 2026", "family_updated": "", "metric_expression": "2.558+", "metric_symbol": "s", "metric_value": 2.558, "reference_text": "s = 2.558+ | Found by Dominik Kamp | in April 2026.", "source_image": "t12.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t12.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.558+ | Found by Dominik Kamp | in April 2026.", "friedman_source_image": "t12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "297283972e34795d", "improvement": -0.0019375473066665094, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5442542765183254e-06, "max_pair_overlap_depth": -1.2119755173123359e-05, "metric_expression": "2.55994", "metric_symbol": "s", "metric_value": 2.5599375473066663, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.558, "setup": "triinsqu", "side": 2.5599375473066663, "solution_path": "data/solutions/generated/triinsqu_012_297283972e34795d.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Dominik Kamp", "svg_path": "data/svg/297283972e34795d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@13", "density": 0.8355745862678572, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t13.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.595+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.595, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@13", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "2.595+", "metric_symbol": "s", "metric_value": 2.595, "reference_text": "s = 2.595+ | Found by David W. Cantrell | in July 2002.", "source_image": "t13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t13.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.595+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "t13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "7628e283de25ee68", "improvement": -0.0005499010705918295, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8264727928819919e-06, "max_pair_overlap_depth": -1.755464108432747e-05, "metric_expression": "2.59555", "metric_symbol": "s", "metric_value": 2.595549901070592, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.595, "setup": "triinsqu", "side": 2.595549901070592, "solution_path": "data/solutions/generated/triinsqu_13_7628e283de25ee68.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/7628e283de25ee68.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@14", "density": 0.8144966370845219, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t14.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.726+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.726, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@14", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "2.726+", "metric_symbol": "s", "metric_value": 2.726, "reference_text": "s = 2.726+ | Found by David W. Cantrell | in July 2002.", "source_image": "t14.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t14.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.726+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "t14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "601dc4458719752d", "improvement": -0.0021590642579867314, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4175513847981196e-06, "max_pair_overlap_depth": -2.112817484914431e-05, "metric_expression": "2.72816", "metric_symbol": "s", "metric_value": 2.7281590642579867, "n": 14, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.726, "setup": "triinsqu", "side": 2.7281590642579867, "solution_path": "data/solutions/generated/triinsqu_014_601dc4458719752d.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/601dc4458719752d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@15", "density": 0.8108430550585584, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t15.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.8299+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.8299, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@15", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.8299+", "metric_symbol": "s", "metric_value": 2.8299, "reference_text": "s = 2.8299+ | Found by Ian Watson | in April 2026.", "source_image": "t15.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t15.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.8299+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "430e70d954ae3759", "improvement": -0.0003679073759270679, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0258685225063818e-06, "max_pair_overlap_depth": -3.8614458098606974e-05, "metric_expression": "2.83027", "metric_symbol": "s", "metric_value": 2.830267907375927, "n": 15, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.8299, "setup": "triinsqu", "side": 2.830267907375927, "solution_path": "data/solutions/generated/triinsqu_015_430e70d954ae3759.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/430e70d954ae3759.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@16", "density": 0.8226134516626242, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t16.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.9000+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.9, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@16", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.9000+", "metric_symbol": "s", "metric_value": 2.9, "reference_text": "s = 2.9000+ | Found by Ian Watson | in April 2026.", "source_image": "t16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t16.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.9000+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "2350476c07d7060c", "improvement": -0.002100218284964317, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0129376531587966e-06, "max_pair_overlap_depth": -5.176872598853383e-06, "metric_expression": "2.90210", "metric_symbol": "s", "metric_value": 2.9021002182849642, "n": 16, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.9, "setup": "triinsqu", "side": 2.9021002182849642, "solution_path": "data/solutions/generated/triinsqu_016_2350476c07d7060c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/2350476c07d7060c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@17", "density": 0.8272516069770434, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t17.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.982+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.982, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@17", "credit": "Maurizio Morandi", "date": "July 2008", "family_updated": "", "metric_expression": "2.982+", "metric_symbol": "s", "metric_value": 2.982, "reference_text": "s = 2.982+ | Found by Maurizio Morandi | in July 2008.", "source_image": "t17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t17.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.982+ | Found by Maurizio Morandi | in July 2008.", "friedman_source_image": "t17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "79a5506fa7f0974c", "improvement": -0.0010186525102788835, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8787966733926961e-06, "max_pair_overlap_depth": -9.562062775358982e-06, "metric_expression": "2.98302", "metric_symbol": "s", "metric_value": 2.983018652510279, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.982, "setup": "triinsqu", "side": 2.983018652510279, "solution_path": "data/solutions/generated/triinsqu_017_79a5506fa7f0974c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/79a5506fa7f0974c.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@18", "density": 0.8369229075763384, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t18.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.051+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.051, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@18", "credit": "Maurizio Morandi", "date": "July 2008", "family_updated": "", "metric_expression": "3.051+", "metric_symbol": "s", "metric_value": 3.051, "reference_text": "s = 3.051+ | Found by Maurizio Morandi | in July 2008.", "source_image": "t18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t18.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 3.051+ | Found by Maurizio Morandi | in July 2008.", "friedman_source_image": "t18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "0d24ad1cca3bbb64", "improvement": -0.0007140908076439523, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6577344713731995e-06, "max_pair_overlap_depth": -7.637757528100586e-06, "metric_expression": "3.05171", "metric_symbol": "s", "metric_value": 3.051714090807644, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.051, "setup": "triinsqu", "side": 3.051714090807644, "solution_path": "data/solutions/generated/triinsqu_018_0d24ad1cca3bbb64.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/0d24ad1cca3bbb64.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@19", "density": 0.8400046005998116, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t19.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.12930+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.1293, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@19", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.12930+", "metric_symbol": "s", "metric_value": 3.1293, "reference_text": "s = 3.12930+ | Found by Ian Watson | in April 2026.", "source_image": "t19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t19.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 3.12930+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "def5af7572515998", "improvement": -0.0002816463880579434, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.93050707669818e-06, "max_pair_overlap_depth": -4.5439275512260124e-06, "metric_expression": "3.12958", "metric_symbol": "s", "metric_value": 3.129581646388058, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.1293, "setup": "triinsqu", "side": 3.129581646388058, "solution_path": "data/solutions/generated/triinsqu_019_def5af7572515998.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/def5af7572515998.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@20", "density": 0.8288697801794981, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t20.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.2305+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.2305, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@20", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.2305+", "metric_symbol": "s", "metric_value": 3.2305, "reference_text": "s = 3.2305+ | Found by Ian Watson | in April 2026.", "source_image": "t20.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t20.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 3.2305+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "43e53f43942a4488", "improvement": -0.0018781535798377824, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.815940991194715e-07, "max_pair_overlap_depth": -1.1219130352735363e-05, "metric_expression": "3.23238", "metric_symbol": "s", "metric_value": 3.232378153579838, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.2305, "setup": "triinsqu", "side": 3.232378153579838, "solution_path": "data/solutions/generated/triinsqu_020_43e53f43942a4488.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/43e53f43942a4488.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@21", "density": 0.8291397947988184, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t21.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.3111+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.3111, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@21", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.3111+", "metric_symbol": "s", "metric_value": 3.3111, "reference_text": "s = 3.3111+ | Found by Ian Watson | in April 2026.", "source_image": "t21.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t21.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 3.3111+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "986572060ddb345a", "improvement": -0.0005626159903786565, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2145623034941338e-06, "max_pair_overlap_depth": -2.9730022631513364e-06, "metric_expression": "3.31166", "metric_symbol": "s", "metric_value": 3.311662615990379, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.3111, "setup": "triinsqu", "side": 3.311662615990379, "solution_path": "data/solutions/generated/triinsqu_021_986572060ddb345a.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/986572060ddb345a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@22", "density": 0.8356286267983878, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t22.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.3757+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.3757, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@22", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.3757+", "metric_symbol": "s", "metric_value": 3.3757, "reference_text": "22. | | s = 3.3757+ | Found by Ian Watson | in April 2026.", "source_image": "t22.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t22.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "22. | | s = 3.3757+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "4d69d20a7a35e585", "improvement": -0.0007086456724314516, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2855965547198588e-06, "max_pair_overlap_depth": -1.8710922459974633e-05, "metric_expression": "3.37641", "metric_symbol": "s", "metric_value": 3.3764086456724316, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.3757, "setup": "triinsqu", "side": 3.3764086456724316, "solution_path": "data/solutions/generated/triinsqu_022_4d69d20a7a35e585.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/4d69d20a7a35e585.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@23", "density": 0.8452320475065837, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t23.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.4311+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.4311, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@23", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.4311+", "metric_symbol": "s", "metric_value": 3.4311, "reference_text": "23. | | s = 3.4311+ | Found by Ian Watson | in April 2026.", "source_image": "t23.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t23.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "23. | | s = 3.4311+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "959574dda9a01133", "improvement": -0.001524187888661821, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.8602101761622194e-08, "max_pair_overlap_depth": -2.5255431334025857e-06, "metric_expression": "3.43262", "metric_symbol": "s", "metric_value": 3.4326241878886616, "n": 23, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.4311, "setup": "triinsqu", "side": 3.4326241878886616, "solution_path": "data/solutions/generated/triinsqu_023_959574dda9a01133.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/959574dda9a01133.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@24", "density": 0.8636751606923448, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "August 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t24.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.468+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.468, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@24", "credit": "Maurizio Morandi", "date": "August 2008", "family_updated": "", "metric_expression": "3.468+", "metric_symbol": "s", "metric_value": 3.468, "reference_text": "24. | | s = 3.468+ | Found by Maurizio Morandi | in August 2008.", "source_image": "t24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t24.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "24. | | s = 3.468+ | Found by Maurizio Morandi | in August 2008.", "friedman_source_image": "t24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "d96b5ad8b1001b9a", "improvement": -0.0008116902395634007, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1658397103314651e-06, "max_pair_overlap_depth": -6.191240536246312e-06, "metric_expression": "3.46881", "metric_symbol": "s", "metric_value": 3.4688116902395634, "n": 24, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.468, "setup": "triinsqu", "side": 3.4688116902395634, "solution_path": "data/solutions/generated/triinsqu_024_d96b5ad8b1001b9a.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/d96b5ad8b1001b9a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@25", "density": 0.8645393394550887, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t25.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.537+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.537, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@25", "credit": "David W. Cantrell", "date": "August 2012", "family_updated": "", "metric_expression": "3.537+", "metric_symbol": "s", "metric_value": 3.537, "reference_text": "25. | | s = 3.537+ | Found by David W. Cantrell | in August 2012.", "source_image": "t25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t25.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "25. | | s = 3.537+ | Found by David W. Cantrell | in August 2012.", "friedman_source_image": "t25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "8da962f4a2ed3f11", "improvement": -0.0015712315473748184, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.272129416634903e-07, "max_pair_overlap_depth": -1.1961418790784606e-05, "metric_expression": "3.53857", "metric_symbol": "s", "metric_value": 3.5385712315473747, "n": 25, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.537, "setup": "triinsqu", "side": 3.5385712315473747, "solution_path": "data/solutions/generated/triinsqu_025_8da962f4a2ed3f11.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/8da962f4a2ed3f11.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@26", "density": 0.8796434821316925, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "September 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t26.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.575+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.575, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@26", "credit": "Maurizio Morandi", "date": "September 2008", "family_updated": "", "metric_expression": "3.575+", "metric_symbol": "s", "metric_value": 3.575, "reference_text": "26. | | s = 3.575+ | Found by Maurizio Morandi | in September 2008.", "source_image": "t26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t26.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "26. | | s = 3.575+ | Found by Maurizio Morandi | in September 2008.", "friedman_source_image": "t26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "c36ce120aca7c2e7", "improvement": -0.002532991640289417, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0954385221262358e-06, "max_pair_overlap_depth": -5.476191583808632e-07, "metric_expression": "3.57753", "metric_symbol": "s", "metric_value": 3.5775329916402896, "n": 26, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.575, "setup": "triinsqu", "side": 3.5775329916402896, "solution_path": "data/solutions/generated/triinsqu_026_c36ce120aca7c2e7.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/c36ce120aca7c2e7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@27", "density": 0.8658956992776567, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t27.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.6726+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.6726, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@27", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.6726+", "metric_symbol": "s", "metric_value": 3.6726, "reference_text": "27. | | s = 3.6726+ | Found by Ian Watson | in April 2026.", "source_image": "t27.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t27.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "27. | | s = 3.6726+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "4c91417b38b8be95", "improvement": -0.0019097898938134072, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.750472262434879e-07, "max_pair_overlap_depth": -1.515528041828862e-05, "metric_expression": "3.67451", "metric_symbol": "s", "metric_value": 3.6745097898938135, "n": 27, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.6726, "setup": "triinsqu", "side": 3.6745097898938135, "solution_path": "data/solutions/generated/triinsqu_027_4c91417b38b8be95.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/4c91417b38b8be95.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@28", "density": 0.8583067572738268, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t28.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.7571+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.7571, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@28", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.7571+", "metric_symbol": "s", "metric_value": 3.7571, "reference_text": "28. | | s = 3.7571+ | Found by Ian Watson | in April 2026.", "source_image": "t28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t28.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "28. | | s = 3.7571+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "9b7285926a894eec", "improvement": -0.0013438626171571322, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7756377540625622e-06, "max_pair_overlap_depth": -5.481622750158621e-06, "metric_expression": "3.75844", "metric_symbol": "s", "metric_value": 3.758443862617157, "n": 28, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.7571, "setup": "triinsqu", "side": 3.758443862617157, "solution_path": "data/solutions/generated/triinsqu_028_9b7285926a894eec.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/9b7285926a894eec.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@29", "density": 0.8613056568479127, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t29.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.818+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.818, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@29", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "3.818+", "metric_symbol": "s", "metric_value": 3.818, "reference_text": "29. | | s = 3.818+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "t29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t29.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "29. | | s = 3.818+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "t29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "a02ae61c838755d7", "improvement": -0.0003054483997071422, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.731563201531232e-06, "max_pair_overlap_depth": -2.352291984759891e-06, "metric_expression": "3.81831", "metric_symbol": "s", "metric_value": 3.818305448399707, "n": 29, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.818, "setup": "triinsqu", "side": 3.818305448399707, "solution_path": "data/solutions/generated/triinsqu_029_a02ae61c838755d7.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/a02ae61c838755d7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@30", "density": 0.868927959047152, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t30.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.8662+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.8662, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@30", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.8662+", "metric_symbol": "s", "metric_value": 3.8662, "reference_text": "30. | | s = 3.8662+ | Found by Ian Watson | in April 2026.", "source_image": "t30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t30.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "30. | | s = 3.8662+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "d3cf7ea5ce255135", "improvement": -0.00030930450574828683, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.0741990212131443e-07, "max_pair_overlap_depth": -7.811820435188466e-06, "metric_expression": "3.86651", "metric_symbol": "s", "metric_value": 3.8665093045057484, "n": 30, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.8662, "setup": "triinsqu", "side": 3.8665093045057484, "solution_path": "data/solutions/generated/triinsqu_030_d3cf7ea5ce255135.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/d3cf7ea5ce255135.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@31", "density": 0.8672905272682435, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t31.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.936+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.936, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.936+", "metric_symbol": "s", "metric_value": 3.936, "reference_text": "31. | | s = 3.936+ | Found by Emerson Connelly | in April 2026.", "source_image": "t31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t31.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "31. | | s = 3.936+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "90f4b9b8010910a7", "improvement": 0.001868580153832955, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1286553127417914e-06, "max_pair_overlap_depth": -1.3206709677193729e-05, "metric_expression": "3.93413", "metric_symbol": "s", "metric_value": 3.934131419846167, "n": 31, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.936, "setup": "triinsqu", "side": 3.934131419846167, "solution_path": "data/solutions/generated/triinsqu_031_90f4b9b8010910a7.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/90f4b9b8010910a7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@32", "density": 0.8675921732131934, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t32.gif", "friedman_metric_expression": "6\u20137\u221a3/6 = 3.979+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.979, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@32", "credit": "Maurizio Morandi", "date": "May 2026", "family_updated": "", "metric_expression": "6\u20137\u221a3/6 = 3.979+", "metric_symbol": "s", "metric_value": 3.979, "reference_text": "32. | | s = 6\u20137\u221a3/6 = 3.979+ | Found by Maurizio Morandi | in May 2026.", "source_image": "t32.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t32.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "32. | | s = 6\u20137\u221a3/6 = 3.979+ | Found by Maurizio Morandi | in May 2026.", "friedman_source_image": "t32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "f42801b713b9238d", "improvement": -0.01738660313705287, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.208596980261035e-06, "max_pair_overlap_depth": -2.3341135757526388e-06, "metric_expression": "3.99639", "metric_symbol": "s", "metric_value": 3.996386603137053, "n": 32, "notes": "Best available verified coordinate reconstruction; the Friedman reference remains the public metric until recovery improves.", "record_type": "verified", "reference_side": 3.979, "setup": "triinsqu", "side": 3.996386603137053, "solution_path": "data/solutions/generated/triinsqu_32_f42801b713b9238d.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/f42801b713b9238d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@33", "density": 0.8675949776849048, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t33.gif", "friedman_metric_expression": "4.058+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.058, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@33", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.058+", "metric_symbol": "s", "metric_value": 4.058, "reference_text": "33. | | s = 4.058+ | Found by Emerson Connelly | in April 2026.", "source_image": "t33.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t33.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "33. | | s = 4.058+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "a6f49f2f24228d1b", "improvement": -0.00034322121207797096, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0747182587902822e-06, "max_pair_overlap_depth": -2.8024820216354662e-05, "metric_expression": "4.05834", "metric_symbol": "s", "metric_value": 4.058343221212078, "n": 33, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.058, "setup": "triinsqu", "side": 4.058343221212078, "solution_path": "data/solutions/generated/triinsqu_33_a6f49f2f24228d1b.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/a6f49f2f24228d1b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@34", "density": 0.8767333914896571, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t34.gif", "friedman_metric_expression": "4.098+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.098, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@34", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.098+", "metric_symbol": "s", "metric_value": 4.098, "reference_text": "34. | | s = 4.098+ | Found by Emerson Connelly | in April 2026.", "source_image": "t34.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t34.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "34. | | s = 4.098+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "61b4d5d6bf22a206", "improvement": 0.00015051348449990343, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6262786795626285e-06, "max_pair_overlap_depth": -1.499292058504409e-05, "metric_expression": "4.09785", "metric_symbol": "s", "metric_value": 4.0978494865155, "n": 34, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.098, "setup": "triinsqu", "side": 4.0978494865155, "solution_path": "data/solutions/generated/triinsqu_34_61b4d5d6bf22a206.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/61b4d5d6bf22a206.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@35", "density": 0.8677502962014568, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t35.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "9(2\u221a3\u20133) = 4.176+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.176, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@35", "credit": "Erich Friedman", "date": "April 2026", "family_updated": "", "metric_expression": "9(2\u221a3\u20133) = 4.176+", "metric_symbol": "s", "metric_value": 4.176, "reference_text": "35. | | s = 9(2\u221a3\u20133) = 4.176+ | Found by Erich Friedman | in April 2026.", "source_image": "t35.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t35.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "35. | | s = 9(2\u221a3\u20133) = 4.176+ | Found by Erich Friedman | in April 2026.", "friedman_source_image": "t35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "ab6aa3f219e08b42", "improvement": -0.0031403345722011977, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.770482464101941e-07, "max_pair_overlap_depth": -3.3297730506198775e-06, "metric_expression": "4.17914", "metric_symbol": "s", "metric_value": 4.179140334572201, "n": 35, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.176, "setup": "triinsqu", "side": 4.179140334572201, "solution_path": "data/solutions/generated/triinsqu_035_ab6aa3f219e08b42.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/ab6aa3f219e08b42.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@36", "density": 0.8790676097280682, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t36.gif", "friedman_metric_expression": "4.214+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.214, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@36", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.214+", "metric_symbol": "s", "metric_value": 4.214, "reference_text": "36. | | s = 4.214+ | Found by Emerson Connelly | in April 2026.", "source_image": "t36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t36.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "36. | | s = 4.214+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "aae755b2130a7a83", "improvement": 0.0029496973593481357, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.160284450281608e-07, "max_pair_overlap_depth": -9.933313802612531e-07, "metric_expression": "4.21105", "metric_symbol": "s", "metric_value": 4.211050302640652, "n": 36, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.214, "setup": "triinsqu", "side": 4.211050302640652, "solution_path": "data/solutions/record_hunt/triinsqu_36_aae755b2130a7a83.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/aae755b2130a7a83.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@37", "density": 0.8782570037473258, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t37.gif", "friedman_metric_expression": "4.27233+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.27233, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@37", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "4.27233+", "metric_symbol": "s", "metric_value": 4.27233, "reference_text": "37. | | s = 4.27233+ | Found by Nathan Roll | in April 2026.", "source_image": "t37.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t37.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "37. | | s = 4.27233+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "t37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "46406dc33c9746a7", "improvement": 0.0012238102051549404, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.826527513923834e-07, "max_pair_overlap_depth": -6.105510335085995e-07, "metric_expression": "4.27111", "metric_symbol": "s", "metric_value": 4.271106189794845, "n": 37, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.27233, "setup": "triinsqu", "side": 4.271106189794845, "solution_path": "data/solutions/record_hunt/triinsqu_37_46406dc33c9746a7.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/46406dc33c9746a7.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@38", "density": 0.87940001283733, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t38.gif", "friedman_metric_expression": "4.327+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.327, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@38", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.327+", "metric_symbol": "s", "metric_value": 4.327, "reference_text": "38. | | s = 4.327+ | Found by Emerson Connelly | in April 2026.", "source_image": "t38.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t38.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "38. | | s = 4.327+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t38.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "f44a4bd75616bc73", "improvement": 0.0013748413573368978, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0707071410413964e-07, "max_pair_overlap_depth": -1.0916553592199563e-06, "metric_expression": "4.32563", "metric_symbol": "s", "metric_value": 4.325625158642663, "n": 38, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.327, "setup": "triinsqu", "side": 4.325625158642663, "solution_path": "data/solutions/record_hunt/triinsqu_38_f44a4bd75616bc73.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/f44a4bd75616bc73.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@39", "density": 0.880118042721229, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t39.gif", "friedman_metric_expression": "4.383+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.383, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@39", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.383+", "metric_symbol": "s", "metric_value": 4.383, "reference_text": "39. | | s = 4.383+ | Found by Emerson Connelly | in April 2026.", "source_image": "t39.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t39.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "39. | | s = 4.383+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "263f7876a25d297b", "improvement": 0.002616247848736819, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.607883972834501e-07, "max_pair_overlap_depth": -1.86255866918561e-06, "metric_expression": "4.38038", "metric_symbol": "s", "metric_value": 4.380383752151263, "n": 39, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.383, "setup": "triinsqu", "side": 4.380383752151264, "solution_path": "data/solutions/record_hunt/triinsqu_39_263f7876a25d297b.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/263f7876a25d297b.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triinsqu@40", "density": 0.878868813713771, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t40.gif", "friedman_metric_expression": "4.437+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.437, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@40", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.437+", "metric_symbol": "s", "metric_value": 4.437, "reference_text": "40. | | s = 4.437+ | Found by Emerson Connelly | in April 2026.", "source_image": "t40.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t40.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "40. | | s = 4.437+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "ccf2360cc8917adc", "improvement": -0.002338759632142029, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.520888573265893e-06, "max_pair_overlap_depth": -1.6780429565876886e-05, "metric_expression": "4.43934", "metric_symbol": "s", "metric_value": 4.439338759632142, "n": 40, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.437, "setup": "triinsqu", "side": 4.439338759632142, "solution_path": "data/solutions/generated/triinsqu_040_ccf2360cc8917adc.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/ccf2360cc8917adc.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@1", "density": 1.0, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt1.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial.", "source_image": "tt1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt1.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "tt1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "b09a5ce4a2c03ad3", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.551115123125788e-17, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00000", "metric_symbol": "s", "metric_value": 1.0, "n": 1, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0, "setup": "triintri", "side": 1.0, "solution_path": "data/solutions/generated/triintri_001_b09a5ce4a2c03ad3.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b09a5ce4a2c03ad3.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "1.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 1.0}
-{"case": "triintri@2", "density": 0.5, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "tt4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "tt4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "2d6f654db11412d8", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.6653345369377346e-16, "max_pair_overlap_depth": 0.0, "metric_expression": "2.00000", "metric_symbol": "s", "metric_value": 2.0, "n": 2, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "triintri", "side": 2.0, "solution_path": "data/solutions/generated/triintri_002_2d6f654db11412d8.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2d6f654db11412d8.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "2.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 2.0}
-{"case": "triintri@3", "density": 0.75, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "tt4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "tt4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "76381d42f8f17f41", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.6653345369377346e-16, "max_pair_overlap_depth": 3.3306690738754696e-16, "metric_expression": "2.00000", "metric_symbol": "s", "metric_value": 2.0, "n": 3, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "triintri", "side": 2.0, "solution_path": "data/solutions/generated/triintri_003_76381d42f8f17f41.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/76381d42f8f17f41.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "2.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 2.0}
-{"case": "triintri@4", "density": 1.0, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "tt4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "tt4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "be1b0b4b1802d8fa", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.6653345369377346e-16, "max_pair_overlap_depth": 3.3306690738754696e-16, "metric_expression": "2.00000", "metric_symbol": "s", "metric_value": 2.0, "n": 4, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "triintri", "side": 2.0, "solution_path": "data/solutions/generated/triintri_004_be1b0b4b1802d8fa.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/be1b0b4b1802d8fa.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "2.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 2.0}
-{"case": "triintri@5", "density": 0.66984024663198, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt5.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "1 + \u221a3 = 2.732+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.732, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@5", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "1 + \u221a3 = 2.732+", "metric_symbol": "s", "metric_value": 2.732, "reference_text": "s = 1 + \u221a3 = 2.732+ | Proved by Erich Friedman | in 1997.", "source_image": "tt5.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt5.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 1 + \u221a3 = 2.732+ | Proved by Erich Friedman | in 1997.", "friedman_source_image": "tt5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "b4ac2c13e986485e", "improvement": -0.00011756309530053244, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.644640909821311e-06, "max_pair_overlap_depth": -6.265774057961264e-06, "metric_expression": "2.73212", "metric_symbol": "s", "metric_value": 2.7321175630953007, "n": 5, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.732, "setup": "triintri", "side": 2.7321175630953007, "solution_path": "data/solutions/generated/triintri_005_b4ac2c13e986485e.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/b4ac2c13e986485e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@6", "density": 0.6768918015337515, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "August 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt6.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "13/8 + 3\u221a13/8 = 2.977+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.977, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@6", "credit": "Maurizio Morandi", "date": "August 2008", "family_updated": "", "metric_expression": "13/8 + 3\u221a13/8 = 2.977+", "metric_symbol": "s", "metric_value": 2.977, "reference_text": "s = 13/8 + 3\u221a13/8 = 2.977+ | Found by Maurizio Morandi | in August 2008.", "source_image": "tt6.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt6.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 13/8 + 3\u221a13/8 = 2.977+ | Found by Maurizio Morandi | in August 2008.", "friedman_source_image": "tt6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "df20f3104adbd6a1", "improvement": -0.00025475946622810497, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3999156171376625e-07, "max_pair_overlap_depth": -1.9147124426771578e-05, "metric_expression": "2.97725", "metric_symbol": "s", "metric_value": 2.977254759466228, "n": 6, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.977, "setup": "triintri", "side": 2.977254759466228, "solution_path": "data/solutions/generated/triintri_006_df20f3104adbd6a1.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/df20f3104adbd6a1.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@7", "density": 0.777777777777778, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "tt9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "tt9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "355f218b5ea40614", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.9605947323337506e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "3.00000", "metric_symbol": "s", "metric_value": 3.0, "n": 7, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "triintri", "side": 3.0, "solution_path": "data/solutions/generated/triintri_007_355f218b5ea40614.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/355f218b5ea40614.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "3.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 3.0}
-{"case": "triintri@8", "density": 0.8888888888888892, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@8", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "tt9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "tt9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "43e3200cf32d77a9", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.9605947323337506e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "3.00000", "metric_symbol": "s", "metric_value": 3.0, "n": 8, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "triintri", "side": 3.0, "solution_path": "data/solutions/generated/triintri_008_43e3200cf32d77a9.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/43e3200cf32d77a9.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "3.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 3.0}
-{"case": "triintri@9", "density": 1.0000000000000004, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@9", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "tt9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "tt9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4242c6a66848e1ea", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.9605947323337516e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "3.00000", "metric_symbol": "s", "metric_value": 3.0, "n": 9, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "triintri", "side": 3.0, "solution_path": "data/solutions/generated/triintri_009_4242c6a66848e1ea.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4242c6a66848e1ea.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "3.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 3.0}
-{"case": "triintri@10", "density": 0.8162770652591925, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt10.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "7/2 = 3.500", "friedman_metric_symbol": "s", "friedman_metric_value": 3.5, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@10", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "7/2 = 3.500", "metric_symbol": "s", "metric_value": 3.5, "reference_text": "s = 7/2 = 3.500 | Found by Erich Friedman | in 1997.", "source_image": "tt10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt10.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 7/2 = 3.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "bcf7fa93627e2e22", "improvement": -0.00010604617003595607, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.918936789115156e-08, "max_pair_overlap_depth": -1.2701382657187565e-05, "metric_expression": "3.50011", "metric_symbol": "s", "metric_value": 3.500106046170036, "n": 10, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.5, "setup": "triintri", "side": 3.500106046170036, "solution_path": "data/solutions/generated/triintri_010_bcf7fa93627e2e22.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/bcf7fa93627e2e22.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@11", "density": 0.792568600317471, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt11.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "9/4 + 9\u221a21/28 = 3.722+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.722, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@11", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "9/4 + 9\u221a21/28 = 3.722+", "metric_symbol": "s", "metric_value": 3.722, "reference_text": "s = 9/4 + 9\u221a21/28 = 3.722+ | Found by Erich Friedman | in 1997.", "source_image": "tt11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt11.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 9/4 + 9\u221a21/28 = 3.722+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "8c37202a929dc807", "improvement": -0.0034428993959463305, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7247990718051686e-06, "max_pair_overlap_depth": -2.4227159337045223e-05, "metric_expression": "3.72544", "metric_symbol": "s", "metric_value": 3.7254428993959463, "n": 11, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.722, "setup": "triintri", "side": 3.7254428993959463, "solution_path": "data/solutions/generated/triintri_11_8c37202a929dc807.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/8c37202a929dc807.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@12", "density": 0.7960259433336867, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2007", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt12.gif", "friedman_metric_expression": "2 + 2cos(\u03c0/9) = 3.879+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.879, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@12", "credit": "David W. Cantrell", "date": "July 2007", "family_updated": "", "metric_expression": "2 + 2cos(\u03c0/9) = 3.879+", "metric_symbol": "s", "metric_value": 3.879, "reference_text": "s = 2 + 2cos(\u03c0/9) = 3.879+ | Found by David W. Cantrell | in July 2007.", "source_image": "tt12.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt12.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 + 2cos(\u03c0/9) = 3.879+ | Found by David W. Cantrell | in July 2007.", "friedman_source_image": "tt12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "7642b05605706588", "improvement": -0.0036389945275763225, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.358547413541538e-08, "max_pair_overlap_depth": -4.590975868801195e-06, "metric_expression": "3.88264", "metric_symbol": "s", "metric_value": 3.8826389945275763, "n": 12, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.879, "setup": "triintri", "side": 3.8826389945275763, "solution_path": "data/solutions/generated/triintri_12_7642b05605706588.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/7642b05605706588.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@13", "density": 0.8146483132163195, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt13.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3.992+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.992, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@13", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "3.992+", "metric_symbol": "s", "metric_value": 3.992, "reference_text": "s = 3.992+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt13.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3.992+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "afe3e3dc6de1bc36", "improvement": -0.0027223080586638027, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.48838412272174e-06, "max_pair_overlap_depth": -3.9197094850562664e-05, "metric_expression": "3.99472", "metric_symbol": "s", "metric_value": 3.994722308058664, "n": 13, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.992, "setup": "triintri", "side": 3.994722308058664, "solution_path": "data/solutions/generated/triintri_013_afe3e3dc6de1bc36.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/afe3e3dc6de1bc36.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@14", "density": 0.875, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@14", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial.", "source_image": "tt16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4 | Trivial.", "friedman_source_image": "tt16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "8676384ad0301449", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.775557561562891e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "4.00000", "metric_symbol": "s", "metric_value": 4.0, "n": 14, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "triintri", "side": 4.0, "solution_path": "data/solutions/generated/triintri_014_8676384ad0301449.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/8676384ad0301449.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "4.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 4.0}
-{"case": "triintri@15", "density": 0.9375, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@15", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial.", "source_image": "tt16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4 | Trivial.", "friedman_source_image": "tt16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "31db3ec76cc0421f", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.775557561562891e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "4.00000", "metric_symbol": "s", "metric_value": 4.0, "n": 15, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "triintri", "side": 4.0, "solution_path": "data/solutions/generated/triintri_015_31db3ec76cc0421f.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/31db3ec76cc0421f.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "4.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 4.0}
-{"case": "triintri@16", "density": 1.0, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@16", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial.", "source_image": "tt16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4 | Trivial.", "friedman_source_image": "tt16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "2ffa329b602032d6", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.775557561562891e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "4.00000", "metric_symbol": "s", "metric_value": 4.0, "n": 16, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "triintri", "side": 4.0, "solution_path": "data/solutions/generated/triintri_016_2ffa329b602032d6.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2ffa329b602032d6.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "4.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 4.0}
-{"case": "triintri@17", "density": 0.8524249732795084, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt17.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4.465+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.465, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@17", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "4.465+", "metric_symbol": "s", "metric_value": 4.465, "reference_text": "s = 4.465+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt17.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4.465+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "ad8aac0e4c06f7d0", "improvement": -0.0007702721912288268, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8241570152012174e-06, "max_pair_overlap_depth": -7.389272764501964e-06, "metric_expression": "4.46577", "metric_symbol": "s", "metric_value": 4.465770272191229, "n": 17, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.465, "setup": "triintri", "side": 4.465770272191229, "solution_path": "data/solutions/generated/triintri_017_ad8aac0e4c06f7d0.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/ad8aac0e4c06f7d0.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@18", "density": 0.8888203671574908, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt18.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "9/2 = 4.500", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@18", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "9/2 = 4.500", "metric_symbol": "s", "metric_value": 4.5, "reference_text": "s = 9/2 = 4.500 | Found by Erich Friedman | in 1997.", "source_image": "tt18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt18.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 9/2 = 4.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "b4cfb14015aca75a", "improvement": -0.00017345566104154386, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.575746594607548e-06, "max_pair_overlap_depth": -5.0550449839370515e-06, "metric_expression": "4.50017", "metric_symbol": "s", "metric_value": 4.5001734556610415, "n": 18, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.5, "setup": "triintri", "side": 4.5001734556610415, "solution_path": "data/solutions/generated/triintri_018_b4cfb14015aca75a.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/b4cfb14015aca75a.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@19", "density": 0.872322152073677, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt19.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "14/3 = 4.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.666, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@19", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "14/3 = 4.666+", "metric_symbol": "s", "metric_value": 4.666, "reference_text": "s = 14/3 = 4.666+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt19.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 14/3 = 4.666+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "bf4e89c9b4dd8ae2", "improvement": -0.0010058992763006813, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.331642222161759e-07, "max_pair_overlap_depth": -9.690072424628493e-06, "metric_expression": "4.66701", "metric_symbol": "s", "metric_value": 4.667005899276301, "n": 19, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.666, "setup": "triintri", "side": 4.667005899276301, "solution_path": "data/solutions/generated/triintri_019_bf4e89c9b4dd8ae2.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/bf4e89c9b4dd8ae2.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@20", "density": 0.845304928058783, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt20.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "33/8 + 9\u221a21/56 = 4.861+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.861, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@20", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "33/8 + 9\u221a21/56 = 4.861+", "metric_symbol": "s", "metric_value": 4.861, "reference_text": "s = 33/8 + 9\u221a21/56 = 4.861+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt20.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt20.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 33/8 + 9\u221a21/56 = 4.861+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4dff2c76e1273753", "improvement": -0.00316498806837906, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.619234998604407e-06, "max_pair_overlap_depth": -7.278337082583519e-06, "metric_expression": "4.86416", "metric_symbol": "s", "metric_value": 4.864164988068379, "n": 20, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.861, "setup": "triintri", "side": 4.864164988068379, "solution_path": "data/solutions/generated/triintri_020_4dff2c76e1273753.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/4dff2c76e1273753.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@21", "density": 0.8650098201747314, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt21.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4.923+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.923, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@21", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "4.923+", "metric_symbol": "s", "metric_value": 4.923, "reference_text": "s = 4.923+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt21.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt21.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4.923+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4dc62e9e9421c0aa", "improvement": -0.004187943781052361, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4013824985741632e-06, "max_pair_overlap_depth": -4.2136074186321926e-06, "metric_expression": "4.92719", "metric_symbol": "s", "metric_value": 4.927187943781052, "n": 21, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.923, "setup": "triintri", "side": 4.927187943781052, "solution_path": "data/solutions/generated/triintri_021_4dc62e9e9421c0aa.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/4dc62e9e9421c0aa.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@22", "density": 0.8799245341930954, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt22.gif", "friedman_metric_expression": "4.996+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.996, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@22", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "4.996+", "metric_symbol": "s", "metric_value": 4.996, "reference_text": "s = 4.996+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt22.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt22.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4.996+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "2122d833c6fcab70", "improvement": -0.004214405286988132, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.380108740330726e-08, "max_pair_overlap_depth": -8.335028905959163e-07, "metric_expression": "5.00021", "metric_symbol": "s", "metric_value": 5.000214405286989, "n": 22, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.996, "setup": "triintri", "side": 5.000214405286989, "solution_path": "data/solutions/generated/triintri_022_2122d833c6fcab70.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/2122d833c6fcab70.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@23", "density": 0.9200000000000002, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@23", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "tt25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "tt25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4f1bdd5d6c07f589", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "5.00000", "metric_symbol": "s", "metric_value": 5.0, "n": 23, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "triintri", "side": 5.0, "solution_path": "data/solutions/generated/triintri_023_4f1bdd5d6c07f589.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4f1bdd5d6c07f589.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "5.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 5.0}
-{"case": "triintri@24", "density": 0.9600000000000002, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@24", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "tt25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "tt25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "64d77fbf13ae0be7", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "5.00000", "metric_symbol": "s", "metric_value": 5.0, "n": 24, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "triintri", "side": 5.0, "solution_path": "data/solutions/generated/triintri_024_64d77fbf13ae0be7.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/64d77fbf13ae0be7.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "5.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 5.0}
-{"case": "triintri@25", "density": 1.0000000000000002, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@25", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "tt25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "tt25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "9c4806b8b37c5d37", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "5.00000", "metric_symbol": "s", "metric_value": 5.0, "n": 25, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "triintri", "side": 5.0, "solution_path": "data/solutions/generated/triintri_025_9c4806b8b37c5d37.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9c4806b8b37c5d37.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "5.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 5.0}
-{"case": "triintri@26", "density": 0.8884053356293824, "friedman_analytical_or_proved": false, "friedman_credit": "Joah Frenzley", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt26.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "5.406+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.406, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@26", "credit": "Joah Frenzley", "date": "April 2026", "family_updated": "", "metric_expression": "5.406+", "metric_symbol": "s", "metric_value": 5.406, "reference_text": "s = 5.406+ | Found by Joah Frenzley | in April 2026.", "source_image": "tt26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt26.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5.406+ | Found by Joah Frenzley | in April 2026.", "friedman_source_image": "tt26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "0af5a3765ecc4828", "improvement": -0.0037985715923722907, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.296466577480461e-06, "max_pair_overlap_depth": -2.0400569136769242e-05, "metric_expression": "5.40980", "metric_symbol": "s", "metric_value": 5.409798571592372, "n": 26, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.406, "setup": "triintri", "side": 5.409798571592372, "solution_path": "data/solutions/generated/triintri_026_0af5a3765ecc4828.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Joah Frenzley", "svg_path": "data/svg/0af5a3765ecc4828.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@27", "density": 0.8924996192164427, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "friedman_metric_expression": "11/2 = 5.500", "friedman_metric_symbol": "s", "friedman_metric_value": 5.5, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@27", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "11/2 = 5.500", "metric_symbol": "s", "metric_value": 5.5, "reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "source_image": "tt28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "f1c2be61e829f1e2", "improvement": -0.00019215549305418023, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8365186496628e-06, "max_pair_overlap_depth": -1.8064988390742442e-06, "metric_expression": "5.50019", "metric_symbol": "s", "metric_value": 5.500192155493054, "n": 27, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.5, "setup": "triintri", "side": 5.500192155493054, "solution_path": "data/solutions/generated/triintri_027_f1c2be61e829f1e2.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f1c2be61e829f1e2.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@28", "density": 0.9255631973649003, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "friedman_metric_expression": "11/2 = 5.500", "friedman_metric_symbol": "s", "friedman_metric_value": 5.5, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@28", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "11/2 = 5.500", "metric_symbol": "s", "metric_value": 5.5, "reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "source_image": "tt28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "9562d7d6e075a347", "improvement": -0.00016827626559567932, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.296352348617567e-06, "max_pair_overlap_depth": -7.147352432124165e-08, "metric_expression": "5.50017", "metric_symbol": "s", "metric_value": 5.500168276265596, "n": 28, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.5, "setup": "triintri", "side": 5.500168276265596, "solution_path": "data/solutions/generated/triintri_028_9562d7d6e075a347.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/9562d7d6e075a347.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@29", "density": 0.9023512759650716, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2007", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt29.gif", "friedman_metric_expression": "17/3 = 5.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.666, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@29", "credit": "David W. Cantrell", "date": "July 2007", "family_updated": "", "metric_expression": "17/3 = 5.666+", "metric_symbol": "s", "metric_value": 5.666, "reference_text": "s = 17/3 = 5.666+ | Found by David W. Cantrell | in July 2007.", "source_image": "tt29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt29.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 17/3 = 5.666+ | Found by David W. Cantrell | in July 2007.", "friedman_source_image": "tt29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "38c7a009f9059275", "improvement": -0.0030616587354490576, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9275914294015247e-06, "max_pair_overlap_depth": -5.088073942505211e-06, "metric_expression": "5.66906", "metric_symbol": "s", "metric_value": 5.669061658735449, "n": 29, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.666, "setup": "triintri", "side": 5.669061658735449, "solution_path": "data/solutions/generated/triintri_29_38c7a009f9059275.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/38c7a009f9059275.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@30", "density": 0.9060985977127352, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt30.gif", "friedman_metric_expression": "23/4 = 5.75", "friedman_metric_symbol": "s", "friedman_metric_value": 5.75, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@30", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "23/4 = 5.75", "metric_symbol": "s", "metric_value": 5.75, "reference_text": "s = 23/4 = 5.75 | Found by Maurizio Morandi | in May 2008.", "source_image": "tt30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt30.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 23/4 = 5.75 | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "149a91e0248df04e", "improvement": -0.004040286099913359, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2882428961650933e-06, "max_pair_overlap_depth": -6.414942533283785e-06, "metric_expression": "5.75404", "metric_symbol": "s", "metric_value": 5.754040286099913, "n": 30, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.75, "setup": "triintri", "side": 5.754040286099913, "solution_path": "data/solutions/generated/triintri_30_149a91e0248df04e.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/149a91e0248df04e.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@31", "density": 0.8852772627280384, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt31.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "5.912+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.912, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "5.912+", "metric_symbol": "s", "metric_value": 5.912, "reference_text": "s = 5.912+ | Found by Emerson Connelly | in April 2026.", "source_image": "tt31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt31.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5.912+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "tt31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "c729ec2b23a32942", "improvement": -0.005539858919105356, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9513696602846134e-06, "max_pair_overlap_depth": -3.7790737070775293e-06, "metric_expression": "5.91754", "metric_symbol": "s", "metric_value": 5.917539858919105, "n": 31, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.912, "setup": "triintri", "side": 5.917539858919105, "solution_path": "data/solutions/generated/triintri_031_c729ec2b23a32942.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/c729ec2b23a32942.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@32", "density": 0.9021138463561595, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt32.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "5.951+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.951, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@32", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "5.951+", "metric_symbol": "s", "metric_value": 5.951, "reference_text": "s = 5.951+ | Found by Ian Watson | in April 2026.", "source_image": "tt32.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt32.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5.951+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "tt32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "fa640909a0850f97", "improvement": -0.004857726462081935, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7950535548187937e-06, "max_pair_overlap_depth": -4.900081183589755e-06, "metric_expression": "5.95586", "metric_symbol": "s", "metric_value": 5.955857726462082, "n": 32, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.951, "setup": "triintri", "side": 5.955857726462082, "solution_path": "data/solutions/generated/triintri_032_fa640909a0850f97.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/fa640909a0850f97.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@33", "density": 0.9166666666666669, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@33", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "34a5032aebc12aff", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.921189464667501e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_symbol": "s", "metric_value": 6.0, "n": 33, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_033_34a5032aebc12aff.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/34a5032aebc12aff.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "6.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 6.0}
-{"case": "triintri@34", "density": 0.9444444444444448, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@34", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "a2ccd0a19cfa1b75", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.921189464667501e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_symbol": "s", "metric_value": 6.0, "n": 34, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_034_a2ccd0a19cfa1b75.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a2ccd0a19cfa1b75.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "6.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 6.0}
-{"case": "triintri@35", "density": 0.9722222222222225, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@35", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "6325f936448cdf94", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.921189464667503e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_symbol": "s", "metric_value": 6.0, "n": 35, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_035_6325f936448cdf94.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6325f936448cdf94.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "6.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 6.0}
-{"case": "triintri@36", "density": 1.0000000000000004, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@36", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4b90836bf07f8a65", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": 6.661338147750939e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_symbol": "s", "metric_value": 6.0, "n": 36, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_036_4b90836bf07f8a65.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:35:32+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4b90836bf07f8a65.svg", "tolerance": 1e-08, "verified": true, "verified_coordinate_metric_expression": "6.00000", "verified_coordinate_metric_symbol": "s", "verified_coordinate_metric_value": 6.0}
-{"case": "triintri@37", "density": 0.9223731676430976, "friedman_analytical_or_proved": true, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt37.gif", "friedman_metric_expression": "19/3 = 6.333+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.333, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@37", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "19/3 = 6.333+", "metric_symbol": "s", "metric_value": 6.333, "reference_text": "s = 19/3 = 6.333+ | Found by Ian Watson | in April 2026.", "source_image": "tt37.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt37.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 19/3 = 6.333+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "tt37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "a824891b660555f5", "improvement": -0.0005547879590457327, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1058084789851627e-06, "max_pair_overlap_depth": -1.0503324543442716e-06, "metric_expression": "6.33355", "metric_symbol": "s", "metric_value": 6.333554787959046, "n": 37, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.333, "setup": "triintri", "side": 6.333554787959046, "solution_path": "data/solutions/generated/triintri_037_a824891b660555f5.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/a824891b660555f5.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@38", "density": 0.9184187453437659, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt38.gif", "friedman_metric_expression": "6.43454+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.43454, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@38", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "6.43454+", "metric_symbol": "s", "metric_value": 6.43454, "reference_text": "s = 6.43454+ | Found by Nathan Roll | in April 2026.", "source_image": "tt38.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt38.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6.43454+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "tt38.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "b1563d97c893e28d", "improvement": 0.002163914058151306, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.8714774987883973e-07, "max_pair_overlap_depth": -1.6883301962034025e-06, "metric_expression": "6.43238", "metric_symbol": "s", "metric_value": 6.432376085941849, "n": 38, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.43454, "setup": "triintri", "side": 6.432376085941849, "solution_path": "data/solutions/record_hunt/triintri_38_b1563d97c893e28d.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/b1563d97c893e28d.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@39", "density": 0.922995893638428, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt40.gif", "friedman_metric_expression": "13/2 = 6.500", "friedman_metric_symbol": "s", "friedman_metric_value": 6.5, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@39", "credit": "Erich Friedman", "date": "April 2026", "family_updated": "", "metric_expression": "13/2 = 6.500", "metric_symbol": "s", "metric_value": 6.5, "reference_text": "s = 13/2 = 6.500 | Found by Erich Friedman | in April 2026.", "source_image": "tt40.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt40.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 13/2 = 6.500 | Found by Erich Friedman | in April 2026.", "friedman_source_image": "tt40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "5ff069bd30790b7f", "improvement": -0.0002853099319573005, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7141701112233934e-06, "max_pair_overlap_depth": -2.524988351737534e-07, "metric_expression": "6.50029", "metric_symbol": "s", "metric_value": 6.500285309931957, "n": 39, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.5, "setup": "triintri", "side": 6.500285309931957, "solution_path": "data/solutions/generated/triintri_039_5ff069bd30790b7f.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/5ff069bd30790b7f.svg", "tolerance": 1e-08, "verified": true}
-{"case": "triintri@40", "density": 0.9466291268019702, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt40.gif", "friedman_metric_expression": "13/2 = 6.500", "friedman_metric_symbol": "s", "friedman_metric_value": 6.5, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@40", "credit": "Erich Friedman", "date": "April 2026", "family_updated": "", "metric_expression": "13/2 = 6.500", "metric_symbol": "s", "metric_value": 6.5, "reference_text": "s = 13/2 = 6.500 | Found by Erich Friedman | in April 2026.", "source_image": "tt40.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt40.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 13/2 = 6.500 | Found by Erich Friedman | in April 2026.", "friedman_source_image": "tt40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "7890a1da940c9143", "improvement": -0.00039973752172350885, "leaderboard_consolidated_at": "2026-05-01T15:18:55+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4343626129297504e-07, "max_pair_overlap_depth": -3.1212060811469655e-06, "metric_expression": "6.50040", "metric_symbol": "s", "metric_value": 6.5003997375217235, "n": 40, "notes": "Verified coordinate recovery within tolerance; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.5, "setup": "triintri", "side": 6.5003997375217235, "solution_path": "data/solutions/generated/triintri_040_7890a1da940c9143.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/7890a1da940c9143.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@1", "density": 0.9999989965161072, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc1.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "1", "friedman_metric_symbol": "r", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "r", "metric_value": 1.0, "reference_text": "r = 1 | Trivial.", "source_image": "ccc1.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc1.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 1 | Trivial.", "friedman_source_image": "ccc1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "d8dddd0385fd067c", "improvement": -5.017423239106478e-07, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1979053238242443e-07, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00000", "metric_symbol": "r", "metric_value": 1.000000501742324, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.0, "setup": "cirincir", "side": 2.000001003484648, "solution_path": "data/solutions/generated/cirincir_001_d8dddd0385fd067c.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d8dddd0385fd067c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@2", "density": 0.4999800974887553, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc2.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "2", "friedman_metric_symbol": "r", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "r", "metric_value": 2.0, "reference_text": "r = 2 | Trivial.", "source_image": "ccc2.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc2.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 2 | Trivial.", "friedman_source_image": "ccc2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "ab9a33067266df01", "improvement": -3.9806210858639446e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.204627911923552e-06, "max_pair_overlap_depth": -5.010172176866945e-05, "metric_expression": "2.00004", "metric_symbol": "r", "metric_value": 2.0000398062108586, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "cirincir", "side": 4.000079612421717, "solution_path": "data/solutions/generated/cirincir_002_ab9a33067266df01.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/ab9a33067266df01.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@3", "density": 0.646133359064875, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc3.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "1 + 2 / \u221a3 = 2.154+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.154, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1 + 2 / \u221a3 = 2.154+", "metric_symbol": "r", "metric_value": 2.154, "reference_text": "r = 1 + 2 / \u221a3 = 2.154+ | Trivial.", "source_image": "ccc3.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc3.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 1 + 2 / \u221a3 = 2.154+ | Trivial.", "friedman_source_image": "ccc3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "fe973ff9a23dc34c", "improvement": -0.000763178385080554, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.918381111833185e-06, "max_pair_overlap_depth": -9.100356622360195e-05, "metric_expression": "2.15476", "metric_symbol": "r", "metric_value": 2.1547631783850805, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.154, "setup": "cirincir", "side": 4.309526356770161, "solution_path": "data/solutions/generated/cirincir_003_fe973ff9a23dc34c.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fe973ff9a23dc34c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@4", "density": 0.6862704996379871, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc4.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "1 + \u221a2 = 2.414+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.414, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1 + \u221a2 = 2.414+", "metric_symbol": "r", "metric_value": 2.414, "reference_text": "r = 1 + \u221a2 = 2.414+ | Trivial.", "source_image": "ccc4.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc4.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 1 + \u221a2 = 2.414+ | Trivial.", "friedman_source_image": "ccc4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "9d3e9e7ea6ba298b", "improvement": -0.00025050219633326876, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.985176337373275e-06, "max_pair_overlap_depth": -4.660490859054178e-05, "metric_expression": "2.41425", "metric_symbol": "r", "metric_value": 2.4142505021963334, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.414, "setup": "cirincir", "side": 4.828501004392667, "solution_path": "data/solutions/generated/cirincir_004_9d3e9e7ea6ba298b.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9d3e9e7ea6ba298b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@5", "density": 0.6851849889863537, "friedman_analytical_or_proved": true, "friedman_credit": "Graham", "friedman_date": "1968", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc5.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "2.701+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.701, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@5", "credit": "Graham", "date": "1968", "family_updated": "", "metric_expression": "2.701+", "metric_symbol": "r", "metric_value": 2.701, "reference_text": "r = 2.701+ | Proved by Graham in 1968.", "source_image": "ccc5.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc5.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 2.701+ | Proved by Graham in 1968.", "friedman_source_image": "ccc5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "9c2ab1f687c06eec", "improvement": -0.0003514001027240532, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.911145805306603e-06, "max_pair_overlap_depth": -4.497481208476373e-05, "metric_expression": "2.70135", "metric_symbol": "r", "metric_value": 2.701351400102724, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.701, "setup": "cirincir", "side": 5.402702800205448, "solution_path": "data/solutions/generated/cirincir_005_9c2ab1f687c06eec.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Graham", "svg_path": "data/svg/9c2ab1f687c06eec.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@6", "density": 0.6666416612940322, "friedman_analytical_or_proved": true, "friedman_credit": "Graham", "friedman_date": "1968", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc6.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3", "friedman_metric_symbol": "r", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@6", "credit": "Graham", "date": "1968", "family_updated": "", "metric_expression": "3", "metric_symbol": "r", "metric_value": 3.0, "reference_text": "r = 3 | Proved by Graham in 1968.", "source_image": "ccc6.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc6.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3 | Proved by Graham in 1968.", "friedman_source_image": "ccc6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "5d3d5beee3c5f33c", "improvement": -5.626367118827247e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.011848064282674e-06, "max_pair_overlap_depth": -4.5345095143201064e-05, "metric_expression": "3.00006", "metric_symbol": "r", "metric_value": 3.0000562636711883, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.0, "setup": "cirincir", "side": 6.0001125273423765, "solution_path": "data/solutions/generated/cirincir_006_5d3d5beee3c5f33c.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Graham", "svg_path": "data/svg/5d3d5beee3c5f33c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@7", "density": 0.77774147574625, "friedman_analytical_or_proved": true, "friedman_credit": "Graham", "friedman_date": "1968", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc7.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3", "friedman_metric_symbol": "r", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@7", "credit": "Graham", "date": "1968", "family_updated": "", "metric_expression": "3", "metric_symbol": "r", "metric_value": 3.0, "reference_text": "r = 3 | Proved by Graham in 1968.", "source_image": "ccc7.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc7.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3 | Proved by Graham in 1968.", "friedman_source_image": "ccc7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "2b1985e4094e6893", "improvement": -7.001351167312819e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.2947164267690425e-06, "max_pair_overlap_depth": -6.231680049051391e-05, "metric_expression": "3.00007", "metric_symbol": "r", "metric_value": 3.000070013511673, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.0, "setup": "cirincir", "side": 6.000140027023346, "solution_path": "data/solutions/generated/cirincir_007_2b1985e4094e6893.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Graham", "svg_path": "data/svg/2b1985e4094e6893.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@8", "density": 0.7324681263291527, "friedman_analytical_or_proved": true, "friedman_credit": "Braaksma", "friedman_date": "1963", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc8.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3.304+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.304, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@8", "credit": "Braaksma", "date": "1963", "family_updated": "", "metric_expression": "3.304+", "metric_symbol": "r", "metric_value": 3.304, "reference_text": "r = 3.304+ | Proved by Braaksma in 1963.", "source_image": "ccc8.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc8.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3.304+ | Proved by Braaksma in 1963.", "friedman_source_image": "ccc8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "0193661647fd3854", "improvement": -0.0008414425517750956, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.96872891495326e-06, "max_pair_overlap_depth": -4.3191348294779885e-05, "metric_expression": "3.30484", "metric_symbol": "r", "metric_value": 3.304841442551775, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.304, "setup": "cirincir", "side": 6.60968288510355, "solution_path": "data/solutions/generated/cirincir_008_0193661647fd3854.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Braaksma", "svg_path": "data/svg/0193661647fd3854.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@9", "density": 0.6893878045655164, "friedman_analytical_or_proved": true, "friedman_credit": "Pirl", "friedman_date": "1969", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc9.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3.613+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.613, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@9", "credit": "Pirl", "date": "1969", "family_updated": "", "metric_expression": "3.613+", "metric_symbol": "r", "metric_value": 3.613, "reference_text": "r = 3.613+ | Proved by Pirl in 1969.", "source_image": "ccc9.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc9.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3.613+ | Proved by Pirl in 1969.", "friedman_source_image": "ccc9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "b51df67181f6833b", "improvement": -0.00017882622267562098, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.656082665643879e-06, "max_pair_overlap_depth": -2.396342999144707e-05, "metric_expression": "3.61318", "metric_symbol": "r", "metric_value": 3.6131788262226756, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.613, "setup": "cirincir", "side": 7.226357652445351, "solution_path": "data/solutions/generated/cirincir_009_b51df67181f6833b.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Pirl", "svg_path": "data/svg/b51df67181f6833b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@10", "density": 0.6877649990272218, "friedman_analytical_or_proved": true, "friedman_credit": "Pirl", "friedman_date": "1969", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc10.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3.813+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.813, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@10", "credit": "Pirl", "date": "1969", "family_updated": "", "metric_expression": "3.813+", "metric_symbol": "r", "metric_value": 3.813, "reference_text": "r = 3.813+ | Proved by Pirl in 1969.", "source_image": "ccc10.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc10.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3.813+ | Proved by Pirl in 1969.", "friedman_source_image": "ccc10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "be14c55679a54b28", "improvement": -0.00011553912424266954, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.081326946625552e-06, "max_pair_overlap_depth": -3.967607896537828e-05, "metric_expression": "3.81312", "metric_symbol": "r", "metric_value": 3.813115539124243, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.813, "setup": "cirincir", "side": 7.626231078248486, "solution_path": "data/solutions/generated/cirincir_10_be14c55679a54b28.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Pirl", "svg_path": "data/svg/be14c55679a54b28.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@11", "density": 0.714418364520267, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1994", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc11.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3.923+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.923, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@11", "credit": "Melissen", "date": "1994", "family_updated": "", "metric_expression": "3.923+", "metric_symbol": "r", "metric_value": 3.923, "reference_text": "r = 3.923+ | Proved by Melissen in 1994.", "source_image": "ccc11.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc11.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3.923+ | Proved by Melissen in 1994.", "friedman_source_image": "ccc11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "e624f391927921fb", "improvement": -0.0009190348380849755, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.92009600810772e-06, "max_pair_overlap_depth": -4.346382940223492e-05, "metric_expression": "3.92392", "metric_symbol": "r", "metric_value": 3.923919034838085, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.923, "setup": "cirincir", "side": 7.84783806967617, "solution_path": "data/solutions/generated/cirincir_011_e624f391927921fb.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Melissen", "svg_path": "data/svg/e624f391927921fb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@12", "density": 0.7389828254275508, "friedman_analytical_or_proved": false, "friedman_credit": "Kravitz", "friedman_date": "1967", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc12.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "4.029+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.029, "friedman_reference": {"analytical_or_proved": false, "case": "cirincir@12", "credit": "Kravitz", "date": "1967", "family_updated": "", "metric_expression": "4.029+", "metric_symbol": "r", "metric_value": 4.029, "reference_text": "r = 4.029+ | Found by Kravitz in 1967.", "source_image": "ccc12.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc12.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 4.029+ | Found by Kravitz in 1967.", "friedman_source_image": "ccc12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "0d7fc8a14b1c4e0e", "improvement": -0.0007068210362959348, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.338373973515445e-06, "max_pair_overlap_depth": -5.019209633871924e-05, "metric_expression": "4.02971", "metric_symbol": "r", "metric_value": 4.029706821036296, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.029, "setup": "cirincir", "side": 8.059413642072592, "solution_path": "data/solutions/generated/cirincir_012_0d7fc8a14b1c4e0e.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Kravitz", "svg_path": "data/svg/0d7fc8a14b1c4e0e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@13", "density": 0.7244255598642273, "friedman_analytical_or_proved": true, "friedman_credit": "F. Fodor", "friedman_date": "2003", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc13.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "2 + \u221a5 = 4.236+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.236, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@13", "credit": "F. Fodor", "date": "2003", "family_updated": "", "metric_expression": "2 + \u221a5 = 4.236+", "metric_symbol": "r", "metric_value": 4.236, "reference_text": "r = 2 + \u221a5 = 4.236+ | Proved by F. Fodor in 2003.", "source_image": "ccc13.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc13.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 2 + \u221a5 = 4.236+ | Proved by F. Fodor in 2003.", "friedman_source_image": "ccc13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "a118f33872e0cb30", "improvement": -0.00018378579659206196, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.324667153277062e-06, "max_pair_overlap_depth": -4.7467215169572086e-05, "metric_expression": "4.23618", "metric_symbol": "r", "metric_value": 4.236183785796592, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.236, "setup": "cirincir", "side": 8.472367571593184, "solution_path": "data/solutions/generated/cirincir_013_a118f33872e0cb30.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "F. Fodor", "svg_path": "data/svg/a118f33872e0cb30.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@14", "density": 0.7472109217319929, "friedman_analytical_or_proved": false, "friedman_credit": "Goldberg", "friedman_date": "1971", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc14.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "4.328+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.328, "friedman_reference": {"analytical_or_proved": false, "case": "cirincir@14", "credit": "Goldberg", "date": "1971", "family_updated": "", "metric_expression": "4.328+", "metric_symbol": "r", "metric_value": 4.328, "reference_text": "r = 4.328+ | Found by Goldberg in 1971.", "source_image": "ccc14.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc14.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 4.328+ | Found by Goldberg in 1971.", "friedman_source_image": "ccc14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "b2ec44e582e5ab93", "improvement": -0.000549738133919675, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.732062089120916e-06, "max_pair_overlap_depth": -4.9839299053555663e-05, "metric_expression": "4.32855", "metric_symbol": "r", "metric_value": 4.32854973813392, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.328, "setup": "cirincir", "side": 8.65709947626784, "solution_path": "data/solutions/generated/cirincir_014_b2ec44e582e5ab93.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Goldberg", "svg_path": "data/svg/b2ec44e582e5ab93.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@15", "density": 0.7334069336541974, "friedman_analytical_or_proved": false, "friedman_credit": "Kravitz", "friedman_date": "1967", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc15.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "4.521+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.521, "friedman_reference": {"analytical_or_proved": false, "case": "cirincir@15", "credit": "Kravitz", "date": "1967", "family_updated": "", "metric_expression": "4.521+", "metric_symbol": "r", "metric_value": 4.521, "reference_text": "r = 4.521+ | Found by Kravitz in 1967.", "source_image": "ccc15.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc15.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 4.521+ | Found by Kravitz in 1967.", "friedman_source_image": "ccc15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "eeff7b918e6d6618", "improvement": -0.00144322894722837, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.2291583220711573e-06, "max_pair_overlap_depth": -2.9276985216153406e-05, "metric_expression": "4.52244", "metric_symbol": "r", "metric_value": 4.522443228947228, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.521, "setup": "cirincir", "side": 9.044886457894457, "solution_path": "data/solutions/generated/cirincir_015_eeff7b918e6d6618.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Kravitz", "svg_path": "data/svg/eeff7b918e6d6618.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@16", "density": 0.7510554296860762, "friedman_analytical_or_proved": false, "friedman_credit": "Goldberg", "friedman_date": "1971", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc16.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "4.615+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.615, "friedman_reference": {"analytical_or_proved": false, "case": "cirincir@16", "credit": "Goldberg", "date": "1971", "family_updated": "", "metric_expression": "4.615+", "metric_symbol": "r", "metric_value": 4.615, "reference_text": "r = 4.615+ | Found by Goldberg in 1971.", "source_image": "ccc16.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc16.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 4.615+ | Found by Goldberg in 1971.", "friedman_source_image": "ccc16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "7824261a02b5ed90", "improvement": -0.0005556989154120728, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.219226596077988e-06, "max_pair_overlap_depth": -4.8324434488655754e-05, "metric_expression": "4.61556", "metric_symbol": "r", "metric_value": 4.615555698915412, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.615, "setup": "cirincir", "side": 9.231111397830825, "solution_path": "data/solutions/generated/cirincir_016_7824261a02b5ed90.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Goldberg", "svg_path": "data/svg/7824261a02b5ed90.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@17", "density": 0.7402604737017356, "friedman_analytical_or_proved": false, "friedman_credit": "Reis", "friedman_date": "1975", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc17.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "4.792+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.792, "friedman_reference": {"analytical_or_proved": false, "case": "cirincir@17", "credit": "Reis", "date": "1975", "family_updated": "", "metric_expression": "4.792+", "metric_symbol": "r", "metric_value": 4.792, "reference_text": "r = 4.792+ | Found by Reis in 1975.", "source_image": "ccc17.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc17.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 4.792+ | Found by Reis in 1975.", "friedman_source_image": "ccc17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "68c41c427924b99a", "improvement": -0.00016960544150546042, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.357123401303568e-06, "max_pair_overlap_depth": -5.05938755308577e-05, "metric_expression": "4.79217", "metric_symbol": "r", "metric_value": 4.792169605441505, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.792, "setup": "cirincir", "side": 9.58433921088301, "solution_path": "data/solutions/generated/cirincir_017_68c41c427924b99a.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Reis", "svg_path": "data/svg/68c41c427924b99a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@18", "density": 0.7608872603215449, "friedman_analytical_or_proved": true, "friedman_credit": "Kravitz", "friedman_date": "1967", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc18.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "1 + \u221a2 + \u221a6 = 4.863+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.863, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@18", "credit": "Kravitz", "date": "1967", "family_updated": "", "metric_expression": "1 + \u221a2 + \u221a6 = 4.863+", "metric_symbol": "r", "metric_value": 4.863, "reference_text": "r = 1 + \u221a2 + \u221a6 = 4.863+ | Found by Kravitz in 1967.", "source_image": "ccc18.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc18.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 1 + \u221a2 + \u221a6 = 4.863+ | Found by Kravitz in 1967.", "friedman_source_image": "ccc18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "6acc78721c0b4ca1", "improvement": -0.000804343250861983, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.509170524220508e-07, "max_pair_overlap_depth": -3.6364882113559815e-05, "metric_expression": "4.86380", "metric_symbol": "r", "metric_value": 4.863804343250862, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.863, "setup": "cirincir", "side": 9.727608686501725, "solution_path": "data/solutions/generated/cirincir_018_6acc78721c0b4ca1.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Kravitz", "svg_path": "data/svg/6acc78721c0b4ca1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@19", "density": 0.8031578398507155, "friedman_analytical_or_proved": true, "friedman_credit": "F. Fodor", "friedman_date": "1999", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc19.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "1 + \u221a2 + \u221a6 = 4.863+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.863, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@19", "credit": "F. Fodor", "date": "1999", "family_updated": "", "metric_expression": "1 + \u221a2 + \u221a6 = 4.863+", "metric_symbol": "r", "metric_value": 4.863, "reference_text": "r = 1 + \u221a2 + \u221a6 = 4.863+ | Proved by F. Fodor in 1999.", "source_image": "ccc19.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc19.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 1 + \u221a2 + \u221a6 = 4.863+ | Proved by F. Fodor in 1999.", "friedman_source_image": "ccc19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "8325b5bff319f74f", "improvement": -0.0008071741580044645, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.170379222960378e-06, "max_pair_overlap_depth": -3.89864445309307e-05, "metric_expression": "4.86381", "metric_symbol": "r", "metric_value": 4.863807174158005, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.863, "setup": "cirincir", "side": 9.72761434831601, "solution_path": "data/solutions/generated/cirincir_019_8325b5bff319f74f.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "F. Fodor", "svg_path": "data/svg/8325b5bff319f74f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirincir@20", "density": 0.7622179661920169, "friedman_analytical_or_proved": false, "friedman_credit": "Goldberg", "friedman_date": "1971", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc20.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "5.122+", "friedman_metric_symbol": "r", "friedman_metric_value": 5.122, "friedman_reference": {"analytical_or_proved": false, "case": "cirincir@20", "credit": "Goldberg", "date": "1971", "family_updated": "", "metric_expression": "5.122+", "metric_symbol": "r", "metric_value": 5.122, "reference_text": "r = 5.122+ | Found by Goldberg in 1971.", "source_image": "ccc20.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc20.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 5.122+ | Found by Goldberg in 1971.", "friedman_source_image": "ccc20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "f0718319f5a05220", "improvement": -0.0004226268079490225, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.474954062416714e-06, "max_pair_overlap_depth": -3.4642111702343215e-05, "metric_expression": "5.12242", "metric_symbol": "r", "metric_value": 5.122422626807949, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.122, "setup": "cirincir", "side": 10.244845253615898, "solution_path": "data/solutions/generated/cirincir_020_f0718319f5a05220.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Goldberg", "svg_path": "data/svg/f0718319f5a05220.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@1", "density": 0.9068802327278791, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc1.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2/\u221a3 = 1.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.154, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2/\u221a3 = 1.154+", "metric_symbol": "s", "metric_value": 1.154, "reference_text": "s = 2/\u221a3 = 1.154+ | Trivial.", "source_image": "hc1.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc1.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2/\u221a3 = 1.154+ | Trivial.", "friedman_source_image": "hc1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "37d86bed2252b572", "improvement": -0.0007129204441553849, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.894559686663413e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.15471", "metric_symbol": "s", "metric_value": 1.1547129204441553, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.154, "setup": "cirinhex", "side": 1.1547129204441553, "solution_path": "data/solutions/generated/cirinhex_001_37d86bed2252b572.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/37d86bed2252b572.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@2", "density": 0.5208916187605724, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc2.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "1 + 2/\u221a3 = 2.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.154, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1 + 2/\u221a3 = 2.154+", "metric_symbol": "s", "metric_value": 2.154, "reference_text": "s = 1 + 2/\u221a3 = 2.154+ | Trivial.", "source_image": "hc2.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc2.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 1 + 2/\u221a3 = 2.154+ | Trivial.", "friedman_source_image": "hc2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "4ba45a45587fd527", "improvement": -0.0007173376388385222, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.535667492477913e-06, "max_pair_overlap_depth": -1.2900139699567603e-05, "metric_expression": "2.15472", "metric_symbol": "s", "metric_value": 2.1547173376388384, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.154, "setup": "cirinhex", "side": 2.1547173376388384, "solution_path": "data/solutions/generated/cirinhex_002_4ba45a45587fd527.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4ba45a45587fd527.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@3", "density": 0.6801656739683151, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc3.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "4/\u221a3 = 2.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.309, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4/\u221a3 = 2.309+", "metric_symbol": "s", "metric_value": 2.309, "reference_text": "s = 4/\u221a3 = 2.309+ | Trivial.", "source_image": "hc3.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc3.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 4/\u221a3 = 2.309+ | Trivial.", "friedman_source_image": "hc3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "fa4a26f729d80904", "improvement": -0.00041650453514563424, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3364075199806535e-06, "max_pair_overlap_depth": -1.818559246702378e-05, "metric_expression": "2.30942", "metric_symbol": "s", "metric_value": 2.309416504535146, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.309, "setup": "cirinhex", "side": 2.309416504535146, "solution_path": "data/solutions/generated/cirinhex_003_fa4a26f729d80904.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fa4a26f729d80904.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@4", "density": 0.6802066538449848, "friedman_analytical_or_proved": true, "friedman_credit": "Shahriar Manzoor", "friedman_date": "April 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc4.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2/\u221a3 + 4/\u221a7 = 2.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.666, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@4", "credit": "Shahriar Manzoor", "date": "April 2002", "family_updated": "", "metric_expression": "2/\u221a3 + 4/\u221a7 = 2.666+", "metric_symbol": "s", "metric_value": 2.666, "reference_text": "s = 2/\u221a3 + 4/\u221a7 = 2.666+ | Found by Shahriar Manzoor | in April 2002.", "source_image": "hc4.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc4.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2/\u221a3 + 4/\u221a7 = 2.666+ | Found by Shahriar Manzoor | in April 2002.", "friedman_source_image": "hc4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "4ea68db180a8d4c4", "improvement": -0.0006041510947536466, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3940250920849735e-07, "max_pair_overlap_depth": -4.6158989302380604e-05, "metric_expression": "2.66660", "metric_symbol": "s", "metric_value": 2.6666041510947536, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.666, "setup": "cirinhex", "side": 2.6666041510947536, "solution_path": "data/solutions/generated/cirinhex_004_4ea68db180a8d4c4.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Shahriar Manzoor", "svg_path": "data/svg/4ea68db180a8d4c4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@5", "density": 0.672041103262025, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1999", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc5.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2.999+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.999, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@5", "credit": "Erich Friedman", "date": "1999", "family_updated": "", "metric_expression": "2.999+", "metric_symbol": "s", "metric_value": 2.999, "reference_text": "s = 2.999+ | Found by Erich Friedman | in 1999.", "source_image": "hc5.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc5.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2.999+ | Found by Erich Friedman | in 1999.", "friedman_source_image": "hc5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "eb88c54ef20a6355", "improvement": -0.00041167259487728813, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6369105032841844e-07, "max_pair_overlap_depth": -3.862655436481788e-05, "metric_expression": "2.99941", "metric_symbol": "s", "metric_value": 2.9994116725948774, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.999, "setup": "cirinhex", "side": 2.9994116725948774, "solution_path": "data/solutions/generated/cirinhex_005_eb88c54ef20a6355.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/eb88c54ef20a6355.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@6", "density": 0.7289750468062609, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc6.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2 + 2/\u221a3 = 3.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.154, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 2/\u221a3 = 3.154+", "metric_symbol": "s", "metric_value": 3.154, "reference_text": "s = 2 + 2/\u221a3 = 3.154+ | Trivial. |", "source_image": "hc6.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc6.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2 + 2/\u221a3 = 3.154+ | Trivial. |", "friedman_source_image": "hc6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "9b7724fab73c49c1", "improvement": -0.0007742481949115643, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.7088474341983044e-06, "max_pair_overlap_depth": -4.5158069021944414e-05, "metric_expression": "3.15477", "metric_symbol": "s", "metric_value": 3.1547742481949115, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.154, "setup": "cirinhex", "side": 3.1547742481949115, "solution_path": "data/solutions/generated/cirinhex_006_9b7724fab73c49c1.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9b7724fab73c49c1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@7", "density": 0.8504825176668179, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc7.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2 + 2/\u221a3 = 3.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.154, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 2/\u221a3 = 3.154+", "metric_symbol": "s", "metric_value": 3.154, "reference_text": "s = 2 + 2/\u221a3 = 3.154+ | Trivial. |", "source_image": "hc7.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc7.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2 + 2/\u221a3 = 3.154+ | Trivial. |", "friedman_source_image": "hc7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "7368090471750227", "improvement": -0.0007526785063349806, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.857578518338343e-06, "max_pair_overlap_depth": -3.893260533160969e-05, "metric_expression": "3.15475", "metric_symbol": "s", "metric_value": 3.154752678506335, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.154, "setup": "cirinhex", "side": 3.154752678506335, "solution_path": "data/solutions/generated/cirinhex_007_7368090471750227.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7368090471750227.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@8", "density": 0.7030218969047355, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc8.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "3.709+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.709, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@8", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "3.709+", "metric_symbol": "s", "metric_value": 3.709, "reference_text": "s = 3.709+ | Found by David W. Cantrell | in July 2002.", "source_image": "hc8.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc8.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 3.709+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "hc8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "afebd04e4720460b", "improvement": -0.0004503331818734857, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1028444985304873e-08, "max_pair_overlap_depth": -2.2199913737530608e-05, "metric_expression": "3.70945", "metric_symbol": "s", "metric_value": 3.7094503331818736, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.709, "setup": "cirinhex", "side": 3.7094503331818736, "solution_path": "data/solutions/generated/cirinhex_008_afebd04e4720460b.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/afebd04e4720460b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@9", "density": 0.6761453195206264, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc9.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "20/7 + 2/\u221a3 = 4.011+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.011, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@9", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "20/7 + 2/\u221a3 = 4.011+", "metric_symbol": "s", "metric_value": 4.011, "reference_text": "s = 20/7 + 2/\u221a3 = 4.011+ | Found by David W. Cantrell | in July 2002.", "source_image": "hc9.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc9.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 20/7 + 2/\u221a3 = 4.011+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "hc9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "64b59a5605599f8e", "improvement": -0.0009011593404411045, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.491846259144452e-06, "max_pair_overlap_depth": -2.6999063512000987e-05, "metric_expression": "4.01190", "metric_symbol": "s", "metric_value": 4.011901159340441, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.011, "setup": "cirinhex", "side": 4.011901159340441, "solution_path": "data/solutions/generated/cirinhex_009_64b59a5605599f8e.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/64b59a5605599f8e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@10", "density": 0.7125686135385136, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc10.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "4.119+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.119, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@10", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "4.119+", "metric_symbol": "s", "metric_value": 4.119, "reference_text": "s = 4.119+ | Found by David W. Cantrell | in July 2002.", "source_image": "hc10.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc10.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 4.119+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "hc10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "cd79fdfeb9e3f24b", "improvement": -0.00041597258721015606, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.240617525155585e-06, "max_pair_overlap_depth": -1.610926851469685e-05, "metric_expression": "4.11942", "metric_symbol": "s", "metric_value": 4.11941597258721, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.119, "setup": "cirinhex", "side": 4.11941597258721, "solution_path": "data/solutions/generated/cirinhex_010_cd79fdfeb9e3f24b.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/cd79fdfeb9e3f24b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@11", "density": 0.7162109607848701, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1999", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc11.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2 + 4/\u221a3 = 4.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.309, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@11", "credit": "Erich Friedman", "date": "1999", "family_updated": "", "metric_expression": "2 + 4/\u221a3 = 4.309+", "metric_symbol": "s", "metric_value": 4.309, "reference_text": "s = 2 + 4/\u221a3 = 4.309+ | Found by Erich Friedman | in 1999.", "source_image": "hc11.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc11.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2 + 4/\u221a3 = 4.309+ | Found by Erich Friedman | in 1999.", "friedman_source_image": "hc11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "cda2ccbbf4de67aa", "improvement": -0.00047984763128994246, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.3670191174924895e-06, "max_pair_overlap_depth": -2.2677850788799248e-05, "metric_expression": "4.30948", "metric_symbol": "s", "metric_value": 4.30947984763129, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.309, "setup": "cirinhex", "side": 4.30947984763129, "solution_path": "data/solutions/generated/cirinhex_011_cda2ccbbf4de67aa.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/cda2ccbbf4de67aa.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@12", "density": 0.7813203208949625, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1999", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc12.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2 + 4/\u221a3 = 4.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.309, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@12", "credit": "Erich Friedman", "date": "1999", "family_updated": "", "metric_expression": "2 + 4/\u221a3 = 4.309+", "metric_symbol": "s", "metric_value": 4.309, "reference_text": "s = 2 + 4/\u221a3 = 4.309+ | Found by Erich Friedman | in 1999.", "source_image": "hc12.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc12.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2 + 4/\u221a3 = 4.309+ | Found by Erich Friedman | in 1999.", "friedman_source_image": "hc12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "c3d795864cfeb74c", "improvement": -0.0004818532104344797, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.009897945360223e-06, "max_pair_overlap_depth": -3.7913329796079864e-05, "metric_expression": "4.30948", "metric_symbol": "s", "metric_value": 4.309481853210435, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.309, "setup": "cirinhex", "side": 4.309481853210435, "solution_path": "data/solutions/generated/cirinhex_012_c3d795864cfeb74c.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/c3d795864cfeb74c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@13", "density": 0.7368031353920613, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1999", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc13.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "8/\u221a3 = 4.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.618, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@13", "credit": "Erich Friedman", "date": "1999", "family_updated": "", "metric_expression": "8/\u221a3 = 4.618+", "metric_symbol": "s", "metric_value": 4.618, "reference_text": "s = 8/\u221a3 = 4.618+ | Found by Erich Friedman | in 1999.", "source_image": "hc13.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc13.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 8/\u221a3 = 4.618+ | Found by Erich Friedman | in 1999.", "friedman_source_image": "hc13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "62c003d981fbe524", "improvement": -0.0009678209334929022, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.589858579651107e-06, "max_pair_overlap_depth": -5.483667702588235e-05, "metric_expression": "4.61897", "metric_symbol": "s", "metric_value": 4.618967820933493, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.618, "setup": "cirinhex", "side": 4.618967820933493, "solution_path": "data/solutions/generated/cirinhex_013_62c003d981fbe524.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/62c003d981fbe524.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@14", "density": 0.7773468150841377, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc14.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "2 + 2/\u221a3 + 4/\u221a7 = 4.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.666, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@14", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "2 + 2/\u221a3 + 4/\u221a7 = 4.666+", "metric_symbol": "s", "metric_value": 4.666, "reference_text": "s = 2 + 2/\u221a3 + 4/\u221a7 = 4.666+ | Found by David W. Cantrell | in July 2002.", "source_image": "hc14.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc14.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 2 + 2/\u221a3 + 4/\u221a7 = 4.666+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "hc14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "d3d8d6f246feeaad", "improvement": -0.0006539683763504556, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4490171600378687e-06, "max_pair_overlap_depth": -3.940553381864831e-05, "metric_expression": "4.66665", "metric_symbol": "s", "metric_value": 4.666653968376351, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.666, "setup": "cirinhex", "side": 4.666653968376351, "solution_path": "data/solutions/generated/cirinhex_014_d3d8d6f246feeaad.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d3d8d6f246feeaad.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@15", "density": 0.7368006709671345, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "January 2006", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc15.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "4.961+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.961, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@15", "credit": "David W. Cantrell", "date": "January 2006", "family_updated": "", "metric_expression": "4.961+", "metric_symbol": "s", "metric_value": 4.961, "reference_text": "s = 4.961+ | Found by David W. Cantrell | in January 2006.", "source_image": "hc15.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc15.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 4.961+ | Found by David W. Cantrell | in January 2006.", "friedman_source_image": "hc15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "d923ba7fbcb56e93", "improvement": -0.0005756365508533534, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.177000685334775e-07, "max_pair_overlap_depth": -3.4253106946380285e-05, "metric_expression": "4.96158", "metric_symbol": "s", "metric_value": 4.961575636550854, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.961, "setup": "cirinhex", "side": 4.961575636550854, "solution_path": "data/solutions/generated/cirinhex_015_d923ba7fbcb56e93.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d923ba7fbcb56e93.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@16", "density": 0.774039756730763, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc16.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "4.999+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.999, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@16", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "4.999+", "metric_symbol": "s", "metric_value": 4.999, "reference_text": "s = 4.999+ | Found by David W. Cantrell | in July 2002.", "source_image": "hc16.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc16.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 4.999+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "hc16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "b6a042d1d6eee5a4", "improvement": -0.0005089546268814971, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.665337648678829e-06, "max_pair_overlap_depth": -4.998313684900779e-05, "metric_expression": "4.99951", "metric_symbol": "s", "metric_value": 4.999508954626881, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.999, "setup": "cirinhex", "side": 4.999508954626881, "solution_path": "data/solutions/generated/cirinhex_016_b6a042d1d6eee5a4.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/b6a042d1d6eee5a4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@17", "density": 0.7801137203230791, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "January 2006", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc17.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "5.133+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.133, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@17", "credit": "David W. Cantrell", "date": "January 2006", "family_updated": "", "metric_expression": "5.133+", "metric_symbol": "s", "metric_value": 5.133, "reference_text": "s = 5.133+ | Found by David W. Cantrell | in January 2006.", "source_image": "hc17.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc17.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 5.133+ | Found by David W. Cantrell | in January 2006.", "friedman_source_image": "hc17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "4927714d5d50ab2b", "improvement": -0.00027458420964432406, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.495950815863736e-06, "max_pair_overlap_depth": -5.684433644859155e-05, "metric_expression": "5.13327", "metric_symbol": "s", "metric_value": 5.133274584209644, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.133, "setup": "cirinhex", "side": 5.133274584209644, "solution_path": "data/solutions/generated/cirinhex_017_4927714d5d50ab2b.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/4927714d5d50ab2b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@18", "density": 0.81913146509039, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc18.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "4 + 2/\u221a3 = 5.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.154, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@18", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4 + 2/\u221a3 = 5.154+", "metric_symbol": "s", "metric_value": 5.154, "reference_text": "s = 4 + 2/\u221a3 = 5.154+ | Trivial. |", "source_image": "hc18.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc18.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 4 + 2/\u221a3 = 5.154+ | Trivial. |", "friedman_source_image": "hc18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "223b1a3f7b7d0e67", "improvement": -0.000759899456760138, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.33856736717064e-06, "max_pair_overlap_depth": -1.657857602177515e-05, "metric_expression": "5.15476", "metric_symbol": "s", "metric_value": 5.15475989945676, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.154, "setup": "cirinhex", "side": 5.15475989945676, "solution_path": "data/solutions/generated/cirinhex_018_223b1a3f7b7d0e67.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/223b1a3f7b7d0e67.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@19", "density": 0.8646358262071194, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc19.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "4 + 2/\u221a3 = 5.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.154, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@19", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4 + 2/\u221a3 = 5.154+", "metric_symbol": "s", "metric_value": 5.154, "reference_text": "s = 4 + 2/\u221a3 = 5.154+ | Trivial. |", "source_image": "hc19.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc19.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 4 + 2/\u221a3 = 5.154+ | Trivial. |", "friedman_source_image": "hc19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "78f9327e80d8ab20", "improvement": -0.0007686707015803762, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.035946609195264e-07, "max_pair_overlap_depth": -1.6136694912294303e-05, "metric_expression": "5.15477", "metric_symbol": "s", "metric_value": 5.15476867070158, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.154, "setup": "cirinhex", "side": 5.15476867070158, "solution_path": "data/solutions/generated/cirinhex_019_78f9327e80d8ab20.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/78f9327e80d8ab20.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@20", "density": 0.7503080962551325, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc20.gif", "friedman_family_updated": "7/19/12", "friedman_metric_expression": "5.676+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.676, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@20", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "5.676+", "metric_symbol": "s", "metric_value": 5.676, "reference_text": "s = 5.676+ | Found by David W. Cantrell | in July 2012.", "source_image": "hc20.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc20.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 5.676+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "hc20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "247c6c06779a7d8b", "improvement": -0.0013304688728235092, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.66649728101143e-06, "max_pair_overlap_depth": -2.983410910761819e-05, "metric_expression": "5.67733", "metric_symbol": "s", "metric_value": 5.677330468872824, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.676, "setup": "cirinhex", "side": 5.677330468872824, "solution_path": "data/solutions/generated/cirinhex_020_247c6c06779a7d8b.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/247c6c06779a7d8b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@21", "density": 0.7789652581067265, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc21.gif", "friedman_metric_expression": "5.709+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.709, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@21", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "5.709+", "metric_symbol": "s", "metric_value": 5.709, "reference_text": "s = 5.709+ | Found by David W. Cantrell | in July 2012.", "source_image": "hc21.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc21.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 5.709+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "hc21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "38040eceeb63c1e3", "improvement": -0.0005199689632231497, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.315374129847953e-06, "max_pair_overlap_depth": -3.751451804223649e-05, "metric_expression": "5.70952", "metric_symbol": "s", "metric_value": 5.709519968963223, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.709, "setup": "cirinhex", "side": 5.709519968963223, "solution_path": "data/solutions/generated/cirinhex_021_38040eceeb63c1e3.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/38040eceeb63c1e3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@22", "density": 0.7509530082141859, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc22.gif", "friedman_metric_expression": "5.952+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.952, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@22", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "5.952+", "metric_symbol": "s", "metric_value": 5.952, "reference_text": "s = 5.952+ | Found by David W. Cantrell | in July 2012.", "source_image": "hc22.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc22.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 5.952+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "hc22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "cd7262f9f7d0d279", "improvement": 0.0001229278734493633, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.663138574891647e-06, "max_pair_overlap_depth": -3.268885699680979e-05, "metric_expression": "5.95188", "metric_symbol": "s", "metric_value": 5.951877072126551, "n": 22, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.952, "setup": "cirinhex", "side": 5.951877072126551, "solution_path": "data/solutions/generated/cirinhex_022_cd7262f9f7d0d279.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/cd7262f9f7d0d279.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@23", "density": 0.7694641578789073, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc23.gif", "friedman_metric_expression": "34/7 + 2/\u221a3 = 6.011+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.011, "friedman_reference": {"analytical_or_proved": true, "case": "cirinhex@23", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "34/7 + 2/\u221a3 = 6.011+", "metric_symbol": "s", "metric_value": 6.011, "reference_text": "s = 34/7 + 2/\u221a3 = 6.011+ | Found by David W. Cantrell | in July 2012.", "source_image": "hc23.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc23.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 34/7 + 2/\u221a3 = 6.011+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "hc23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "de2d2aabdc650ad2", "improvement": -0.0009964252593954726, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.1931154748950945e-06, "max_pair_overlap_depth": -3.920500316656472e-05, "metric_expression": "6.01200", "metric_symbol": "s", "metric_value": 6.011996425259396, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.011, "setup": "cirinhex", "side": 6.011996425259396, "solution_path": "data/solutions/generated/cirinhex_023_de2d2aabdc650ad2.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/de2d2aabdc650ad2.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinhex@24", "density": 0.7747882438352293, "friedman_analytical_or_proved": false, "friedman_credit": "Rob Pratt", "friedman_date": "March 2020", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc24.gif", "friedman_metric_expression": "6.117+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.117, "friedman_reference": {"analytical_or_proved": false, "case": "cirinhex@24", "credit": "Rob Pratt", "date": "March 2020", "family_updated": "", "metric_expression": "6.117+", "metric_symbol": "s", "metric_value": 6.117, "reference_text": "s = 6.117+ | Found by Rob Pratt | in March 2020.", "source_image": "hc24.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinhex/hc24.gif", "source_page": "https://erich-friedman.github.io/packing/cirinhex/", "title": "Circles in Hexagons"}, "friedman_reference_text": "s = 6.117+ | Found by Rob Pratt | in March 2020.", "friedman_source_image": "hc24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinhex/", "friedman_title": "Circles in Hexagons", "frontend_seed": true, "id": "61a86ae12328c1ca", "improvement": -0.003164603994216364, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4542492504119338e-06, "max_pair_overlap_depth": -2.04098607499148e-05, "metric_expression": "6.12016", "metric_symbol": "s", "metric_value": 6.120164603994216, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.117, "setup": "cirinhex", "side": 6.120164603994216, "solution_path": "data/solutions/generated/cirinhex_024_61a86ae12328c1ca.json", "source_url": "https://erich-friedman.github.io/packing/cirinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Rob Pratt", "svg_path": "data/svg/61a86ae12328c1ca.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@1", "density": 0.8647965566225578, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.1.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "1.453+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.453, "friedman_reference": {"analytical_or_proved": true, "case": "cirinpen@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1.453+", "metric_symbol": "s", "metric_value": 1.453, "reference_text": "s = 2tan(36\u00b0) | s = 1.453+ | Trivial.", "source_image": "pent.1.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.1.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 2tan(36\u00b0) | s = 1.453+ | Trivial.", "friedman_source_image": "pent.1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "2249439da521383e", "improvement": -9.321311914245456e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.798939718275719e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.45309", "metric_symbol": "s", "metric_value": 1.4530932131191425, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.453, "setup": "cirinpen", "side": 1.4530932131191425, "solution_path": "data/solutions/generated/cirinpen_001_2249439da521383e.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2249439da521383e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@2", "density": 0.5049856179230741, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.2.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "2.689+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.689, "friedman_reference": {"analytical_or_proved": true, "case": "cirinpen@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2.689+", "metric_symbol": "s", "metric_value": 2.689, "reference_text": "s = 2tan(36\u00b0) + \u221a5 - 1 | s = 2.689+ | Trivial.", "source_image": "pent.2.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.2.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 2tan(36\u00b0) + \u221a5 - 1 | s = 2.689+ | Trivial.", "friedman_source_image": "pent.2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "042df6c460b769f5", "improvement": -0.00021758992828191381, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.904067621218203e-06, "max_pair_overlap_depth": -5.9422400474407056e-05, "metric_expression": "2.68922", "metric_symbol": "s", "metric_value": 2.689217589928282, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.689, "setup": "cirinpen", "side": 2.689217589928282, "solution_path": "data/solutions/generated/cirinpen_002_042df6c460b769f5.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/042df6c460b769f5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@3", "density": 0.6022625992350803, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.3.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "3.015+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.015, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@3", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "3.015+", "metric_symbol": "s", "metric_value": 3.015, "reference_text": "s = 3.015+ | Found by George Sicherman | in July 2012.", "source_image": "pent.3.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.3.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 3.015+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "7afe63c4f78d1c3c", "improvement": -0.0009082861357154748, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.348511423333477e-06, "max_pair_overlap_depth": -3.760319804513301e-05, "metric_expression": "3.01591", "metric_symbol": "s", "metric_value": 3.0159082861357156, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.015, "setup": "cirinpen", "side": 3.0159082861357156, "solution_path": "data/solutions/generated/cirinpen_003_7afe63c4f78d1c3c.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/7afe63c4f78d1c3c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@4", "density": 0.6648380416307991, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.4.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "3.314+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.314, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@4", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "3.314+", "metric_symbol": "s", "metric_value": 3.314, "reference_text": "s = 3.314+ | Found by George Sicherman | in July 2012.", "source_image": "pent.4.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.4.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 3.314+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "3b6f81450d56967c", "improvement": -0.0005342806983965609, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4291241632899983e-06, "max_pair_overlap_depth": -5.729132607923404e-05, "metric_expression": "3.31453", "metric_symbol": "s", "metric_value": 3.3145342806983966, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.314, "setup": "cirinpen", "side": 3.3145342806983966, "solution_path": "data/solutions/generated/cirinpen_004_3b6f81450d56967c.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/3b6f81450d56967c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@5", "density": 0.7656755676390014, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.5.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "3.453+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.453, "friedman_reference": {"analytical_or_proved": true, "case": "cirinpen@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3.453+", "metric_symbol": "s", "metric_value": 3.453, "reference_text": "s = 2tan(36\u00b0) + 2 | s = 3.453+ | Trivial.", "source_image": "pent.5.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.5.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 2tan(36\u00b0) + 2 | s = 3.453+ | Trivial.", "friedman_source_image": "pent.5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "6a788c9a1ab05f1f", "improvement": -0.00013102756244531832, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4842628671150067e-07, "max_pair_overlap_depth": -2.3617760550376232e-05, "metric_expression": "3.45313", "metric_symbol": "s", "metric_value": 3.453131027562445, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.453, "setup": "cirinpen", "side": 3.453131027562445, "solution_path": "data/solutions/generated/cirinpen_005_6a788c9a1ab05f1f.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6a788c9a1ab05f1f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@6", "density": 0.7570230558933833, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.6.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "3.804+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.804, "friedman_reference": {"analytical_or_proved": true, "case": "cirinpen@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3.804+", "metric_symbol": "s", "metric_value": 3.804, "reference_text": "s = 2tan(36\u00b0) + 4sin(36\u00b0) | s = 3.804+ | Trivial.", "source_image": "pent.6.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.6.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 2tan(36\u00b0) + 4sin(36\u00b0) | s = 3.804+ | Trivial.", "friedman_source_image": "pent.6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "a73fe466a153c6dc", "improvement": -0.0002716609440795992, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.235659036235575e-06, "max_pair_overlap_depth": -1.860019961119619e-05, "metric_expression": "3.80427", "metric_symbol": "s", "metric_value": 3.8042716609440794, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.804, "setup": "cirinpen", "side": 3.8042716609440794, "solution_path": "data/solutions/generated/cirinpen_006_a73fe466a153c6dc.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a73fe466a153c6dc.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@7", "density": 0.7122554139396908, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.7.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "4.236+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.236, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@7", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "4.236+", "metric_symbol": "s", "metric_value": 4.236, "reference_text": "s = 4.236+ | Found by George Sicherman | in July 2012.", "source_image": "pent.7.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.7.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 4.236+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "f2fb4964063d721f", "improvement": -0.0002500861768837126, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.256168874145928e-06, "max_pair_overlap_depth": -7.389596920859276e-05, "metric_expression": "4.23625", "metric_symbol": "s", "metric_value": 4.2362500861768835, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.236, "setup": "cirinpen", "side": 4.2362500861768835, "solution_path": "data/solutions/generated/cirinpen_007_f2fb4964063d721f.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/f2fb4964063d721f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@8", "density": 0.7010960627351681, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.8.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "4.564+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.564, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@8", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "4.564+", "metric_symbol": "s", "metric_value": 4.564, "reference_text": "s = 4.564+ | Found by George Sicherman | in July 2012.", "source_image": "pent.8.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.8.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 4.564+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "7550c7a4a15ba971", "improvement": -0.0006415548993858877, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.759741664666194e-07, "max_pair_overlap_depth": -2.5138721591932978e-05, "metric_expression": "4.56464", "metric_symbol": "s", "metric_value": 4.564641554899386, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.564, "setup": "cirinpen", "side": 4.564641554899386, "solution_path": "data/solutions/generated/cirinpen_008_7550c7a4a15ba971.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/7550c7a4a15ba971.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@9", "density": 0.7058543729480167, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.9.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "4.825+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.825, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@9", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "4.825+", "metric_symbol": "s", "metric_value": 4.825, "reference_text": "s = 4.825+ | Found by George Sicherman | in July 2012.", "source_image": "pent.9.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.9.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 4.825+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "44a9c06211e4a669", "improvement": -0.0001870114230673181, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.33965970525874e-06, "max_pair_overlap_depth": -6.87249875483964e-05, "metric_expression": "4.82519", "metric_symbol": "s", "metric_value": 4.8251870114230675, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.825, "setup": "cirinpen", "side": 4.8251870114230675, "solution_path": "data/solutions/generated/cirinpen_009_44a9c06211e4a669.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/44a9c06211e4a669.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@10", "density": 0.7033183837462447, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.10.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "5.095+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.095, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@10", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "5.095+", "metric_symbol": "s", "metric_value": 5.095, "reference_text": "s = 5.095+ | Found by George Sicherman | in July 2012.", "source_image": "pent.10.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.10.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 5.095+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "38c046e395ee0204", "improvement": -0.0003552139806179966, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.918153382491795e-06, "max_pair_overlap_depth": -5.2545257565039805e-05, "metric_expression": "5.09536", "metric_symbol": "s", "metric_value": 5.095355213980618, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.095, "setup": "cirinpen", "side": 5.095355213980618, "solution_path": "data/solutions/generated/cirinpen_010_38c046e395ee0204.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/38c046e395ee0204.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@11", "density": 0.7314145264425335, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.11.gif", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "5.240+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.24, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@11", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "5.240+", "metric_symbol": "s", "metric_value": 5.24, "reference_text": "s = 5.240+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.11.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.11.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 5.240+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "3c4a20ceb03b166b", "improvement": -0.0004068061718305316, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.25538255039082e-06, "max_pair_overlap_depth": -4.98370515433777e-05, "metric_expression": "5.24041", "metric_symbol": "s", "metric_value": 5.240406806171831, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.24, "setup": "cirinpen", "side": 5.240406806171831, "solution_path": "data/solutions/generated/cirinpen_011_3c4a20ceb03b166b.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/3c4a20ceb03b166b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@12", "density": 0.7492511392398852, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.12.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "5.407+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.407, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@12", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "5.407+", "metric_symbol": "s", "metric_value": 5.407, "reference_text": "s = 5.407+ | Found by George Sicherman | in July 2012.", "source_image": "pent.12.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.12.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 5.407+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "1156bc3b50d815fe", "improvement": -0.0008839986719992865, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7054584944851143e-06, "max_pair_overlap_depth": -3.0020938044383882e-05, "metric_expression": "5.40788", "metric_symbol": "s", "metric_value": 5.407883998671999, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.407, "setup": "cirinpen", "side": 5.407883998671999, "solution_path": "data/solutions/generated/cirinpen_012_1156bc3b50d815fe.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/1156bc3b50d815fe.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@13", "density": 0.7430341365754948, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.13.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "5.651+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.651, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@13", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "5.651+", "metric_symbol": "s", "metric_value": 5.651, "reference_text": "s = 5.651+ | Found by George Sicherman | in July 2012.", "source_image": "pent.13.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.13.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 5.651+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "213cdbfbbdb494b1", "improvement": -0.0012028933276413412, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.063538795673338e-06, "max_pair_overlap_depth": -3.936589445130423e-05, "metric_expression": "5.65220", "metric_symbol": "s", "metric_value": 5.652202893327641, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.651, "setup": "cirinpen", "side": 5.652202893327641, "solution_path": "data/solutions/generated/cirinpen_013_213cdbfbbdb494b1.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/213cdbfbbdb494b1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@14", "density": 0.7576628256268029, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.14.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "5.808+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.808, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@14", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "5.808+", "metric_symbol": "s", "metric_value": 5.808, "reference_text": "s = 5.808+ | Found by George Sicherman | in July 2012.", "source_image": "pent.14.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.14.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 5.808+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "44805be876ac7787", "improvement": -0.0006669796446772125, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.282572655998294e-06, "max_pair_overlap_depth": -2.37953924338008e-05, "metric_expression": "5.80867", "metric_symbol": "s", "metric_value": 5.808666979644677, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.808, "setup": "cirinpen", "side": 5.808666979644677, "solution_path": "data/solutions/generated/cirinpen_014_44805be876ac7787.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/44805be876ac7787.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@15", "density": 0.7748926471130506, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.15.gif", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "5.945+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.945, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@15", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "5.945+", "metric_symbol": "s", "metric_value": 5.945, "reference_text": "s = 5.945+ | Found by George Sicherman | in July 2012.", "source_image": "pent.15.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.15.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 5.945+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "36fcf02cb6fd9f92", "improvement": -0.0003210236521935528, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.123112133491702e-06, "max_pair_overlap_depth": -5.711461050461608e-05, "metric_expression": "5.94532", "metric_symbol": "s", "metric_value": 5.945321023652194, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.945, "setup": "cirinpen", "side": 5.945321023652194, "solution_path": "data/solutions/generated/cirinpen_015_36fcf02cb6fd9f92.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/36fcf02cb6fd9f92.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@16", "density": 0.7775664873268765, "friedman_analytical_or_proved": false, "friedman_credit": "George Sicherman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.16.png", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "6.129+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.129, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@16", "credit": "George Sicherman", "date": "July 2012", "family_updated": "", "metric_expression": "6.129+", "metric_symbol": "s", "metric_value": 6.129, "reference_text": "s = 6.129+ | Found by George Sicherman | in July 2012.", "source_image": "pent.16.png", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.16.png", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 6.129+ | Found by George Sicherman | in July 2012.", "friedman_source_image": "pent.16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "39b6519b6fb7eef4", "improvement": -0.0007346433642512196, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.515317749508867e-06, "max_pair_overlap_depth": -2.965957000666819e-05, "metric_expression": "6.12973", "metric_symbol": "s", "metric_value": 6.129734643364251, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.129, "setup": "cirinpen", "side": 6.129734643364251, "solution_path": "data/solutions/generated/cirinpen_016_39b6519b6fb7eef4.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "George Sicherman", "svg_path": "data/svg/39b6519b6fb7eef4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@17", "density": 0.7527556084558986, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.17.gif", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "6.419+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.419, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@17", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "6.419+", "metric_symbol": "s", "metric_value": 6.419, "reference_text": "s = 6.419+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.17.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.17.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 6.419+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "5e677c639e143e85", "improvement": -0.0026689184768269314, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.2234530213948176e-06, "max_pair_overlap_depth": -4.317787105012627e-05, "metric_expression": "6.42167", "metric_symbol": "s", "metric_value": 6.4216689184768265, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.419, "setup": "cirinpen", "side": 6.4216689184768265, "solution_path": "data/solutions/generated/cirinpen_017_5e677c639e143e85.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/5e677c639e143e85.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@18", "density": 0.7574722105185439, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.18.gif", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "6.586+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.586, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@18", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "6.586+", "metric_symbol": "s", "metric_value": 6.586, "reference_text": "s = 6.586+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.18.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.18.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 6.586+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "34fda06736bbaaf7", "improvement": -0.0012379322999080955, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.803338162222119e-06, "max_pair_overlap_depth": -4.5158559001112764e-05, "metric_expression": "6.58724", "metric_symbol": "s", "metric_value": 6.587237932299908, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.586, "setup": "cirinpen", "side": 6.587237932299908, "solution_path": "data/solutions/generated/cirinpen_018_34fda06736bbaaf7.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/34fda06736bbaaf7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@19", "density": 0.7551620068495595, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.19.gif", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "6.777+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.777, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@19", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "6.777+", "metric_symbol": "s", "metric_value": 6.777, "reference_text": "s = 6.777+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.19.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.19.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 6.777+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "dfaba7cef14e53f5", "improvement": -0.0010877146457399789, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.911905938958071e-06, "max_pair_overlap_depth": -5.528697090007029e-05, "metric_expression": "6.77809", "metric_symbol": "s", "metric_value": 6.77808771464574, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.777, "setup": "cirinpen", "side": 6.77808771464574, "solution_path": "data/solutions/generated/cirinpen_019_dfaba7cef14e53f5.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/dfaba7cef14e53f5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@20", "density": 0.7572204985441312, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.20.gif", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "6.944+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.944, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@20", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "6.944+", "metric_symbol": "s", "metric_value": 6.944, "reference_text": "s = 6.944+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.20.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.20.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 6.944+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "d513055d785b23f7", "improvement": -0.0007124298362715109, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4085836122521584e-06, "max_pair_overlap_depth": -2.3984136221066876e-05, "metric_expression": "6.94471", "metric_symbol": "s", "metric_value": 6.9447124298362715, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.944, "setup": "cirinpen", "side": 6.9447124298362715, "solution_path": "data/solutions/generated/cirinpen_020_d513055d785b23f7.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d513055d785b23f7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@21", "density": 0.7590915818467144, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.21.gif", "friedman_family_updated": "7/10/12", "friedman_metric_expression": "7.107+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.107, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@21", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "7.107+", "metric_symbol": "s", "metric_value": 7.107, "reference_text": "s = 7.107+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.21.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.21.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 7.107+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "71add6eacc6c4fb9", "improvement": -0.0004368549852129533, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.952484924405695e-06, "max_pair_overlap_depth": -3.840822292655588e-05, "metric_expression": "7.10744", "metric_symbol": "s", "metric_value": 7.107436854985213, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.107, "setup": "cirinpen", "side": 7.107436854985213, "solution_path": "data/solutions/generated/cirinpen_021_71add6eacc6c4fb9.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/71add6eacc6c4fb9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@22", "density": 0.7595290345154431, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.22.gif", "friedman_metric_expression": "7.271+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.271, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@22", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "7.271+", "metric_symbol": "s", "metric_value": 7.271, "reference_text": "s = 7.271+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.22.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.22.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 7.271+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "ebdb05232e56526e", "improvement": -0.0015983061562394596, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.413438632810411e-06, "max_pair_overlap_depth": -6.856237147889743e-05, "metric_expression": "7.27260", "metric_symbol": "s", "metric_value": 7.272598306156239, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.271, "setup": "cirinpen", "side": 7.272598306156239, "solution_path": "data/solutions/generated/cirinpen_022_ebdb05232e56526e.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/ebdb05232e56526e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@23", "density": 0.7675010590620791, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.23.gif", "friedman_metric_expression": "7.397+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.397, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@23", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "7.397+", "metric_symbol": "s", "metric_value": 7.397, "reference_text": "s = 7.397+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.23.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.23.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 7.397+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "0115bd21477c01e6", "improvement": -0.0003280102549396702, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.583943801847681e-07, "max_pair_overlap_depth": -3.4243400256706025e-05, "metric_expression": "7.39733", "metric_symbol": "s", "metric_value": 7.39732801025494, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.397, "setup": "cirinpen", "side": 7.39732801025494, "solution_path": "data/solutions/generated/cirinpen_023_0115bd21477c01e6.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/0115bd21477c01e6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinpen@24", "density": 0.7719374015643087, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.24.gif", "friedman_metric_expression": "7.534+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.534, "friedman_reference": {"analytical_or_proved": false, "case": "cirinpen@24", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "7.534+", "metric_symbol": "s", "metric_value": 7.534, "reference_text": "s = 7.534+ | Found by David W. Cantrell | in July 2012.", "source_image": "pent.24.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinpen/pent.24.gif", "source_page": "https://erich-friedman.github.io/packing/cirinpen/", "title": "Circles in Pentagons"}, "friedman_reference_text": "s = 7.534+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "pent.24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinpen/", "friedman_title": "Circles in Pentagons", "frontend_seed": true, "id": "9eb26a95274f1d1d", "improvement": -0.0006837609794780874, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.468927518428643e-06, "max_pair_overlap_depth": -2.3713548386705696e-05, "metric_expression": "7.53468", "metric_symbol": "s", "metric_value": 7.534683760979478, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.534, "setup": "cirinpen", "side": 7.534683760979478, "solution_path": "data/solutions/generated/cirinpen_024_9eb26a95274f1d1d.json", "source_url": "https://erich-friedman.github.io/packing/cirinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/9eb26a95274f1d1d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@1", "density": 0.7853971453379307, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs1.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "cs1.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs1.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "cs1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "329c377d412c468f", "improvement": -1.2962348967526793e-06, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0795596527835016e-07, "max_pair_overlap_depth": 0.0, "metric_expression": "2.00000", "metric_symbol": "s", "metric_value": 2.0000012962348968, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "cirinsqu", "side": 2.0000012962348968, "solution_path": "data/solutions/generated/cirinsqu_001_329c377d412c468f.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/329c377d412c468f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@2", "density": 0.5390070210000131, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs2.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + \u221a2 = 3.414+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.414, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + \u221a2 = 3.414+", "metric_symbol": "s", "metric_value": 3.414, "reference_text": "s = 2 + \u221a2 = 3.414+ | Trivial.", "source_image": "cs2.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs2.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + \u221a2 = 3.414+ | Trivial.", "friedman_source_image": "cs2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "e3bab8445b304ebe", "improvement": -0.00022959896202801033, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1788679802625666e-06, "max_pair_overlap_depth": -1.6252235923275293e-05, "metric_expression": "3.41423", "metric_symbol": "s", "metric_value": 3.414229598962028, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.414, "setup": "cirinsqu", "side": 3.414229598962028, "solution_path": "data/solutions/generated/cirinsqu_002_e3bab8445b304ebe.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/e3bab8445b304ebe.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@3", "density": 0.6096071430698954, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs3.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + 1/\u221a2 + \u221a6/2 = 3.931+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.931, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 1/\u221a2 + \u221a6/2 = 3.931+", "metric_symbol": "s", "metric_value": 3.931, "reference_text": "s = 2 + 1/\u221a2 + \u221a6/2 = 3.931+ | Trivial.", "source_image": "cs3.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs3.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + 1/\u221a2 + \u221a6/2 = 3.931+ | Trivial.", "friedman_source_image": "cs3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "63c314af1da3741c", "improvement": -0.0009731189575252763, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.540614135605807e-06, "max_pair_overlap_depth": -7.777950709542125e-05, "metric_expression": "3.93197", "metric_symbol": "s", "metric_value": 3.9319731189575253, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.931, "setup": "cirinsqu", "side": 3.9319731189575253, "solution_path": "data/solutions/generated/cirinsqu_003_63c314af1da3741c.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/63c314af1da3741c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@4", "density": 0.7853846601471004, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs4.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial.", "source_image": "cs4.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs4.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 4 | Trivial.", "friedman_source_image": "cs4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "9917d423eeeb95cb", "improvement": -3.438618804896265e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3561316251363706e-06, "max_pair_overlap_depth": -2.9461600919766795e-05, "metric_expression": "4.00003", "metric_symbol": "s", "metric_value": 4.000034386188049, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.0, "setup": "cirinsqu", "side": 4.000034386188049, "solution_path": "data/solutions/generated/cirinsqu_004_9917d423eeeb95cb.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9917d423eeeb95cb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@5", "density": 0.6737513675455145, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs5.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + 2\u221a2 = 4.828+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.828, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 2\u221a2 = 4.828+", "metric_symbol": "s", "metric_value": 4.828, "reference_text": "s = 2 + 2\u221a2 = 4.828+ | Trivial.", "source_image": "cs5.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs5.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + 2\u221a2 = 4.828+ | Trivial.", "friedman_source_image": "cs5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "f91bd37c36f9320b", "improvement": -0.000476351133615438, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7157366511154635e-07, "max_pair_overlap_depth": -2.294469225416762e-05, "metric_expression": "4.82848", "metric_symbol": "s", "metric_value": 4.828476351133616, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.828, "setup": "cirinsqu", "side": 4.828476351133616, "solution_path": "data/solutions/generated/cirinsqu_005_f91bd37c36f9320b.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/f91bd37c36f9320b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@6", "density": 0.6639170340529372, "friedman_analytical_or_proved": true, "friedman_credit": "Graham", "friedman_date": "1963", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs6.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + 12/\u221a13 = 5.328+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.328, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@6", "credit": "Graham", "date": "1963", "family_updated": "", "metric_expression": "2 + 12/\u221a13 = 5.328+", "metric_symbol": "s", "metric_value": 5.328, "reference_text": "s = 2 + 12/\u221a13 = 5.328+ | Proved by Graham in 1963.", "source_image": "cs6.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs6.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + 12/\u221a13 = 5.328+ | Proved by Graham in 1963.", "friedman_source_image": "cs6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "96dda57e28868925", "improvement": -0.0003611830538092775, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.928735258591388e-06, "max_pair_overlap_depth": -7.200734794432506e-05, "metric_expression": "5.32836", "metric_symbol": "s", "metric_value": 5.32836118305381, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.328, "setup": "cirinsqu", "side": 5.32836118305381, "solution_path": "data/solutions/generated/cirinsqu_006_96dda57e28868925.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Graham", "svg_path": "data/svg/96dda57e28868925.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@7", "density": 0.669268877528072, "friedman_analytical_or_proved": true, "friedman_credit": "Schaer", "friedman_date": "1964", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs7.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "4 + \u221a3 = 5.732+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.732, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@7", "credit": "Schaer", "date": "1964", "family_updated": "", "metric_expression": "4 + \u221a3 = 5.732+", "metric_symbol": "s", "metric_value": 5.732, "reference_text": "s = 4 + \u221a3 = 5.732+ | Proved by Schaer in 1964.", "source_image": "cs7.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs7.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 4 + \u221a3 = 5.732+ | Proved by Schaer in 1964.", "friedman_source_image": "cs7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "7e4d3361af467e5d", "improvement": -0.0002304452542292168, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.33238067148811e-06, "max_pair_overlap_depth": -6.633595249638802e-05, "metric_expression": "5.73223", "metric_symbol": "s", "metric_value": 5.732230445254229, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.732, "setup": "cirinsqu", "side": 5.732230445254229, "solution_path": "data/solutions/generated/cirinsqu_007_7e4d3361af467e5d.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Schaer", "svg_path": "data/svg/7e4d3361af467e5d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@8", "density": 0.7309263604724501, "friedman_analytical_or_proved": true, "friedman_credit": "Schaer/Meir", "friedman_date": "1964", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs8.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + \u221a2 + \u221a6 = 5.863+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.863, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@8", "credit": "Schaer/Meir", "date": "1964", "family_updated": "", "metric_expression": "2 + \u221a2 + \u221a6 = 5.863+", "metric_symbol": "s", "metric_value": 5.863, "reference_text": "s = 2 + \u221a2 + \u221a6 = 5.863+ | Proved by Schaer/Meir in 1964.", "source_image": "cs8.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs8.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + \u221a2 + \u221a6 = 5.863+ | Proved by Schaer/Meir in 1964.", "friedman_source_image": "cs8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "67f6750a9af94d2c", "improvement": -0.0008535799033406732, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.247320970871769e-06, "max_pair_overlap_depth": -5.202375940349668e-05, "metric_expression": "5.86385", "metric_symbol": "s", "metric_value": 5.863853579903341, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.863, "setup": "cirinsqu", "side": 5.863853579903341, "solution_path": "data/solutions/generated/cirinsqu_008_67f6750a9af94d2c.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Schaer/Meir", "svg_path": "data/svg/67f6750a9af94d2c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@9", "density": 0.7853720614609764, "friedman_analytical_or_proved": true, "friedman_credit": "Schaer", "friedman_date": "1964", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs9.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@9", "credit": "Schaer", "date": "1964", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Proved by Schaer in 1964.", "source_image": "cs9.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs9.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 6 | Proved by Schaer in 1964.", "friedman_source_image": "cs9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "a35561fa5a518c8b", "improvement": -9.970453832330861e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.26657331825875e-06, "max_pair_overlap_depth": -3.06706256441025e-05, "metric_expression": "6.00010", "metric_symbol": "s", "metric_value": 6.000099704538323, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.0, "setup": "cirinsqu", "side": 6.000099704538323, "solution_path": "data/solutions/generated/cirinsqu_009_a35561fa5a518c8b.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Schaer", "svg_path": "data/svg/a35561fa5a518c8b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@10", "density": 0.6900075339157021, "friedman_analytical_or_proved": true, "friedman_credit": "De Groot", "friedman_date": "1990", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs10.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "6.747+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.747, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@10", "credit": "De Groot", "date": "1990", "family_updated": "", "metric_expression": "6.747+", "metric_symbol": "s", "metric_value": 6.747, "reference_text": "s = 6.747+ | Proved by De Groot in 1990.", "source_image": "cs10.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs10.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 6.747+ | Proved by De Groot in 1990.", "friedman_source_image": "cs10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "522948d614943781", "improvement": -0.0005796538825402209, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6772848518975536e-06, "max_pair_overlap_depth": -4.0928067794876455e-05, "metric_expression": "6.74758", "metric_symbol": "s", "metric_value": 6.74757965388254, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.747, "setup": "cirinsqu", "side": 6.74757965388254, "solution_path": "data/solutions/generated/cirinsqu_10_522948d614943781.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "De Groot", "svg_path": "data/svg/522948d614943781.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@11", "density": 0.7006956988559802, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs11.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "7.022+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.022, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@11", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "7.022+", "metric_symbol": "s", "metric_value": 7.022, "reference_text": "s = 7.022+ | Proved by Peikert in 1991.", "source_image": "cs11.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs11.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 7.022+ | Proved by Peikert in 1991.", "friedman_source_image": "cs11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "b0ae38086de8cdf9", "improvement": -0.0007394033277021705, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.667419736983149e-06, "max_pair_overlap_depth": -2.585792898823769e-05, "metric_expression": "7.02274", "metric_symbol": "s", "metric_value": 7.022739403327702, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.022, "setup": "cirinsqu", "side": 7.022739403327702, "solution_path": "data/solutions/generated/cirinsqu_011_b0ae38086de8cdf9.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/b0ae38086de8cdf9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@12", "density": 0.7384288108172983, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs12.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + 15\u221a(2/17) = 7.144+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.144, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@12", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "2 + 15\u221a(2/17) = 7.144+", "metric_symbol": "s", "metric_value": 7.144, "reference_text": "s = 2 + 15\u221a(2/17) = 7.144+ | Proved by Peikert in 1991.", "source_image": "cs12.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs12.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + 15\u221a(2/17) = 7.144+ | Proved by Peikert in 1991.", "friedman_source_image": "cs12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "e58733a8646795ab", "improvement": -0.0011482300231175913, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.939770909988098e-06, "max_pair_overlap_depth": -5.93949466169974e-05, "metric_expression": "7.14515", "metric_symbol": "s", "metric_value": 7.145148230023118, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.144, "setup": "cirinsqu", "side": 7.145148230023118, "solution_path": "data/solutions/generated/cirinsqu_012_e58733a8646795ab.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/e58733a8646795ab.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@13", "density": 0.7332320892098975, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs13.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "7.463+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.463, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@13", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "7.463+", "metric_symbol": "s", "metric_value": 7.463, "reference_text": "s = 7.463+ | Proved by Peikert in 1991.", "source_image": "cs13.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs13.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 7.463+ | Proved by Peikert in 1991.", "friedman_source_image": "cs13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "60997371c312edb4", "improvement": -0.0002137620977080701, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.323292483705444e-06, "max_pair_overlap_depth": -3.8715255959331785e-05, "metric_expression": "7.46321", "metric_symbol": "s", "metric_value": 7.463213762097708, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.463, "setup": "cirinsqu", "side": 7.463213762097708, "solution_path": "data/solutions/generated/cirinsqu_013_60997371c312edb4.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/60997371c312edb4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@14", "density": 0.7356508336971848, "friedman_analytical_or_proved": true, "friedman_credit": "Wengerodt", "friedman_date": "1987", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs14.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "6 + \u221a3 = 7.732+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.732, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@14", "credit": "Wengerodt", "date": "1987", "family_updated": "", "metric_expression": "6 + \u221a3 = 7.732+", "metric_symbol": "s", "metric_value": 7.732, "reference_text": "s = 6 + \u221a3 = 7.732+ | Proved by Wengerodt in 1987.", "source_image": "cs14.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs14.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 6 + \u221a3 = 7.732+ | Proved by Wengerodt in 1987.", "friedman_source_image": "cs14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "2ccdf5f3d4bee77b", "improvement": -0.000200169889159163, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.480532338595026e-07, "max_pair_overlap_depth": -3.0206326999682176e-05, "metric_expression": "7.73220", "metric_symbol": "s", "metric_value": 7.732200169889159, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.732, "setup": "cirinsqu", "side": 7.732200169889159, "solution_path": "data/solutions/generated/cirinsqu_014_2ccdf5f3d4bee77b.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Wengerodt", "svg_path": "data/svg/2ccdf5f3d4bee77b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@15", "density": 0.7620259985870776, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs15.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "4 + \u221a2 + \u221a6 = 7.863+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.863, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@15", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "4 + \u221a2 + \u221a6 = 7.863+", "metric_symbol": "s", "metric_value": 7.863, "reference_text": "s = 4 + \u221a2 + \u221a6 = 7.863+ | Proved by Peikert in 1991.", "source_image": "cs15.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs15.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 4 + \u221a2 + \u221a6 = 7.863+ | Proved by Peikert in 1991.", "friedman_source_image": "cs15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "edc16b42d64e4d82", "improvement": -0.0008581593284961997, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.610873706565542e-06, "max_pair_overlap_depth": -3.564256081922679e-05, "metric_expression": "7.86386", "metric_symbol": "s", "metric_value": 7.863858159328497, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.863, "setup": "cirinsqu", "side": 7.863858159328497, "solution_path": "data/solutions/generated/cirinsqu_015_edc16b42d64e4d82.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/edc16b42d64e4d82.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@16", "density": 0.7853654831135446, "friedman_analytical_or_proved": true, "friedman_credit": "Wengerodt", "friedman_date": "1983", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs16.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "8", "friedman_metric_symbol": "s", "friedman_metric_value": 8.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@16", "credit": "Wengerodt", "date": "1983", "family_updated": "", "metric_expression": "8", "metric_symbol": "s", "metric_value": 8.0, "reference_text": "s = 8 | Proved by Wengerodt in 1983.", "source_image": "cs16.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs16.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 8 | Proved by Wengerodt in 1983.", "friedman_source_image": "cs16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "6ef91613d0db44e8", "improvement": -0.00016644451351055523, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.383802090528576e-06, "max_pair_overlap_depth": -3.451016213729119e-05, "metric_expression": "8.00017", "metric_symbol": "s", "metric_value": 8.00016644451351, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.0, "setup": "cirinsqu", "side": 8.00016644451351, "solution_path": "data/solutions/generated/cirinsqu_016_6ef91613d0db44e8.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Wengerodt", "svg_path": "data/svg/6ef91613d0db44e8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@17", "density": 0.7333955245290497, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs17.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "8.532+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.532, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@17", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "8.532+", "metric_symbol": "s", "metric_value": 8.532, "reference_text": "s = 8.532+ | Proved by Peikert in 1991.", "source_image": "cs17.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs17.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 8.532+ | Proved by Peikert in 1991.", "friedman_source_image": "cs17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "9211255584acb7a4", "improvement": -0.001560450998908891, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6768059695481128e-06, "max_pair_overlap_depth": -3.5588125128604986e-05, "metric_expression": "8.53356", "metric_symbol": "s", "metric_value": 8.533560450998909, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.532, "setup": "cirinsqu", "side": 8.533560450998909, "solution_path": "data/solutions/generated/cirinsqu_017_9211255584acb7a4.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/9211255584acb7a4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@18", "density": 0.7546080214690231, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs18.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + 24/\u221a13 = 8.656+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.656, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@18", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "2 + 24/\u221a13 = 8.656+", "metric_symbol": "s", "metric_value": 8.656, "reference_text": "s = 2 + 24/\u221a13 = 8.656+ | Proved by Peikert in 1991.", "source_image": "cs18.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs18.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + 24/\u221a13 = 8.656+ | Proved by Peikert in 1991.", "friedman_source_image": "cs18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "4872462eca76de82", "improvement": -0.0006623865813519814, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.585640835636951e-06, "max_pair_overlap_depth": -5.7251545610981225e-05, "metric_expression": "8.65666", "metric_symbol": "s", "metric_value": 8.656662386581353, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.656, "setup": "cirinsqu", "side": 8.656662386581353, "solution_path": "data/solutions/generated/cirinsqu_018_4872462eca76de82.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/4872462eca76de82.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@19", "density": 0.752224113091453, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs19.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "8.907+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.907, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@19", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "8.907+", "metric_symbol": "s", "metric_value": 8.907, "reference_text": "s = 8.907+ | Proved by Peikert in 1991.", "source_image": "cs19.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs19.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 8.907+ | Proved by Peikert in 1991.", "friedman_source_image": "cs19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "72fa8cf8e074fd32", "improvement": -0.0009569875134012307, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.019564082723974e-06, "max_pair_overlap_depth": -6.564483433546542e-05, "metric_expression": "8.90796", "metric_symbol": "s", "metric_value": 8.907956987513401, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.907, "setup": "cirinsqu", "side": 8.907956987513401, "solution_path": "data/solutions/generated/cirinsqu_019_72fa8cf8e074fd32.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/72fa8cf8e074fd32.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@20", "density": 0.7794478091956536, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs20.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "130/17 + 16/17\u221a2 = 8.978+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.978, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@20", "credit": "Peikert", "date": "1991", "family_updated": "", "metric_expression": "130/17 + 16/17\u221a2 = 8.978+", "metric_symbol": "s", "metric_value": 8.978, "reference_text": "s = 130/17 + 16/17\u221a2 = 8.978+ | Proved by Peikert in 1991.", "source_image": "cs20.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs20.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 130/17 + 16/17\u221a2 = 8.978+ | Proved by Peikert in 1991.", "friedman_source_image": "cs20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "ed16784370181dab", "improvement": -0.0003475703211961445, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3975782322539487e-06, "max_pair_overlap_depth": -5.140241392753708e-05, "metric_expression": "8.97835", "metric_symbol": "s", "metric_value": 8.978347570321196, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.978, "setup": "cirinsqu", "side": 8.978347570321196, "solution_path": "data/solutions/generated/cirinsqu_020_ed16784370181dab.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Peikert", "svg_path": "data/svg/ed16784370181dab.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@21", "density": 0.7531827030031223, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman Packing Center", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs21.gif", "friedman_metric_expression": "9.358+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.358, "friedman_reference": {"analytical_or_proved": false, "case": "cirinsqu@21", "credit": "Erich Friedman Packing Center", "date": "", "family_updated": "", "metric_expression": "9.358+", "metric_symbol": "s", "metric_value": 9.358, "reference_text": "s = 9.358+", "source_image": "cs21.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs21.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 9.358+", "friedman_source_image": "cs21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "65afc927a4d3cb10", "improvement": -0.0011070505635846217, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.857662265602471e-06, "max_pair_overlap_depth": -3.291297220409106e-05, "metric_expression": "9.35911", "metric_symbol": "s", "metric_value": 9.359107050563585, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.358, "setup": "cirinsqu", "side": 9.359107050563585, "solution_path": "data/solutions/generated/cirinsqu_021_65afc927a4d3cb10.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman Packing Center", "svg_path": "data/svg/65afc927a4d3cb10.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@22", "density": 0.7716499915329439, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman Packing Center", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs22.gif", "friedman_metric_expression": "9.463+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.463, "friedman_reference": {"analytical_or_proved": false, "case": "cirinsqu@22", "credit": "Erich Friedman Packing Center", "date": "", "family_updated": "", "metric_expression": "9.463+", "metric_symbol": "s", "metric_value": 9.463, "reference_text": "s = 9.463+", "source_image": "cs22.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs22.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 9.463+", "friedman_source_image": "cs22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "27d48d864449f174", "improvement": -0.0010297949122062278, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1647620365984324e-06, "max_pair_overlap_depth": -2.796224374757017e-05, "metric_expression": "9.46403", "metric_symbol": "s", "metric_value": 9.464029794912205, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.463, "setup": "cirinsqu", "side": 9.464029794912205, "solution_path": "data/solutions/generated/cirinsqu_022_27d48d864449f174.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman Packing Center", "svg_path": "data/svg/27d48d864449f174.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@23", "density": 0.7628623175184639, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman Packing Center", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs23.gif", "friedman_metric_expression": "2 + 2\u221a2 + 2\u221a6 = 9.727+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.727, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@23", "credit": "Erich Friedman Packing Center", "date": "", "family_updated": "", "metric_expression": "2 + 2\u221a2 + 2\u221a6 = 9.727+", "metric_symbol": "s", "metric_value": 9.727, "reference_text": "s = 2 + 2\u221a2 + 2\u221a6 = 9.727+", "source_image": "cs23.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs23.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + 2\u221a2 + 2\u221a6 = 9.727+", "friedman_source_image": "cs23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "cfa99060ee2203a1", "improvement": -0.005306391434194424, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8298221013957061e-06, "max_pair_overlap_depth": -2.6081243190478887e-05, "metric_expression": "9.73231", "metric_symbol": "s", "metric_value": 9.732306391434195, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.727, "setup": "cirinsqu", "side": 9.732306391434195, "solution_path": "data/solutions/generated/cirinsqu_023_cfa99060ee2203a1.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman Packing Center", "svg_path": "data/svg/cfa99060ee2203a1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirinsqu@24", "density": 0.7749295071503051, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman Packing Center", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs24.gif", "friedman_metric_expression": "6 + \u221a2 + \u221a6 = 9.863+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.863, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@24", "credit": "Erich Friedman Packing Center", "date": "", "family_updated": "", "metric_expression": "6 + \u221a2 + \u221a6 = 9.863+", "metric_symbol": "s", "metric_value": 9.863, "reference_text": "s = 6 + \u221a2 + \u221a6 = 9.863+", "source_image": "cs24.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs24.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 6 + \u221a2 + \u221a6 = 9.863+", "friedman_source_image": "cs24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "e2eca298edbdb88c", "improvement": -0.0009181131343805049, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.506144878683841e-06, "max_pair_overlap_depth": -4.0021533562306644e-05, "metric_expression": "9.86392", "metric_symbol": "s", "metric_value": 9.86391811313438, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.863, "setup": "cirinsqu", "side": 9.86391811313438, "solution_path": "data/solutions/generated/cirinsqu_024_e2eca298edbdb88c.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman Packing Center", "svg_path": "data/svg/e2eca298edbdb88c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@1", "density": 0.6045962978262086, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct1.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "2\u221a3 = 3.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2\u221a3 = 3.464+", "metric_symbol": "s", "metric_value": 3.464, "reference_text": "s = 2\u221a3 = 3.464+ | Trivial.", "source_image": "ct1.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct1.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 2\u221a3 = 3.464+ | Trivial.", "friedman_source_image": "ct1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "10a7a380c7f5376a", "improvement": -0.00011161401610815247, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7091087705090047e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "3.46411", "metric_symbol": "s", "metric_value": 3.464111614016108, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.464, "setup": "cirintri", "side": 3.464111614016108, "solution_path": "data/solutions/generated/cirintri_001_10a7a380c7f5376a.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/10a7a380c7f5376a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@2", "density": 0.48598475821399, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct2.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "2 + 2\u221a3 = 5.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 2\u221a3 = 5.464+", "metric_symbol": "s", "metric_value": 5.464, "reference_text": "s = 2 + 2\u221a3 = 5.464+ | Trivial.", "source_image": "ct2.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct2.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 2 + 2\u221a3 = 5.464+ | Trivial.", "friedman_source_image": "ct2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "f5cde1b8295da3ad", "improvement": -0.00022144929157175142, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7587277048611867e-06, "max_pair_overlap_depth": -7.068795172848752e-05, "metric_expression": "5.46422", "metric_symbol": "s", "metric_value": 5.464221449291572, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.464, "setup": "cirintri", "side": 5.464221449291572, "solution_path": "data/solutions/generated/cirintri_002_f5cde1b8295da3ad.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/f5cde1b8295da3ad.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@3", "density": 0.7289907317557431, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct3.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "2 + 2\u221a3 = 5.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 2\u221a3 = 5.464+", "metric_symbol": "s", "metric_value": 5.464, "reference_text": "s = 2 + 2\u221a3 = 5.464+ | Trivial.", "source_image": "ct3.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct3.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 2 + 2\u221a3 = 5.464+ | Trivial.", "friedman_source_image": "ct3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "fcee9d6e8b7c5ce0", "improvement": -0.000170499845251193, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0755666179917966e-07, "max_pair_overlap_depth": -5.0474041760395494e-05, "metric_expression": "5.46417", "metric_symbol": "s", "metric_value": 5.464170499845252, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.464, "setup": "cirintri", "side": 5.464170499845252, "solution_path": "data/solutions/generated/cirintri_003_fcee9d6e8b7c5ce0.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fcee9d6e8b7c5ce0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@4", "density": 0.6045810458812538, "friedman_analytical_or_proved": true, "friedman_credit": "Milano", "friedman_date": "1987", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct4.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "4\u221a3 = 6.928+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.928, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@4", "credit": "Milano", "date": "1987", "family_updated": "", "metric_expression": "4\u221a3 = 6.928+", "metric_symbol": "s", "metric_value": 6.928, "reference_text": "s = 4\u221a3 = 6.928+ | Proved by Milano in 1987.", "source_image": "ct4.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct4.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 4\u221a3 = 6.928+ | Proved by Milano in 1987.", "friedman_source_image": "ct4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "9e0c39beb3cc8aa7", "improvement": -0.0003106176498706148, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7330840119143005e-06, "max_pair_overlap_depth": -4.49553106092182e-05, "metric_expression": "6.92831", "metric_symbol": "s", "metric_value": 6.9283106176498706, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.928, "setup": "cirintri", "side": 6.9283106176498706, "solution_path": "data/solutions/generated/cirintri_004_9e0c39beb3cc8aa7.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Milano", "svg_path": "data/svg/9e0c39beb3cc8aa7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@5", "density": 0.6510985676665063, "friedman_analytical_or_proved": true, "friedman_credit": "Milano", "friedman_date": "1987", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct5.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "4 + 2\u221a3 = 7.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@5", "credit": "Milano", "date": "1987", "family_updated": "", "metric_expression": "4 + 2\u221a3 = 7.464+", "metric_symbol": "s", "metric_value": 7.464, "reference_text": "s = 4 + 2\u221a3 = 7.464+ | Proved by Milano in 1987.", "source_image": "ct5.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct5.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 4 + 2\u221a3 = 7.464+ | Proved by Milano in 1987.", "friedman_source_image": "ct5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "ad32c3b8e4c714b5", "improvement": -0.00025126705611278055, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.12146637612598e-06, "max_pair_overlap_depth": -3.650753517803196e-05, "metric_expression": "7.46425", "metric_symbol": "s", "metric_value": 7.464251267056113, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.464, "setup": "cirintri", "side": 7.464251267056113, "solution_path": "data/solutions/generated/cirintri_005_ad32c3b8e4c714b5.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Milano", "svg_path": "data/svg/ad32c3b8e4c714b5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@6", "density": 0.7813295579764934, "friedman_analytical_or_proved": true, "friedman_credit": "Oler/Groemer", "friedman_date": "1961", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct6.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "4 + 2\u221a3 = 7.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@6", "credit": "Oler/Groemer", "date": "1961", "family_updated": "", "metric_expression": "4 + 2\u221a3 = 7.464+", "metric_symbol": "s", "metric_value": 7.464, "reference_text": "s = 4 + 2\u221a3 = 7.464+ | Proved by Oler/Groemer in 1961.", "source_image": "ct6.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct6.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 4 + 2\u221a3 = 7.464+ | Proved by Oler/Groemer in 1961.", "friedman_source_image": "ct6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "09bddcb97a85d368", "improvement": -0.00019740182247307558, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6105130511423482e-06, "max_pair_overlap_depth": -3.3410124574029254e-05, "metric_expression": "7.46420", "metric_symbol": "s", "metric_value": 7.4641974018224735, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.464, "setup": "cirintri", "side": 7.4641974018224735, "solution_path": "data/solutions/generated/cirintri_006_09bddcb97a85d368.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Oler/Groemer", "svg_path": "data/svg/09bddcb97a85d368.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@7", "density": 0.6370913423934993, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1993", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct7.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "2 + 4\u221a3 = 8.928+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.928, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@7", "credit": "Melissen", "date": "1993", "family_updated": "", "metric_expression": "2 + 4\u221a3 = 8.928+", "metric_symbol": "s", "metric_value": 8.928, "reference_text": "s = 2 + 4\u221a3 = 8.928+ | Proved by Melissen in 1993.", "source_image": "ct7.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct7.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 2 + 4\u221a3 = 8.928+ | Proved by Melissen in 1993.", "friedman_source_image": "ct7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "1d479488f6f25560", "improvement": -0.0003824812590771387, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.6242968528307244e-06, "max_pair_overlap_depth": -4.768068922000879e-05, "metric_expression": "8.92838", "metric_symbol": "s", "metric_value": 8.928382481259078, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.928, "setup": "cirintri", "side": 8.928382481259078, "solution_path": "data/solutions/generated/cirintri_007_1d479488f6f25560.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Melissen", "svg_path": "data/svg/1d479488f6f25560.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@8", "density": 0.6718185539703104, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1993", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct8.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "2 + 2\u221a3 + 2\u221a33/3 = 9.293+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.293, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@8", "credit": "Melissen", "date": "1993", "family_updated": "", "metric_expression": "2 + 2\u221a3 + 2\u221a33/3 = 9.293+", "metric_symbol": "s", "metric_value": 9.293, "reference_text": "s = 2 + 2\u221a3 + 2\u221a33/3 = 9.293+ | Proved by Melissen in 1993.", "source_image": "ct8.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct8.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 2 + 2\u221a3 + 2\u221a33/3 = 9.293+ | Proved by Melissen in 1993.", "friedman_source_image": "ct8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "ad6e4ca149e6364f", "improvement": -0.0018763116025901638, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.090428418316975e-06, "max_pair_overlap_depth": -4.461887232398709e-05, "metric_expression": "9.29488", "metric_symbol": "s", "metric_value": 9.29487631160259, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.293, "setup": "cirintri", "side": 9.29487631160259, "solution_path": "data/solutions/generated/cirintri_008_ad6e4ca149e6364f.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Melissen", "svg_path": "data/svg/ad6e4ca149e6364f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@9", "density": 0.728985395977573, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1993", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct9.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "6 + 2\u221a3 = 9.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@9", "credit": "Melissen", "date": "1993", "family_updated": "", "metric_expression": "6 + 2\u221a3 = 9.464+", "metric_symbol": "s", "metric_value": 9.464, "reference_text": "s = 6 + 2\u221a3 = 9.464+ | Proved by Melissen in 1993.", "source_image": "ct9.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct9.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 6 + 2\u221a3 = 9.464+ | Proved by Melissen in 1993.", "friedman_source_image": "ct9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "108d2156f1819ef5", "improvement": -0.0002555633725904727, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.43897889048894e-06, "max_pair_overlap_depth": -2.63301951002326e-05, "metric_expression": "9.46426", "metric_symbol": "s", "metric_value": 9.46425556337259, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.464, "setup": "cirintri", "side": 9.46425556337259, "solution_path": "data/solutions/generated/cirintri_009_108d2156f1819ef5.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Melissen", "svg_path": "data/svg/108d2156f1819ef5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@10", "density": 0.8099824382101458, "friedman_analytical_or_proved": true, "friedman_credit": "Oler/Groemer", "friedman_date": "1961", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct10.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "6 + 2\u221a3 = 9.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@10", "credit": "Oler/Groemer", "date": "1961", "family_updated": "", "metric_expression": "6 + 2\u221a3 = 9.464+", "metric_symbol": "s", "metric_value": 9.464, "reference_text": "s = 6 + 2\u221a3 = 9.464+ | Proved by Oler/Groemer in 1961.", "source_image": "ct10.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct10.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 6 + 2\u221a3 = 9.464+ | Proved by Oler/Groemer in 1961.", "friedman_source_image": "ct10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "abf88e494059d012", "improvement": -0.0002633633602293628, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.885918628489861e-06, "max_pair_overlap_depth": -4.501428401626484e-05, "metric_expression": "9.46426", "metric_symbol": "s", "metric_value": 9.46426336336023, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.464, "setup": "cirintri", "side": 9.46426336336023, "solution_path": "data/solutions/generated/cirintri_010_abf88e494059d012.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Oler/Groemer", "svg_path": "data/svg/abf88e494059d012.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@11", "density": 0.693141460882906, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1993", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct11.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "4 + 2\u221a3 + 4\u221a6/3 = 10.730+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.73, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@11", "credit": "Melissen", "date": "1993", "family_updated": "", "metric_expression": "4 + 2\u221a3 + 4\u221a6/3 = 10.730+", "metric_symbol": "s", "metric_value": 10.73, "reference_text": "s = 4 + 2\u221a3 + 4\u221a6/3 = 10.730+ | Proved by Melissen in 1993.", "source_image": "ct11.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct11.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 4 + 2\u221a3 + 4\u221a6/3 = 10.730+ | Proved by Melissen in 1993.", "friedman_source_image": "ct11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "64f835afcfaa2245", "improvement": -0.00025448918442982915, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7228747402064926e-06, "max_pair_overlap_depth": -3.113241300978231e-05, "metric_expression": "10.73025", "metric_symbol": "s", "metric_value": 10.73025448918443, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 10.73, "setup": "cirintri", "side": 10.73025448918443, "solution_path": "data/solutions/generated/cirintri_011_64f835afcfaa2245.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Melissen", "svg_path": "data/svg/64f835afcfaa2245.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@12", "density": 0.7289832647467578, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1994", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct12.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "4 + 4\u221a3 = 10.928+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.928, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@12", "credit": "Melissen", "date": "1994", "family_updated": "", "metric_expression": "4 + 4\u221a3 = 10.928+", "metric_symbol": "s", "metric_value": 10.928, "reference_text": "s = 4 + 4\u221a3 = 10.928+ | Proved by Melissen in 1994.", "source_image": "ct12.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct12.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 4 + 4\u221a3 = 10.928+ | Proved by Melissen in 1994.", "friedman_source_image": "ct12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "c94cd40d17cad41e", "improvement": -0.00039696929599308817, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.790000236816084e-06, "max_pair_overlap_depth": -3.824380327710486e-05, "metric_expression": "10.92840", "metric_symbol": "s", "metric_value": 10.928396969295994, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 10.928, "setup": "cirintri", "side": 10.928396969295994, "solution_path": "data/solutions/generated/cirintri_012_c94cd40d17cad41e.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Melissen", "svg_path": "data/svg/c94cd40d17cad41e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@13", "density": 0.7248761583522253, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1993", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct13.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "4 + 2\u221a6/3 + 10\u221a3/3 = 11.406+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.406, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@13", "credit": "Melissen", "date": "1993", "family_updated": "", "metric_expression": "4 + 2\u221a6/3 + 10\u221a3/3 = 11.406+", "metric_symbol": "s", "metric_value": 11.406, "reference_text": "s = 4 + 2\u221a6/3 + 10\u221a3/3 = 11.406+ | Found by Melissen in 1993.", "source_image": "ct13.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct13.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 4 + 2\u221a6/3 + 10\u221a3/3 = 11.406+ | Found by Melissen in 1993.", "friedman_source_image": "ct13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "4ca74ad9707b79ad", "improvement": -0.0008146992920448071, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.27136224545599e-06, "max_pair_overlap_depth": -4.371332125208127e-05, "metric_expression": "11.40681", "metric_symbol": "s", "metric_value": 11.406814699292045, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 11.406, "setup": "cirintri", "side": 11.406814699292045, "solution_path": "data/solutions/generated/cirintri_013_4ca74ad9707b79ad.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Melissen", "svg_path": "data/svg/4ca74ad9707b79ad.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@14", "density": 0.7728234005113342, "friedman_analytical_or_proved": true, "friedman_credit": "Erd\u0151s/Oler", "friedman_date": "1961", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct14.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "8 + 2\u221a3 = 11.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@14", "credit": "Erd\u0151s/Oler", "date": "1961", "family_updated": "", "metric_expression": "8 + 2\u221a3 = 11.464+", "metric_symbol": "s", "metric_value": 11.464, "reference_text": "s = 8 + 2\u221a3 = 11.464+ | Found by Erd\u0151s/Oler in 1961.", "source_image": "ct14.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct14.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 8 + 2\u221a3 = 11.464+ | Found by Erd\u0151s/Oler in 1961.", "friedman_source_image": "ct14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "67743153538612b8", "improvement": -0.00032539636501560665, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.0042126467086945e-06, "max_pair_overlap_depth": -3.7763457817252544e-05, "metric_expression": "11.46433", "metric_symbol": "s", "metric_value": 11.464325396365016, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 11.464, "setup": "cirintri", "side": 11.464325396365016, "solution_path": "data/solutions/generated/cirintri_014_67743153538612b8.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erd\u0151s/Oler", "svg_path": "data/svg/67743153538612b8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "cirintri@15", "density": 0.82802371463932, "friedman_analytical_or_proved": true, "friedman_credit": "Erd\u0151s/Groemer", "friedman_date": "1961", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirintri/ct15.gif", "friedman_family_updated": "7/19/02", "friedman_metric_expression": "8 + 2\u221a3 = 11.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.464, "friedman_reference": {"analytical_or_proved": true, "case": "cirintri@15", "credit": "Erd\u0151s/Groemer", "date": "1961", "family_updated": "", "metric_expression": "8 + 2\u221a3 = 11.464+", "metric_symbol": "s", "metric_value": 11.464, "reference_text": "s = 8 + 2\u221a3 = 11.464+ | Proved by Erd\u0151s/Groemer in 1961.", "source_image": "ct15.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirintri/ct15.gif", "source_page": "https://erich-friedman.github.io/packing/cirintri/", "title": "Circles in Triangles"}, "friedman_reference_text": "s = 8 + 2\u221a3 = 11.464+ | Proved by Erd\u0151s/Groemer in 1961.", "friedman_source_image": "ct15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirintri/", "friedman_title": "Circles in Triangles", "frontend_seed": true, "id": "2a26f9d337f635e0", "improvement": -0.00033479280357262553, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.377565219179871e-06, "max_pair_overlap_depth": -4.0761680023315705e-05, "metric_expression": "11.46433", "metric_symbol": "s", "metric_value": 11.464334792803573, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 11.464, "setup": "cirintri", "side": 11.464334792803573, "solution_path": "data/solutions/generated/cirintri_015_2a26f9d337f635e0.json", "source_url": "https://erich-friedman.github.io/packing/cirintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erd\u0151s/Groemer", "svg_path": "data/svg/2a26f9d337f635e0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@1", "density": 0.9999892980292665, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/1.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "1. | | s = 1 | Trivial.", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/1.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "1. | | s = 1 | Trivial.", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "6b50d5741feee5dc", "improvement": -5.351028316669115e-06, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6755141583345576e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00001", "metric_symbol": "s", "metric_value": 1.0000053510283167, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.0, "setup": "domindom", "side": 2.0000107020566333, "solution_path": "data/solutions/generated/domindom_001_6b50d5741feee5dc.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6b50d5741feee5dc.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@2", "density": 0.5146978614404703, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/2.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "(2+\u221a2)/\u221a3 = 1.971+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.971, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@2", "credit": "Erich Friedman", "date": "April 2026", "family_updated": "", "metric_expression": "(2+\u221a2)/\u221a3 = 1.971+", "metric_symbol": "s", "metric_value": 1.971, "reference_text": "2. | | s = (2+\u221a2)/\u221a3 = 1.971+ | Found by Erich Friedman in April 2026.", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/2.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "2. | | s = (2+\u221a2)/\u221a3 = 1.971+ | Found by Erich Friedman in April 2026.", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "e1760976ec64bd64", "improvement": -0.000236880650400062, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9155926735958673e-06, "max_pair_overlap_depth": -6.0125633082330765e-06, "metric_expression": "1.97124", "metric_symbol": "s", "metric_value": 1.9712368806504001, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.971, "setup": "domindom", "side": 3.9424737613008003, "solution_path": "data/solutions/generated/domindom_002_e1760976ec64bd64.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/e1760976ec64bd64.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@3", "density": 0.7499858951262421, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/4.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "3.-4. | | s = 2 | Trivial.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/4.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "3.-4. | | s = 2 | Trivial.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "5194ab3dddc343bb", "improvement": -1.8806763611323163e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3349574317089719e-06, "max_pair_overlap_depth": -6.50598393479697e-06, "metric_expression": "2.00002", "metric_symbol": "s", "metric_value": 2.0000188067636113, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "domindom", "side": 4.000037613527223, "solution_path": "data/solutions/generated/domindom_003_5194ab3dddc343bb.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/5194ab3dddc343bb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@4", "density": 0.9999632147512098, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/4.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "3.-4. | | s = 2 | Trivial.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/4.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "3.-4. | | s = 2 | Trivial.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "14a703d2c83d5b43", "improvement": -3.6786263687194065e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.672168442576563e-06, "max_pair_overlap_depth": -2.5740489017265176e-05, "metric_expression": "2.00004", "metric_symbol": "s", "metric_value": 2.000036786263687, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "domindom", "side": 4.000073572527374, "solution_path": "data/solutions/generated/domindom_004_14a703d2c83d5b43.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/14a703d2c83d5b43.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@5", "density": 0.7999239260919105, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/5.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "5/2 = 2.5", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5/2 = 2.5", "metric_symbol": "s", "metric_value": 2.5, "reference_text": "5. | | s = 5/2 = 2.5 | Trivial.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/5.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "5. | | s = 5/2 = 2.5 | Trivial.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "af0995e19f0dc8ce", "improvement": -0.00011887395946308388, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.6075584103633105e-07, "max_pair_overlap_depth": -2.297221660940707e-05, "metric_expression": "2.50012", "metric_symbol": "s", "metric_value": 2.500118873959463, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.5, "setup": "domindom", "side": 5.000237747918926, "solution_path": "data/solutions/generated/domindom_005_af0995e19f0dc8ce.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/af0995e19f0dc8ce.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@6", "density": 0.7051360525077088, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/6.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "2.916+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.916, "friedman_reference": {"analytical_or_proved": false, "case": "domindom@6", "credit": "Erich Friedman", "date": "April 2026", "family_updated": "", "metric_expression": "2.916+", "metric_symbol": "s", "metric_value": 2.916, "reference_text": "6. | | s = 2.916+ | Found by Erich Friedman in April 2026.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/6.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "6. | | s = 2.916+ | Found by Erich Friedman in April 2026.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "0f3d5defa1e042a8", "improvement": -0.001018377106842383, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.01302900812739e-06, "max_pair_overlap_depth": -7.57681649177222e-06, "metric_expression": "2.91702", "metric_symbol": "s", "metric_value": 2.9170183771068423, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.916, "setup": "domindom", "side": 5.834036754213685, "solution_path": "data/solutions/generated/domindom_006_0f3d5defa1e042a8.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/0f3d5defa1e042a8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@7", "density": 0.777726224180711, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "7.-9. | | s = 3 | Trivial.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "7.-9. | | s = 3 | Trivial.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "7b5335d8b8c4a5db", "improvement": -9.942973726095872e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3331344280185253e-06, "max_pair_overlap_depth": -2.1961883454491016e-05, "metric_expression": "3.00010", "metric_symbol": "s", "metric_value": 3.000099429737261, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.0, "setup": "domindom", "side": 6.000198859474522, "solution_path": "data/solutions/generated/domindom_007_7b5335d8b8c4a5db.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7b5335d8b8c4a5db.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@8", "density": 0.8888442639489805, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@8", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "7.-9. | | s = 3 | Trivial.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "7.-9. | | s = 3 | Trivial.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "953d10b3e926402a", "improvement": -7.530742160444248e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.80012014980602e-06, "max_pair_overlap_depth": -1.8965164302886528e-05, "metric_expression": "3.00008", "metric_symbol": "s", "metric_value": 3.0000753074216044, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.0, "setup": "domindom", "side": 6.000150614843209, "solution_path": "data/solutions/generated/domindom_008_953d10b3e926402a.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/953d10b3e926402a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@9", "density": 0.9999668695349745, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@9", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "7.-9. | | s = 3 | Trivial.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/9.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "7.-9. | | s = 3 | Trivial.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "43b9630822b40990", "improvement": -4.9696932403531235e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7542337882889854e-07, "max_pair_overlap_depth": -3.0527855267026283e-06, "metric_expression": "3.00005", "metric_symbol": "s", "metric_value": 3.0000496969324035, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.0, "setup": "domindom", "side": 6.000099393864807, "solution_path": "data/solutions/generated/domindom_009_43b9630822b40990.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/43b9630822b40990.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@10", "density": 0.8162099861183555, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/10.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "7/2 = 3.5", "friedman_metric_symbol": "s", "friedman_metric_value": 3.5, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@10", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7/2 = 3.5", "metric_symbol": "s", "metric_value": 3.5, "reference_text": "10. | | s = 7/2 = 3.5 | Trivial.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/10.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "10. | | s = 7/2 = 3.5 | Trivial.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "d7508c45c3f18a7c", "improvement": -0.0002498690138819093, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5739397918324016e-07, "max_pair_overlap_depth": -9.291197802940854e-05, "metric_expression": "3.50025", "metric_symbol": "s", "metric_value": 3.500249869013882, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.5, "setup": "domindom", "side": 7.000499738027764, "solution_path": "data/solutions/generated/domindom_010_d7508c45c3f18a7c.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d7508c45c3f18a7c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@11", "density": 0.7611589259525644, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/11.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "(5+\u221a41)/3 = 3.801+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.801, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@11", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "(5+\u221a41)/3 = 3.801+", "metric_symbol": "s", "metric_value": 3.801, "reference_text": "11. | | s = (5+\u221a41)/3 = 3.801+ | Found by Maurizio Morandi in April 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/11.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "11. | | s = (5+\u221a41)/3 = 3.801+ | Found by Maurizio Morandi in April 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "0fa9d3de612fbf03", "improvement": -0.0005321723096423092, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0757791549262947e-06, "max_pair_overlap_depth": -0.0001019023981195799, "metric_expression": "3.80153", "metric_symbol": "s", "metric_value": 3.8015321723096425, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.801, "setup": "domindom", "side": 7.603064344619285, "solution_path": "data/solutions/generated/domindom_011_0fa9d3de612fbf03.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/0fa9d3de612fbf03.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@12", "density": 0.7822558234911755, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/12.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "5/2+\u221a2 = 3.914+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.914, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@12", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "5/2+\u221a2 = 3.914+", "metric_symbol": "s", "metric_value": 3.914, "reference_text": "12. | | s = 5/2+\u221a2 = 3.914+ | Found by Maurizio Morandi in April 2026.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/12.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "12. | | s = 5/2+\u221a2 = 3.914+ | Found by Maurizio Morandi in April 2026.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "3a75d33bd56228b0", "improvement": -0.002663137487304734, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.0673284962267076e-07, "max_pair_overlap_depth": -1.4217486933421064e-05, "metric_expression": "3.91666", "metric_symbol": "s", "metric_value": 3.916663137487305, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.914, "setup": "domindom", "side": 7.83332627497461, "solution_path": "data/solutions/generated/domindom_012_3a75d33bd56228b0.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T03:07:58+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/3a75d33bd56228b0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@14", "density": 0.8749484114386541, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@14", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "14.-16 | | s = 4 | Trivial.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "14.-16 | | s = 4 | Trivial.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "8be6cbfdbea40909", "improvement": -0.00011792192603543583, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.966859551203129e-06, "max_pair_overlap_depth": -1.688780940467538e-05, "metric_expression": "4.00012", "metric_symbol": "s", "metric_value": 4.000117921926035, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.0, "setup": "domindom", "side": 8.00023584385207, "solution_path": "data/solutions/generated/domindom_014_8be6cbfdbea40909.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/8be6cbfdbea40909.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@15", "density": 0.9374471956142247, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@15", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "14.-16 | | s = 4 | Trivial.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "14.-16 | | s = 4 | Trivial.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "d2b8e4757dde349c", "improvement": -0.00011265411524785662, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.415030228686277e-06, "max_pair_overlap_depth": -1.0456826571081379e-05, "metric_expression": "4.00011", "metric_symbol": "s", "metric_value": 4.000112654115248, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.0, "setup": "domindom", "side": 8.000225308230496, "solution_path": "data/solutions/generated/domindom_015_d2b8e4757dde349c.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d2b8e4757dde349c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@16", "density": 0.9999595560012681, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@16", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "14.-16 | | s = 4 | Trivial.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/16.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "14.-16 | | s = 4 | Trivial.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "5102d8436800a7ff", "improvement": -8.08904511222508e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.23688533190969e-06, "max_pair_overlap_depth": -4.621969549978289e-06, "metric_expression": "4.00008", "metric_symbol": "s", "metric_value": 4.000080890451122, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.0, "setup": "domindom", "side": 8.000161780902245, "solution_path": "data/solutions/generated/domindom_016_5102d8436800a7ff.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/5102d8436800a7ff.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@17", "density": 0.8394293686085971, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/18.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "9/2 = 4.5", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@17", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "9/2 = 4.5", "metric_symbol": "s", "metric_value": 4.5, "reference_text": "17.-18 | | s = 9/2 = 4.5 | Trivial.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "17.-18 | | s = 9/2 = 4.5 | Trivial.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "3b388ddd068a843f", "improvement": -0.00020586075893191946, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.953267209728551e-07, "max_pair_overlap_depth": -5.000451830872166e-05, "metric_expression": "4.50021", "metric_symbol": "s", "metric_value": 4.500205860758932, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.5, "setup": "domindom", "side": 9.000411721517864, "solution_path": "data/solutions/generated/domindom_017_3b388ddd068a843f.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3b388ddd068a843f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domindom@18", "density": 0.887184224626392, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domindom/18.gif", "friedman_family_updated": "4/3/26", "friedman_metric_expression": "9/2 = 4.5", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_reference": {"analytical_or_proved": true, "case": "domindom@18", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "9/2 = 4.5", "metric_symbol": "s", "metric_value": 4.5, "reference_text": "17.-18 | | s = 9/2 = 4.5 | Trivial.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/domindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/domindom/", "title": "Dominoes in Dominoes"}, "friedman_reference_text": "17.-18 | | s = 9/2 = 4.5 | Trivial.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/domindom/", "friedman_title": "Dominoes in Dominoes", "frontend_seed": true, "id": "6a5fcb6ec4e55187", "improvement": -0.004321147560441219, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4781434060372476e-06, "max_pair_overlap_depth": -4.894490537976637e-05, "metric_expression": "4.50432", "metric_symbol": "s", "metric_value": 4.504321147560441, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.5, "setup": "domindom", "side": 9.008642295120882, "solution_path": "data/solutions/generated/domindom_018_6a5fcb6ec4e55187.json", "source_url": "https://erich-friedman.github.io/packing/domindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6a5fcb6ec4e55187.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@1", "density": 0.46408935997723705, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/1.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "2+2/\u221a3 = 3.15470+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.1547, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2+2/\u221a3 = 3.15470+", "metric_symbol": "s", "metric_value": 3.1547, "reference_text": "s = 2+2/\u221a3 = 3.15470+ | Trivial.", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/1.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 2+2/\u221a3 = 3.15470+ | Trivial.", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "6121ec989c6c813b", "improvement": -4.219103236957622e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.613940413552034e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "3.15474", "metric_symbol": "s", "metric_value": 3.1547421910323696, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.1547, "setup": "domintri", "side": 3.1547421910323696, "solution_path": "data/solutions/generated/domintri_001_6121ec989c6c813b.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6121ec989c6c813b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@2", "density": 0.4973974214603669, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/2.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "2+4/\u221a3 = 4.30940+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.3094, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2+4/\u221a3 = 4.30940+", "metric_symbol": "s", "metric_value": 4.3094, "reference_text": "s = 2+4/\u221a3 = 4.30940+ | Trivial.", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/2.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 2+4/\u221a3 = 4.30940+ | Trivial.", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "75b0ecc0320e01de", "improvement": -0.00011019921439459068, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.546312299587745e-06, "max_pair_overlap_depth": -2.318487220109855e-05, "metric_expression": "4.30951", "metric_symbol": "s", "metric_value": 4.309510199214395, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.3094, "setup": "domintri", "side": 4.309510199214395, "solution_path": "data/solutions/generated/domintri_002_75b0ecc0320e01de.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/75b0ecc0320e01de.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@3", "density": 0.746073641317824, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/3.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "2+4/\u221a3 = 4.30934+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.30934, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2+4/\u221a3 = 4.30934+", "metric_symbol": "s", "metric_value": 4.30934, "reference_text": "s = 2+4/\u221a3 = 4.30934+ | Trivial.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/3.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 2+4/\u221a3 = 4.30934+ | Trivial.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "ecf54a038a2d7184", "improvement": -0.00023515521175365706, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2511235262659097e-06, "max_pair_overlap_depth": -8.934604283755698e-05, "metric_expression": "4.30958", "metric_symbol": "s", "metric_value": 4.309575155211753, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.30934, "setup": "domintri", "side": 4.309575155211753, "solution_path": "data/solutions/generated/domintri_003_ecf54a038a2d7184.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/ecf54a038a2d7184.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@4", "density": 0.6377999539199563, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/4.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "5.38188+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.38188, "friedman_reference": {"analytical_or_proved": false, "case": "domintri@4", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.38188+", "metric_symbol": "s", "metric_value": 5.38188, "reference_text": "s = 5.38188+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/4.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 5.38188+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "6f3730492b03ce4f", "improvement": -0.0002283152146249634, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.401205447247945e-07, "max_pair_overlap_depth": -4.238554796887062e-05, "metric_expression": "5.38211", "metric_symbol": "s", "metric_value": 5.382108315214625, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.38188, "setup": "domintri", "side": 5.382108315214625, "solution_path": "data/solutions/generated/domintri_004_6f3730492b03ce4f.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/6f3730492b03ce4f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@5", "density": 0.7734497127773695, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/5.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "2+2\u221a3 = 5.46410+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.4641, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@5", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2+2\u221a3 = 5.46410+", "metric_symbol": "s", "metric_value": 5.4641, "reference_text": "s = 2+2\u221a3 = 5.46410+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/5.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 2+2\u221a3 = 5.46410+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "272a5ede0aa9a415", "improvement": -0.00018874969044357925, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.315600305266359e-07, "max_pair_overlap_depth": -2.2346020378205544e-05, "metric_expression": "5.46429", "metric_symbol": "s", "metric_value": 5.464288749690444, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.4641, "setup": "domintri", "side": 5.464288749690444, "solution_path": "data/solutions/generated/domintri_005_272a5ede0aa9a415.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/272a5ede0aa9a415.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@6", "density": 0.6961017329945272, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/6.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "4+4/\u221a3 = 6.30940+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.3094, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@6", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4+4/\u221a3 = 6.30940+", "metric_symbol": "s", "metric_value": 6.3094, "reference_text": "s = 4+4/\u221a3 = 6.30940+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/6.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 4+4/\u221a3 = 6.30940+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "5709487003259292", "improvement": -0.0002307959840699425, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1058985619703807e-06, "max_pair_overlap_depth": -5.695799924376743e-05, "metric_expression": "6.30963", "metric_symbol": "s", "metric_value": 6.30963079598407, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.3094, "setup": "domintri", "side": 6.30963079598407, "solution_path": "data/solutions/generated/domintri_006_5709487003259292.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/5709487003259292.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@7", "density": 0.7379982685276676, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/7.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "6.61409+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.61409, "friedman_reference": {"analytical_or_proved": false, "case": "domintri@7", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.61409+", "metric_symbol": "s", "metric_value": 6.61409, "reference_text": "s = 6.61409+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/7.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 6.61409+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "4f3a41ae14776742", "improvement": -0.004813180227946745, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.155481361422797e-07, "max_pair_overlap_depth": -2.185552101252597e-05, "metric_expression": "6.61890", "metric_symbol": "s", "metric_value": 6.618903180227947, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.61409, "setup": "domintri", "side": 6.618903180227947, "solution_path": "data/solutions/generated/domintri_007_4f3a41ae14776742.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/4f3a41ae14776742.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@8", "density": 0.7218039663224785, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/8.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "6+2/\u221a3 = 7.15470+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.1547, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@8", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6+2/\u221a3 = 7.15470+", "metric_symbol": "s", "metric_value": 7.1547, "reference_text": "s = 6+2/\u221a3 = 7.15470+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/8.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 6+2/\u221a3 = 7.15470+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "6c005b2a9a930ec2", "improvement": -0.00014189325334523772, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4194624753243232e-06, "max_pair_overlap_depth": -3.0722639065561275e-05, "metric_expression": "7.15484", "metric_symbol": "s", "metric_value": 7.154841893253345, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.1547, "setup": "domintri", "side": 7.154841893253345, "solution_path": "data/solutions/generated/domintri_008_6c005b2a9a930ec2.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/6c005b2a9a930ec2.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@9", "density": 0.7778159590517906, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/9.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "5+4/\u221a3 = 7.30940+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.3094, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@9", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5+4/\u221a3 = 7.30940+", "metric_symbol": "s", "metric_value": 7.3094, "reference_text": "s = 5+4/\u221a3 = 7.30940+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/9.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 5+4/\u221a3 = 7.30940+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "efd3c01a911fbe39", "improvement": -0.0011072089155117482, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4249809133477843e-06, "max_pair_overlap_depth": -9.12510211048545e-05, "metric_expression": "7.31051", "metric_symbol": "s", "metric_value": 7.310507208915512, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.3094, "setup": "domintri", "side": 7.310507208915512, "solution_path": "data/solutions/generated/domintri_009_efd3c01a911fbe39.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/efd3c01a911fbe39.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@10", "density": 0.7955429934788115, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/10.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "3+8/\u221a3 = 7.61885+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.61885, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@10", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3+8/\u221a3 = 7.61885+", "metric_symbol": "s", "metric_value": 7.61885, "reference_text": "s = 3+8/\u221a3 = 7.61885+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/10.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 3+8/\u221a3 = 7.61885+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "0ce8e56f538c019b", "improvement": -0.000761914714127343, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.832683484389887e-07, "max_pair_overlap_depth": -3.7959726496283874e-06, "metric_expression": "7.61961", "metric_symbol": "s", "metric_value": 7.6196119147141275, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.61885, "setup": "domintri", "side": 7.6196119147141275, "solution_path": "data/solutions/generated/domintri_010_0ce8e56f538c019b.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/0ce8e56f538c019b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@11", "density": 0.7639001254938724, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/11.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "7+2/\u221a3 = 8.15470+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.1547, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "7+2/\u221a3 = 8.15470+", "metric_symbol": "s", "metric_value": 8.1547, "reference_text": "s = 7+2/\u221a3 = 8.15470+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/11.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 7+2/\u221a3 = 8.15470+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "9dea260c7a23e4ab", "improvement": -0.0006523939405695955, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1114403341932665e-06, "max_pair_overlap_depth": -5.305705658820159e-06, "metric_expression": "8.15535", "metric_symbol": "s", "metric_value": 8.15535239394057, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.1547, "setup": "domintri", "side": 8.15535239394057, "solution_path": "data/solutions/generated/domintri_011_9dea260c7a23e4ab.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/9dea260c7a23e4ab.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@12", "density": 0.8025727620592626, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/12.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "6+4/\u221a3 = 8.30940+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.3094, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@12", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6+4/\u221a3 = 8.30940+", "metric_symbol": "s", "metric_value": 8.3094, "reference_text": "s = 6+4/\u221a3 = 8.30940+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/12.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 6+4/\u221a3 = 8.30940+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "c7b6e34fcb0ca6b3", "improvement": -0.0008309724073285452, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.6999836404354686e-06, "max_pair_overlap_depth": -5.386448125865861e-06, "metric_expression": "8.31023", "metric_symbol": "s", "metric_value": 8.310230972407329, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.3094, "setup": "domintri", "side": 8.310230972407329, "solution_path": "data/solutions/generated/domintri_012_c7b6e34fcb0ca6b3.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/c7b6e34fcb0ca6b3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@13", "density": 0.8080926405411509, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/13.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "4+8/\u221a3 = 8.61885+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.61885, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4+8/\u221a3 = 8.61885+", "metric_symbol": "s", "metric_value": 8.61885, "reference_text": "s = 4+8/\u221a3 = 8.61885+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/13.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 4+8/\u221a3 = 8.61885+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "9f9d19c6f117401b", "improvement": -0.0011205125576889685, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5415762652602807e-06, "max_pair_overlap_depth": -3.765847597092531e-05, "metric_expression": "8.61997", "metric_symbol": "s", "metric_value": 8.619970512557689, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.61885, "setup": "domintri", "side": 8.619970512557689, "solution_path": "data/solutions/generated/domintri_013_9f9d19c6f117401b.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/9f9d19c6f117401b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@14", "density": 0.8398945758370188, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/14.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "3+10/\u221a3 = 8.77350+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.7735, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@14", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "3+10/\u221a3 = 8.77350+", "metric_symbol": "s", "metric_value": 8.7735, "reference_text": "s = 3+10/\u221a3 = 8.77350+ | Found by Maurizio Morandi | in April 2026.", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/14.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 3+10/\u221a3 = 8.77350+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "333566d5c4b3657c", "improvement": -0.0008773842173539492, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.796172321890858e-09, "max_pair_overlap_depth": -3.2457310312317134e-05, "metric_expression": "8.77438", "metric_symbol": "s", "metric_value": 8.774377384217354, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.7735, "setup": "domintri", "side": 8.774377384217354, "solution_path": "data/solutions/generated/domintri_014_333566d5c4b3657c.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/333566d5c4b3657c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@15", "density": 0.7991843102868493, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/15.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "7+4/\u221a3 = 9.30940+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.3094, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@15", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "7+4/\u221a3 = 9.30940+", "metric_symbol": "s", "metric_value": 9.3094, "reference_text": "s = 7+4/\u221a3 = 9.30940+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/15.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 7+4/\u221a3 = 9.30940+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "2da1308626a02461", "improvement": -0.0013965021666475508, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5011317742776256e-06, "max_pair_overlap_depth": -8.262234862044249e-05, "metric_expression": "9.31080", "metric_symbol": "s", "metric_value": 9.310796502166648, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.3094, "setup": "domintri", "side": 9.310796502166648, "solution_path": "data/solutions/generated/domintri_015_2da1308626a02461.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/2da1308626a02461.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@16", "density": 0.8249213221645568, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/16.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "6+2\u221a3 = 9.46410+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.4641, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@16", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "6+2\u221a3 = 9.46410+", "metric_symbol": "s", "metric_value": 9.4641, "reference_text": "s = 6+2\u221a3 = 9.46410+ | Found by Maurizio Morandi | in April 2026.", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/16.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 6+2\u221a3 = 9.46410+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "cc2a76b6a75c7120", "improvement": -0.000851797392776632, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.428695004104409e-06, "max_pair_overlap_depth": -2.0153462060745753e-05, "metric_expression": "9.46495", "metric_symbol": "s", "metric_value": 9.464951797392777, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.4641, "setup": "domintri", "side": 9.464951797392777, "solution_path": "data/solutions/generated/domintri_016_cc2a76b6a75c7120.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/cc2a76b6a75c7120.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@17", "density": 0.8218638425440753, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/17.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "4+10/\u221a3 = 9.77350+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.7735, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4+10/\u221a3 = 9.77350+", "metric_symbol": "s", "metric_value": 9.7735, "reference_text": "s = 4+10/\u221a3 = 9.77350+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/17.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 4+10/\u221a3 = 9.77350+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "842ace76b15a5c6d", "improvement": -0.0008796443209906357, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.8870325675374086e-06, "max_pair_overlap_depth": -2.297706007370337e-05, "metric_expression": "9.77438", "metric_symbol": "s", "metric_value": 9.774379644320991, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.7735, "setup": "domintri", "side": 9.774379644320991, "solution_path": "data/solutions/generated/domintri_017_842ace76b15a5c6d.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/842ace76b15a5c6d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@18", "density": 0.8433917909368376, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/18.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "3+4\u221a3 = 9.92820+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.9282, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@18", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3+4\u221a3 = 9.92820+", "metric_symbol": "s", "metric_value": 9.9282, "reference_text": "s = 3+4\u221a3 = 9.92820+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/18.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 3+4\u221a3 = 9.92820+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "286272298adf99a4", "improvement": -0.00035960973454685075, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.814371048432239e-06, "max_pair_overlap_depth": -1.8972863250255045e-05, "metric_expression": "9.92856", "metric_symbol": "s", "metric_value": 9.928559609734547, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.9282, "setup": "domintri", "side": 9.928559609734547, "solution_path": "data/solutions/generated/domintri_018_286272298adf99a4.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/286272298adf99a4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@19", "density": 0.8255562955017095, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/19.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "8+4/\u221a3 = 10.30940+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.3094, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@19", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "8+4/\u221a3 = 10.30940+", "metric_symbol": "s", "metric_value": 10.3094, "reference_text": "s = 8+4/\u221a3 = 10.30940+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/19.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 8+4/\u221a3 = 10.30940+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "77c1eb98f0e7fed4", "improvement": -0.0008247746809697531, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.73171847830086e-06, "max_pair_overlap_depth": -2.8002623395084747e-06, "metric_expression": "10.31022", "metric_symbol": "s", "metric_value": 10.31022477468097, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 10.3094, "setup": "domintri", "side": 10.31022477468097, "solution_path": "data/solutions/generated/domintri_019_77c1eb98f0e7fed4.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/77c1eb98f0e7fed4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@20", "density": 0.843501980040664, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/20.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "7+2\u221a3 = 10.46410+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.4641, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@20", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "7+2\u221a3 = 10.46410+", "metric_symbol": "s", "metric_value": 10.4641, "reference_text": "s = 7+2\u221a3 = 10.46410+ | Found by Maurizio Morandi | in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/20.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 7+2\u221a3 = 10.46410+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "19214c32485a435f", "improvement": -0.0008371510213134314, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8019716271983103e-06, "max_pair_overlap_depth": -2.85508650477162e-06, "metric_expression": "10.46494", "metric_symbol": "s", "metric_value": 10.464937151021314, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 10.4641, "setup": "domintri", "side": 10.464937151021314, "solution_path": "data/solutions/generated/domintri_020_19214c32485a435f.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/19214c32485a435f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@21", "density": 0.8533496545178485, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/21.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "2+5\u221a3 = 10.66025+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.66025, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@21", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "2+5\u221a3 = 10.66025+", "metric_symbol": "s", "metric_value": 10.66025, "reference_text": "s = 2+5\u221a3 = 10.66025+ | Found by Maurizio Morandi | in April 2026.", "source_image": "21.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/21.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 2+5\u221a3 = 10.66025+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "21.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "797030ebf0f8c4e4", "improvement": -0.0010660665758290833, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.564559491289197e-06, "max_pair_overlap_depth": -2.4360684990565318e-05, "metric_expression": "10.66132", "metric_symbol": "s", "metric_value": 10.661316066575829, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 10.66025, "setup": "domintri", "side": 10.661316066575829, "solution_path": "data/solutions/generated/domintri_021_797030ebf0f8c4e4.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/797030ebf0f8c4e4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@22", "density": 0.8506833122891716, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/22.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "4(1+\u221a3) = 10.92820+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.9282, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@22", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "4(1+\u221a3) = 10.92820+", "metric_symbol": "s", "metric_value": 10.9282, "reference_text": "s = 4(1+\u221a3) = 10.92820+ | Found by Maurizio Morandi | in April 2026.", "source_image": "22.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/22.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 4(1+\u221a3) = 10.92820+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "22.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "90061b49654f08af", "improvement": -0.001092862620856394, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.631830815268069e-06, "max_pair_overlap_depth": -1.6046632975275088e-05, "metric_expression": "10.92929", "metric_symbol": "s", "metric_value": 10.929292862620857, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 10.9282, "setup": "domintri", "side": 10.929292862620857, "solution_path": "data/solutions/generated/domintri_022_90061b49654f08af.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/90061b49654f08af.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@23", "density": 0.8536884993646849, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/23.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "10+2/\u221a3 = 11.15470+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.1547, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@23", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "10+2/\u221a3 = 11.15470+", "metric_symbol": "s", "metric_value": 11.1547, "reference_text": "s = 10+2/\u221a3 = 11.15470+ | Found by Maurizio Morandi | in April 2026.", "source_image": "23.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/23.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 10+2/\u221a3 = 11.15470+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "23.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "8ca8e97a5947c876", "improvement": -0.0005390769738546908, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.068904299595512e-06, "max_pair_overlap_depth": -1.1497281311445562e-05, "metric_expression": "11.15524", "metric_symbol": "s", "metric_value": 11.155239076973855, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 11.1547, "setup": "domintri", "side": 11.155239076973855, "solution_path": "data/solutions/generated/domintri_023_8ca8e97a5947c876.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/8ca8e97a5947c876.svg", "tolerance": 1e-08, "verified": true}
+{"case": "domintri@24", "density": 0.8664947344465272, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/domintri/24.png", "friedman_family_updated": "4/23/26", "friedman_metric_expression": "9+4/\u221a3 = 11.30940+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.3094, "friedman_reference": {"analytical_or_proved": true, "case": "domintri@24", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "9+4/\u221a3 = 11.30940+", "metric_symbol": "s", "metric_value": 11.3094, "reference_text": "s = 9+4/\u221a3 = 11.30940+ | Found by Maurizio Morandi | in April 2026.", "source_image": "24.png", "source_image_url": "https://erich-friedman.github.io/packing/domintri/24.png", "source_page": "https://erich-friedman.github.io/packing/domintri/", "title": "Dominoes in Triangles"}, "friedman_reference_text": "s = 9+4/\u221a3 = 11.30940+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "24.png", "friedman_source_url": "https://erich-friedman.github.io/packing/domintri/", "friedman_title": "Dominoes in Triangles", "frontend_seed": true, "id": "f8436aa831ae57e5", "improvement": -0.0012440894174083894, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.95572835399557e-06, "max_pair_overlap_depth": -5.492290923159171e-05, "metric_expression": "11.31064", "metric_symbol": "s", "metric_value": 11.310644089417409, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 11.3094, "setup": "domintri", "side": 11.310644089417409, "solution_path": "data/solutions/generated/domintri_024_f8436aa831ae57e5.json", "source_url": "https://erich-friedman.github.io/packing/domintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/f8436aa831ae57e5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@1", "density": 0.826977987261634, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/1.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "r", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "r", "metric_value": 1.0, "reference_text": "r = 1 | Trivial.", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/1.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 1 | Trivial.", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "48f5aa37c90550c8", "improvement": -9.284285680699611e-06, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.547655283739573e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00001", "metric_symbol": "r", "metric_value": 1.0000092842856807, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.0, "setup": "hexincir", "side": 2.0000185685713614, "solution_path": "data/solutions/generated/hexincir_001_48f5aa37c90550c8.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/48f5aa37c90550c8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@2", "density": 0.508894613048892, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/2.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a13/2 = 1.802+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.802, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a13/2 = 1.802+", "metric_symbol": "r", "metric_value": 1.802, "reference_text": "r = \u221a13/2 = 1.802+ | Trivial.", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/2.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a13/2 = 1.802+ | Trivial.", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "3b6bd2554e997adc", "improvement": -0.0008187982614582534, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4989872811899119e-06, "max_pair_overlap_depth": -3.954921521903687e-05, "metric_expression": "1.80282", "metric_symbol": "r", "metric_value": 1.8028187982614583, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.802, "setup": "hexincir", "side": 3.6056375965229166, "solution_path": "data/solutions/generated/hexincir_002_3b6bd2554e997adc.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3b6bd2554e997adc.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@3", "density": 0.6202101203168955, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/3.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "r", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "r", "metric_value": 2.0, "reference_text": "r = 2 | Trivial.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/3.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 2 | Trivial.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "194e619a74e29452", "improvement": -5.624955299809997e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.970405754344824e-06, "max_pair_overlap_depth": -7.230141190991447e-05, "metric_expression": "2.00006", "metric_symbol": "r", "metric_value": 2.000056249552998, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "hexincir", "side": 4.000112499105996, "solution_path": "data/solutions/generated/hexincir_003_194e619a74e29452.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/194e619a74e29452.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@4", "density": 0.6594633839026793, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/4.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(21\u2013\u221a145)/4 = 2.239+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.239, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@4", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "(21\u2013\u221a145)/4 = 2.239+", "metric_symbol": "r", "metric_value": 2.239, "reference_text": "r = (21\u2013\u221a145)/4 = 2.239+ | Found by Maurizio Morandi | in March 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/4.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = (21\u2013\u221a145)/4 = 2.239+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "355b11dfc363f5ba", "improvement": -0.0006783874729134887, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.544625404889757e-07, "max_pair_overlap_depth": -5.057462804924917e-05, "metric_expression": "2.23968", "metric_symbol": "r", "metric_value": 2.2396783874729134, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.239, "setup": "hexincir", "side": 4.479356774945827, "solution_path": "data/solutions/generated/hexincir_004_355b11dfc363f5ba.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/355b11dfc363f5ba.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@5", "density": 0.6766813833384198, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/5.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2.471+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.471, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@5", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "2.471+", "metric_symbol": "r", "metric_value": 2.471, "reference_text": "r = 2.471+ | Found by Maurizio Morandi | in March 2026.", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/5.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 2.471+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "9427b00c077aafc3", "improvement": -0.0009739785189006867, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.673907177263573e-06, "max_pair_overlap_depth": -5.983134966366932e-05, "metric_expression": "2.47197", "metric_symbol": "r", "metric_value": 2.4719739785189008, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.471, "setup": "hexincir", "side": 4.9439479570378015, "solution_path": "data/solutions/generated/hexincir_005_9427b00c077aafc3.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/9427b00c077aafc3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@6", "density": 0.7087991567669789, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/7.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a7 = 2.645+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.645, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a7 = 2.645+", "metric_symbol": "r", "metric_value": 2.645, "reference_text": "r = \u221a7 = 2.645+ | Trivial. |", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/7.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a7 = 2.645+ | Trivial. |", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "d7d900dd0bba32a3", "improvement": -0.0008488830398025726, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0221671226394733e-07, "max_pair_overlap_depth": -3.6674166167594e-05, "metric_expression": "2.64585", "metric_symbol": "r", "metric_value": 2.6458488830398026, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.645, "setup": "hexincir", "side": 5.291697766079605, "solution_path": "data/solutions/generated/hexincir_006_d7d900dd0bba32a3.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d7d900dd0bba32a3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@7", "density": 0.8269668154291873, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/7.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a7 = 2.645+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.645, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a7 = 2.645+", "metric_symbol": "r", "metric_value": 2.645, "reference_text": "r = \u221a7 = 2.645+ | Trivial. |", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/7.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a7 = 2.645+ | Trivial. |", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "5cd27a740cc9b2a5", "improvement": -0.0007937463486662466, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.370930279076532e-06, "max_pair_overlap_depth": -2.6143388138111234e-05, "metric_expression": "2.64579", "metric_symbol": "r", "metric_value": 2.6457937463486663, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.645, "setup": "hexincir", "side": 5.2915874926973325, "solution_path": "data/solutions/generated/hexincir_007_5cd27a740cc9b2a5.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/5cd27a740cc9b2a5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@8", "density": 0.7426247287224678, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/8.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2.984+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.984, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@8", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "2.984+", "metric_symbol": "r", "metric_value": 2.984, "reference_text": "r = 2.984+ | Found by Maurizio Morandi | in March 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/8.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 2.984+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "7d27528d78a05a43", "improvement": -0.0007729280237529501, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.6889173538057207e-06, "max_pair_overlap_depth": -1.732806256671271e-05, "metric_expression": "2.98477", "metric_symbol": "r", "metric_value": 2.984772928023753, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.984, "setup": "hexincir", "side": 5.969545856047506, "solution_path": "data/solutions/generated/hexincir_008_7d27528d78a05a43.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/7d27528d78a05a43.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@9", "density": 0.682777017123973, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/9.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3.301+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.301, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@9", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "3.301+", "metric_symbol": "r", "metric_value": 3.301, "reference_text": "r = 3.301+ | Found by Maurizio Morandi | in March 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/9.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 3.301+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "5f9496662ac0b661", "improvement": -0.000663545504778984, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.641836701819614e-06, "max_pair_overlap_depth": -2.219136600895011e-05, "metric_expression": "3.30166", "metric_symbol": "r", "metric_value": 3.301663545504779, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.301, "setup": "hexincir", "side": 6.603327091009558, "solution_path": "data/solutions/generated/hexincir_009_5f9496662ac0b661.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/5f9496662ac0b661.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@10", "density": 0.6804997422869841, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/10.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "7/2 = 3.5", "friedman_metric_symbol": "r", "friedman_metric_value": 3.5, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@10", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "7/2 = 3.5", "metric_symbol": "r", "metric_value": 3.5, "reference_text": "r = 7/2 = 3.5 | Found by Maurizio Morandi | in March 2026.", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/10.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 7/2 = 3.5 | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": false, "id": "8fccd3008565aabb", "improvement": 0.013922608209857135, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.016042661651568e-06, "max_pair_overlap_depth": -3.440098626628085e-05, "metric_expression": "3.48608", "metric_symbol": "r", "metric_value": 3.486077391790143, "n": 10, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 3.5, "setup": "hexincir", "side": 6.972154783580286, "solution_path": "data/solutions/generated/hexincir_010_8fccd3008565aabb.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/8fccd3008565aabb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@11", "density": 0.7038726397318542, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/12.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a13 = 3.605+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.605, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@11", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "\u221a13 = 3.605+", "metric_symbol": "r", "metric_value": 3.605, "reference_text": "r = \u221a13 = 3.605+ | Found by Maurizio Morandi | in March 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/12.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a13 = 3.605+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": false, "id": "2d2e76ec34c49fb5", "improvement": 0.009988305972369105, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.0630450586220945e-06, "max_pair_overlap_depth": -3.6222629933746475e-05, "metric_expression": "3.59501", "metric_symbol": "r", "metric_value": 3.595011694027631, "n": 11, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 3.605, "setup": "hexincir", "side": 7.190023388055262, "solution_path": "data/solutions/generated/hexincir_011_2d2e76ec34c49fb5.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/2d2e76ec34c49fb5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@12", "density": 0.7629156641677376, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/12.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a13 = 3.605+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.605, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@12", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "\u221a13 = 3.605+", "metric_symbol": "r", "metric_value": 3.605, "reference_text": "r = \u221a13 = 3.605+ | Found by Maurizio Morandi | in March 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/12.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a13 = 3.605+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "5515c10818a71365", "improvement": -0.001644724814655607, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.2444141653265035e-06, "max_pair_overlap_depth": -3.49998284549164e-05, "metric_expression": "3.60664", "metric_symbol": "r", "metric_value": 3.6066447248146556, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.605, "setup": "hexincir", "side": 7.213289449629311, "solution_path": "data/solutions/generated/hexincir_012_5515c10818a71365.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/5515c10818a71365.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@13", "density": 0.7347845850822048, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/13.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3.82298+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.82298, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.82298+", "metric_symbol": "r", "metric_value": 3.82298, "reference_text": "r = 3.82298+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/13.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 3.82298+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "defdebc1228b1d63", "improvement": -0.002118865563556671, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3642191814564626e-06, "max_pair_overlap_depth": -3.899276946051078e-05, "metric_expression": "3.82510", "metric_symbol": "r", "metric_value": 3.8250988655635565, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.82298, "setup": "hexincir", "side": 7.650197731127113, "solution_path": "data/solutions/generated/hexincir_013_defdebc1228b1d63.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/defdebc1228b1d63.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@14", "density": 0.7696929179154312, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/14.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a(109\u20139\u221a109) = 3.877+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.877, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@14", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "\u221a(109\u20139\u221a109) = 3.877+", "metric_symbol": "r", "metric_value": 3.877, "reference_text": "r = \u221a(109\u20139\u221a109) = 3.877+ | Found by Maurizio Morandi | in March 2026.", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/14.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a(109\u20139\u221a109) = 3.877+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "cf6592c7009ff83d", "improvement": -0.0014328751718628574, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.009541618976158e-06, "max_pair_overlap_depth": -5.416099891136783e-05, "metric_expression": "3.87843", "metric_symbol": "r", "metric_value": 3.8784328751718626, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.877, "setup": "hexincir", "side": 7.756865750343725, "solution_path": "data/solutions/generated/hexincir_014_cf6592c7009ff83d.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/cf6592c7009ff83d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@15", "density": 0.7439908773476231, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/15.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.08174+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.08174, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@15", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.08174+", "metric_symbol": "r", "metric_value": 4.08174, "reference_text": "r = 4.08174+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/15.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 4.08174+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "ea523b99f3bf81fb", "improvement": -0.001574502888137097, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.330196611616714e-06, "max_pair_overlap_depth": -5.918641434021765e-05, "metric_expression": "4.08331", "metric_symbol": "r", "metric_value": 4.083314502888137, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.08174, "setup": "hexincir", "side": 8.166629005776274, "solution_path": "data/solutions/generated/hexincir_015_ea523b99f3bf81fb.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/ea523b99f3bf81fb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@16", "density": 0.7714178714001686, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/16.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.13966+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.13966, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@16", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.13966+", "metric_symbol": "r", "metric_value": 4.13966, "reference_text": "r = 4.13966+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/16.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 4.13966+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "928ee7a851bce6e5", "improvement": -0.0019209106018989175, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.8109720468001456e-06, "max_pair_overlap_depth": -4.18854370716204e-05, "metric_expression": "4.14158", "metric_symbol": "r", "metric_value": 4.141580910601899, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.13966, "setup": "hexincir", "side": 8.283161821203798, "solution_path": "data/solutions/generated/hexincir_016_928ee7a851bce6e5.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/928ee7a851bce6e5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@17", "density": 0.7591563034967428, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/17.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.30170+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.3017, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.30170+", "metric_symbol": "r", "metric_value": 4.3017, "reference_text": "r = 4.30170+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/17.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 4.30170+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "e582b760dc061cd3", "improvement": -0.001681676380650643, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.855951803468429e-06, "max_pair_overlap_depth": -4.543663468803416e-05, "metric_expression": "4.30338", "metric_symbol": "r", "metric_value": 4.303381676380651, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.3017, "setup": "hexincir", "side": 8.606763352761302, "solution_path": "data/solutions/generated/hexincir_017_e582b760dc061cd3.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T03:07:58+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/e582b760dc061cd3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@18", "density": 0.7834006630179079, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/19.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a19 = 4.35889+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.35889, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@18", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a19 = 4.35889+", "metric_symbol": "r", "metric_value": 4.35889, "reference_text": "r = \u221a19 = 4.35889+ | Trivial.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/19.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a19 = 4.35889+ | Trivial.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "12eaac899664a007", "improvement": -0.0001945423028058002, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.696366953105269e-06, "max_pair_overlap_depth": -1.0490734119961154e-05, "metric_expression": "4.35908", "metric_symbol": "r", "metric_value": 4.3590845423028055, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.35889, "setup": "hexincir", "side": 8.718169084605611, "solution_path": "data/solutions/generated/hexincir_018_12eaac899664a007.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/12eaac899664a007.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@19", "density": 0.8269599994475847, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/19.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "\u221a19 = 4.35889+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.35889, "friedman_reference": {"analytical_or_proved": true, "case": "hexincir@19", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a19 = 4.35889+", "metric_symbol": "r", "metric_value": 4.35889, "reference_text": "r = \u221a19 = 4.35889+ | Trivial.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/19.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = \u221a19 = 4.35889+ | Trivial.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "b4bf200670caf09c", "improvement": -9.681979153608467e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.636539265552187e-06, "max_pair_overlap_depth": -6.973398227638761e-06, "metric_expression": "4.35899", "metric_symbol": "r", "metric_value": 4.358986819791536, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.35889, "setup": "hexincir", "side": 8.717973639583072, "solution_path": "data/solutions/generated/hexincir_019_b4bf200670caf09c.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b4bf200670caf09c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@20", "density": 0.7723857667872704, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/20.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.62598+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.62598, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@20", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.62598+", "metric_symbol": "r", "metric_value": 4.62598, "reference_text": "r = 4.62598+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/20.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 4.62598+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "30108dc5d1ea9a14", "improvement": -0.0015460646281777457, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8288872105642895e-06, "max_pair_overlap_depth": -4.792351042670617e-06, "metric_expression": "4.62753", "metric_symbol": "r", "metric_value": 4.627526064628178, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.62598, "setup": "hexincir", "side": 9.255052129256356, "solution_path": "data/solutions/generated/hexincir_020_30108dc5d1ea9a14.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/30108dc5d1ea9a14.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexincir@21", "density": 0.7917677222663352, "friedman_analytical_or_proved": false, "friedman_credit": "Enea Victor", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexincir/21.png", "friedman_metric_expression": "4.68339+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.68339, "friedman_reference": {"analytical_or_proved": false, "case": "hexincir@21", "credit": "Enea Victor", "date": "April 2026", "family_updated": "", "metric_expression": "4.68339+", "metric_symbol": "r", "metric_value": 4.68339, "reference_text": "r = 4.68339+ | Found by Enea Victor | in April 2026.", "source_image": "21.png", "source_image_url": "https://erich-friedman.github.io/packing/hexincir/21.png", "source_page": "https://erich-friedman.github.io/packing/hexincir/", "title": "Hexagons in Circles"}, "friedman_reference_text": "r = 4.68339+ | Found by Enea Victor | in April 2026.", "friedman_source_image": "21.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexincir/", "friedman_title": "Hexagons in Circles", "frontend_seed": true, "id": "49a047c29bb184c3", "improvement": -1.5457789629635954e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.498160996568856e-06, "max_pair_overlap_depth": -3.8704072840012316e-06, "metric_expression": "4.68341", "metric_symbol": "r", "metric_value": 4.68340545778963, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.68339, "setup": "hexincir", "side": 9.36681091557926, "solution_path": "data/solutions/generated/hexincir_021_49a047c29bb184c3.json", "source_url": "https://erich-friedman.github.io/packing/hexincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Enea Victor", "svg_path": "data/svg/49a047c29bb184c3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@1", "density": 0.433010701548144, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/1.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "\u221a3 = 1.73205+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.73205, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3 = 1.73205+", "metric_symbol": "s", "metric_value": 1.73205, "reference_text": "1. | | s = \u221a3 = 1.73205+ | Trivial.", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/1.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "1. | | s = \u221a3 = 1.73205+ | Trivial.", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "899222ca3c83cd8e", "improvement": -4.8082708896668436e-06, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0003297862271054e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.73205", "metric_symbol": "s", "metric_value": 1.7320548082708898, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.73205, "setup": "hexindom", "side": 3.4641096165417795, "solution_path": "data/solutions/generated/hexindom_001_899222ca3c83cd8e.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/899222ca3c83cd8e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@2", "density": 0.7237057542591544, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/2.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "1.8946+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.8946, "friedman_reference": {"analytical_or_proved": false, "case": "hexindom@2", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.8946+", "metric_symbol": "s", "metric_value": 1.8946, "reference_text": "2. | | s = 1.8946+ | Found by Jonathan Viquerat in April 2026.", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/2.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "2. | | s = 1.8946+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "419ae881803808e0", "improvement": -0.00011952413629479075, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.154031481298006e-06, "max_pair_overlap_depth": -4.420983116036614e-05, "metric_expression": "1.89472", "metric_symbol": "s", "metric_value": 1.8947195241362949, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.8946, "setup": "hexindom", "side": 3.7894390482725897, "solution_path": "data/solutions/generated/hexindom_002_419ae881803808e0.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/419ae881803808e0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@3", "density": 0.6059845041872388, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/3.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "6\u20132\u221a3 = 2.53589+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.53589, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@3", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6\u20132\u221a3 = 2.53589+", "metric_symbol": "s", "metric_value": 2.53589, "reference_text": "3. | | s = 6\u20132\u221a3 = 2.53589+ | Found by Jonathan Viquerat in April 2026.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/3.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "3. | | s = 6\u20132\u221a3 = 2.53589+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "b8984cccc8c8714d", "improvement": -6.074031198854968e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3815485928603266e-06, "max_pair_overlap_depth": -3.5690372763075295e-05, "metric_expression": "2.53595", "metric_symbol": "s", "metric_value": 2.5359507403119887, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.53589, "setup": "hexindom", "side": 5.0719014806239775, "solution_path": "data/solutions/generated/hexindom_003_b8984cccc8c8714d.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b8984cccc8c8714d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@4", "density": 0.5981044188175139, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/4.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "22\u201311\u221a3 = 2.94744+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.94744, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@4", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "22\u201311\u221a3 = 2.94744+", "metric_symbol": "s", "metric_value": 2.94744, "reference_text": "4. | | s = 22\u201311\u221a3 = 2.94744+ | Found by Jonathan Viquerat in April 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/4.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "4. | | s = 22\u201311\u221a3 = 2.94744+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "de86d3aed3d6a439", "improvement": -5.064613954619546e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.326043343385379e-06, "max_pair_overlap_depth": -1.3607159264461544e-05, "metric_expression": "2.94749", "metric_symbol": "s", "metric_value": 2.947490646139546, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.94744, "setup": "hexindom", "side": 5.894981292279092, "solution_path": "data/solutions/generated/hexindom_004_de86d3aed3d6a439.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/de86d3aed3d6a439.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@5", "density": 0.6153950410217744, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/5.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "7(2\u221a3\u20133) = 3.24871+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.24871, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@5", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "7(2\u221a3\u20133) = 3.24871+", "metric_symbol": "s", "metric_value": 3.24871, "reference_text": "5. | | s = 7(2\u221a3\u20133) = 3.24871+ | Found by Jonathan Viquerat in April 2026.", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/5.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "5. | | s = 7(2\u221a3\u20133) = 3.24871+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "6c9e9b8c4daf8e17", "improvement": -5.988993836103518e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.879985772620616e-06, "max_pair_overlap_depth": -1.7392941288552066e-05, "metric_expression": "3.24877", "metric_symbol": "s", "metric_value": 3.248769889938361, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.24871, "setup": "hexindom", "side": 6.497539779876722, "solution_path": "data/solutions/generated/hexindom_005_6c9e9b8c4daf8e17.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/6c9e9b8c4daf8e17.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@6", "density": 0.669460522582725, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/6.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "21(15\u20132\u221a3)/71 = 3.41202+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.41202, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@6", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "21(15\u20132\u221a3)/71 = 3.41202+", "metric_symbol": "s", "metric_value": 3.41202, "reference_text": "6. | | s = 21(15\u20132\u221a3)/71 = 3.41202+ | Found by Jonathan Viquerat in April 2026.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/6.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "6. | | s = 21(15\u20132\u221a3)/71 = 3.41202+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "3331b621360b9c32", "improvement": -9.83352674857052e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.54326023255247e-06, "max_pair_overlap_depth": -5.983710998713754e-05, "metric_expression": "3.41212", "metric_symbol": "s", "metric_value": 3.4121183352674858, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.41202, "setup": "hexindom", "side": 6.8242366705349715, "solution_path": "data/solutions/generated/hexindom_006_3331b621360b9c32.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/3331b621360b9c32.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@7", "density": 0.74772405053785, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/7.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "3.4873+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.4873, "friedman_reference": {"analytical_or_proved": false, "case": "hexindom@7", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.4873+", "metric_symbol": "s", "metric_value": 3.4873, "reference_text": "7. | | s = 3.4873+ | Found by Jonathan Viquerat in April 2026.", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/7.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "7. | | s = 3.4873+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "1cf3d5c48cc9bce3", "improvement": 1.5563940758411832e-07, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.731889618803109e-06, "max_pair_overlap_depth": -2.9943125185782904e-05, "metric_expression": "3.48730", "metric_symbol": "s", "metric_value": 3.4872998443605923, "n": 7, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.4873, "setup": "hexindom", "side": 6.9745996887211845, "solution_path": "data/solutions/generated/hexindom_007_1cf3d5c48cc9bce3.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/1cf3d5c48cc9bce3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@8", "density": 0.7629201456996986, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/8.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "3.6906+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.6906, "friedman_reference": {"analytical_or_proved": false, "case": "hexindom@8", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.6906+", "metric_symbol": "s", "metric_value": 3.6906, "reference_text": "8. | | s = 3.6906+ | Found by Jonathan Viquerat in April 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/8.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "8. | | s = 3.6906+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "645a3046b41a06e7", "improvement": -0.00016503281621149668, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.30145200672294e-06, "max_pair_overlap_depth": -4.406733776018967e-05, "metric_expression": "3.69077", "metric_symbol": "s", "metric_value": 3.6907650328162114, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.6906, "setup": "hexindom", "side": 7.381530065632423, "solution_path": "data/solutions/generated/hexindom_008_645a3046b41a06e7.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/645a3046b41a06e7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@9", "density": 0.7092437911682054, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/9.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "(36+5\u221a3)/11 = 4.06002+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.06002, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@9", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "(36+5\u221a3)/11 = 4.06002+", "metric_symbol": "s", "metric_value": 4.06002, "reference_text": "9. | | s = (36+5\u221a3)/11 = 4.06002+ | Found by Jonathan Viquerat in April 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/9.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "9. | | s = (36+5\u221a3)/11 = 4.06002+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "412a5c4dab2378db", "improvement": -5.848340764558202e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.8730699123165095e-06, "max_pair_overlap_depth": -8.397966625661724e-06, "metric_expression": "4.06008", "metric_symbol": "s", "metric_value": 4.060078483407645, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.06002, "setup": "hexindom", "side": 8.12015696681529, "solution_path": "data/solutions/generated/hexindom_009_412a5c4dab2378db.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/412a5c4dab2378db.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@10", "density": 0.7445544573306387, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/10.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "9(2\u221a3\u20133) = 4.17691+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.17691, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@10", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "9(2\u221a3\u20133) = 4.17691+", "metric_symbol": "s", "metric_value": 4.17691, "reference_text": "10. | | s = 9(2\u221a3\u20133) = 4.17691+ | Found by Jonathan Viquerat in April 2026.", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/10.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "10. | | s = 9(2\u221a3\u20133) = 4.17691+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "b404bfce93e4b698", "improvement": -7.303556637161535e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.152317509946272e-07, "max_pair_overlap_depth": -2.262161166943777e-05, "metric_expression": "4.17698", "metric_symbol": "s", "metric_value": 4.176983035566372, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.17691, "setup": "hexindom", "side": 8.353966071132744, "solution_path": "data/solutions/generated/hexindom_010_b404bfce93e4b698.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b404bfce93e4b698.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@11", "density": 0.7041255860017571, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/11.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "4.5036+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5036, "friedman_reference": {"analytical_or_proved": false, "case": "hexindom@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.5036+", "metric_symbol": "s", "metric_value": 4.5036, "reference_text": "11. | | s = 4.5036+ | Found by Jonathan Viquerat in April 2026.", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/11.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "11. | | s = 4.5036+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "de36b527dfa03258", "improvement": -0.0012696093619926074, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.31456043970391e-06, "max_pair_overlap_depth": -1.8725366454197534e-05, "metric_expression": "4.50487", "metric_symbol": "s", "metric_value": 4.504869609361992, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.5036, "setup": "hexindom", "side": 9.009739218723984, "solution_path": "data/solutions/generated/hexindom_011_de36b527dfa03258.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/de36b527dfa03258.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@12", "density": 0.7322405469378734, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/12.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "41(18\u20135\u221a3)/83 = 4.6136+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.6136, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@12", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "41(18\u20135\u221a3)/83 = 4.6136+", "metric_symbol": "s", "metric_value": 4.6136, "reference_text": "12. | | s = 41(18\u20135\u221a3)/83 = 4.6136+ | Found by Jonathan Viquerat in April 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/12.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "12. | | s = 41(18\u20135\u221a3)/83 = 4.6136+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "32dafbf1407bbd72", "improvement": -0.0003690981500499291, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.932113721409451e-07, "max_pair_overlap_depth": -3.449528945953517e-05, "metric_expression": "4.61397", "metric_symbol": "s", "metric_value": 4.61396909815005, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.6136, "setup": "hexindom", "side": 9.2279381963001, "solution_path": "data/solutions/generated/hexindom_012_32dafbf1407bbd72.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/32dafbf1407bbd72.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@13", "density": 0.7287142254379715, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/13.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "4.8137+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.8137, "friedman_reference": {"analytical_or_proved": false, "case": "hexindom@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.8137+", "metric_symbol": "s", "metric_value": 4.8137, "reference_text": "13. | | s = 4.8137+ | Found by Jonathan Viquerat in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/13.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "13. | | s = 4.8137+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "4d59f64640558812", "improvement": -0.000276853025948931, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.288235630449776e-07, "max_pair_overlap_depth": -5.701790221179781e-05, "metric_expression": "4.81398", "metric_symbol": "s", "metric_value": 4.813976853025949, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.8137, "setup": "hexindom", "side": 9.627953706051898, "solution_path": "data/solutions/generated/hexindom_013_4d59f64640558812.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/4d59f64640558812.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@14", "density": 0.7731048613358414, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/14.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "25(6\u2013\u221a3)/22 = 4.84994+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.84994, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@14", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "25(6\u2013\u221a3)/22 = 4.84994+", "metric_symbol": "s", "metric_value": 4.84994, "reference_text": "14. | | s = 25(6\u2013\u221a3)/22 = 4.84994+ | Found by Jonathan Viquerat in April 2026.", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/14.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "14. | | s = 25(6\u2013\u221a3)/22 = 4.84994+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "1f0ed34d05c35380", "improvement": -0.00021659973266263677, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.312817282834061e-06, "max_pair_overlap_depth": -5.706330400911952e-05, "metric_expression": "4.85016", "metric_symbol": "s", "metric_value": 4.850156599732663, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.84994, "setup": "hexindom", "side": 9.700313199465326, "solution_path": "data/solutions/generated/hexindom_014_1f0ed34d05c35380.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/1f0ed34d05c35380.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@15", "density": 0.7946102881582437, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/15.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "56(27\u20134\u221a3)/227 = 4.95163+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.95163, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@15", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "56(27\u20134\u221a3)/227 = 4.95163+", "metric_symbol": "s", "metric_value": 4.95163, "reference_text": "15. | | s = 56(27\u20134\u221a3)/227 = 4.95163+ | Found by Jonathan Viquerat in April 2026.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/15.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "15. | | s = 56(27\u20134\u221a3)/227 = 4.95163+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "1b00a25adb5cc2e4", "improvement": -0.0003568532282001158, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.185835293301636e-06, "max_pair_overlap_depth": -6.286085527285046e-05, "metric_expression": "4.95199", "metric_symbol": "s", "metric_value": 4.9519868532282, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.95163, "setup": "hexindom", "side": 9.9039737064564, "solution_path": "data/solutions/generated/hexindom_015_1b00a25adb5cc2e4.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/1b00a25adb5cc2e4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@16", "density": 0.7697888241254879, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/17.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "3\u221a3 = 5.19615+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.19615, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@16", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3\u221a3 = 5.19615+", "metric_symbol": "s", "metric_value": 5.19615, "reference_text": "16.-17. | | s = 3\u221a3 = 5.19615+ | Trivial.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/17.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "16.-17. | | s = 3\u221a3 = 5.19615+ | Trivial.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "f690e0f9c2986401", "improvement": -4.135307393138277e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.00251676779817e-06, "max_pair_overlap_depth": -1.257294679668508e-05, "metric_expression": "5.19619", "metric_symbol": "s", "metric_value": 5.196191353073932, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.19615, "setup": "hexindom", "side": 10.392382706147863, "solution_path": "data/solutions/generated/hexindom_016_f690e0f9c2986401.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/f690e0f9c2986401.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@17", "density": 0.8178722029268405, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/17.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "3\u221a3 = 5.19615+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.19615, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@17", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3\u221a3 = 5.19615+", "metric_symbol": "s", "metric_value": 5.19615, "reference_text": "16.-17. | | s = 3\u221a3 = 5.19615+ | Trivial.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/17.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "16.-17. | | s = 3\u221a3 = 5.19615+ | Trivial.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "fe395a0426baa674", "improvement": -0.00013164134761289858, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.712543075160227e-06, "max_pair_overlap_depth": -2.5961203853341885e-08, "metric_expression": "5.19628", "metric_symbol": "s", "metric_value": 5.196281641347613, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.19615, "setup": "hexindom", "side": 10.392563282695226, "solution_path": "data/solutions/generated/hexindom_017_fe395a0426baa674.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fe395a0426baa674.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@18", "density": 0.7993930048940154, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/18.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "31(4\u2013\u221a3)/13 = 5.40818+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.40818, "friedman_reference": {"analytical_or_proved": true, "case": "hexindom@18", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "31(4\u2013\u221a3)/13 = 5.40818+", "metric_symbol": "s", "metric_value": 5.40818, "reference_text": "18. | | s = 31(4\u2013\u221a3)/13 = 5.40818+ | Found by Jonathan Viquerat in April 2026.", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/18.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "18. | | s = 31(4\u2013\u221a3)/13 = 5.40818+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "24be62ab2c11bd5c", "improvement": -0.0001978579530232949, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.006386418315856e-06, "max_pair_overlap_depth": -4.2927090152522496e-05, "metric_expression": "5.40838", "metric_symbol": "s", "metric_value": 5.408377857953023, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.40818, "setup": "hexindom", "side": 10.816755715906046, "solution_path": "data/solutions/generated/hexindom_018_24be62ab2c11bd5c.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/24be62ab2c11bd5c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@19", "density": 0.7722825822258855, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/19.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "5.6530+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.653, "friedman_reference": {"analytical_or_proved": false, "case": "hexindom@19", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.6530+", "metric_symbol": "s", "metric_value": 5.653, "reference_text": "19. | | s = 5.6530+ | Found by Jonathan Viquerat in April 2026.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/19.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "19. | | s = 5.6530+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "ca10088586ed125c", "improvement": -0.0002686721147275861, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.885998928396475e-06, "max_pair_overlap_depth": -2.764579666703071e-05, "metric_expression": "5.65327", "metric_symbol": "s", "metric_value": 5.653268672114727, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.653, "setup": "hexindom", "side": 11.306537344229454, "solution_path": "data/solutions/generated/hexindom_019_ca10088586ed125c.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/ca10088586ed125c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexindom@20", "density": 0.7718745532343245, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexindom/20.png", "friedman_family_updated": "4/6/26", "friedman_metric_expression": "5.8012+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.8012, "friedman_reference": {"analytical_or_proved": false, "case": "hexindom@20", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.8012+", "metric_symbol": "s", "metric_value": 5.8012, "reference_text": "20. | | s = 5.8012+ | Found by Jonathan Viquerat in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/hexindom/20.png", "source_page": "https://erich-friedman.github.io/packing/hexindom/", "title": "Hexagons in Dominoes"}, "friedman_reference_text": "20. | | s = 5.8012+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexindom/", "friedman_title": "Hexagons in Dominoes", "frontend_seed": true, "id": "49ce661db444ec25", "improvement": -0.0004641087628032281, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.575652109486185e-06, "max_pair_overlap_depth": -1.0063478294242145e-05, "metric_expression": "5.80166", "metric_symbol": "s", "metric_value": 5.801664108762803, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.8012, "setup": "hexindom", "side": 11.603328217525606, "solution_path": "data/solutions/generated/hexindom_020_49ce661db444ec25.json", "source_url": "https://erich-friedman.github.io/packing/hexindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/49ce661db444ec25.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@1", "density": 0.9999864500659439, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/1.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "1. | | s = 1 | Trivial.", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/1.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "1. | | s = 1 | Trivial.", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "fd6c6df7b6a9fd4f", "improvement": -6.7750358792029175e-06, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.374323059598708e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00001", "metric_symbol": "s", "metric_value": 1.0000067750358792, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.0, "setup": "hexinhex", "side": 1.0000067750358792, "solution_path": "data/solutions/generated/hexinhex_001_fd6c6df7b6a9fd4f.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fd6c6df7b6a9fd4f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@2", "density": 0.49998387949392886, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/2.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "2. | | s = 2 | Trivial.", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/2.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "2. | | s = 2 | Trivial.", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "4c0cc36e970c84b1", "improvement": -3.2241791775522444e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.177681809212628e-06, "max_pair_overlap_depth": -3.6384174144801684e-05, "metric_expression": "2.00003", "metric_symbol": "s", "metric_value": 2.0000322417917755, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "hexinhex", "side": 2.0000322417917755, "solution_path": "data/solutions/generated/hexinhex_002_4c0cc36e970c84b1.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4c0cc36e970c84b1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@3", "density": 0.7499882422262695, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/3.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "3. | | s = 2 | Trivial.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/3.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "3. | | s = 2 | Trivial.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "2e22e12b6d11de4d", "improvement": -1.5677215970288927e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.457810528986454e-06, "max_pair_overlap_depth": -9.916736621090457e-06, "metric_expression": "2.00002", "metric_symbol": "s", "metric_value": 2.0000156772159703, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "hexinhex", "side": 2.0000156772159703, "solution_path": "data/solutions/generated/hexinhex_003_2e22e12b6d11de4d.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2e22e12b6d11de4d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@4", "density": 0.7346808358825446, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/4.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "7/3 = 2.333+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.333, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7/3 = 2.333+", "metric_symbol": "s", "metric_value": 2.333, "reference_text": "4. | | s = 7/3 = 2.333+ | Trivial.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/4.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "4. | | s = 7/3 = 2.333+ | Trivial.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "73ddab3f27a95ffe", "improvement": -0.00035404329556776304, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.969618748468479e-06, "max_pair_overlap_depth": -1.5832978702556844e-05, "metric_expression": "2.33335", "metric_symbol": "s", "metric_value": 2.333354043295568, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.333, "setup": "hexinhex", "side": 2.333354043295568, "solution_path": "data/solutions/generated/hexinhex_004_73ddab3f27a95ffe.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/73ddab3f27a95ffe.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@5", "density": 0.7031111089487749, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "March 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/5.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "8/3 = 2.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.666, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@5", "credit": "Erich Friedman", "date": "March 2015", "family_updated": "", "metric_expression": "8/3 = 2.666+", "metric_symbol": "s", "metric_value": 2.666, "reference_text": "5. | | s = 8/3 = 2.666+ | Found by Erich Friedman | in March 2015.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/5.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "5. | | s = 8/3 = 2.666+ | Found by Erich Friedman | in March 2015.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "f45300c86d046ee7", "improvement": -0.0006930086059693252, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5309295562371306e-06, "max_pair_overlap_depth": -6.88869177611906e-06, "metric_expression": "2.66669", "metric_symbol": "s", "metric_value": 2.6666930086059693, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.666, "setup": "hexinhex", "side": 2.6666930086059693, "solution_path": "data/solutions/generated/hexinhex_005_f45300c86d046ee7.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f45300c86d046ee7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@6", "density": 0.7199337426480286, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/7.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5/\u221a3 = 2.886+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.886, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@6", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "5/\u221a3 = 2.886+", "metric_symbol": "s", "metric_value": 2.886, "reference_text": "6-7. | | s = 5/\u221a3 = 2.886+ | Found by Maurizio Morandi | in April 2015.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/7.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "6-7. | | s = 5/\u221a3 = 2.886+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "06b9c79146f8c119", "improvement": -0.0008841804633918926, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.09670685382364e-06, "max_pair_overlap_depth": -5.069396536394599e-05, "metric_expression": "2.88688", "metric_symbol": "s", "metric_value": 2.886884180463392, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.886, "setup": "hexinhex", "side": 2.886884180463392, "solution_path": "data/solutions/generated/hexinhex_006_06b9c79146f8c119.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/06b9c79146f8c119.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@7", "density": 0.8399730426517898, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/7.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5/\u221a3 = 2.886+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.886, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@7", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "5/\u221a3 = 2.886+", "metric_symbol": "s", "metric_value": 2.886, "reference_text": "6-7. | | s = 5/\u221a3 = 2.886+ | Found by Maurizio Morandi | in April 2015.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/7.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "6-7. | | s = 5/\u221a3 = 2.886+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "d2a43d4ca2836467", "improvement": -0.0007976679923635821, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.6806989039887556e-06, "max_pair_overlap_depth": -2.9189085207215193e-05, "metric_expression": "2.88680", "metric_symbol": "s", "metric_value": 2.8867976679923637, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.886, "setup": "hexinhex", "side": 2.8867976679923637, "solution_path": "data/solutions/generated/hexinhex_007_d2a43d4ca2836467.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/d2a43d4ca2836467.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@8", "density": 0.728171041468435, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/8.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "(51+5\u221a3)/18 = 3.314+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.314, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@8", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "(51+5\u221a3)/18 = 3.314+", "metric_symbol": "s", "metric_value": 3.314, "reference_text": "8. | | s = (51+5\u221a3)/18 = 3.314+ | Found by Maurizio Morandi | in April 2015.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/8.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "8. | | s = (51+5\u221a3)/18 = 3.314+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "8d9e02005a596470", "improvement": -0.0005783682856219663, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.71875974277703e-06, "max_pair_overlap_depth": -2.9528072121864923e-05, "metric_expression": "3.31458", "metric_symbol": "s", "metric_value": 3.314578368285622, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.314, "setup": "hexinhex", "side": 3.314578368285622, "solution_path": "data/solutions/generated/hexinhex_008_8d9e02005a596470.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/8d9e02005a596470.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@9", "density": 0.683335999337671, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/9.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "(24+5\u221a3)/9 = 3.628+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.628, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@9", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "(24+5\u221a3)/9 = 3.628+", "metric_symbol": "s", "metric_value": 3.628, "reference_text": "9. | | s = (24+5\u221a3)/9 = 3.628+ | Found by Maurizio Morandi | in April 2015.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/9.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "9. | | s = (24+5\u221a3)/9 = 3.628+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "21c530fbe612f27c", "improvement": -0.0011431939654920065, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.454488069704802e-06, "max_pair_overlap_depth": -2.836888994439679e-05, "metric_expression": "3.62914", "metric_symbol": "s", "metric_value": 3.629143193965492, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.628, "setup": "hexinhex", "side": 3.629143193965492, "solution_path": "data/solutions/generated/hexinhex_009_21c530fbe612f27c.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/21c530fbe612f27c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@10", "density": 0.7179574622038186, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/10.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "2 + \u221a3 = 3.732+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.732, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@10", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "2 + \u221a3 = 3.732+", "metric_symbol": "s", "metric_value": 3.732, "reference_text": "10. | | s = 2 + \u221a3 = 3.732+ | Found by Maurizio Morandi | in April 2015.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/10.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "10. | | s = 2 + \u221a3 = 3.732+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "fd5a10d8a6354174", "improvement": -7.740911509834802e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.847623197892974e-06, "max_pair_overlap_depth": -2.6880918478910942e-06, "metric_expression": "3.73208", "metric_symbol": "s", "metric_value": 3.7320774091150986, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.732, "setup": "hexinhex", "side": 3.7320774091150986, "solution_path": "data/solutions/generated/hexinhex_010_fd5a10d8a6354174.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/fd5a10d8a6354174.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@11", "density": 0.7131589068275569, "friedman_analytical_or_proved": true, "friedman_credit": "Johann Schellhorn", "friedman_date": "July 2025", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/11.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "2+10/3\u221a3 = 3.92450+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.9245, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@11", "credit": "Johann Schellhorn", "date": "July 2025", "family_updated": "", "metric_expression": "2+10/3\u221a3 = 3.92450+", "metric_symbol": "s", "metric_value": 3.9245, "reference_text": "11. | | s = 2+10/3\u221a3 = 3.92450+ | Found by Johann Schellhorn | in July 2025. | | | |", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/11.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "11. | | s = 2+10/3\u221a3 = 3.92450+ | Found by Johann Schellhorn | in July 2025. | | | |", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "cf10a9116bb04a72", "improvement": -0.0028823793751930893, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3732870796469545e-06, "max_pair_overlap_depth": -1.938912890975253e-05, "metric_expression": "3.92738", "metric_symbol": "s", "metric_value": 3.927382379375193, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.9245, "setup": "hexinhex", "side": 3.927382379375193, "solution_path": "data/solutions/generated/hexinhex_011_cf10a9116bb04a72.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Johann Schellhorn", "svg_path": "data/svg/cf10a9116bb04a72.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@12", "density": 0.7723090107470407, "friedman_analytical_or_proved": true, "friedman_credit": "Alexander Novikov et al", "friedman_date": "June 2025", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/12.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "3.94164+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.94164, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@12", "credit": "Alexander Novikov et al", "date": "June 2025", "family_updated": "", "metric_expression": "3.94164+", "metric_symbol": "s", "metric_value": 3.94164, "reference_text": "12. | | s = 3.94164+ | Found by Alexander Novikov et al | in June 2025. | Improved slightly by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/12.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "12. | | s = 3.94164+ | Found by Alexander Novikov et al | in June 2025. | Improved slightly by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "bdc671b076ad0ffe", "improvement": -0.0001644193364005453, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.69077332163481e-06, "max_pair_overlap_depth": -4.133319357890741e-07, "metric_expression": "3.94180", "metric_symbol": "s", "metric_value": 3.9418044193364006, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.94164, "setup": "hexinhex", "side": 3.9418044193364006, "solution_path": "data/solutions/generated/hexinhex_012_bdc671b076ad0ffe.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Alexander Novikov et al", "svg_path": "data/svg/bdc671b076ad0ffe.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@13", "density": 0.8124849152838151, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/13.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@13", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "13 | | s = 4 | Trivial. | | | | |", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/13.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "13 | | s = 4 | Trivial. | | | | |", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "1ac594ff5f07e3d9", "improvement": -3.7132126112560115e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.027960154271181e-06, "max_pair_overlap_depth": -5.440029557712833e-06, "metric_expression": "4.00004", "metric_symbol": "s", "metric_value": 4.000037132126113, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.0, "setup": "hexinhex", "side": 4.000037132126113, "solution_path": "data/solutions/generated/hexinhex_013_1ac594ff5f07e3d9.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/1ac594ff5f07e3d9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@14", "density": 0.7678833113230449, "friedman_analytical_or_proved": false, "friedman_credit": "Dominik Kamp", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/14.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "4.26899+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.26899, "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@14", "credit": "Dominik Kamp", "date": "April 2026", "family_updated": "", "metric_expression": "4.26899+", "metric_symbol": "s", "metric_value": 4.26899, "reference_text": "14. | | s = 4.26899+ | Found by Dominik Kamp | in April 2026.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/14.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "14. | | s = 4.26899+ | Found by Dominik Kamp | in April 2026.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "f86af09d863723a2", "improvement": -0.0008972113481453547, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.52825412539728e-06, "max_pair_overlap_depth": -2.1736642138669993e-05, "metric_expression": "4.26989", "metric_symbol": "s", "metric_value": 4.269887211348145, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.26899, "setup": "hexinhex", "side": 4.269887211348145, "solution_path": "data/solutions/generated/hexinhex_014_f86af09d863723a2.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Dominik Kamp", "svg_path": "data/svg/f86af09d863723a2.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@15", "density": 0.7579263950656321, "friedman_analytical_or_proved": true, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/15.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "4.44727+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.44727, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@15", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "4.44727+", "metric_symbol": "s", "metric_value": 4.44727, "reference_text": "15. | | s = 4.44727+ | Found by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/15.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "15. | | s = 4.44727+ | Found by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "536c412ab623afbc", "improvement": -0.0014196922804980971, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.564398033090082e-06, "max_pair_overlap_depth": -2.5468766324809877e-06, "metric_expression": "4.44869", "metric_symbol": "s", "metric_value": 4.448689692280498, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.44727, "setup": "hexinhex", "side": 4.448689692280498, "solution_path": "data/solutions/generated/hexinhex_015_536c412ab623afbc.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/536c412ab623afbc.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@16", "density": 0.7803167098453238, "friedman_analytical_or_proved": true, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/16.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "4.52746+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.52746, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@16", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "4.52746+", "metric_symbol": "s", "metric_value": 4.52746, "reference_text": "16. | | s = 4.52746+ | Found by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/16.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "16. | | s = 4.52746+ | Found by Timo Berthold et al | in January 2026. | Slightly improved by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "17cb432379a18c18", "improvement": -0.0007289208955771898, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.541715273786282e-06, "max_pair_overlap_depth": -2.3206185233659582e-05, "metric_expression": "4.52819", "metric_symbol": "s", "metric_value": 4.528188920895577, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.52746, "setup": "hexinhex", "side": 4.528188920895577, "solution_path": "data/solutions/generated/hexinhex_016_17cb432379a18c18.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/17cb432379a18c18.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@17", "density": 0.7985097849440846, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "February 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/17.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "4.6136+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.6136, "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@17", "credit": "Timo Berthold et al", "date": "February 2026", "family_updated": "", "metric_expression": "4.6136+", "metric_symbol": "s", "metric_value": 4.6136, "reference_text": "17. | | s = 4.6136+ | Found by Timo Berthold et al | and Alex Kravatsky et al | in February 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/17.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "17. | | s = 4.6136+ | Found by Timo Berthold et al | and Alex Kravatsky et al | in February 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "df809b176b000a6d", "improvement": -0.0004717062665458016, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.290243996089003e-06, "max_pair_overlap_depth": -4.642493336670661e-05, "metric_expression": "4.61407", "metric_symbol": "s", "metric_value": 4.614071706266546, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.6136, "setup": "hexinhex", "side": 4.614071706266546, "solution_path": "data/solutions/generated/hexinhex_017_df809b176b000a6d.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/df809b176b000a6d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@18", "density": 0.8437116836701195, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/19.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "8/\u221a3 = 4.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.618, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@18", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "8/\u221a3 = 4.618+", "metric_symbol": "s", "metric_value": 4.618, "reference_text": "18-19. | | s = 8/\u221a3 = 4.618+ | Found by Maurizio Morandi | in April 2015. |", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/19.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "18-19. | | s = 8/\u221a3 = 4.618+ | Found by Maurizio Morandi | in April 2015. |", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "e320d39029a817c1", "improvement": -0.0009070314872570151, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.0443685916789865e-09, "max_pair_overlap_depth": -3.080790913889686e-05, "metric_expression": "4.61891", "metric_symbol": "s", "metric_value": 4.618907031487257, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.618, "setup": "hexinhex", "side": 4.618907031487257, "solution_path": "data/solutions/generated/hexinhex_018_e320d39029a817c1.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/e320d39029a817c1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@19", "density": 0.8905854861904853, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/19.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "8/\u221a3 = 4.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.618, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@19", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "8/\u221a3 = 4.618+", "metric_symbol": "s", "metric_value": 4.618, "reference_text": "18-19. | | s = 8/\u221a3 = 4.618+ | Found by Maurizio Morandi | in April 2015. |", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/19.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "18-19. | | s = 8/\u221a3 = 4.618+ | Found by Maurizio Morandi | in April 2015. |", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "4b11a1a09f2a5b70", "improvement": -0.0009046166982376036, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.772589960788204e-06, "max_pair_overlap_depth": -3.612300978728111e-05, "metric_expression": "4.61890", "metric_symbol": "s", "metric_value": 4.618904616698238, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.618, "setup": "hexinhex", "side": 4.618904616698238, "solution_path": "data/solutions/generated/hexinhex_019_4b11a1a09f2a5b70.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/4b11a1a09f2a5b70.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@20", "density": 0.7999496824068819, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/21.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@20", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "20-21. | | s = 5 | Trivial. | |", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/21.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "20-21. | | s = 5 | Trivial. | |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "2830e2dec38749ef", "improvement": -0.00015724989644283482, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.445035556179186e-06, "max_pair_overlap_depth": -1.6551811964360397e-06, "metric_expression": "5.00016", "metric_symbol": "s", "metric_value": 5.000157249896443, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.0, "setup": "hexinhex", "side": 5.000157249896443, "solution_path": "data/solutions/generated/hexinhex_020_2830e2dec38749ef.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2830e2dec38749ef.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@21", "density": 0.8399773177134815, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/21.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@21", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "20-21. | | s = 5 | Trivial. | |", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/21.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "20-21. | | s = 5 | Trivial. | |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "4a431b677be571c9", "improvement": -6.750817229672634e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5389132812752791e-06, "max_pair_overlap_depth": -3.519050301936666e-06, "metric_expression": "5.00007", "metric_symbol": "s", "metric_value": 5.000067508172297, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.0, "setup": "hexinhex", "side": 5.000067508172297, "solution_path": "data/solutions/generated/hexinhex_021_4a431b677be571c9.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4a431b677be571c9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@22", "density": 0.7873804951730353, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/22.gif", "friedman_metric_expression": "(39+8\u221a3)/10 = 5.285+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.285, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@22", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "(39+8\u221a3)/10 = 5.285+", "metric_symbol": "s", "metric_value": 5.285, "reference_text": "22. | | s = (39+8\u221a3)/10 = 5.285+ | Found by Maurizio Morandi | in April 2015.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/22.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "22. | | s = (39+8\u221a3)/10 = 5.285+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "003a713177cb171c", "improvement": -0.0009008698048820207, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.575542655386671e-06, "max_pair_overlap_depth": -1.0001604064946434e-05, "metric_expression": "5.28590", "metric_symbol": "s", "metric_value": 5.285900869804882, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.285, "setup": "hexinhex", "side": 5.285900869804882, "solution_path": "data/solutions/generated/hexinhex_022_003a713177cb171c.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/003a713177cb171c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@23", "density": 0.7886255260759908, "friedman_analytical_or_proved": false, "friedman_credit": "Alex Kravatsky et al", "friedman_date": "February 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/23.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5.400+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.4, "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@23", "credit": "Alex Kravatsky et al", "date": "February 2026", "family_updated": "", "metric_expression": "5.400+", "metric_symbol": "s", "metric_value": 5.4, "reference_text": "23. | | s = 5.400+ | Found by Alex Kravatsky et al | in February 2026.", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/23.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "23. | | s = 5.400+ | Found by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "e0a82406671652d2", "improvement": -0.00043201188015196124, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4507055603926188e-06, "max_pair_overlap_depth": -1.2502329530761358e-05, "metric_expression": "5.40043", "metric_symbol": "s", "metric_value": 5.400432011880152, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.4, "setup": "hexinhex", "side": 5.400432011880152, "solution_path": "data/solutions/generated/hexinhex_023_e0a82406671652d2.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Alex Kravatsky et al", "svg_path": "data/svg/e0a82406671652d2.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@24", "density": 0.797738483588153, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/24.gif", "friedman_metric_expression": "19\u221a3/6 = 5.484+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.484, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@24", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "19\u221a3/6 = 5.484+", "metric_symbol": "s", "metric_value": 5.484, "reference_text": "24. | | s = 19\u221a3/6 = 5.484+ | Found by Maurizio Morandi | in April 2015.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/24.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "24. | | s = 19\u221a3/6 = 5.484+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "c96649ff013a9455", "improvement": -0.00098379990619879, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.588312731744032e-06, "max_pair_overlap_depth": -4.485524381903083e-05, "metric_expression": "5.48498", "metric_symbol": "s", "metric_value": 5.484983799906199, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.484, "setup": "hexinhex", "side": 5.484983799906199, "solution_path": "data/solutions/generated/hexinhex_024_c96649ff013a9455.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/c96649ff013a9455.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@25", "density": 0.7898226009433017, "friedman_analytical_or_proved": false, "friedman_credit": "Alex Kravatsky et al", "friedman_date": "February 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/25.gif", "friedman_metric_expression": "5.62390+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.6239, "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@25", "credit": "Alex Kravatsky et al", "date": "February 2026", "family_updated": "", "metric_expression": "5.62390+", "metric_symbol": "s", "metric_value": 5.6239, "reference_text": "25. | | s = 5.62390+ | Found by Alex Kravatsky et al | in February 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/25.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "25. | | s = 5.62390+ | Found by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "4ac680367830872b", "improvement": -0.002171223468265815, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8833146044517865e-07, "max_pair_overlap_depth": -2.7316686140865798e-05, "metric_expression": "5.62607", "metric_symbol": "s", "metric_value": 5.626071223468266, "n": 25, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.6239, "setup": "hexinhex", "side": 5.626071223468266, "solution_path": "data/solutions/generated/hexinhex_025_4ac680367830872b.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Alex Kravatsky et al", "svg_path": "data/svg/4ac680367830872b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@26", "density": 0.7965297187067493, "friedman_analytical_or_proved": false, "friedman_credit": "Alex Kravatsky et al", "friedman_date": "February 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/26.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5.70972+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.70972, "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@26", "credit": "Alex Kravatsky et al", "date": "February 2026", "family_updated": "", "metric_expression": "5.70972+", "metric_symbol": "s", "metric_value": 5.70972, "reference_text": "26. | | s = 5.70972+ | Found by Alex Kravatsky et al | in February 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/26.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "26. | | s = 5.70972+ | Found by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "8ed8c9fb94f18119", "improvement": -0.0035622784284790043, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9624619447323227e-06, "max_pair_overlap_depth": -1.2105377510118753e-05, "metric_expression": "5.71328", "metric_symbol": "s", "metric_value": 5.713282278428479, "n": 26, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.70972, "setup": "hexinhex", "side": 5.713282278428479, "solution_path": "data/solutions/generated/hexinhex_026_8ed8c9fb94f18119.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Alex Kravatsky et al", "svg_path": "data/svg/8ed8c9fb94f18119.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@27", "density": 0.82621496088318, "friedman_analytical_or_proved": false, "friedman_credit": "Alex Kravatsky et al", "friedman_date": "February 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/27.gif", "friedman_metric_expression": "5.71415+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.71415, "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@27", "credit": "Alex Kravatsky et al", "date": "February 2026", "family_updated": "", "metric_expression": "5.71415+", "metric_symbol": "s", "metric_value": 5.71415, "reference_text": "27. | | s = 5.71415+ | Found by Alex Kravatsky et al | in February 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/27.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "27. | | s = 5.71415+ | Found by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "4e9628a9f49def85", "improvement": -0.0024177469948583052, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.56387472302006e-06, "max_pair_overlap_depth": -5.6693937127594296e-05, "metric_expression": "5.71657", "metric_symbol": "s", "metric_value": 5.716567746994858, "n": 27, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.71415, "setup": "hexinhex", "side": 5.716567746994858, "solution_path": "data/solutions/generated/hexinhex_027_4e9628a9f49def85.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Alex Kravatsky et al", "svg_path": "data/svg/4e9628a9f49def85.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@28", "density": 0.8011452289129106, "friedman_analytical_or_proved": false, "friedman_credit": "Alex Kravatsky et al", "friedman_date": "February 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/28.gif", "friedman_family_updated": "4/15/26", "friedman_metric_expression": "5.90893+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.90893, "friedman_reference": {"analytical_or_proved": false, "case": "hexinhex@28", "credit": "Alex Kravatsky et al", "date": "February 2026", "family_updated": "", "metric_expression": "5.90893+", "metric_symbol": "s", "metric_value": 5.90893, "reference_text": "28. | | s = 5.90893+ | Found by Alex Kravatsky et al | in February 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/28.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "28. | | s = 5.90893+ | Found by Alex Kravatsky et al | in February 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "ba522f97744aed1a", "improvement": -0.002919783094952777, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.424277796084301e-06, "max_pair_overlap_depth": -3.209757705180394e-05, "metric_expression": "5.91185", "metric_symbol": "s", "metric_value": 5.911849783094953, "n": 28, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.90893, "setup": "hexinhex", "side": 5.911849783094953, "solution_path": "data/solutions/generated/hexinhex_028_ba522f97744aed1a.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Alex Kravatsky et al", "svg_path": "data/svg/ba522f97744aed1a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@29", "density": 0.8055194808067645, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@29", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "29-31. | | s = 6 | Trivial. |", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "29-31. | | s = 6 | Trivial. |", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "dba25a591b8992bc", "improvement": -0.00013435185279675466, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.231638156342025e-06, "max_pair_overlap_depth": -6.307567709251671e-06, "metric_expression": "6.00013", "metric_symbol": "s", "metric_value": 6.000134351852797, "n": 29, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.0, "setup": "hexinhex", "side": 6.000134351852797, "solution_path": "data/solutions/generated/hexinhex_029_dba25a591b8992bc.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/dba25a591b8992bc.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@30", "density": 0.8333115277207207, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@30", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "29-31. | | s = 6 | Trivial. |", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "29-31. | | s = 6 | Trivial. |", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "100db5f9f28894cc", "improvement": -7.850174601031767e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7272811687230685e-07, "max_pair_overlap_depth": -4.232627726974414e-06, "metric_expression": "6.00008", "metric_symbol": "s", "metric_value": 6.00007850174601, "n": 30, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.0, "setup": "hexinhex", "side": 6.00007850174601, "solution_path": "data/solutions/generated/hexinhex_030_100db5f9f28894cc.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/100db5f9f28894cc.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinhex@31", "density": 0.8610814974425702, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexinhex@31", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "29-31. | | s = 6 | Trivial. |", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexinhex/31.gif", "source_page": "https://erich-friedman.github.io/packing/hexinhex/", "title": "Hexagons in Hexagons"}, "friedman_reference_text": "29-31. | | s = 6 | Trivial. |", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinhex/", "friedman_title": "Hexagons in Hexagons", "frontend_seed": true, "id": "d503e2b08764a35e", "improvement": -0.0001031728611780025, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0312532034243658e-06, "max_pair_overlap_depth": -8.80446118189937e-06, "metric_expression": "6.00010", "metric_symbol": "s", "metric_value": 6.000103172861178, "n": 31, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.0, "setup": "hexinhex", "side": 6.000103172861178, "solution_path": "data/solutions/generated/hexinhex_031_d503e2b08764a35e.json", "source_url": "https://erich-friedman.github.io/packing/hexinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d503e2b08764a35e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@1", "density": 0.6961402306608654, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/1.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(1+\u221a3)/\u221a2 = 1.931+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.931, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(1+\u221a3)/\u221a2 = 1.931+", "metric_symbol": "s", "metric_value": 1.931, "reference_text": "s = (1+\u221a3)/\u221a2 = 1.931+ | Trivial. |", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/1.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = (1+\u221a3)/\u221a2 = 1.931+ | Trivial. |", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "1cfff36edc20de72", "improvement": -0.000868569514983486, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8875760125229138e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.93187", "metric_symbol": "s", "metric_value": 1.9318685695149835, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.931, "setup": "hexinsqu", "side": 1.9318685695149835, "solution_path": "data/solutions/generated/hexinsqu_001_1cfff36edc20de72.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/1cfff36edc20de72.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@2", "density": 0.5214616218348717, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/2.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(1+2\u221a3)/\u221a2 = 3.156+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.156, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(1+2\u221a3)/\u221a2 = 3.156+", "metric_symbol": "s", "metric_value": 3.156, "reference_text": "s = (1+2\u221a3)/\u221a2 = 3.156+ | Trivial. |", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/2.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = (1+2\u221a3)/\u221a2 = 3.156+ | Trivial. |", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "ff43d04cda84c856", "improvement": -0.0006742348964126066, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2405599436706566e-05, "max_pair_overlap_depth": -1.629836664562411e-05, "metric_expression": "3.15667", "metric_symbol": "s", "metric_value": 3.1566742348964127, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.156, "setup": "hexinsqu", "side": 3.1566742348964127, "solution_path": "data/solutions/generated/hexinsqu_002_ff43d04cda84c856.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/ff43d04cda84c856.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@3", "density": 0.6391474518535116, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/3.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "9(6\u2013\u221a3)/11 = 3.491+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.491, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@3", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "9(6\u2013\u221a3)/11 = 3.491+", "metric_symbol": "s", "metric_value": 3.491, "reference_text": "s = 9(6\u2013\u221a3)/11 = 3.491+ | Found by Maurizio Morandi | in March 2026.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/3.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 9(6\u2013\u221a3)/11 = 3.491+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "9525d376eafe3c9c", "improvement": -0.0010949178753740796, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.084317521962771e-06, "max_pair_overlap_depth": -8.912711465047796e-05, "metric_expression": "3.49209", "metric_symbol": "s", "metric_value": 3.492094917875374, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.491, "setup": "hexinsqu", "side": 3.492094917875374, "solution_path": "data/solutions/generated/hexinsqu_003_9525d376eafe3c9c.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/9525d376eafe3c9c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@4", "density": 0.748920266367969, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/4.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(7\u2013\u221a3)/\u221a2 = 3.725+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.725, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@4", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "(7\u2013\u221a3)/\u221a2 = 3.725+", "metric_symbol": "s", "metric_value": 3.725, "reference_text": "s = (7\u2013\u221a3)/\u221a2 = 3.725+ | Found by Maurizio Morandi | in March 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/4.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = (7\u2013\u221a3)/\u221a2 = 3.725+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "33ddf042dd02d191", "improvement": -0.00010181398140396425, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.577436704193743e-06, "max_pair_overlap_depth": -5.3695636117151446e-05, "metric_expression": "3.72510", "metric_symbol": "s", "metric_value": 3.725101813981404, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.725, "setup": "hexinsqu", "side": 3.725101813981404, "solution_path": "data/solutions/generated/hexinsqu_004_33ddf042dd02d191.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/33ddf042dd02d191.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@5", "density": 0.6595778437263831, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/5.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "7(3\u2013\u221a3)/2 = 4.437+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.437, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@5", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "7(3\u2013\u221a3)/2 = 4.437+", "metric_symbol": "s", "metric_value": 4.437, "reference_text": "s = 7(3\u2013\u221a3)/2 = 4.437+ | Found by Maurizio Morandi | in March 2026.", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/5.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 7(3\u2013\u221a3)/2 = 4.437+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "38f2dc66ba941978", "improvement": -0.0009041295527705273, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.5429089870357805e-06, "max_pair_overlap_depth": -1.5874388830661168e-05, "metric_expression": "4.43790", "metric_symbol": "s", "metric_value": 4.437904129552771, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.437, "setup": "hexinsqu", "side": 4.437904129552771, "solution_path": "data/solutions/generated/hexinsqu_005_38f2dc66ba941978.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/38f2dc66ba941978.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@6", "density": 0.6726426228424899, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/6.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "20(9\u20132\u221a3)/23 = 4.813+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.813, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@6", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "20(9\u20132\u221a3)/23 = 4.813+", "metric_symbol": "s", "metric_value": 4.813, "reference_text": "s = 20(9\u20132\u221a3)/23 = 4.813+ | Found by Maurizio Morandi | in March 2026.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/6.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 20(9\u20132\u221a3)/23 = 4.813+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "31291850c6887128", "improvement": -0.0010364766922137392, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6841191118288626e-06, "max_pair_overlap_depth": -6.696342223433405e-05, "metric_expression": "4.81404", "metric_symbol": "s", "metric_value": 4.8140364766922135, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.813, "setup": "hexinsqu", "side": 4.8140364766922135, "solution_path": "data/solutions/generated/hexinsqu_006_31291850c6887128.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/31291850c6887128.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@7", "density": 0.6735519886722757, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/8.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3\u221a3 = 5.196+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.196, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3\u221a3 = 5.196+", "metric_symbol": "s", "metric_value": 5.196, "reference_text": "s = 3\u221a3 = 5.196+ | Trivial. |", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/8.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 3\u221a3 = 5.196+ | Trivial. |", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "a4c696c014daab3c", "improvement": -0.00024239437505713113, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.667089398456013e-06, "max_pair_overlap_depth": -2.5433886154724306e-05, "metric_expression": "5.19624", "metric_symbol": "s", "metric_value": 5.196242394375057, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.196, "setup": "hexinsqu", "side": 5.196242394375057, "solution_path": "data/solutions/generated/hexinsqu_007_a4c696c014daab3c.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a4c696c014daab3c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@8", "density": 0.7697897324076285, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/8.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3\u221a3 = 5.196+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.196, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@8", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3\u221a3 = 5.196+", "metric_symbol": "s", "metric_value": 5.196, "reference_text": "s = 3\u221a3 = 5.196+ | Trivial. |", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/8.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 3\u221a3 = 5.196+ | Trivial. |", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "b1252a56565d2ec0", "improvement": -0.00018828755551858478, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.033193436423119e-06, "max_pair_overlap_depth": -1.0846799477937097e-05, "metric_expression": "5.19619", "metric_symbol": "s", "metric_value": 5.196188287555518, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.196, "setup": "hexinsqu", "side": 5.196188287555518, "solution_path": "data/solutions/generated/hexinsqu_008_b1252a56565d2ec0.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b1252a56565d2ec0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@9", "density": 0.7678652278206223, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/9.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(13\u20133\u221a3)/\u221a2 = 5.518+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.518, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@9", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "(13\u20133\u221a3)/\u221a2 = 5.518+", "metric_symbol": "s", "metric_value": 5.518, "reference_text": "s = (13\u20133\u221a3)/\u221a2 = 5.518+ | Found by Maurizio Morandi | in March 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/9.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = (13\u20133\u221a3)/\u221a2 = 5.518+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "5b4ef4c36dcfba05", "improvement": -0.0002922521392276778, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.874395903155008e-06, "max_pair_overlap_depth": -4.408050364324012e-05, "metric_expression": "5.51829", "metric_symbol": "s", "metric_value": 5.5182922521392275, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.518, "setup": "hexinsqu", "side": 5.5182922521392275, "solution_path": "data/solutions/generated/hexinsqu_009_5b4ef4c36dcfba05.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/5b4ef4c36dcfba05.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@10", "density": 0.6970336920701942, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/10.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.10469+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.10469, "friedman_reference": {"analytical_or_proved": false, "case": "hexinsqu@10", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.10469+", "metric_symbol": "s", "metric_value": 6.10469, "reference_text": "s = 6.10469+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/10.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 6.10469+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "031f63d864cce10c", "improvement": -0.0004982231227232958, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5872663706407347e-08, "max_pair_overlap_depth": -2.0702587337384504e-05, "metric_expression": "6.10519", "metric_symbol": "s", "metric_value": 6.105188223122723, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.10469, "setup": "hexinsqu", "side": 6.105188223122723, "solution_path": "data/solutions/generated/hexinsqu_010_031f63d864cce10c.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/031f63d864cce10c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@11", "density": 0.7126826634184014, "friedman_analytical_or_proved": false, "friedman_credit": "Maksymilian Jankowski", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/11.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.33218+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.33218, "friedman_reference": {"analytical_or_proved": false, "case": "hexinsqu@11", "credit": "Maksymilian Jankowski", "date": "April 2026", "family_updated": "", "metric_expression": "6.33218+", "metric_symbol": "s", "metric_value": 6.33218, "reference_text": "s = 6.33218+ | Found by Maksymilian Jankowski | in April 2026.", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/11.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 6.33218+ | Found by Maksymilian Jankowski | in April 2026.", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "f0fe574f6ed3708e", "improvement": -0.0003052751802421483, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.734623780299186e-06, "max_pair_overlap_depth": -4.342994577477555e-05, "metric_expression": "6.33249", "metric_symbol": "s", "metric_value": 6.332485275180242, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.33218, "setup": "hexinsqu", "side": 6.332485275180242, "solution_path": "data/solutions/generated/hexinsqu_011_f0fe574f6ed3708e.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maksymilian Jankowski", "svg_path": "data/svg/f0fe574f6ed3708e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@12", "density": 0.7640646951117188, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/12.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "43(5\u2013\u221a3)/22 = 6.387+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.387, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@12", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "43(5\u2013\u221a3)/22 = 6.387+", "metric_symbol": "s", "metric_value": 6.387, "reference_text": "s = 43(5\u2013\u221a3)/22 = 6.387+ | Found by Maurizio Morandi | in March 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/12.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 43(5\u2013\u221a3)/22 = 6.387+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "70f372d7c2d3b859", "improvement": -0.0008027909827861279, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6841159701879252e-07, "max_pair_overlap_depth": -5.12694605685482e-05, "metric_expression": "6.38780", "metric_symbol": "s", "metric_value": 6.387802790982786, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.387, "setup": "hexinsqu", "side": 6.387802790982786, "solution_path": "data/solutions/generated/hexinsqu_012_70f372d7c2d3b859.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/70f372d7c2d3b859.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@13", "density": 0.7399851108638961, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/13.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.75558+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.75558, "friedman_reference": {"analytical_or_proved": false, "case": "hexinsqu@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.75558+", "metric_symbol": "s", "metric_value": 6.75558, "reference_text": "s = 6.75558+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/13.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 6.75558+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "96118c364cb8b05d", "improvement": -0.0003653626017898759, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.852098428599495e-06, "max_pair_overlap_depth": -2.731772979069902e-05, "metric_expression": "6.75595", "metric_symbol": "s", "metric_value": 6.75594536260179, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.75558, "setup": "hexinsqu", "side": 6.75594536260179, "solution_path": "data/solutions/generated/hexinsqu_013_96118c364cb8b05d.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/96118c364cb8b05d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@14", "density": 0.7577523835410583, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/14.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4\u221a3 = 6.928+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.928, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@14", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4\u221a3 = 6.928+", "metric_symbol": "s", "metric_value": 6.928, "reference_text": "s = 4\u221a3 = 6.928+ | Trivial. |", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/14.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 4\u221a3 = 6.928+ | Trivial. |", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "8b7d0912d343dc8b", "improvement": -0.00029395100744089575, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.277327734003356e-06, "max_pair_overlap_depth": -1.698575172759398e-05, "metric_expression": "6.92829", "metric_symbol": "s", "metric_value": 6.928293951007441, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.928, "setup": "hexinsqu", "side": 6.928293951007441, "solution_path": "data/solutions/generated/hexinsqu_014_8b7d0912d343dc8b.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/8b7d0912d343dc8b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@15", "density": 0.8035851614122442, "friedman_analytical_or_proved": false, "friedman_credit": "Maksymilian Jankowski", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/15.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.96367+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.96367, "friedman_reference": {"analytical_or_proved": false, "case": "hexinsqu@15", "credit": "Maksymilian Jankowski", "date": "April 2026", "family_updated": "", "metric_expression": "6.96367+", "metric_symbol": "s", "metric_value": 6.96367, "reference_text": "s = 6.96367+ | Found by Maksymilian Jankowski | in April 2026.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/15.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 6.96367+ | Found by Maksymilian Jankowski | in April 2026.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "0691341c684bd112", "improvement": -0.00027958460374311045, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.156990403080692e-07, "max_pair_overlap_depth": -5.813914172492218e-05, "metric_expression": "6.96395", "metric_symbol": "s", "metric_value": 6.963949584603743, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.96367, "setup": "hexinsqu", "side": 6.963949584603743, "solution_path": "data/solutions/generated/hexinsqu_015_0691341c684bd112.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maksymilian Jankowski", "svg_path": "data/svg/0691341c684bd112.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@16", "density": 0.7776171365327941, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/16.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(19\u20135\u221a3)/\u221a2 = 7.311+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.311, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@16", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "(19\u20135\u221a3)/\u221a2 = 7.311+", "metric_symbol": "s", "metric_value": 7.311, "reference_text": "s = (19\u20135\u221a3)/\u221a2 = 7.311+ | Found by Maurizio Morandi | in April 2026.", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/16.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = (19\u20135\u221a3)/\u221a2 = 7.311+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "03465bdf660197b5", "improvement": -0.000441730863443901, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8577837974051795e-06, "max_pair_overlap_depth": -3.069264188360066e-05, "metric_expression": "7.31144", "metric_symbol": "s", "metric_value": 7.311441730863444, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.311, "setup": "hexinsqu", "side": 7.311441730863444, "solution_path": "data/solutions/generated/hexinsqu_016_03465bdf660197b5.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/03465bdf660197b5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@17", "density": 0.7624212140097384, "friedman_analytical_or_proved": true, "friedman_credit": "Maksymilian Jankowski", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/17.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6(3\u2013\u221a3) = 7.60383+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.60383, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@17", "credit": "Maksymilian Jankowski", "date": "April 2026", "family_updated": "", "metric_expression": "6(3\u2013\u221a3) = 7.60383+", "metric_symbol": "s", "metric_value": 7.60383, "reference_text": "s = 6(3\u2013\u221a3) = 7.60383+ | Found by Maksymilian Jankowski | in April 2026.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/17.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 6(3\u2013\u221a3) = 7.60383+ | Found by Maksymilian Jankowski | in April 2026.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "080714e7afd39346", "improvement": -0.007366212799500005, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.378937853940429e-07, "max_pair_overlap_depth": -2.071443269358042e-05, "metric_expression": "7.61120", "metric_symbol": "s", "metric_value": 7.6111962127995, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.60383, "setup": "hexinsqu", "side": 7.6111962127995, "solution_path": "data/solutions/generated/hexinsqu_017_080714e7afd39346.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maksymilian Jankowski", "svg_path": "data/svg/080714e7afd39346.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@18", "density": 0.7882207738949305, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/18.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "32(9\u20132\u221a3)/23 = 7.702+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.702, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@18", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "32(9\u20132\u221a3)/23 = 7.702+", "metric_symbol": "s", "metric_value": 7.702, "reference_text": "s = 32(9\u20132\u221a3)/23 = 7.702+ | Found by Maurizio Morandi | in April 2026.", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/18.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 32(9\u20132\u221a3)/23 = 7.702+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "39a9a99a50ac7cba", "improvement": -0.0006161769446162424, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5800172874233264e-06, "max_pair_overlap_depth": -3.4078889656230515e-05, "metric_expression": "7.70262", "metric_symbol": "s", "metric_value": 7.702616176944616, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.702, "setup": "hexinsqu", "side": 7.702616176944616, "solution_path": "data/solutions/generated/hexinsqu_018_39a9a99a50ac7cba.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/39a9a99a50ac7cba.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@19", "density": 0.7859807737367643, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/19.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "25(3\u2013\u221a3)/4 = 7.924+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.924, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@19", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "25(3\u2013\u221a3)/4 = 7.924+", "metric_symbol": "s", "metric_value": 7.924, "reference_text": "s = 25(3\u2013\u221a3)/4 = 7.924+ | Found by Maurizio Morandi | in April 2026.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/19.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 25(3\u2013\u221a3)/4 = 7.924+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "10d9f5dde53d25ba", "improvement": -0.0009545892016307178, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.893895229540556e-06, "max_pair_overlap_depth": -3.6221248433054853e-05, "metric_expression": "7.92495", "metric_symbol": "s", "metric_value": 7.924954589201631, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.924, "setup": "hexinsqu", "side": 7.924954589201631, "solution_path": "data/solutions/generated/hexinsqu_019_10d9f5dde53d25ba.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/10d9f5dde53d25ba.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@20", "density": 0.8222377903171622, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/20.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "74(21\u20134\u221a3)/131 = 7.948+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.948, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@20", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "74(21\u20134\u221a3)/131 = 7.948+", "metric_symbol": "s", "metric_value": 7.948, "reference_text": "s = 74(21\u20134\u221a3)/131 = 7.948+ | Found by Maurizio Morandi | in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/20.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 74(21\u20134\u221a3)/131 = 7.948+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "3d530c0ac6e7e3b1", "improvement": -0.0015441274741458955, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.121785608292394e-06, "max_pair_overlap_depth": -4.480534023287852e-05, "metric_expression": "7.94954", "metric_symbol": "s", "metric_value": 7.949544127474146, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.948, "setup": "hexinsqu", "side": 7.949544127474146, "solution_path": "data/solutions/generated/hexinsqu_020_3d530c0ac6e7e3b1.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/3d530c0ac6e7e3b1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@21", "density": 0.7762795555424945, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/21.png", "friedman_metric_expression": "8.381+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.381, "friedman_reference": {"analytical_or_proved": false, "case": "hexinsqu@21", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "8.381+", "metric_symbol": "s", "metric_value": 8.381, "reference_text": "s = 8.381+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "21.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/21.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 8.381+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "21.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "5d369a2740fde606", "improvement": -0.002522153681731254, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.85221561134309e-06, "max_pair_overlap_depth": -3.989714052887727e-05, "metric_expression": "8.38352", "metric_symbol": "s", "metric_value": 8.383522153681731, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.381, "setup": "hexinsqu", "side": 8.383522153681731, "solution_path": "data/solutions/generated/hexinsqu_021_5d369a2740fde606.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/5d369a2740fde606.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@22", "density": 0.7770815845496467, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/22.png", "friedman_metric_expression": "8.568+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.568, "friedman_reference": {"analytical_or_proved": false, "case": "hexinsqu@22", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "8.568+", "metric_symbol": "s", "metric_value": 8.568, "reference_text": "s = 8.568+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "22.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/22.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 8.568+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "22.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "2f0b0033f0c66708", "improvement": -0.008379204155273712, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.913077650512531e-06, "max_pair_overlap_depth": -1.9097490104869053e-05, "metric_expression": "8.57638", "metric_symbol": "s", "metric_value": 8.576379204155273, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.568, "setup": "hexinsqu", "side": 8.576379204155273, "solution_path": "data/solutions/generated/hexinsqu_022_2f0b0033f0c66708.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/2f0b0033f0c66708.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@23", "density": 0.7967168154089415, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/23.png", "friedman_metric_expression": "5\u221a3 = 8.660+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.66, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@23", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5\u221a3 = 8.660+", "metric_symbol": "s", "metric_value": 8.66, "reference_text": "s = 5\u221a3 = 8.660+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "23.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/23.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 5\u221a3 = 8.660+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "23.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "31cc17034bd25d7b", "improvement": -0.0003983679346664104, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.345963802137002e-06, "max_pair_overlap_depth": -2.1349502021816846e-07, "metric_expression": "8.66040", "metric_symbol": "s", "metric_value": 8.660398367934667, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.66, "setup": "hexinsqu", "side": 8.660398367934667, "solution_path": "data/solutions/generated/hexinsqu_023_31cc17034bd25d7b.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/31cc17034bd25d7b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@24", "density": 0.8058676683577897, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/24.png", "friedman_metric_expression": "8.795+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.795, "friedman_reference": {"analytical_or_proved": false, "case": "hexinsqu@24", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "8.795+", "metric_symbol": "s", "metric_value": 8.795, "reference_text": "s = 8.795+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "24.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/24.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 8.795+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "24.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "941fbce4f50492c6", "improvement": -0.0012932219182459903, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0969884292392119e-07, "max_pair_overlap_depth": -1.611027152126354e-05, "metric_expression": "8.79629", "metric_symbol": "s", "metric_value": 8.796293221918246, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.795, "setup": "hexinsqu", "side": 8.796293221918246, "solution_path": "data/solutions/generated/hexinsqu_024_941fbce4f50492c6.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/941fbce4f50492c6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexinsqu@25", "density": 0.7898968080720588, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexinsqu/25.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "30(4\u221a3\u20133)/13 = 9.065+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.065, "friedman_reference": {"analytical_or_proved": true, "case": "hexinsqu@25", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "30(4\u221a3\u20133)/13 = 9.065+", "metric_symbol": "s", "metric_value": 9.065, "reference_text": "s = 30(4\u221a3\u20133)/13 = 9.065+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "25.png", "source_image_url": "https://erich-friedman.github.io/packing/hexinsqu/25.png", "source_page": "https://erich-friedman.github.io/packing/hexinsqu/", "title": "Hexagons in Squares"}, "friedman_reference_text": "s = 30(4\u221a3\u20133)/13 = 9.065+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "25.png", "friedman_source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "friedman_title": "Hexagons in Squares", "frontend_seed": true, "id": "a62e2508884336f7", "improvement": -0.00298447907218069, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.02442885507961e-06, "max_pair_overlap_depth": -5.091125714784539e-06, "metric_expression": "9.06798", "metric_symbol": "s", "metric_value": 9.06798447907218, "n": 25, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.065, "setup": "hexinsqu", "side": 9.06798447907218, "solution_path": "data/solutions/generated/hexinsqu_025_a62e2508884336f7.json", "source_url": "https://erich-friedman.github.io/packing/hexinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/a62e2508884336f7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@1", "density": 0.6666490309389511, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/1.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/1.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "c2fff75544b88239", "improvement": -3.9681174644368156e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.67464904574664e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "3.00004", "metric_symbol": "s", "metric_value": 3.0000396811746444, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.0, "setup": "hexintri", "side": 3.0000396811746444, "solution_path": "data/solutions/generated/hexintri_001_c2fff75544b88239.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/c2fff75544b88239.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@2", "density": 0.4799802064426444, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/3.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/3.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "b77426ecb97f3ce6", "improvement": -0.00010309463302426991, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3591084715906373e-06, "max_pair_overlap_depth": -4.2178351187804886e-05, "metric_expression": "5.00010", "metric_symbol": "s", "metric_value": 5.000103094633024, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.0, "setup": "hexintri", "side": 5.000103094633024, "solution_path": "data/solutions/generated/hexintri_002_b77426ecb97f3ce6.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b77426ecb97f3ce6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@3", "density": 0.7199659208954582, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/3.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/3.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "748bfe1ef47ea18c", "improvement": -0.00011833442488207169, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3474135610435156e-06, "max_pair_overlap_depth": -4.4955482604913666e-05, "metric_expression": "5.00012", "metric_symbol": "s", "metric_value": 5.000118334424882, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.0, "setup": "hexintri", "side": 5.000118334424882, "solution_path": "data/solutions/generated/hexintri_003_748bfe1ef47ea18c.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/748bfe1ef47ea18c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@4", "density": 0.6666571046827545, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/4.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/4.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 6 | Trivial.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "4bf531bb9fa044b6", "improvement": -4.302939048272947e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.615149234551692e-07, "max_pair_overlap_depth": -1.1132001730862484e-05, "metric_expression": "6.00004", "metric_symbol": "s", "metric_value": 6.000043029390483, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.0, "setup": "hexintri", "side": 6.000043029390483, "solution_path": "data/solutions/generated/hexintri_004_4bf531bb9fa044b6.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4bf531bb9fa044b6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@5", "density": 0.6430466587435534, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/5.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "5(1+\u221a3)/2 = 6.8301+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.8301, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@5", "credit": "Maurizio Morandi", "date": "April 2015", "family_updated": "", "metric_expression": "5(1+\u221a3)/2 = 6.8301+", "metric_symbol": "s", "metric_value": 6.8301, "reference_text": "s = 5(1+\u221a3)/2 = 6.8301+ | Found by Maurizio Morandi | in April 2015.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/5.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 5(1+\u221a3)/2 = 6.8301+ | Found by Maurizio Morandi | in April 2015.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "92331a16a75cf6c4", "improvement": -0.00019379102171246387, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4626103815685519e-06, "max_pair_overlap_depth": -3.302865088408366e-05, "metric_expression": "6.83029", "metric_symbol": "s", "metric_value": 6.830293791021712, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.8301, "setup": "hexintri", "side": 6.830293791021712, "solution_path": "data/solutions/generated/hexintri_005_92331a16a75cf6c4.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/92331a16a75cf6c4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@6", "density": 0.7560505607557761, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/6.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "2(1+\u221a6) = 6.89897+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.89897, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@6", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2(1+\u221a6) = 6.89897+", "metric_symbol": "s", "metric_value": 6.89897, "reference_text": "s = 2(1+\u221a6) = 6.89897+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/6.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 2(1+\u221a6) = 6.89897+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "f3621ba5529eef9a", "improvement": -0.0014548494669490708, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.849296155859392e-06, "max_pair_overlap_depth": -8.276209533908396e-05, "metric_expression": "6.90042", "metric_symbol": "s", "metric_value": 6.900424849466949, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.89897, "setup": "hexintri", "side": 6.900424849466949, "solution_path": "data/solutions/generated/hexintri_006_f3621ba5529eef9a.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/f3621ba5529eef9a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@7", "density": 0.6562369627570366, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/7.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "8", "friedman_metric_symbol": "s", "friedman_metric_value": 8.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "8", "metric_symbol": "s", "metric_value": 8.0, "reference_text": "s = 8 | Trivial. |", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/7.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 8 | Trivial. |", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "635b0cc9d15ec460", "improvement": -7.946628399579936e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.24931067292516e-06, "max_pair_overlap_depth": -1.4685785256318695e-05, "metric_expression": "8.00008", "metric_symbol": "s", "metric_value": 8.000079466283996, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.0, "setup": "hexintri", "side": 8.000079466283996, "solution_path": "data/solutions/generated/hexintri_007_635b0cc9d15ec460.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/635b0cc9d15ec460.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@8", "density": 0.6907428320047944, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2015", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/8.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "25/3 = 8.3333+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.3333, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@8", "credit": "Erich Friedman", "date": "April 2015", "family_updated": "", "metric_expression": "25/3 = 8.3333+", "metric_symbol": "s", "metric_value": 8.3333, "reference_text": "s = 25/3 = 8.3333+ | Found by Erich Friedman | in April 2015.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/8.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 25/3 = 8.3333+ | Found by Erich Friedman | in April 2015.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "adf439e6f7b2afb5", "improvement": -0.002790584626882975, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.227185763431761e-07, "max_pair_overlap_depth": -7.882545446008526e-05, "metric_expression": "8.33609", "metric_symbol": "s", "metric_value": 8.336090584626882, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.3333, "setup": "hexintri", "side": 8.336090584626882, "solution_path": "data/solutions/generated/hexintri_008_adf439e6f7b2afb5.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/adf439e6f7b2afb5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@9", "density": 0.7214446032247255, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/10.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "5\u221a3 = 8.6602+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.6602, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@9", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5\u221a3 = 8.6602+", "metric_symbol": "s", "metric_value": 8.6602, "reference_text": "s = 5\u221a3 = 8.6602+ | Trivial. |", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/10.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 5\u221a3 = 8.6602+ | Trivial. |", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "936a184a58977242", "improvement": 0.008620848542102877, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0939511862432736e-06, "max_pair_overlap_depth": -5.506633970053798e-05, "metric_expression": "8.65158", "metric_symbol": "s", "metric_value": 8.651579151457897, "n": 9, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 8.6602, "setup": "hexintri", "side": 8.651579151457897, "solution_path": "data/solutions/generated/hexintri_009_936a184a58977242.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/936a184a58977242.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@10", "density": 0.7999538247696885, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/10.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "5\u221a3 = 8.6602+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.6602, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@10", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5\u221a3 = 8.6602+", "metric_symbol": "s", "metric_value": 8.6602, "reference_text": "s = 5\u221a3 = 8.6602+ | Trivial. |", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/10.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 5\u221a3 = 8.6602+ | Trivial. |", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "5df6cec7a90f52fe", "improvement": -0.00030397942970239455, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2645686941820143e-06, "max_pair_overlap_depth": -4.384611814706174e-05, "metric_expression": "8.66050", "metric_symbol": "s", "metric_value": 8.660503979429702, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.6602, "setup": "hexintri", "side": 8.660503979429702, "solution_path": "data/solutions/generated/hexintri_010_5df6cec7a90f52fe.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/5df6cec7a90f52fe.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@11", "density": 0.7237571814877352, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/11.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "9.54823+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.54823, "friedman_reference": {"analytical_or_proved": false, "case": "hexintri@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "9.54823+", "metric_symbol": "s", "metric_value": 9.54823, "reference_text": "s = 9.54823+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/11.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 9.54823+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "d582cb032b9c575f", "improvement": -0.0011576763865779327, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.6199514834868076e-06, "max_pair_overlap_depth": -2.6419225191420814e-05, "metric_expression": "9.54939", "metric_symbol": "s", "metric_value": 9.549387676386578, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.54823, "setup": "hexintri", "side": 9.549387676386578, "solution_path": "data/solutions/generated/hexintri_011_d582cb032b9c575f.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/d582cb032b9c575f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@12", "density": 0.7199169251480841, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/12.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "10", "friedman_metric_symbol": "s", "friedman_metric_value": 10.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@12", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "10", "metric_symbol": "s", "metric_value": 10.0, "reference_text": "s = 10 | Trivial. |", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/12.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 10 | Trivial. |", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "b6e20361c3af5615", "improvement": -0.0005769586222079681, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.006918638920166e-07, "max_pair_overlap_depth": -2.082412255655086e-05, "metric_expression": "10.00058", "metric_symbol": "s", "metric_value": 10.000576958622208, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 10.0, "setup": "hexintri", "side": 10.000576958622208, "solution_path": "data/solutions/generated/hexintri_012_b6e20361c3af5615.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b6e20361c3af5615.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@13", "density": 0.7221828032992709, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "6\u221a3 = 10.3923+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.3923, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@13", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6\u221a3 = 10.3923+", "metric_symbol": "s", "metric_value": 10.3923, "reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "da55bd8ffef5f50e", "improvement": -0.00028846326752507423, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7733151165742505e-06, "max_pair_overlap_depth": -4.4289278687159594e-05, "metric_expression": "10.39259", "metric_symbol": "s", "metric_value": 10.392588463267526, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 10.3923, "setup": "hexintri", "side": 10.392588463267526, "solution_path": "data/solutions/generated/hexintri_013_da55bd8ffef5f50e.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/da55bd8ffef5f50e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@14", "density": 0.7777237680646841, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "6\u221a3 = 10.3923+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.3923, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@14", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6\u221a3 = 10.3923+", "metric_symbol": "s", "metric_value": 10.3923, "reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "272118e9284c579f", "improvement": -0.0003656905370288399, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.442529134834328e-06, "max_pair_overlap_depth": -4.275788154761706e-05, "metric_expression": "10.39267", "metric_symbol": "s", "metric_value": 10.39266569053703, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 10.3923, "setup": "hexintri", "side": 10.39266569053703, "solution_path": "data/solutions/generated/hexintri_014_272118e9284c579f.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/272118e9284c579f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@15", "density": 0.833304219754554, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "6\u221a3 = 10.3923+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.3923, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@15", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6\u221a3 = 10.3923+", "metric_symbol": "s", "metric_value": 10.3923, "reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/15.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 6\u221a3 = 10.3923+ | Trivial. |", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "83460d360e93ff66", "improvement": -0.00018638448149665976, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.672744116747984e-07, "max_pair_overlap_depth": -2.8784199847526563e-05, "metric_expression": "10.39249", "metric_symbol": "s", "metric_value": 10.392486384481497, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 10.3923, "setup": "hexintri", "side": 10.392486384481497, "solution_path": "data/solutions/generated/hexintri_015_83460d360e93ff66.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/83460d360e93ff66.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@16", "density": 0.7429369875675369, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/16.gif", "friedman_metric_expression": "11.36599+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.36599, "friedman_reference": {"analytical_or_proved": false, "case": "hexintri@16", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "11.36599+", "metric_symbol": "s", "metric_value": 11.36599, "reference_text": "s = 11.36599+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/16.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 11.36599+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "89d5bef670451c83", "improvement": -0.0013703170956897992, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.657075146936789e-07, "max_pair_overlap_depth": -2.5616124350591463e-06, "metric_expression": "11.36736", "metric_symbol": "s", "metric_value": 11.36736031709569, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 11.36599, "setup": "hexintri", "side": 11.36736031709569, "solution_path": "data/solutions/generated/hexintri_16_89d5bef670451c83.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/89d5bef670451c83.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@17", "density": 0.7525303158572544, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/17.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "11.63457+", "friedman_metric_symbol": "s", "friedman_metric_value": 11.63457, "friedman_reference": {"analytical_or_proved": false, "case": "hexintri@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "11.63457+", "metric_symbol": "s", "metric_value": 11.63457, "reference_text": "s = 11.63457+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/17.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 11.63457+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "8137d039fd339f48", "improvement": -0.007711226979866481, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.447216764354318e-05, "max_pair_overlap_depth": -1.3408176938511929e-05, "metric_expression": "11.64228", "metric_symbol": "s", "metric_value": 11.642281226979867, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 11.63457, "setup": "hexintri", "side": 11.642281226979867, "solution_path": "data/solutions/generated/hexintri_017_8137d039fd339f48.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T03:07:58+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/8137d039fd339f48.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@18", "density": 0.7507767554452202, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/19.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "12", "friedman_metric_symbol": "s", "friedman_metric_value": 12.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@18", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "12", "metric_symbol": "s", "metric_value": 12.0, "reference_text": "s = 12 | Trivial. |", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/19.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 12 | Trivial. |", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "8f10cd5e01be9029", "improvement": 0.0062092209316340075, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.734105056962567e-06, "max_pair_overlap_depth": -5.8920104462956147e-05, "metric_expression": "11.99379", "metric_symbol": "s", "metric_value": 11.993790779068366, "n": 18, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 12.0, "setup": "hexintri", "side": 11.993790779068366, "solution_path": "data/solutions/generated/hexintri_018_8f10cd5e01be9029.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/8f10cd5e01be9029.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@19", "density": 0.7866173060867477, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/19.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "12", "friedman_metric_symbol": "s", "friedman_metric_value": 12.0, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@19", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "12", "metric_symbol": "s", "metric_value": 12.0, "reference_text": "s = 12 | Trivial. |", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/19.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 12 | Trivial. |", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "18178ad9efaf7172", "improvement": -0.0384528795380632, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.242681221171444e-06, "max_pair_overlap_depth": -3.13088544121598e-05, "metric_expression": "12.03845", "metric_symbol": "s", "metric_value": 12.038452879538063, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 12.0, "setup": "hexintri", "side": 12.038452879538063, "solution_path": "data/solutions/generated/hexintri_019_18178ad9efaf7172.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T03:07:58+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/18178ad9efaf7172.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@20", "density": 0.8162942117377381, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/21.gif", "friedman_family_updated": "4/5/26", "friedman_metric_expression": "7\u221a3 = 12.1243+", "friedman_metric_symbol": "s", "friedman_metric_value": 12.1243, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@20", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7\u221a3 = 12.1243+", "metric_symbol": "s", "metric_value": 12.1243, "reference_text": "s = 7\u221a3 = 12.1243+ | Trivial. |", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/21.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 7\u221a3 = 12.1243+ | Trivial. |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "f59179c5b6665dc4", "improvement": -0.00029566549525483765, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.116403950810266e-06, "max_pair_overlap_depth": -3.364591626842106e-05, "metric_expression": "12.12460", "metric_symbol": "s", "metric_value": 12.124595665495255, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 12.1243, "setup": "hexintri", "side": 12.124595665495255, "solution_path": "data/solutions/generated/hexintri_020_f59179c5b6665dc4.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/f59179c5b6665dc4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "hexintri@21", "density": 0.8571035425372456, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/hexintri/21.gif", "friedman_metric_expression": "7\u221a3 = 12.1243+", "friedman_metric_symbol": "s", "friedman_metric_value": 12.1243, "friedman_reference": {"analytical_or_proved": true, "case": "hexintri@21", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7\u221a3 = 12.1243+", "metric_symbol": "s", "metric_value": 12.1243, "reference_text": "s = 7\u221a3 = 12.1243+ | Trivial. |", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/hexintri/21.gif", "source_page": "https://erich-friedman.github.io/packing/hexintri/", "title": "Hexagons in Triangles"}, "friedman_reference_text": "s = 7\u221a3 = 12.1243+ | Trivial. |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/hexintri/", "friedman_title": "Hexagons in Triangles", "frontend_seed": true, "id": "6cf726ac83ba7d7f", "improvement": -0.00033371669977810825, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7607980864287318e-06, "max_pair_overlap_depth": -2.977721456209892e-05, "metric_expression": "12.12463", "metric_symbol": "s", "metric_value": 12.124633716699778, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 12.1243, "setup": "hexintri", "side": 12.124633716699778, "solution_path": "data/solutions/generated/hexintri_021_6cf726ac83ba7d7f.json", "source_url": "https://erich-friedman.github.io/packing/hexintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6cf726ac83ba7d7f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@1", "density": 0.9999940796878448, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/1.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial. |", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/1.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 1 | Trivial. |", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "13c63959e19cfeb5", "improvement": -2.96016922152198e-06, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1903479056983753e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00000", "metric_symbol": "s", "metric_value": 1.0000029601692215, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.0, "setup": "octinoct", "side": 1.0000029601692215, "solution_path": "data/solutions/generated/octinoct_001_13c63959e19cfeb5.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/13c63959e19cfeb5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@2", "density": 0.4999889107582135, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/2.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial. |", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/2.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 2 | Trivial. |", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "937fb450c622d9b9", "improvement": -2.2178852493670576e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.909847599495863e-06, "max_pair_overlap_depth": -3.0343321494846753e-05, "metric_expression": "2.00002", "metric_symbol": "s", "metric_value": 2.0000221788524937, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "octinoct", "side": 2.0000221788524937, "solution_path": "data/solutions/generated/octinoct_002_937fb450c622d9b9.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/937fb450c622d9b9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@3", "density": 0.6361550822369931, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/3.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "5\u20132\u221a2 = 2.17157+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.17157, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@3", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5\u20132\u221a2 = 2.17157+", "metric_symbol": "s", "metric_value": 2.17157, "reference_text": "s = 5\u20132\u221a2 = 2.17157+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/3.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5\u20132\u221a2 = 2.17157+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "4cb37df94277ed6e", "improvement": -2.6468916854760494e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.359798521217012e-07, "max_pair_overlap_depth": -2.9093807346031175e-05, "metric_expression": "2.17160", "metric_symbol": "s", "metric_value": 2.1715964689168548, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.17157, "setup": "octinoct", "side": 2.1715964689168548, "solution_path": "data/solutions/generated/octinoct_003_4cb37df94277ed6e.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/4cb37df94277ed6e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@4", "density": 0.7008534259385343, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/4.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "\u221a(5+1/\u221a2) = 2.38895+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.38895, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@4", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "\u221a(5+1/\u221a2) = 2.38895+", "metric_symbol": "s", "metric_value": 2.38895, "reference_text": "s = \u221a(5+1/\u221a2) = 2.38895+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/4.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = \u221a(5+1/\u221a2) = 2.38895+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "8759832408431e84", "improvement": -5.1351046062464434e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.873179406841027e-06, "max_pair_overlap_depth": -4.8382172928693526e-05, "metric_expression": "2.38900", "metric_symbol": "s", "metric_value": 2.3890013510460624, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.38895, "setup": "octinoct", "side": 2.3890013510460624, "solution_path": "data/solutions/generated/octinoct_004_8759832408431e84.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/8759832408431e84.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@5", "density": 0.6822471038460284, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/5.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "2+1/\u221a2 = 2.70710+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.7071, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@5", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2+1/\u221a2 = 2.70710+", "metric_symbol": "s", "metric_value": 2.7071, "reference_text": "s = 2+1/\u221a2 = 2.70710+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/5.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 2+1/\u221a2 = 2.70710+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "d3468761b0fe9922", "improvement": -6.14172351678377e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3153662665326975e-06, "max_pair_overlap_depth": -4.586217379215496e-05, "metric_expression": "2.70716", "metric_symbol": "s", "metric_value": 2.707161417235168, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.7071, "setup": "octinoct", "side": 2.707161417235168, "solution_path": "data/solutions/generated/octinoct_005_d3468761b0fe9922.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/d3468761b0fe9922.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@6", "density": 0.6715848857993428, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/6.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "2.98915+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.98915, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@6", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.98915+", "metric_symbol": "s", "metric_value": 2.98915, "reference_text": "s = 2.98915+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/6.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 2.98915+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "3c4b2267adcca69f", "improvement": 0.0001551389501637246, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.973975728535522e-06, "max_pair_overlap_depth": -2.153911208435666e-05, "metric_expression": "2.98899", "metric_symbol": "s", "metric_value": 2.9889948610498362, "n": 6, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.98915, "setup": "octinoct", "side": 2.9889948610498362, "solution_path": "data/solutions/generated/octinoct_006_3c4b2267adcca69f.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/3c4b2267adcca69f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@7", "density": 0.7777522978920184, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/7.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial. |", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/7.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 3 | Trivial. |", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "a0053fa5c22778e9", "improvement": -4.914098707065051e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2079023857814054e-06, "max_pair_overlap_depth": -2.6395744560092638e-05, "metric_expression": "3.00005", "metric_symbol": "s", "metric_value": 3.0000491409870707, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.0, "setup": "octinoct", "side": 3.0000491409870707, "solution_path": "data/solutions/generated/octinoct_007_a0053fa5c22778e9.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a0053fa5c22778e9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@8", "density": 0.7457363085593237, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/8.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "3.27526+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.27526, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@8", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3.27526+", "metric_symbol": "s", "metric_value": 3.27526, "reference_text": "s = 3.27526+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/8.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 3.27526+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "697fa4dfd3dbc6eb", "improvement": -4.953250458550329e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.188513199476399e-06, "max_pair_overlap_depth": -7.758148462100323e-06, "metric_expression": "3.27531", "metric_symbol": "s", "metric_value": 3.2753095325045853, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.27526, "setup": "octinoct", "side": 3.2753095325045853, "solution_path": "data/solutions/generated/octinoct_008_697fa4dfd3dbc6eb.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/697fa4dfd3dbc6eb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@9", "density": 0.7720692797565871, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/9.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "2+\u221a2 = 3.41421+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.41421, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@9", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2+\u221a2 = 3.41421+", "metric_symbol": "s", "metric_value": 3.41421, "reference_text": "s = 2+\u221a2 = 3.41421+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/9.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 2+\u221a2 = 3.41421+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "7face58a13302eec", "improvement": -2.270780627799951e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3803938489295472e-05, "max_pair_overlap_depth": -2.6349120965107886e-06, "metric_expression": "3.41423", "metric_symbol": "s", "metric_value": 3.414232707806278, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.41421, "setup": "octinoct", "side": 3.414232707806278, "solution_path": "data/solutions/generated/octinoct_009_7face58a13302eec.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/7face58a13302eec.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@10", "density": 0.7276473732898788, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/10.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "3+1/\u221a2 = 3.70710+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.7071, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@10", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3+1/\u221a2 = 3.70710+", "metric_symbol": "s", "metric_value": 3.7071, "reference_text": "s = 3+1/\u221a2 = 3.70710+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/10.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 3+1/\u221a2 = 3.70710+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "bcfc9e294bb15f21", "improvement": -4.452047150094174e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.744023326611903e-06, "max_pair_overlap_depth": -8.094414775450254e-06, "metric_expression": "3.70714", "metric_symbol": "s", "metric_value": 3.707144520471501, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.7071, "setup": "octinoct", "side": 3.707144520471501, "solution_path": "data/solutions/generated/octinoct_010_bcfc9e294bb15f21.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/bcfc9e294bb15f21.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@11", "density": 0.7111917967389362, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/11.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "3.93270+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.9327, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@11", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3.93270+", "metric_symbol": "s", "metric_value": 3.9327, "reference_text": "s = 3.93270+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/11.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 3.93270+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "5cff1aa2d330c49e", "improvement": -0.00011007046297972778, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.479089320941585e-06, "max_pair_overlap_depth": -8.811356971838435e-06, "metric_expression": "3.93281", "metric_symbol": "s", "metric_value": 3.93281007046298, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.9327, "setup": "octinoct", "side": 3.93281007046298, "solution_path": "data/solutions/generated/octinoct_011_5cff1aa2d330c49e.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/5cff1aa2d330c49e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@12", "density": 0.7499773041118397, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/12.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@12", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial. |", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/12.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 4 | Trivial. |", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "7bdd4832407bd3be", "improvement": -6.052374207055067e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.082066348113342e-06, "max_pair_overlap_depth": -1.2861422511445397e-05, "metric_expression": "4.00006", "metric_symbol": "s", "metric_value": 4.0000605237420706, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.0, "setup": "octinoct", "side": 4.0000605237420706, "solution_path": "data/solutions/generated/octinoct_012_7bdd4832407bd3be.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7bdd4832407bd3be.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@13", "density": 0.7469920374756177, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/13.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "7\u20132\u221a2 = 4.17157+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.17157, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@13", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "7\u20132\u221a2 = 4.17157+", "metric_symbol": "s", "metric_value": 4.17157, "reference_text": "s = 7\u20132\u221a2 = 4.17157+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/13.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 7\u20132\u221a2 = 4.17157+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "867ad172957f0ddb", "improvement": -0.00013596034935225987, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.683174531654637e-06, "max_pair_overlap_depth": -2.596769201668181e-06, "metric_expression": "4.17171", "metric_symbol": "s", "metric_value": 4.171705960349352, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.17157, "setup": "octinoct", "side": 4.171705960349352, "solution_path": "data/solutions/generated/octinoct_013_867ad172957f0ddb.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/867ad172957f0ddb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@14", "density": 0.7637114154056861, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/14.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "4.28155+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.28155, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@14", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "4.28155+", "metric_symbol": "s", "metric_value": 4.28155, "reference_text": "s = 4.28155+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/14.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 4.28155+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "190d6c8f3f4e37d6", "improvement": 1.6200963535162316e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9636658184049817e-06, "max_pair_overlap_depth": -1.2788583332312697e-05, "metric_expression": "4.28153", "metric_symbol": "s", "metric_value": 4.281533799036465, "n": 14, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.28155, "setup": "octinoct", "side": 4.281533799036465, "solution_path": "data/solutions/generated/octinoct_014_190d6c8f3f4e37d6.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/190d6c8f3f4e37d6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@15", "density": 0.7554665958519668, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/15.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "4.45594+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.45594, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@15", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "4.45594+", "metric_symbol": "s", "metric_value": 4.45594, "reference_text": "s = 4.45594+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/15.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 4.45594+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "60843ec4ea56ad88", "improvement": 1.3726756218801484e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.050759161282626e-06, "max_pair_overlap_depth": -3.648956153723404e-05, "metric_expression": "4.45593", "metric_symbol": "s", "metric_value": 4.455926273243781, "n": 15, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.45594, "setup": "octinoct", "side": 4.455926273243781, "solution_path": "data/solutions/generated/octinoct_015_60843ec4ea56ad88.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/60843ec4ea56ad88.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@16", "density": 0.7566625545620705, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/16.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "4.59824+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.59824, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@16", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "4.59824+", "metric_symbol": "s", "metric_value": 4.59824, "reference_text": "s = 4.59824+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/16.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 4.59824+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "1178c6831eb46264", "improvement": -0.00018248651533525617, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.297640975948298e-06, "max_pair_overlap_depth": -9.126693151007004e-06, "metric_expression": "4.59842", "metric_symbol": "s", "metric_value": 4.598422486515335, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.59824, "setup": "octinoct", "side": 4.598422486515335, "solution_path": "data/solutions/generated/octinoct_016_1178c6831eb46264.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/1178c6831eb46264.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@17", "density": 0.7511087688524583, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/17.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "4.75741+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.75741, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@17", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "4.75741+", "metric_symbol": "s", "metric_value": 4.75741, "reference_text": "s = 4.75741+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/17.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 4.75741+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "ebcfc45448cb8c5f", "improvement": -2.6985802159096295e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.580862819791944e-06, "max_pair_overlap_depth": -2.6446396224955038e-05, "metric_expression": "4.75744", "metric_symbol": "s", "metric_value": 4.757436985802159, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.75741, "setup": "octinoct", "side": 4.757436985802159, "solution_path": "data/solutions/generated/octinoct_017_ebcfc45448cb8c5f.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/ebcfc45448cb8c5f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@18", "density": 0.7629999765077309, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/18.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "4.85717+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.85717, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@18", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "4.85717+", "metric_symbol": "s", "metric_value": 4.85717, "reference_text": "s = 4.85717+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/18.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 4.85717+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "46f5ac176d150ffa", "improvement": 0.00010416375427002578, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.225773906749903e-06, "max_pair_overlap_depth": -2.521618462036912e-05, "metric_expression": "4.85707", "metric_symbol": "s", "metric_value": 4.85706583624573, "n": 18, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.85717, "setup": "octinoct", "side": 4.85706583624573, "solution_path": "data/solutions/generated/octinoct_018_46f5ac176d150ffa.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/46f5ac176d150ffa.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@19", "density": 0.7795195170060211, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/19.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "4.93586+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.93586, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@19", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "4.93586+", "metric_symbol": "s", "metric_value": 4.93586, "reference_text": "s = 4.93586+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/19.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 4.93586+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "4a7169be4bda1825", "improvement": -0.0011420082689239663, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1928573301213418e-05, "max_pair_overlap_depth": -1.491417372267989e-06, "metric_expression": "4.93700", "metric_symbol": "s", "metric_value": 4.937002008268924, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.93586, "setup": "octinoct", "side": 4.937002008268924, "solution_path": "data/solutions/generated/octinoct_019_4a7169be4bda1825.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/4a7169be4bda1825.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@20", "density": 0.778654925537014, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/20.gif", "friedman_family_updated": "3/27/26", "friedman_metric_expression": "5.06781+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.06781, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@20", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.06781+", "metric_symbol": "s", "metric_value": 5.06781, "reference_text": "s = 5.06781+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "20.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/20.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.06781+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "ef9ffbe57f51321a", "improvement": -0.0002585475900360734, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.202696065204785e-06, "max_pair_overlap_depth": -3.692189855719619e-05, "metric_expression": "5.06807", "metric_symbol": "s", "metric_value": 5.068068547590036, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.06781, "setup": "octinoct", "side": 5.068068547590036, "solution_path": "data/solutions/generated/octinoct_020_ef9ffbe57f51321a.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/ef9ffbe57f51321a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@21", "density": 0.7774838310416734, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/21.gif", "friedman_metric_expression": "5.19715+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.19715, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@21", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.19715+", "metric_symbol": "s", "metric_value": 5.19715, "reference_text": "s = 5.19715+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/21.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.19715+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "92c4bcf9428983c2", "improvement": 1.540400287325383e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.1696825424876755e-06, "max_pair_overlap_depth": -2.317393259165801e-05, "metric_expression": "5.19713", "metric_symbol": "s", "metric_value": 5.1971345959971265, "n": 21, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.19715, "setup": "octinoct", "side": 5.1971345959971265, "solution_path": "data/solutions/generated/octinoct_021_92c4bcf9428983c2.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/92c4bcf9428983c2.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@22", "density": 0.7663269933124698, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/22.gif", "friedman_metric_expression": "5.35750+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.3575, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@22", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.35750+", "metric_symbol": "s", "metric_value": 5.3575, "reference_text": "s = 5.35750+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/22.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.35750+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "b8f2befaa2d4a75e", "improvement": -0.0005193617687213603, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.99203724676962e-06, "max_pair_overlap_depth": -1.57400449714018e-05, "metric_expression": "5.35802", "metric_symbol": "s", "metric_value": 5.358019361768721, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.3575, "setup": "octinoct", "side": 5.358019361768721, "solution_path": "data/solutions/generated/octinoct_022_b8f2befaa2d4a75e.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/b8f2befaa2d4a75e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@23", "density": 0.7845835529808874, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/23.gif", "friedman_metric_expression": "5.41440+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.4144, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@23", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.41440+", "metric_symbol": "s", "metric_value": 5.4144, "reference_text": "s = 5.41440+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/23.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.41440+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "d6528e8801e36ce0", "improvement": 7.503853147827755e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3646686331794102e-06, "max_pair_overlap_depth": -4.536567227386712e-05, "metric_expression": "5.41432", "metric_symbol": "s", "metric_value": 5.414324961468521, "n": 23, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.4144, "setup": "octinoct", "side": 5.414324961468521, "solution_path": "data/solutions/generated/octinoct_023_d6528e8801e36ce0.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/d6528e8801e36ce0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@24", "density": 0.777859618157, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/24.gif", "friedman_metric_expression": "5.55464+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.55464, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@24", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.55464+", "metric_symbol": "s", "metric_value": 5.55464, "reference_text": "s = 5.55464+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/24.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.55464+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "a34ac6a3c52f70df", "improvement": 1.1631965286973411e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.721783559078116e-06, "max_pair_overlap_depth": -2.4974900467711336e-05, "metric_expression": "5.55463", "metric_symbol": "s", "metric_value": 5.554628368034713, "n": 24, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.55464, "setup": "octinoct", "side": 5.554628368034713, "solution_path": "data/solutions/generated/octinoct_024_a34ac6a3c52f70df.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/a34ac6a3c52f70df.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@25", "density": 0.7805673327264805, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/25.gif", "friedman_metric_expression": "5.65593+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.65593, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@25", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.65593+", "metric_symbol": "s", "metric_value": 5.65593, "reference_text": "s = 5.65593+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/25.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.65593+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "113492960b7e7dce", "improvement": -0.003397389940226425, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.281212189125961e-06, "max_pair_overlap_depth": -2.6024797291412938e-05, "metric_expression": "5.65933", "metric_symbol": "s", "metric_value": 5.659327389940226, "n": 25, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.65593, "setup": "octinoct", "side": 5.659327389940226, "solution_path": "data/solutions/generated/octinoct_025_113492960b7e7dce.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/113492960b7e7dce.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@26", "density": 0.7881657239941322, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/26.gif", "friedman_metric_expression": "5.73940+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.7394, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@26", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.73940+", "metric_symbol": "s", "metric_value": 5.7394, "reference_text": "s = 5.73940+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/26.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.73940+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "581188c4895e4037", "improvement": -0.004116881359766289, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.43705423351296e-06, "max_pair_overlap_depth": -2.686615288460814e-05, "metric_expression": "5.74352", "metric_symbol": "s", "metric_value": 5.743516881359766, "n": 26, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.7394, "setup": "octinoct", "side": 5.743516881359766, "solution_path": "data/solutions/generated/octinoct_026_581188c4895e4037.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/581188c4895e4037.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@27", "density": 0.7989711846148311, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/27.gif", "friedman_metric_expression": "5.81238+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.81238, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@27", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "5.81238+", "metric_symbol": "s", "metric_value": 5.81238, "reference_text": "s = 5.81238+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/27.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 5.81238+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "25c7f85bc0b15209", "improvement": -0.0008341744667159645, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.851030556516347e-06, "max_pair_overlap_depth": -2.4661009629944886e-05, "metric_expression": "5.81321", "metric_symbol": "s", "metric_value": 5.813214174466716, "n": 27, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.81238, "setup": "octinoct", "side": 5.813214174466716, "solution_path": "data/solutions/generated/octinoct_027_25c7f85bc0b15209.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/25c7f85bc0b15209.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@28", "density": 0.8242204012359605, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/28.gif", "friedman_metric_expression": "3+2\u221a2 = 5.82842+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.82842, "friedman_reference": {"analytical_or_proved": true, "case": "octinoct@28", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3+2\u221a2 = 5.82842+", "metric_symbol": "s", "metric_value": 5.82842, "reference_text": "s = 3+2\u221a2 = 5.82842+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/28.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 3+2\u221a2 = 5.82842+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "f8d9c8a9f9a38d45", "improvement": -8.71778893589692e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.739062384519148e-06, "max_pair_overlap_depth": -9.673739088000843e-06, "metric_expression": "5.82851", "metric_symbol": "s", "metric_value": 5.828507177889359, "n": 28, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.82842, "setup": "octinoct", "side": 5.828507177889359, "solution_path": "data/solutions/generated/octinoct_028_f8d9c8a9f9a38d45.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/f8d9c8a9f9a38d45.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@29", "density": 0.7902111886471865, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/29.gif", "friedman_metric_expression": "6.05746+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.05746, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@29", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "6.05746+", "metric_symbol": "s", "metric_value": 6.05746, "reference_text": "s = 6.05746+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/29.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 6.05746+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "30789c6bb8fd99b7", "improvement": -0.0005140918720876186, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.805265150873157e-06, "max_pair_overlap_depth": -5.6301265520808386e-05, "metric_expression": "6.05797", "metric_symbol": "s", "metric_value": 6.0579740918720875, "n": 29, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.05746, "setup": "octinoct", "side": 6.0579740918720875, "solution_path": "data/solutions/generated/octinoct_029_30789c6bb8fd99b7.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/30789c6bb8fd99b7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "octinoct@30", "density": 0.7919434557655346, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/octinoct/30.gif", "friedman_metric_expression": "6.15432+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.15432, "friedman_reference": {"analytical_or_proved": false, "case": "octinoct@30", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "6.15432+", "metric_symbol": "s", "metric_value": 6.15432, "reference_text": "s = 6.15432+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/octinoct/30.gif", "source_page": "https://erich-friedman.github.io/packing/octinoct/", "title": "Octagons in Octagons"}, "friedman_reference_text": "s = 6.15432+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/octinoct/", "friedman_title": "Octagons in Octagons", "frontend_seed": true, "id": "1a16dc8a4d741008", "improvement": -0.00047426130353134255, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1243912272535679e-05, "max_pair_overlap_depth": -2.8046648627544357e-05, "metric_expression": "6.15479", "metric_symbol": "s", "metric_value": 6.154794261303532, "n": 30, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.15432, "setup": "octinoct", "side": 6.154794261303532, "solution_path": "data/solutions/generated/octinoct_030_1a16dc8a4d741008.json", "source_url": "https://erich-friedman.github.io/packing/octinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/1a16dc8a4d741008.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@1", "density": 0.7568185573097284, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/1.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a((5+\u221a5)/10) = .86065+", "friedman_metric_symbol": "r", "friedman_metric_value": 0.86065, "friedman_reference": {"analytical_or_proved": true, "case": "penincir@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a((5+\u221a5)/10) = .86065+", "metric_symbol": "r", "metric_value": 0.86065, "reference_text": "r = \u221a((5+\u221a5)/10) = .86065+ | Trivial.", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/1.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = \u221a((5+\u221a5)/10) = .86065+ | Trivial.", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": false, "id": "ca75a5cd7cad481c", "improvement": 0.009994599443903796, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0926430426282394e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.85066", "metric_symbol": "r", "metric_value": 0.8506554005560962, "n": 1, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 0.86065, "setup": "penincir", "side": 1.7013108011121925, "solution_path": "data/solutions/generated/penincir_001_ca75a5cd7cad481c.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/ca75a5cd7cad481c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@2", "density": 0.462509898701161, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/2.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a(5+2\u221a5)/2 = 1.53884+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.53884, "friedman_reference": {"analytical_or_proved": true, "case": "penincir@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a(5+2\u221a5)/2 = 1.53884+", "metric_symbol": "r", "metric_value": 1.53884, "reference_text": "r = \u221a(5+2\u221a5)/2 = 1.53884+ | Trivial.", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/2.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = \u221a(5+2\u221a5)/2 = 1.53884+ | Trivial.", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "44da10d1d5897aae", "improvement": -3.743542237466535e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.684702690331392e-08, "max_pair_overlap_depth": -5.994543612077171e-05, "metric_expression": "1.53888", "metric_symbol": "r", "metric_value": 1.5388774354223747, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.53884, "setup": "penincir", "side": 3.0777548708447493, "solution_path": "data/solutions/generated/penincir_002_44da10d1d5897aae.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/44da10d1d5897aae.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@3", "density": 0.6275054691586099, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/3.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "(1+\u221a5)/2 = 1.61803+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.61803, "friedman_reference": {"analytical_or_proved": true, "case": "penincir@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(1+\u221a5)/2 = 1.61803+", "metric_symbol": "r", "metric_value": 1.61803, "reference_text": "r = (1+\u221a5)/2 = 1.61803+ | Trivial.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/3.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = (1+\u221a5)/2 = 1.61803+ | Trivial.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "350d35e2110a508b", "improvement": -5.532631868709892e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.919906156253262e-06, "max_pair_overlap_depth": -5.073652219536114e-05, "metric_expression": "1.61809", "metric_symbol": "r", "metric_value": 1.6180853263186872, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.61803, "setup": "penincir", "side": 3.2361706526373744, "solution_path": "data/solutions/generated/penincir_003_350d35e2110a508b.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/350d35e2110a508b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@4", "density": 0.6542873961171766, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/4.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "1.82971+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.82971, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@4", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.82971+", "metric_symbol": "r", "metric_value": 1.82971, "reference_text": "r = 1.82971+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/4.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 1.82971+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "d03dba2c6efc98dd", "improvement": -5.479863407176211e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.02075710251404e-06, "max_pair_overlap_depth": -3.1570603938990693e-05, "metric_expression": "1.82976", "metric_symbol": "r", "metric_value": 1.8297647986340717, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.82971, "setup": "penincir", "side": 3.6595295972681434, "solution_path": "data/solutions/generated/penincir_004_d03dba2c6efc98dd.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/d03dba2c6efc98dd.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@5", "density": 0.6626607161574516, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/5.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.03271+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.03271, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@5", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.03271+", "metric_symbol": "r", "metric_value": 2.03271, "reference_text": "r = 2.03271+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/5.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 2.03271+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "97027d267cb23b7f", "improvement": -6.326016348978669e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.209947377376011e-06, "max_pair_overlap_depth": -3.7991648266566536e-05, "metric_expression": "2.03277", "metric_symbol": "r", "metric_value": 2.0327732601634896, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.03271, "setup": "penincir", "side": 4.065546520326979, "solution_path": "data/solutions/generated/penincir_005_97027d267cb23b7f.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/97027d267cb23b7f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@6", "density": 0.6624931586206462, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/6.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a((25+11\u221a5)/10) = 2.22703+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.22703, "friedman_reference": {"analytical_or_proved": true, "case": "penincir@6", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "\u221a((25+11\u221a5)/10) = 2.22703+", "metric_symbol": "r", "metric_value": 2.22703, "reference_text": "r = \u221a((25+11\u221a5)/10) = 2.22703+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/6.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = \u221a((25+11\u221a5)/10) = 2.22703+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "4b5f96ff666b1595", "improvement": -4.3119632947785647e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3229349565289112e-06, "max_pair_overlap_depth": -2.759876285962637e-05, "metric_expression": "2.22707", "metric_symbol": "r", "metric_value": 2.227073119632948, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.22703, "setup": "penincir", "side": 4.454146239265896, "solution_path": "data/solutions/generated/penincir_006_4b5f96ff666b1595.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/4b5f96ff666b1595.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@7", "density": 0.7279626259462092, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/7.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.29474+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.29474, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@7", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.29474+", "metric_symbol": "r", "metric_value": 2.29474, "reference_text": "r = 2.29474+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/7.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 2.29474+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "ef28e611d9e22eb0", "improvement": -5.570482584049685e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.717800656082403e-06, "max_pair_overlap_depth": -4.1591260455220436e-05, "metric_expression": "2.29480", "metric_symbol": "r", "metric_value": 2.2947957048258405, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.29474, "setup": "penincir", "side": 4.589591409651681, "solution_path": "data/solutions/generated/penincir_007_ef28e611d9e22eb0.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/ef28e611d9e22eb0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@8", "density": 0.7115852708183573, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/8.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.48128+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.48128, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@8", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.48128+", "metric_symbol": "r", "metric_value": 2.48128, "reference_text": "r = 2.48128+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/8.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 2.48128+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "1a134d3903b2814e", "improvement": -3.0242757913079288e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.375187333127144e-07, "max_pair_overlap_depth": -8.34480700673268e-06, "metric_expression": "2.48131", "metric_symbol": "r", "metric_value": 2.481310242757913, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.48128, "setup": "penincir", "side": 4.962620485515826, "solution_path": "data/solutions/generated/penincir_008_1a134d3903b2814e.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/1a134d3903b2814e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@9", "density": 0.674675474238072, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/9.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.70171+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.70171, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@9", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.70171+", "metric_symbol": "r", "metric_value": 2.70171, "reference_text": "r = 2.70171+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/9.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 2.70171+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "1ca74640a9bafbff", "improvement": -0.0011487048015377255, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.541032137428402e-06, "max_pair_overlap_depth": -3.412963059545859e-05, "metric_expression": "2.70286", "metric_symbol": "r", "metric_value": 2.7028587048015376, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.70171, "setup": "penincir", "side": 5.405717409603075, "solution_path": "data/solutions/generated/penincir_009_1ca74640a9bafbff.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/1ca74640a9bafbff.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@10", "density": 0.6983515169709827, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/10.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.79768+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.79768, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@10", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.79768+", "metric_symbol": "r", "metric_value": 2.79768, "reference_text": "r = 2.79768+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/10.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 2.79768+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "b50c304b68607cc4", "improvement": -0.002671254854186955, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.377527520027513e-06, "max_pair_overlap_depth": -1.3481998292097508e-05, "metric_expression": "2.80035", "metric_symbol": "r", "metric_value": 2.800351254854187, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.79768, "setup": "penincir", "side": 5.600702509708374, "solution_path": "data/solutions/generated/penincir_010_b50c304b68607cc4.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b50c304b68607cc4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@11", "density": 0.7152777059461288, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/11.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.90180+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.9018, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.90180+", "metric_symbol": "r", "metric_value": 2.9018, "reference_text": "r = 2.90180+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/11.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 2.90180+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "37f9123e1b2153ff", "improvement": -0.0002744381126040807, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.569993058365583e-07, "max_pair_overlap_depth": -3.207817623795961e-05, "metric_expression": "2.90207", "metric_symbol": "r", "metric_value": 2.9020744381126042, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.9018, "setup": "penincir", "side": 5.8041488762252085, "solution_path": "data/solutions/generated/penincir_011_37f9123e1b2153ff.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/37f9123e1b2153ff.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@12", "density": 0.7213016172544706, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/12.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.01826+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.01826, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@12", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.01826+", "metric_symbol": "r", "metric_value": 3.01826, "reference_text": "r = 3.01826+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/12.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.01826+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "a564353850abf267", "improvement": -0.00017424806894128508, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.481723353017998e-06, "max_pair_overlap_depth": -2.751801632361328e-05, "metric_expression": "3.01843", "metric_symbol": "r", "metric_value": 3.0184342480689415, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.01826, "setup": "penincir", "side": 6.036868496137883, "solution_path": "data/solutions/generated/penincir_012_a564353850abf267.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/a564353850abf267.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@13", "density": 0.7515432650569983, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/13.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a(5+2\u221a5)/2 =3.07769+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.07769, "friedman_reference": {"analytical_or_proved": true, "case": "penincir@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "\u221a(5+2\u221a5)/2 =3.07769+", "metric_symbol": "r", "metric_value": 3.07769, "reference_text": "r = \u221a(5+2\u221a5)/2 =3.07769+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/13.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = \u221a(5+2\u221a5)/2 =3.07769+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "c39e9475d5bde4a6", "improvement": -0.0001371927275344298, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.94574749318366e-06, "max_pair_overlap_depth": -1.7086792766463077e-05, "metric_expression": "3.07783", "metric_symbol": "r", "metric_value": 3.0778271927275345, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.07769, "setup": "penincir", "side": 6.155654385455069, "solution_path": "data/solutions/generated/penincir_013_c39e9475d5bde4a6.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/c39e9475d5bde4a6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@14", "density": 0.7320837719347534, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/14.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.23385+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.23385, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@14", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.23385+", "metric_symbol": "r", "metric_value": 3.23385, "reference_text": "r = 3.23385+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/14.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.23385+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "4798ec354812edfb", "improvement": -0.0023337860780814346, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.2470094596680497e-07, "max_pair_overlap_depth": -1.086583313658629e-05, "metric_expression": "3.23618", "metric_symbol": "r", "metric_value": 3.2361837860780813, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.23385, "setup": "penincir", "side": 6.472367572156163, "solution_path": "data/solutions/generated/penincir_014_4798ec354812edfb.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/4798ec354812edfb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@15", "density": 0.7364932523686155, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/15.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.33911+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.33911, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@15", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.33911+", "metric_symbol": "r", "metric_value": 3.33911, "reference_text": "r = 3.33911+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/15.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.33911+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "5fb8f24380ed14ba", "improvement": -0.0006156585665682002, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.531307800791922e-07, "max_pair_overlap_depth": -1.652091133119704e-05, "metric_expression": "3.33973", "metric_symbol": "r", "metric_value": 3.339725658566568, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.33911, "setup": "penincir", "side": 6.679451317133136, "solution_path": "data/solutions/generated/penincir_015_5fb8f24380ed14ba.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/5fb8f24380ed14ba.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@16", "density": 0.7415495303728211, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/16.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.43657+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.43657, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@16", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.43657+", "metric_symbol": "r", "metric_value": 3.43657, "reference_text": "r = 3.43657+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/16.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.43657+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "eb41133064f5f2da", "improvement": -0.0009042932667306935, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3274146969676224e-06, "max_pair_overlap_depth": -1.0162839398697443e-05, "metric_expression": "3.43747", "metric_symbol": "r", "metric_value": 3.437474293266731, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.43657, "setup": "penincir", "side": 6.874948586533462, "solution_path": "data/solutions/generated/penincir_016_eb41133064f5f2da.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/eb41133064f5f2da.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@17", "density": 0.7557819084683968, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/17.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.50958+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.50958, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.50958+", "metric_symbol": "r", "metric_value": 3.50958, "reference_text": "r = 3.50958+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/17.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.50958+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "9469ef74fc89a75b", "improvement": -0.00016662106909848262, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3042585254605683e-06, "max_pair_overlap_depth": -2.7118152433835974e-05, "metric_expression": "3.50975", "metric_symbol": "r", "metric_value": 3.5097466210690986, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.50958, "setup": "penincir", "side": 7.019493242138197, "solution_path": "data/solutions/generated/penincir_017_9469ef74fc89a75b.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/9469ef74fc89a75b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@18", "density": 0.7533573813986445, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/18.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.62080+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.6208, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@18", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.62080+", "metric_symbol": "r", "metric_value": 3.6208, "reference_text": "r = 3.62080+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/18.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.62080+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "767a627015948f32", "improvement": 0.00349375267338381, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7774333678797234e-06, "max_pair_overlap_depth": -3.109065453799076e-05, "metric_expression": "3.61731", "metric_symbol": "r", "metric_value": 3.617306247326616, "n": 18, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.6208, "setup": "penincir", "side": 7.234612494653232, "solution_path": "data/solutions/generated/penincir_018_767a627015948f32.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/767a627015948f32.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@19", "density": 0.771366675320165, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/19.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.67287+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.67287, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@19", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.67287+", "metric_symbol": "r", "metric_value": 3.67287, "reference_text": "r = 3.67287+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/19.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.67287+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "9d482c8012998879", "improvement": 8.154982387598153e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.817340790579493e-06, "max_pair_overlap_depth": -1.6517110137281854e-05, "metric_expression": "3.67279", "metric_symbol": "r", "metric_value": 3.672788450176124, "n": 19, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.67287, "setup": "penincir", "side": 7.345576900352248, "solution_path": "data/solutions/generated/penincir_019_9d482c8012998879.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/9d482c8012998879.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@20", "density": 0.7575056542556782, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/20.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.80415+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.80415, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@20", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.80415+", "metric_symbol": "r", "metric_value": 3.80415, "reference_text": "r = 3.80415+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/20.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.80415+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "5271226652403829", "improvement": 0.0016291138375779113, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.231424091092009e-06, "max_pair_overlap_depth": -3.071782635011289e-05, "metric_expression": "3.80252", "metric_symbol": "r", "metric_value": 3.802520886162422, "n": 20, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.80415, "setup": "penincir", "side": 7.605041772324844, "solution_path": "data/solutions/generated/penincir_020_5271226652403829.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/5271226652403829.svg", "tolerance": 1e-08, "verified": true}
+{"case": "penincir@21", "density": 0.7521173669587212, "friedman_analytical_or_proved": false, "friedman_credit": "Cyril Summerfield", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/penincir/21.png", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.90735+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.90735, "friedman_reference": {"analytical_or_proved": false, "case": "penincir@21", "credit": "Cyril Summerfield", "date": "April 2026", "family_updated": "", "metric_expression": "3.90735+", "metric_symbol": "r", "metric_value": 3.90735, "reference_text": "r = 3.90735+ | Found by Cyril Summerfield | in April 2026.", "source_image": "21.png", "source_image_url": "https://erich-friedman.github.io/packing/penincir/21.png", "source_page": "https://erich-friedman.github.io/packing/penincir/", "title": "Pentagons in Circles"}, "friedman_reference_text": "r = 3.90735+ | Found by Cyril Summerfield | in April 2026.", "friedman_source_image": "21.png", "friedman_source_url": "https://erich-friedman.github.io/packing/penincir/", "friedman_title": "Pentagons in Circles", "frontend_seed": true, "id": "4625f732ad11a859", "improvement": -0.0030068209876064067, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9813425364011437e-06, "max_pair_overlap_depth": -1.4218243721247914e-05, "metric_expression": "3.91036", "metric_symbol": "r", "metric_value": 3.9103568209876065, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.90735, "setup": "penincir", "side": 7.820713641975213, "solution_path": "data/solutions/generated/penincir_021_4625f732ad11a859.json", "source_url": "https://erich-friedman.github.io/packing/penincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Cyril Summerfield", "svg_path": "data/svg/4625f732ad11a859.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@1", "density": 0.9999780359352667, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/1.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "peninpen@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial.", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/1.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "b3173648630fcdeb", "improvement": -1.0982213277443975e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.256503348572871e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00001", "metric_symbol": "s", "metric_value": 1.0000109822132774, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.0, "setup": "peninpen", "side": 1.0000109822132774, "solution_path": "data/solutions/generated/peninpen_001_b3173648630fcdeb.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b3173648630fcdeb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@2", "density": 0.4999891232484648, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/2.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "peninpen@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/2.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "edac094186a54129", "improvement": -2.175385798786067e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0537211348702708e-06, "max_pair_overlap_depth": -1.9731507571563167e-05, "metric_expression": "2.00002", "metric_symbol": "s", "metric_value": 2.000021753857988, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "peninpen", "side": 2.000021753857988, "solution_path": "data/solutions/generated/peninpen_002_edac094186a54129.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/edac094186a54129.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@3", "density": 0.599978840427638, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/3.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a5 = 2.23606+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.23606, "friedman_reference": {"analytical_or_proved": true, "case": "peninpen@3", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "\u221a5 = 2.23606+", "metric_symbol": "s", "metric_value": 2.23606, "reference_text": "s = \u221a5 = 2.23606+ | Found by Jonathan Viquerat in April 2026.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/3.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = \u221a5 = 2.23606+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "bf261a4dce178a90", "improvement": -4.7407077830463606e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7662857614262655e-06, "max_pair_overlap_depth": -2.7755970794667603e-05, "metric_expression": "2.23611", "metric_symbol": "s", "metric_value": 2.2361074070778306, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.23606, "setup": "peninpen", "side": 2.2361074070778306, "solution_path": "data/solutions/generated/peninpen_003_bf261a4dce178a90.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/bf261a4dce178a90.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@4", "density": 0.6678717661629443, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/4.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.44719+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.44719, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@4", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.44719+", "metric_symbol": "s", "metric_value": 2.44719, "reference_text": "s = 2.44719+ | Found by Jonathan Viquerat in April 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/4.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 2.44719+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "85435733af14070a", "improvement": -8.88306059834143e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0634336001709255e-06, "max_pair_overlap_depth": -3.575601486860425e-05, "metric_expression": "2.44728", "metric_symbol": "s", "metric_value": 2.4472788306059834, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.44719, "setup": "peninpen", "side": 2.4472788306059834, "solution_path": "data/solutions/generated/peninpen_004_85435733af14070a.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/85435733af14070a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@5", "density": 0.7294617516267693, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/6.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(3+\u221a5)/2 = 2.61803+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.61803, "friedman_reference": {"analytical_or_proved": true, "case": "peninpen@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(3+\u221a5)/2 = 2.61803+", "metric_symbol": "s", "metric_value": 2.61803, "reference_text": "s = (3+\u221a5)/2 = 2.61803+ | Trivial.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/6.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = (3+\u221a5)/2 = 2.61803+ | Trivial.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "28fa9f787595145b", "improvement": -5.4982701188421856e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.513169742800457e-06, "max_pair_overlap_depth": -2.4501936757292953e-06, "metric_expression": "2.61808", "metric_symbol": "s", "metric_value": 2.6180849827011885, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.61803, "setup": "peninpen", "side": 2.6180849827011885, "solution_path": "data/solutions/generated/peninpen_005_28fa9f787595145b.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/28fa9f787595145b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@6", "density": 0.8753627744848066, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/6.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(3+\u221a5)/2 = 2.61803+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.61803, "friedman_reference": {"analytical_or_proved": true, "case": "peninpen@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(3+\u221a5)/2 = 2.61803+", "metric_symbol": "s", "metric_value": 2.61803, "reference_text": "s = (3+\u221a5)/2 = 2.61803+ | Trivial.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/6.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = (3+\u221a5)/2 = 2.61803+ | Trivial.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "198bd12691483df1", "improvement": -4.201351602306502e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4414928963119693e-06, "max_pair_overlap_depth": -1.898562579671914e-05, "metric_expression": "2.61807", "metric_symbol": "s", "metric_value": 2.618072013516023, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.61803, "setup": "peninpen", "side": 2.618072013516023, "solution_path": "data/solutions/generated/peninpen_006_198bd12691483df1.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/198bd12691483df1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@7", "density": 0.7384663216832239, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/7.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.07866+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.07866, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@7", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.07866+", "metric_symbol": "s", "metric_value": 3.07866, "reference_text": "s = 3.07866+ | Found by Jonathan Viquerat in April 2026.", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/7.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 3.07866+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "5b6283ab3dc4c3b5", "improvement": -0.00015556249375014986, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.964189962454e-06, "max_pair_overlap_depth": -1.5079265303863387e-05, "metric_expression": "3.07882", "metric_symbol": "s", "metric_value": 3.0788155624937503, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.07866, "setup": "peninpen", "side": 3.0788155624937503, "solution_path": "data/solutions/generated/peninpen_007_5b6283ab3dc4c3b5.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/5b6283ab3dc4c3b5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@8", "density": 0.7191427449408644, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/8.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.33514+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.33514, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@8", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.33514+", "metric_symbol": "s", "metric_value": 3.33514, "reference_text": "s = 3.33514+ | Found by Jonathan Viquerat in April 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/8.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 3.33514+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "de5f4ff90a59b8bf", "improvement": -0.00017949381007964504, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.630985964589326e-06, "max_pair_overlap_depth": -3.1607294870367175e-05, "metric_expression": "3.33532", "metric_symbol": "s", "metric_value": 3.3353194938100796, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.33514, "setup": "peninpen", "side": 3.3353194938100796, "solution_path": "data/solutions/generated/peninpen_008_de5f4ff90a59b8bf.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/de5f4ff90a59b8bf.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@9", "density": 0.756208451025979, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/9.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.44956+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.44956, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@9", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.44956+", "metric_symbol": "s", "metric_value": 3.44956, "reference_text": "s = 3.44956+ | Found by Jonathan Viquerat in April 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/9.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 3.44956+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "291eb24a57cfcd61", "improvement": -0.00029221756483766015, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.66255661877786e-06, "max_pair_overlap_depth": -4.9483298464991776e-05, "metric_expression": "3.44985", "metric_symbol": "s", "metric_value": 3.4498522175648376, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.44956, "setup": "peninpen", "side": 3.4498522175648376, "solution_path": "data/solutions/generated/peninpen_009_291eb24a57cfcd61.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/291eb24a57cfcd61.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@10", "density": 0.7370589618267208, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/10.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.68324+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.68324, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@10", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.68324+", "metric_symbol": "s", "metric_value": 3.68324, "reference_text": "s = 3.68324+ | Found by Jonathan Viquerat in April 2026.", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/10.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 3.68324+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "90badee8fc213075", "improvement": -0.0001600058005193361, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0386948383243063e-06, "max_pair_overlap_depth": -3.744974033881299e-05, "metric_expression": "3.68340", "metric_symbol": "s", "metric_value": 3.6834000058005194, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.68324, "setup": "peninpen", "side": 3.6834000058005194, "solution_path": "data/solutions/generated/peninpen_010_90badee8fc213075.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/90badee8fc213075.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@11", "density": 0.7628515256681525, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/11.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.79704+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.79704, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.79704+", "metric_symbol": "s", "metric_value": 3.79704, "reference_text": "s = 3.79704+ | Found by Jonathan Viquerat in April 2026.", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/11.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 3.79704+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "e62e9f2d5556f0f5", "improvement": -0.0002724485549276956, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.930866570871943e-06, "max_pair_overlap_depth": -3.185052846443881e-05, "metric_expression": "3.79731", "metric_symbol": "s", "metric_value": 3.7973124485549277, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.79704, "setup": "peninpen", "side": 3.7973124485549277, "solution_path": "data/solutions/generated/peninpen_011_e62e9f2d5556f0f5.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/e62e9f2d5556f0f5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@12", "density": 0.7615744792401016, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/12.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.96926+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.96926, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@12", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.96926+", "metric_symbol": "s", "metric_value": 3.96926, "reference_text": "s = 3.96926+ | Found by Jonathan Viquerat in April 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/12.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 3.96926+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "6f961579a76d6be8", "improvement": -0.00022743831780980273, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.8385847579674985e-06, "max_pair_overlap_depth": -3.1875868486519465e-05, "metric_expression": "3.96949", "metric_symbol": "s", "metric_value": 3.9694874383178096, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.96926, "setup": "peninpen", "side": 3.9694874383178096, "solution_path": "data/solutions/generated/peninpen_012_6f961579a76d6be8.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/6f961579a76d6be8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@13", "density": 0.7865963698125578, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/13.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "4.06523+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.06523, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.06523+", "metric_symbol": "s", "metric_value": 4.06523, "reference_text": "s = 4.06523+ | Found by Jonathan Viquerat in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/13.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 4.06523+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "ce66c878ffee73fc", "improvement": -9.908623501608815e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.805365438846161e-06, "max_pair_overlap_depth": -6.344819103887289e-06, "metric_expression": "4.06533", "metric_symbol": "s", "metric_value": 4.065329086235016, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.06523, "setup": "peninpen", "side": 4.065329086235016, "solution_path": "data/solutions/generated/peninpen_013_ce66c878ffee73fc.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/ce66c878ffee73fc.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@14", "density": 0.7892057892472966, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/14.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "4.21120+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.2112, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@14", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.21120+", "metric_symbol": "s", "metric_value": 4.2112, "reference_text": "s = 4.21120+ | Found by Jonathan Viquerat in April 2026.", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/14.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 4.21120+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "367de42acefe1aa2", "improvement": -0.0006111178171677167, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.1042346422399235e-06, "max_pair_overlap_depth": -2.557915411749434e-05, "metric_expression": "4.21181", "metric_symbol": "s", "metric_value": 4.2118111178171675, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.2112, "setup": "peninpen", "side": 4.2118111178171675, "solution_path": "data/solutions/generated/peninpen_014_367de42acefe1aa2.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/367de42acefe1aa2.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@15", "density": 0.8358892308226911, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/15.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2+\u221a5 = 4.23606+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.23606, "friedman_reference": {"analytical_or_proved": true, "case": "peninpen@15", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2+\u221a5 = 4.23606+", "metric_symbol": "s", "metric_value": 4.23606, "reference_text": "s = 2+\u221a5 = 4.23606+ | Trivial.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/15.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 2+\u221a5 = 4.23606+ | Trivial.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "251c50a18682e896", "improvement": -8.936254454106063e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.6294493786472997e-06, "max_pair_overlap_depth": -5.5218241392651635e-06, "metric_expression": "4.23615", "metric_symbol": "s", "metric_value": 4.236149362544541, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.23606, "setup": "peninpen", "side": 4.236149362544541, "solution_path": "data/solutions/generated/peninpen_015_251c50a18682e896.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/251c50a18682e896.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@16", "density": 0.8202808577467057, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/16.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "4.41636+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.41636, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@16", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.41636+", "metric_symbol": "s", "metric_value": 4.41636, "reference_text": "s = 4.41636+ | Found by Jonathan Viquerat in April 2026.", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/16.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 4.41636+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "e2bb776d24739328", "improvement": -0.00014476044476463557, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7250867795500517e-06, "max_pair_overlap_depth": -1.097689597839846e-05, "metric_expression": "4.41650", "metric_symbol": "s", "metric_value": 4.416504760444765, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.41636, "setup": "peninpen", "side": 4.416504760444765, "solution_path": "data/solutions/generated/peninpen_016_e2bb776d24739328.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/e2bb776d24739328.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@17", "density": 0.7834157819221014, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/17.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "4.65561+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.65561, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.65561+", "metric_symbol": "s", "metric_value": 4.65561, "reference_text": "s = 4.65561+ | Found by Jonathan Viquerat in April 2026.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/17.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 4.65561+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "54856cb865a77d57", "improvement": -0.0026991003814726966, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9034050721792276e-06, "max_pair_overlap_depth": -2.8148707974851028e-05, "metric_expression": "4.65831", "metric_symbol": "s", "metric_value": 4.658309100381473, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.65561, "setup": "peninpen", "side": 4.658309100381473, "solution_path": "data/solutions/generated/peninpen_017_54856cb865a77d57.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/54856cb865a77d57.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@18", "density": 0.7892194761771266, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/18.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "4.77541+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.77541, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@18", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.77541+", "metric_symbol": "s", "metric_value": 4.77541, "reference_text": "s = 4.77541+ | Found by Jonathan Viquerat in April 2026.", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/18.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 4.77541+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "9c442356ca78c3de", "improvement": -0.0002934971525236918, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.1318004154280254e-06, "max_pair_overlap_depth": -1.258487199295999e-05, "metric_expression": "4.77570", "metric_symbol": "s", "metric_value": 4.775703497152524, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.77541, "setup": "peninpen", "side": 4.775703497152524, "solution_path": "data/solutions/generated/peninpen_018_9c442356ca78c3de.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/9c442356ca78c3de.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@19", "density": 0.7735353200149236, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/19.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "4.95269+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.95269, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@19", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "4.95269+", "metric_symbol": "s", "metric_value": 4.95269, "reference_text": "s = 4.95269+ | Found by Ian Watson in April 2026.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/19.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 4.95269+ | Found by Ian Watson in April 2026.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "b5440311f7b9b10b", "improvement": -0.0033719480383975764, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.8067945532274755e-06, "max_pair_overlap_depth": -5.844604614307913e-06, "metric_expression": "4.95606", "metric_symbol": "s", "metric_value": 4.956061948038397, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.95269, "setup": "peninpen", "side": 4.956061948038397, "solution_path": "data/solutions/generated/peninpen_019_b5440311f7b9b10b.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/b5440311f7b9b10b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@20", "density": 0.7890584950748962, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/20.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "5.04154+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.04154, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@20", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.04154+", "metric_symbol": "s", "metric_value": 5.04154, "reference_text": "s = 5.04154+ | Found by Jonathan Viquerat in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/20.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 5.04154+ | Found by Jonathan Viquerat in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "74f61b73dca76fef", "improvement": 0.0069930192193110585, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.063547145356236e-06, "max_pair_overlap_depth": -1.8870971417950244e-05, "metric_expression": "5.03455", "metric_symbol": "s", "metric_value": 5.034546980780689, "n": 20, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.04154, "setup": "peninpen", "side": 5.034546980780689, "solution_path": "data/solutions/generated/peninpen_020_74f61b73dca76fef.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/74f61b73dca76fef.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@21", "density": 0.7887631772097712, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/21.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "5.15948+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.15948, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@21", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "5.15948+", "metric_symbol": "s", "metric_value": 5.15948, "reference_text": "s = 5.15948+ | Found by Ignacio Vallejo in April 2026.", "source_image": "21.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/21.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 5.15948+ | Found by Ignacio Vallejo in April 2026.", "friedman_source_image": "21.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "b731ea055eb53ce1", "improvement": -0.00036117133928748046, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7271978645132776e-06, "max_pair_overlap_depth": -5.857494896677196e-06, "metric_expression": "5.15984", "metric_symbol": "s", "metric_value": 5.159841171339288, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.15948, "setup": "peninpen", "side": 5.159841171339288, "solution_path": "data/solutions/generated/peninpen_021_b731ea055eb53ce1.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/b731ea055eb53ce1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@22", "density": 0.7927891464448104, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/22.png", "friedman_metric_expression": "5.26740+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.2674, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@22", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "5.26740+", "metric_symbol": "s", "metric_value": 5.2674, "reference_text": "s = 5.26740+ | Found by Ignacio Vallejo in April 2026.", "source_image": "22.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/22.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 5.26740+ | Found by Ignacio Vallejo in April 2026.", "friedman_source_image": "22.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "07e217f200bb677e", "improvement": -0.0004389908125848052, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.647717325013132e-07, "max_pair_overlap_depth": -3.201222129578518e-05, "metric_expression": "5.26784", "metric_symbol": "s", "metric_value": 5.267838990812585, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.2674, "setup": "peninpen", "side": 5.267838990812585, "solution_path": "data/solutions/generated/peninpen_022_07e217f200bb677e.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/07e217f200bb677e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@23", "density": 0.8042579677659262, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/23.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "5.34794+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.34794, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@23", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "5.34794+", "metric_symbol": "s", "metric_value": 5.34794, "reference_text": "s = 5.34794+ | Found by Ignacio Vallejo in April 2026.", "source_image": "23.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/23.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 5.34794+ | Found by Ignacio Vallejo in April 2026.", "friedman_source_image": "23.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "4286cb7dfd8b00d7", "improvement": 0.00024989883471082663, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.435117490986131e-07, "max_pair_overlap_depth": -1.717758867814112e-05, "metric_expression": "5.34769", "metric_symbol": "s", "metric_value": 5.3476901011652895, "n": 23, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.34794, "setup": "peninpen", "side": 5.3476901011652895, "solution_path": "data/solutions/generated/peninpen_023_4286cb7dfd8b00d7.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/4286cb7dfd8b00d7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@24", "density": 0.7956104957168404, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/24.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "5.49241+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.49241, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@24", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "5.49241+", "metric_symbol": "s", "metric_value": 5.49241, "reference_text": "s = 5.49241+ | Found by Ignacio Vallejo in April 2026.", "source_image": "24.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/24.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 5.49241+ | Found by Ignacio Vallejo in April 2026.", "friedman_source_image": "24.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "8859e051020078b2", "improvement": 9.586393159466411e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.446346359580623e-06, "max_pair_overlap_depth": -9.458275242596459e-07, "metric_expression": "5.49231", "metric_symbol": "s", "metric_value": 5.492314136068405, "n": 24, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.49241, "setup": "peninpen", "side": 5.492314136068405, "solution_path": "data/solutions/generated/peninpen_024_8859e051020078b2.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/8859e051020078b2.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninpen@25", "density": 0.8019460079699491, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninpen/25.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "5.58370+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.5837, "friedman_reference": {"analytical_or_proved": false, "case": "peninpen@25", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "5.58370+", "metric_symbol": "s", "metric_value": 5.5837, "reference_text": "s = 5.58370+ | Found by Ignacio Vallejo in April 2026.", "source_image": "25.png", "source_image_url": "https://erich-friedman.github.io/packing/peninpen/25.png", "source_page": "https://erich-friedman.github.io/packing/peninpen/", "title": "Pentagons in Pentagons"}, "friedman_reference_text": "s = 5.58370+ | Found by Ignacio Vallejo in April 2026.", "friedman_source_image": "25.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninpen/", "friedman_title": "Pentagons in Pentagons", "frontend_seed": true, "id": "0f0645180a7be991", "improvement": 0.00031674927347591364, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.618192580437412e-06, "max_pair_overlap_depth": -2.8739793660470525e-06, "metric_expression": "5.58338", "metric_symbol": "s", "metric_value": 5.583383250726524, "n": 25, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.5837, "setup": "peninpen", "side": 5.583383250726524, "solution_path": "data/solutions/generated/peninpen_025_0f0645180a7be991.json", "source_url": "https://erich-friedman.github.io/packing/peninpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/0f0645180a7be991.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@1", "density": 0.6736263863631768, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/1.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "(3\u221a2+\u221a10+2\u221a(5+\u221a5))/8 = 1.59811+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.59811, "friedman_reference": {"analytical_or_proved": true, "case": "peninsqu@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(3\u221a2+\u221a10+2\u221a(5+\u221a5))/8 = 1.59811+", "metric_symbol": "s", "metric_value": 1.59811, "reference_text": "s = (3\u221a2+\u221a10+2\u221a(5+\u221a5))/8 = 1.59811+ | Trivial. |", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/1.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = (3\u221a2+\u221a10+2\u221a(5+\u221a5))/8 = 1.59811+ | Trivial. |", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "ec216f3e37e16785", "improvement": -3.043902908794749e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.733207856432108e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.59814", "metric_symbol": "s", "metric_value": 1.5981404390290879, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.59811, "setup": "peninsqu", "side": 1.5981404390290879, "solution_path": "data/solutions/generated/peninsqu_001_ec216f3e37e16785.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/ec216f3e37e16785.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@2", "density": 0.5669967061443146, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/2.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2.46345+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.46345, "friedman_reference": {"analytical_or_proved": true, "case": "peninsqu@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2.46345+", "metric_symbol": "s", "metric_value": 2.46345, "reference_text": "s = 2.46345+ | Trivial. |", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/2.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 2.46345+ | Trivial. |", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "bb5e10eb4917963f", "improvement": -3.0994037563569776e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.416866607183818e-07, "max_pair_overlap_depth": -1.473425980974552e-05, "metric_expression": "2.46348", "metric_symbol": "s", "metric_value": 2.4634809940375635, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.46345, "setup": "peninsqu", "side": 2.4634809940375635, "solution_path": "data/solutions/generated/peninsqu_002_bb5e10eb4917963f.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/bb5e10eb4917963f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@3", "density": 0.6102674136126431, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/3.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2.90812+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.90812, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@3", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.90812+", "metric_symbol": "s", "metric_value": 2.90812, "reference_text": "s = 2.90812+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/3.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 2.90812+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "52b96ec8c7f6b741", "improvement": -8.50430757179943e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.362613813108376e-06, "max_pair_overlap_depth": -5.5799797820554886e-05, "metric_expression": "2.90821", "metric_symbol": "s", "metric_value": 2.908205043075718, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.90812, "setup": "peninsqu", "side": 2.908205043075718, "solution_path": "data/solutions/generated/peninsqu_003_52b96ec8c7f6b741.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/52b96ec8c7f6b741.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@4", "density": 0.7148924907955395, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/4.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3.10252+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.10252, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@4", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.10252+", "metric_symbol": "s", "metric_value": 3.10252, "reference_text": "s = 3.10252+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/4.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 3.10252+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "a2fcd0b0beef7a6f", "improvement": -0.00013948165686938907, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2690386136380924e-06, "max_pair_overlap_depth": -3.0975557991252956e-05, "metric_expression": "3.10266", "metric_symbol": "s", "metric_value": 3.1026594816568696, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.10252, "setup": "peninsqu", "side": 3.1026594816568696, "solution_path": "data/solutions/generated/peninsqu_004_a2fcd0b0beef7a6f.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/a2fcd0b0beef7a6f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@5", "density": 0.6829538263136812, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/5.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3.54903+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.54903, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@5", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.54903+", "metric_symbol": "s", "metric_value": 3.54903, "reference_text": "s = 3.54903+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/5.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 3.54903+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "bda3fbac20c8a756", "improvement": -3.388580996466217e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.104606408908601e-07, "max_pair_overlap_depth": -5.251218256785428e-06, "metric_expression": "3.54906", "metric_symbol": "s", "metric_value": 3.549063885809965, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.54903, "setup": "peninsqu", "side": 3.549063885809965, "solution_path": "data/solutions/generated/peninsqu_005_bda3fbac20c8a756.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/bda3fbac20c8a756.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@6", "density": 0.6845927654826146, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/6.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3.88300+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.883, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@6", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.88300+", "metric_symbol": "s", "metric_value": 3.883, "reference_text": "s = 3.88300+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/6.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 3.88300+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "a267ec5809e79964", "improvement": -0.00014813728762552714, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0190677678703185e-06, "max_pair_overlap_depth": -5.678337215275775e-05, "metric_expression": "3.88315", "metric_symbol": "s", "metric_value": 3.8831481372876255, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.883, "setup": "peninsqu", "side": 3.8831481372876255, "solution_path": "data/solutions/generated/peninsqu_006_a267ec5809e79964.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/a267ec5809e79964.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@7", "density": 0.6891282499986281, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/7.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.18038+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.18038, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@7", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.18038+", "metric_symbol": "s", "metric_value": 4.18038, "reference_text": "s = 4.18038+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/7.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 4.18038+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "b5565b951637dc85", "improvement": -7.429190517971307e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4725627916323898e-06, "max_pair_overlap_depth": -1.919059115845778e-05, "metric_expression": "4.18045", "metric_symbol": "s", "metric_value": 4.18045429190518, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.18038, "setup": "peninsqu", "side": 4.18045429190518, "solution_path": "data/solutions/generated/peninsqu_007_b5565b951637dc85.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b5565b951637dc85.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@8", "density": 0.7167004559270932, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/8.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.38191+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.38191, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@8", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.38191+", "metric_symbol": "s", "metric_value": 4.38191, "reference_text": "s = 4.38191+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/8.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 4.38191+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "eac74e93d82a54dd", "improvement": -0.00037521167052823046, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2846178652549832e-06, "max_pair_overlap_depth": -2.0945039418324152e-05, "metric_expression": "4.38229", "metric_symbol": "s", "metric_value": 4.382285211670529, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.38191, "setup": "peninsqu", "side": 4.382285211670529, "solution_path": "data/solutions/generated/peninsqu_008_eac74e93d82a54dd.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/eac74e93d82a54dd.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@9", "density": 0.7314097601583741, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/9.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.60033+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.60033, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@9", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.60033+", "metric_symbol": "s", "metric_value": 4.60033, "reference_text": "s = 4.60033+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/9.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 4.60033+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "0270d827693b6f8d", "improvement": -0.0008091240419938472, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8644555136003204e-06, "max_pair_overlap_depth": -1.0559011279553943e-05, "metric_expression": "4.60114", "metric_symbol": "s", "metric_value": 4.601139124041993, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.60033, "setup": "peninsqu", "side": 4.601139124041993, "solution_path": "data/solutions/generated/peninsqu_009_0270d827693b6f8d.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/0270d827693b6f8d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@10", "density": 0.7142869465585481, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/10.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4.90584+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.90584, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@10", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "4.90584+", "metric_symbol": "s", "metric_value": 4.90584, "reference_text": "s = 4.90584+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/10.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 4.90584+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "fba502b92276222e", "improvement": -0.0019743867107484547, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.710048632437797e-06, "max_pair_overlap_depth": -3.5387419318078095e-05, "metric_expression": "4.90781", "metric_symbol": "s", "metric_value": 4.907814386710749, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.90584, "setup": "peninsqu", "side": 4.907814386710749, "solution_path": "data/solutions/generated/peninsqu_010_fba502b92276222e.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/fba502b92276222e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@11", "density": 0.7230188560400136, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/11.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5.11557+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.11557, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.11557+", "metric_symbol": "s", "metric_value": 5.11557, "reference_text": "s = 5.11557+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/11.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 5.11557+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "3596998e61e47a27", "improvement": -0.0006123708648555493, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.520631561500943e-07, "max_pair_overlap_depth": -2.3531878005744744e-05, "metric_expression": "5.11618", "metric_symbol": "s", "metric_value": 5.1161823708648555, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.11557, "setup": "peninsqu", "side": 5.1161823708648555, "solution_path": "data/solutions/generated/peninsqu_011_3596998e61e47a27.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/3596998e61e47a27.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@12", "density": 0.7539256100231233, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/12.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5.23250+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.2325, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@12", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.23250+", "metric_symbol": "s", "metric_value": 5.2325, "reference_text": "s = 5.23250+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/12.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 5.23250+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "72fe010d0104fc76", "improvement": -0.0005015230417875216, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6940168948787563e-07, "max_pair_overlap_depth": -3.660774509528508e-05, "metric_expression": "5.23300", "metric_symbol": "s", "metric_value": 5.2330015230417875, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.2325, "setup": "peninsqu", "side": 5.2330015230417875, "solution_path": "data/solutions/generated/peninsqu_012_72fe010d0104fc76.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/72fe010d0104fc76.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@13", "density": 0.7325239130137876, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/13.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5.52183+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.52183, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.52183+", "metric_symbol": "s", "metric_value": 5.52183, "reference_text": "s = 5.52183+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/13.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 5.52183+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "b54e12045e01b08b", "improvement": -0.0038439980668076856, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7653030405329436e-07, "max_pair_overlap_depth": -2.666929117722816e-05, "metric_expression": "5.52567", "metric_symbol": "s", "metric_value": 5.525673998066807, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.52183, "setup": "peninsqu", "side": 5.525673998066807, "solution_path": "data/solutions/generated/peninsqu_013_b54e12045e01b08b.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b54e12045e01b08b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@14", "density": 0.7420924387932526, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/14.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5.69775+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.69775, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@14", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.69775+", "metric_symbol": "s", "metric_value": 5.69775, "reference_text": "s = 5.69775+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/14.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 5.69775+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "70191e649c0c34a4", "improvement": 0.0005757762276479284, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.376635005869759e-06, "max_pair_overlap_depth": -1.761308907194392e-05, "metric_expression": "5.69717", "metric_symbol": "s", "metric_value": 5.697174223772352, "n": 14, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.69775, "setup": "peninsqu", "side": 5.697174223772352, "solution_path": "data/solutions/generated/peninsqu_014_70191e649c0c34a4.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/70191e649c0c34a4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@15", "density": 0.737588729235034, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/15.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5.91106+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.91106, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@15", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "5.91106+", "metric_symbol": "s", "metric_value": 5.91106, "reference_text": "s = 5.91106+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/15.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 5.91106+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "f5d70026e15060e9", "improvement": -0.004052102776268995, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.031797802365455e-06, "max_pair_overlap_depth": -3.458627910934986e-05, "metric_expression": "5.91511", "metric_symbol": "s", "metric_value": 5.915112102776269, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.91106, "setup": "peninsqu", "side": 5.915112102776269, "solution_path": "data/solutions/generated/peninsqu_015_f5d70026e15060e9.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/f5d70026e15060e9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@16", "density": 0.7479800847585042, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/16.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.11021+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.11021, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@16", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.11021+", "metric_symbol": "s", "metric_value": 6.11021, "reference_text": "s = 6.11021+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/16.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 6.11021+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": false, "id": "4644b3e97f0bdea2", "improvement": 0.04369242522759276, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5325677563252472e-06, "max_pair_overlap_depth": -5.091262792966678e-06, "metric_expression": "6.06652", "metric_symbol": "s", "metric_value": 6.066517574772408, "n": 16, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 6.11021, "setup": "peninsqu", "side": 6.066517574772408, "solution_path": "data/solutions/generated/peninsqu_016_4644b3e97f0bdea2.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/4644b3e97f0bdea2.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@17", "density": 0.7514950307673967, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/17.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.26466+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.26466, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.26466+", "metric_symbol": "s", "metric_value": 6.26466, "reference_text": "s = 6.26466+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/17.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 6.26466+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": false, "id": "a6cd0104bc26d2a0", "improvement": 0.026077965234885703, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.35268853525978e-06, "max_pair_overlap_depth": -1.565351530530279e-05, "metric_expression": "6.23858", "metric_symbol": "s", "metric_value": 6.238582034765114, "n": 17, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 6.26466, "setup": "peninsqu", "side": 6.238582034765114, "solution_path": "data/solutions/generated/peninsqu_017_a6cd0104bc26d2a0.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/a6cd0104bc26d2a0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@18", "density": 0.7710926756280118, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/18.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.33632+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.33632, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@18", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.33632+", "metric_symbol": "s", "metric_value": 6.33632, "reference_text": "s = 6.33632+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/18.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 6.33632+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "9705d17615c32308", "improvement": -0.0010264494540361824, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.1888526227077803e-06, "max_pair_overlap_depth": -2.3019703547788595e-06, "metric_expression": "6.33735", "metric_symbol": "s", "metric_value": 6.337346449454036, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.33632, "setup": "peninsqu", "side": 6.337346449454036, "solution_path": "data/solutions/generated/peninsqu_018_9705d17615c32308.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/9705d17615c32308.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@19", "density": 0.759469213458349, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/19.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.55939+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.55939, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@19", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.55939+", "metric_symbol": "s", "metric_value": 6.55939, "reference_text": "s = 6.55939+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/19.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 6.55939+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "3b1113745987c1d6", "improvement": -0.0012498840705594816, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9916513391919655e-06, "max_pair_overlap_depth": -2.545361094719034e-05, "metric_expression": "6.56064", "metric_symbol": "s", "metric_value": 6.560639884070559, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.55939, "setup": "peninsqu", "side": 6.560639884070559, "solution_path": "data/solutions/generated/peninsqu_019_3b1113745987c1d6.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/3b1113745987c1d6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@20", "density": 0.7723902783602073, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/20.png", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.69588+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.69588, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@20", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "6.69588+", "metric_symbol": "s", "metric_value": 6.69588, "reference_text": "s = 6.69588+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/20.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 6.69588+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": false, "id": "eafd28d28fb8e254", "improvement": 0.021343927884186265, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.197045603091071e-06, "max_pair_overlap_depth": -2.9426570577584243e-05, "metric_expression": "6.67454", "metric_symbol": "s", "metric_value": 6.674536072115814, "n": 20, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 6.69588, "setup": "peninsqu", "side": 6.674536072115814, "solution_path": "data/solutions/generated/peninsqu_020_eafd28d28fb8e254.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/eafd28d28fb8e254.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@21", "density": 0.7688621226676746, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/21.png", "friedman_metric_expression": "6.85540+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.8554, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@21", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "6.85540+", "metric_symbol": "s", "metric_value": 6.8554, "reference_text": "s = 6.85540+ | Found by Emerson Connelly | in April 2026.", "source_image": "21.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/21.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 6.85540+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "21.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "9d320f91f05e2f0e", "improvement": 0.0003614666962281632, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5104768148309566e-06, "max_pair_overlap_depth": -4.3073173047414315e-05, "metric_expression": "6.85504", "metric_symbol": "s", "metric_value": 6.855038533303772, "n": 21, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.8554, "setup": "peninsqu", "side": 6.855038533303772, "solution_path": "data/solutions/generated/peninsqu_21_9d320f91f05e2f0e.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/9d320f91f05e2f0e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@22", "density": 0.7625061251104759, "friedman_analytical_or_proved": false, "friedman_credit": "Teodors Smith", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/22.png", "friedman_metric_expression": "7.04532+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.04532, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@22", "credit": "Teodors Smith", "date": "April 2026", "family_updated": "", "metric_expression": "7.04532+", "metric_symbol": "s", "metric_value": 7.04532, "reference_text": "s = 7.04532+ | Found by Teodors Smith | in April 2026.", "source_image": "22.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/22.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 7.04532+ | Found by Teodors Smith | in April 2026.", "friedman_source_image": "22.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "70964f50e662283c", "improvement": -0.0002179452221122702, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.550099486392989e-07, "max_pair_overlap_depth": -3.276547527986118e-05, "metric_expression": "7.04554", "metric_symbol": "s", "metric_value": 7.0455379452221125, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.04532, "setup": "peninsqu", "side": 7.0455379452221125, "solution_path": "data/solutions/generated/peninsqu_22_70964f50e662283c.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "Teodors Smith", "svg_path": "data/svg/70964f50e662283c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@23", "density": 0.7617923823536098, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/23.png", "friedman_metric_expression": "7.20731+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.20731, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@23", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "7.20731+", "metric_symbol": "s", "metric_value": 7.20731, "reference_text": "s = 7.20731+ | Found by Emerson Connelly | in April 2026.", "source_image": "23.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/23.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 7.20731+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "23.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "5b526f4bb0fda83c", "improvement": 5.162671385594564e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.635959726042516e-06, "max_pair_overlap_depth": -3.224520897293637e-05, "metric_expression": "7.20726", "metric_symbol": "s", "metric_value": 7.207258373286144, "n": 23, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.20731, "setup": "peninsqu", "side": 7.207258373286144, "solution_path": "data/solutions/generated/peninsqu_23_5b526f4bb0fda83c.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/5b526f4bb0fda83c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "peninsqu@24", "density": 0.766901141845129, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/peninsqu/24.png", "friedman_metric_expression": "7.33823+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.33823, "friedman_reference": {"analytical_or_proved": false, "case": "peninsqu@24", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "7.33823+", "metric_symbol": "s", "metric_value": 7.33823, "reference_text": "s = 7.33823+ | Found by Emerson Connelly | in April 2026.", "source_image": "24.png", "source_image_url": "https://erich-friedman.github.io/packing/peninsqu/24.png", "source_page": "https://erich-friedman.github.io/packing/peninsqu/", "title": "Pentagons in Squares"}, "friedman_reference_text": "s = 7.33823+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "24.png", "friedman_source_url": "https://erich-friedman.github.io/packing/peninsqu/", "friedman_title": "Pentagons in Squares", "frontend_seed": true, "id": "3e42e9e085ed4c16", "improvement": 0.0005221765418985669, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.294217705264547e-06, "max_pair_overlap_depth": -1.6096651148655994e-05, "metric_expression": "7.33771", "metric_symbol": "s", "metric_value": 7.337707823458102, "n": 24, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 7.33823, "setup": "peninsqu", "side": 7.337707823458102, "solution_path": "data/solutions/generated/peninsqu_24_3e42e9e085ed4c16.json", "source_url": "https://erich-friedman.github.io/packing/peninsqu/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/3e42e9e085ed4c16.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@1", "density": 0.6366152339868586, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/1.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a2 / 2 = .707+", "friedman_metric_symbol": "r", "friedman_metric_value": 0.707, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a2 / 2 = .707+", "metric_symbol": "r", "metric_value": 0.707, "reference_text": "r = \u221a2 / 2 = .707+ | Trivial.", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/1.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a2 / 2 = .707+ | Trivial.", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "e1e16b7d18b1243e", "improvement": -0.00010930163680866745, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.331866965270656e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.70711", "metric_symbol": "r", "metric_value": 0.7071093016368086, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.707, "setup": "squincir", "side": 1.4142186032736173, "solution_path": "data/solutions/generated/squincir_001_e1e16b7d18b1243e.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/e1e16b7d18b1243e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@2", "density": 0.5092878238610891, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/2.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a5 / 2 = 1.118+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.118, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a5 / 2 = 1.118+", "metric_symbol": "r", "metric_value": 1.118, "reference_text": "r = \u221a5 / 2 = 1.118+ | Trivial.", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/2.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a5 / 2 = 1.118+ | Trivial.", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "e0e32383f27cd27d", "improvement": -4.2763322036343254e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.769902571588602e-06, "max_pair_overlap_depth": -3.18210820038356e-06, "metric_expression": "1.11804", "metric_symbol": "r", "metric_value": 1.1180427633220364, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.118, "setup": "squincir", "side": 2.236085526644073, "solution_path": "data/solutions/generated/squincir_002_e0e32383f27cd27d.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/e0e32383f27cd27d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@3", "density": 0.5751857401746373, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/3.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "5 \u221a17 / 16 = 1.288+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.288, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@3", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 \u221a17 / 16 = 1.288+", "metric_symbol": "r", "metric_value": 1.288, "reference_text": "r = 5 \u221a17 / 16 = 1.288+ | Found by Erich Friedman | in 1997.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/3.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 5 \u221a17 / 16 = 1.288+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "226172dcb69c3daf", "improvement": -0.0004917307904808155, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4552703365164916e-07, "max_pair_overlap_depth": -2.6483206490832134e-05, "metric_expression": "1.28849", "metric_symbol": "r", "metric_value": 1.2884917307904808, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.288, "setup": "squincir", "side": 2.5769834615809617, "solution_path": "data/solutions/generated/squincir_003_226172dcb69c3daf.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/226172dcb69c3daf.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@4", "density": 0.6365934661039774, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/4.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a2 = 1.414+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.414, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@4", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a2 = 1.414+", "metric_symbol": "r", "metric_value": 1.414, "reference_text": "r = \u221a2 = 1.414+ | Found by Erich Friedman | in 1997.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/4.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a2 = 1.414+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "59645297dac888b6", "improvement": -0.0002427821910928607, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.7659081458272254e-06, "max_pair_overlap_depth": -2.7879728913877792e-05, "metric_expression": "1.41424", "metric_symbol": "r", "metric_value": 1.4142427821910928, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.414, "setup": "squincir", "side": 2.8284855643821856, "solution_path": "data/solutions/generated/squincir_004_59645297dac888b6.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/59645297dac888b6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@5", "density": 0.6366006254695611, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/5.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a10 / 2 = 1.581+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.581, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@5", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a10 / 2 = 1.581+", "metric_symbol": "r", "metric_value": 1.581, "reference_text": "r = \u221a10 / 2 = 1.581+ | Found by Erich Friedman | in 1997.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/5.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a10 / 2 = 1.581+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "ee29f4bb5ac19bc4", "improvement": -0.00016260768909037715, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.383463508896625e-07, "max_pair_overlap_depth": -1.4376414261496606e-05, "metric_expression": "1.58116", "metric_symbol": "r", "metric_value": 1.5811626076890903, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.581, "setup": "squincir", "side": 3.1623252153781807, "solution_path": "data/solutions/generated/squincir_005_ee29f4bb5ac19bc4.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/ee29f4bb5ac19bc4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@6", "density": 0.6697921489003816, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/6.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "1.688+ *", "friedman_metric_symbol": "r", "friedman_metric_value": 1.688, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@6", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "1.688+ *", "metric_symbol": "r", "metric_value": 1.688, "reference_text": "r = 1.688+ * | Found by Erich Friedman | in 1997.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/6.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 1.688+ * | Found by Erich Friedman | in 1997.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "00816b044d90c514", "improvement": -0.0006150768345383195, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.526575301033731e-06, "max_pair_overlap_depth": -2.575413327610132e-05, "metric_expression": "1.68862", "metric_symbol": "r", "metric_value": 1.6886150768345383, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.688, "setup": "squincir", "side": 3.3772301536690765, "solution_path": "data/solutions/generated/squincir_006_00816b044d90c514.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/00816b044d90c514.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@7", "density": 0.685554128713642, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/7.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a13 / 2 = 1.802+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.802, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@7", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a13 / 2 = 1.802+", "metric_symbol": "r", "metric_value": 1.802, "reference_text": "r = \u221a13 / 2 = 1.802+ | Found by Erich Friedman | in 1997.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/7.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a13 / 2 = 1.802+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "f533c6ece1865b92", "improvement": -0.0008234908634709992, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3338190876808227e-06, "max_pair_overlap_depth": -2.5625121306882548e-05, "metric_expression": "1.80282", "metric_symbol": "r", "metric_value": 1.802823490863471, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.802, "setup": "squincir", "side": 3.605646981726942, "solution_path": "data/solutions/generated/squincir_007_f533c6ece1865b92.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f533c6ece1865b92.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@8", "density": 0.6501060351379704, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "March 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/8.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "1.978+ *", "friedman_metric_symbol": "r", "friedman_metric_value": 1.978, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@8", "credit": "David W. Cantrell", "date": "March 2002", "family_updated": "", "metric_expression": "1.978+ *", "metric_symbol": "r", "metric_value": 1.978, "reference_text": "r = 1.978+ * | Found by David W. Cantrell | in March 2002.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/8.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 1.978+ * | Found by David W. Cantrell | in March 2002.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "056dfca89502965a", "improvement": -0.0011465711618996455, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7255803199667241e-06, "max_pair_overlap_depth": -1.3772848531545812e-05, "metric_expression": "1.97915", "metric_symbol": "r", "metric_value": 1.9791465711618996, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.978, "setup": "squincir", "side": 3.9582931423237993, "solution_path": "data/solutions/generated/squincir_008_056dfca89502965a.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/056dfca89502965a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@9", "density": 0.6635107369929176, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/9.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a1105 / 16 = 2.077+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.077, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@9", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a1105 / 16 = 2.077+", "metric_symbol": "r", "metric_value": 2.077, "reference_text": "r = \u221a1105 / 16 = 2.077+ | Found by Erich Friedman | in 1997.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/9.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a1105 / 16 = 2.077+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "a0eb38ed275fd3c7", "improvement": -0.0008889848991340266, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.939578609861826e-06, "max_pair_overlap_depth": -1.5145046883935365e-05, "metric_expression": "2.07789", "metric_symbol": "r", "metric_value": 2.077888984899134, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.077, "setup": "squincir", "side": 4.155777969798268, "solution_path": "data/solutions/generated/squincir_009_a0eb38ed275fd3c7.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a0eb38ed275fd3c7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@10", "density": 0.7072738827790132, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/10.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3 \u221a2 / 2 = 2.121+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.121, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@10", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 \u221a2 / 2 = 2.121+", "metric_symbol": "r", "metric_value": 2.121, "reference_text": "r = 3 \u221a2 / 2 = 2.121+ | Found by Erich Friedman | in 1997.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/10.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3 \u221a2 / 2 = 2.121+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "eacd7cf43d9f9c99", "improvement": -0.00044244096873002903, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.093937328557672e-07, "max_pair_overlap_depth": -1.6684797704336063e-05, "metric_expression": "2.12144", "metric_symbol": "r", "metric_value": 2.12144244096873, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.121, "setup": "squincir", "side": 4.24288488193746, "solution_path": "data/solutions/generated/squincir_010_eacd7cf43d9f9c99.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/eacd7cf43d9f9c99.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@11", "density": 0.7141684249475255, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "March 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/11.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.214+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.214, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@11", "credit": "David W. Cantrell", "date": "March 2002", "family_updated": "", "metric_expression": "2.214+", "metric_symbol": "r", "metric_value": 2.214, "reference_text": "r = 2.214+ | Found by David W. Cantrell | in March 2002.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/11.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 2.214+ | Found by David W. Cantrell | in March 2002.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "ed2b520c292f3457", "improvement": -0.00022160321374320446, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.710372663145449e-07, "max_pair_overlap_depth": -1.7996528127950384e-05, "metric_expression": "2.21422", "metric_symbol": "r", "metric_value": 2.214221603213743, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.214, "setup": "squincir", "side": 4.428443206427486, "solution_path": "data/solutions/generated/squincir_011_ed2b520c292f3457.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/ed2b520c292f3457.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@12", "density": 0.7638630053365897, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/12.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a5 = 2.236+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.236, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@12", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a5 = 2.236+", "metric_symbol": "r", "metric_value": 2.236, "reference_text": "r = \u221a5 = 2.236+ | Found by Erich Friedman | in 1997.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/12.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a5 = 2.236+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "064666a02cb4bcc6", "improvement": -0.00018612303226639426, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.505810228203842e-06, "max_pair_overlap_depth": -1.897198109612752e-05, "metric_expression": "2.23619", "metric_symbol": "r", "metric_value": 2.2361861230322666, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.236, "setup": "squincir", "side": 4.472372246064533, "solution_path": "data/solutions/generated/squincir_012_064666a02cb4bcc6.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/064666a02cb4bcc6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@13", "density": 0.7419440056386433, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "November 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/13.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.3607+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.3607, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@13", "credit": "David W. Cantrell", "date": "November 2010", "family_updated": "", "metric_expression": "2.3607+", "metric_symbol": "r", "metric_value": 2.3607, "reference_text": "r = 2.3607+ | Found by David W. Cantrell | in November 2010.", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/13.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 2.3607+ | Found by David W. Cantrell | in November 2010.", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "1ff894504e55415a", "improvement": -0.0009262889169745492, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0049576295578788e-06, "max_pair_overlap_depth": -2.3549111853871096e-05, "metric_expression": "2.36163", "metric_symbol": "r", "metric_value": 2.3616262889169746, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.3607, "setup": "squincir", "side": 4.723252577833949, "solution_path": "data/solutions/generated/squincir_013_1ff894504e55415a.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/1ff894504e55415a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@14", "density": 0.7129258170438343, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/14.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.5", "friedman_metric_symbol": "r", "friedman_metric_value": 2.5, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@14", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "2.5", "metric_symbol": "r", "metric_value": 2.5, "reference_text": "r = 2.5 | Found by Erich Friedman | in 1997.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/14.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 2.5 | Found by Erich Friedman | in 1997.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "16de07d86777c846", "improvement": -0.0001548640652071498, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.318191377805647e-07, "max_pair_overlap_depth": -1.9980447390599565e-05, "metric_expression": "2.50015", "metric_symbol": "r", "metric_value": 2.500154864065207, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.5, "setup": "squincir", "side": 5.000309728130414, "solution_path": "data/solutions/generated/squincir_014_16de07d86777c846.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/16de07d86777c846.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@15", "density": 0.7434807535174716, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "June 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/15.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.533+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.533, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@15", "credit": "David W. Cantrell", "date": "June 2002", "family_updated": "", "metric_expression": "2.533+", "metric_symbol": "r", "metric_value": 2.533, "reference_text": "r = 2.533+ | Found by by David W. Cantrell | in June 2002.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/15.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 2.533+ | Found by by David W. Cantrell | in June 2002.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "4a2ad28ea2097ba3", "improvement": -0.001170482100864234, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5068887214985125e-06, "max_pair_overlap_depth": -8.103900655004326e-06, "metric_expression": "2.53417", "metric_symbol": "r", "metric_value": 2.534170482100864, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.533, "setup": "squincir", "side": 5.068340964201728, "solution_path": "data/solutions/generated/squincir_015_4a2ad28ea2097ba3.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/4a2ad28ea2097ba3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@16", "density": 0.7401412241351131, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/16.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a11009 / 40 = 2.623+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.623, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@16", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a11009 / 40 = 2.623+", "metric_symbol": "r", "metric_value": 2.623, "reference_text": "r = \u221a11009 / 40 = 2.623+ | Found by Erich Friedman | in 1997.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/16.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a11009 / 40 = 2.623+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "94d0a09f4c75dcc4", "improvement": -0.00017797945661435222, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.109474628753105e-06, "max_pair_overlap_depth": -5.84557896199378e-06, "metric_expression": "2.62318", "metric_symbol": "r", "metric_value": 2.6231779794566146, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.623, "setup": "squincir", "side": 5.246355958913229, "solution_path": "data/solutions/generated/squincir_016_94d0a09f4c75dcc4.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/94d0a09f4c75dcc4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@17", "density": 0.7544956917261334, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "December 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/17.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.677+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.677, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@17", "credit": "Maurizio Morandi", "date": "December 2010", "family_updated": "", "metric_expression": "2.677+", "metric_symbol": "r", "metric_value": 2.677, "reference_text": "r = 2.677+ | Found by Maurizio Morandi | in December 2010.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/17.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 2.677+ | Found by Maurizio Morandi | in December 2010.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "036910f6c0550e2f", "improvement": -0.0010651760341882088, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2297138575488873e-06, "max_pair_overlap_depth": -2.14353819976143e-05, "metric_expression": "2.67807", "metric_symbol": "r", "metric_value": 2.6780651760341883, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.677, "setup": "squincir", "side": 5.3561303520683765, "solution_path": "data/solutions/generated/squincir_017_036910f6c0550e2f.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/036910f6c0550e2f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@18", "density": 0.7617085472368383, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "March 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/18.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a481 / 8 = 2.741+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.741, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@18", "credit": "David W. Cantrell", "date": "March 2002", "family_updated": "", "metric_expression": "\u221a481 / 8 = 2.741+", "metric_symbol": "r", "metric_value": 2.741, "reference_text": "r = \u221a481 / 8 = 2.741+ | Found by David W. Cantrell | in March 2002.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/18.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a481 / 8 = 2.741+ | Found by David W. Cantrell | in March 2002.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "d726e310dc0e9eaa", "improvement": -0.001627976420446764, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.727763966399067e-06, "max_pair_overlap_depth": -2.573974201414142e-05, "metric_expression": "2.74263", "metric_symbol": "r", "metric_value": 2.742627976420447, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.741, "setup": "squincir", "side": 5.485255952840894, "solution_path": "data/solutions/generated/squincir_018_d726e310dc0e9eaa.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d726e310dc0e9eaa.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@19", "density": 0.76918194004105, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/19.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.802+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.802, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@19", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.802+", "metric_symbol": "r", "metric_value": 2.802, "reference_text": "r = 2.802+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/19.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 2.802+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "1966e8e14b94f055", "improvement": -0.002060145605657926, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.987997947874078e-06, "max_pair_overlap_depth": -1.7618196217927817e-05, "metric_expression": "2.80406", "metric_symbol": "r", "metric_value": 2.804060145605658, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.802, "setup": "squincir", "side": 5.608120291211316, "solution_path": "data/solutions/generated/squincir_019_1966e8e14b94f055.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/1966e8e14b94f055.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@20", "density": 0.7600583693203081, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "June 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/20.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "2.893+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.893, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@20", "credit": "David W. Cantrell", "date": "June 2002", "family_updated": "", "metric_expression": "2.893+", "metric_symbol": "r", "metric_value": 2.893, "reference_text": "r = 2.893+ | Found by David W. Cantrell | in June 2002.", "source_image": "20.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/20.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 2.893+ | Found by David W. Cantrell | in June 2002.", "friedman_source_image": "20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "d2a3e673398d1cd8", "improvement": -0.001120361357695998, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.0691232615918125e-06, "max_pair_overlap_depth": -2.4937171242911393e-05, "metric_expression": "2.89412", "metric_symbol": "r", "metric_value": 2.894120361357696, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.893, "setup": "squincir", "side": 5.788240722715392, "solution_path": "data/solutions/generated/squincir_020_d2a3e673398d1cd8.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d2a3e673398d1cd8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@21", "density": 0.7859297816487661, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/21.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a34 / 2 = 2.915+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.915, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@21", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a34 / 2 = 2.915+", "metric_symbol": "r", "metric_value": 2.915, "reference_text": "r = \u221a34 / 2 = 2.915+ | Found by Erich Friedman | in 1997.", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/21.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a34 / 2 = 2.915+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "af376e36d9f212b7", "improvement": -0.0013714498117400886, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.2933372016152873e-06, "max_pair_overlap_depth": -5.039957310160048e-06, "metric_expression": "2.91637", "metric_symbol": "r", "metric_value": 2.91637144981174, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.915, "setup": "squincir", "side": 5.83274289962348, "solution_path": "data/solutions/generated/squincir_021_af376e36d9f212b7.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/af376e36d9f212b7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@22", "density": 0.7599553171633651, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/22.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.033+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.033, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@22", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.033+", "metric_symbol": "r", "metric_value": 3.033, "reference_text": "r = 3.033+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/22.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.033+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "d08f4c9050539106", "improvement": -0.002584838809432277, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.8140931262198308e-06, "max_pair_overlap_depth": -1.4316626561150514e-05, "metric_expression": "3.03558", "metric_symbol": "r", "metric_value": 3.035584838809432, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.033, "setup": "squincir", "side": 6.071169677618864, "solution_path": "data/solutions/generated/squincir_022_d08f4c9050539106.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/d08f4c9050539106.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@23", "density": 0.7743799032674085, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "June 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/23.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.073+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.073, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@23", "credit": "David W. Cantrell", "date": "June 2002", "family_updated": "", "metric_expression": "3.073+", "metric_symbol": "r", "metric_value": 3.073, "reference_text": "r = 3.073+ | Found by David W. Cantrell | in June 2002.", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/23.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.073+ | Found by David W. Cantrell | in June 2002.", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "3cacab4fa2a4614f", "improvement": -0.0017651132247276102, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.255819924494972e-06, "max_pair_overlap_depth": -2.255301194975523e-05, "metric_expression": "3.07477", "metric_symbol": "r", "metric_value": 3.0747651132247276, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.073, "setup": "squincir", "side": 6.149530226449455, "solution_path": "data/solutions/generated/squincir_023_3cacab4fa2a4614f.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/3cacab4fa2a4614f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@24", "density": 0.7887737589513796, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "October 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/24.gif", "friedman_metric_expression": "3.110+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.11, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@24", "credit": "Maurizio Morandi", "date": "October 2010", "family_updated": "", "metric_expression": "3.110+", "metric_symbol": "r", "metric_value": 3.11, "reference_text": "r = 3.110+ | Found by Maurizio Morandi | in October 2010.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/24.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.110+ | Found by Maurizio Morandi | in October 2010.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "35e22c7b35c718bd", "improvement": -0.0021065491772924716, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.225275177878075e-06, "max_pair_overlap_depth": -2.5024692080766187e-05, "metric_expression": "3.11211", "metric_symbol": "r", "metric_value": 3.1121065491772923, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.11, "setup": "squincir", "side": 6.224213098354585, "solution_path": "data/solutions/generated/squincir_024_35e22c7b35c718bd.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/35e22c7b35c718bd.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@25", "density": 0.7888590017487276, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "April 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/25.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a5809 / 24 = 3.175+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.175, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@25", "credit": "David W. Cantrell", "date": "April 2002", "family_updated": "", "metric_expression": "\u221a5809 / 24 = 3.175+", "metric_symbol": "r", "metric_value": 3.175, "reference_text": "r = \u221a5809 / 24 = 3.175+ | Found by David W. Cantrell | in April 2002.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/25.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a5809 / 24 = 3.175+ | Found by David W. Cantrell | in April 2002.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "0cabb86031dba8fd", "improvement": -0.0011088296688002153, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4103138832897173e-06, "max_pair_overlap_depth": -3.0578471952869e-05, "metric_expression": "3.17611", "metric_symbol": "r", "metric_value": 3.1761088296688, "n": 25, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.175, "setup": "squincir", "side": 6.3522176593376, "solution_path": "data/solutions/generated/squincir_025_0cabb86031dba8fd.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/0cabb86031dba8fd.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@26", "density": 0.806871444276765, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/26.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a41 / 2 = 3.201+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.201, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@26", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a41 / 2 = 3.201+", "metric_symbol": "r", "metric_value": 3.201, "reference_text": "r = \u221a41 / 2 = 3.201+ | Found by Erich Friedman | in 1997.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/26.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a41 / 2 = 3.201+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "d3c528742ca2afa0", "improvement": -0.0016506279557328973, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.80804369340504e-06, "max_pair_overlap_depth": -4.195014817909293e-06, "metric_expression": "3.20265", "metric_symbol": "r", "metric_value": 3.202650627955733, "n": 26, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.201, "setup": "squincir", "side": 6.405301255911466, "solution_path": "data/solutions/generated/squincir_026_d3c528742ca2afa0.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/d3c528742ca2afa0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@27", "density": 0.8080261870592086, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "September 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/27.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "13\u221a145/48 = 3.261+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.261, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@27", "credit": "Maurizio Morandi", "date": "September 2010", "family_updated": "", "metric_expression": "13\u221a145/48 = 3.261+", "metric_symbol": "r", "metric_value": 3.261, "reference_text": "r = 13\u221a145/48 = 3.261+ | Found by Maurizio Morandi | in September 2010.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/27.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 13\u221a145/48 = 3.261+ | Found by Maurizio Morandi | in September 2010.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "24101c6f5ef6d134", "improvement": -0.0003261127181617418, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.239249729032224e-06, "max_pair_overlap_depth": -1.3019517370316258e-05, "metric_expression": "3.26133", "metric_symbol": "r", "metric_value": 3.261326112718162, "n": 27, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.261, "setup": "squincir", "side": 6.522652225436324, "solution_path": "data/solutions/generated/squincir_027_24101c6f5ef6d134.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/24101c6f5ef6d134.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@28", "density": 0.8008141779745183, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/28.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.333+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.333, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@28", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.333+", "metric_symbol": "r", "metric_value": 3.333, "reference_text": "r = 3.333+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/28.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.333+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "33c1007b5e868482", "improvement": -0.0030934121555343808, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.8010138785924426e-06, "max_pair_overlap_depth": -3.2305275887267726e-05, "metric_expression": "3.33609", "metric_symbol": "r", "metric_value": 3.3360934121555346, "n": 28, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.333, "setup": "squincir", "side": 6.672186824311069, "solution_path": "data/solutions/generated/squincir_028_33c1007b5e868482.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T03:19:00+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/33c1007b5e868482.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@29", "density": 0.7983676158390189, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/29.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.401+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.401, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@29", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.401+", "metric_symbol": "r", "metric_value": 3.401, "reference_text": "r = 3.401+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/29.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.401+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "ab533ab0ad8e3eeb", "improvement": 0.000657947854124874, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.686355668135576e-06, "max_pair_overlap_depth": -1.5593607360919748e-05, "metric_expression": "3.40034", "metric_symbol": "r", "metric_value": 3.400342052145875, "n": 29, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.401, "setup": "squincir", "side": 6.80068410429175, "solution_path": "data/solutions/generated/squincir_029_ab533ab0ad8e3eeb.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/ab533ab0ad8e3eeb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@30", "density": 0.7941259454921153, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/30.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.472+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.472, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@30", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.472+", "metric_symbol": "r", "metric_value": 3.472, "reference_text": "r = 3.472+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/30.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.472+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "d6d4b17471528bfa", "improvement": 0.004304153213534612, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2997799895563276e-06, "max_pair_overlap_depth": -1.7959424514213396e-05, "metric_expression": "3.46770", "metric_symbol": "r", "metric_value": 3.4676958467864654, "n": 30, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.472, "setup": "squincir", "side": 6.935391693572931, "solution_path": "data/solutions/generated/squincir_030_d6d4b17471528bfa.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/d6d4b17471528bfa.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@31", "density": 0.7887915525098625, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/31.gif", "friedman_metric_expression": "5 \u221a2 / 2 = 3.535+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.535, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@31", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 \u221a2 / 2 = 3.535+", "metric_symbol": "r", "metric_value": 3.535, "reference_text": "r = 5 \u221a2 / 2 = 3.535+ | Found by Erich Friedman | in 1997.", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/31.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 5 \u221a2 / 2 = 3.535+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "fdf9170d30584f4f", "improvement": -0.0019163212872146396, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.62119976746078e-06, "max_pair_overlap_depth": -1.7231292489316985e-05, "metric_expression": "3.53692", "metric_symbol": "r", "metric_value": 3.536916321287215, "n": 31, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.535, "setup": "squincir", "side": 7.07383264257443, "solution_path": "data/solutions/generated/squincir_031_fdf9170d30584f4f.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/fdf9170d30584f4f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@32", "density": 0.7916012707303632, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/32.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "3.584+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.584, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@32", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3.584+", "metric_symbol": "r", "metric_value": 3.584, "reference_text": "r = 3.584+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/32.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.584+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "b8bda371acb2b492", "improvement": -0.0031274839526354015, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.2509087862826505e-06, "max_pair_overlap_depth": -6.52310694349012e-09, "metric_expression": "3.58713", "metric_symbol": "r", "metric_value": 3.5871274839526355, "n": 32, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.584, "setup": "squincir", "side": 7.174254967905271, "solution_path": "data/solutions/generated/squincir_032_b8bda371acb2b492.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b8bda371acb2b492.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@33", "density": 0.8076634866746639, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/33.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a13 = 3.605+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.605, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@33", "credit": "Maurizio Morandi", "date": "June 2010", "family_updated": "", "metric_expression": "\u221a13 = 3.605+", "metric_symbol": "r", "metric_value": 3.605, "reference_text": "r = \u221a13 = 3.605+ | Found by Maurizio Morandi | in June 2010.", "source_image": "33.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/33.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a13 = 3.605+ | Found by Maurizio Morandi | in June 2010.", "friedman_source_image": "33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "e5a9fc9a4b7235a0", "improvement": -0.0013411620456880158, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.493268955270025e-06, "max_pair_overlap_depth": -1.6481229587750335e-06, "metric_expression": "3.60634", "metric_symbol": "r", "metric_value": 3.606341162045688, "n": 33, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.605, "setup": "squincir", "side": 7.212682324091376, "solution_path": "data/solutions/generated/squincir_033_e5a9fc9a4b7235a0.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/e5a9fc9a4b7235a0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@34", "density": 0.819454861319337, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "December 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/34.gif", "friedman_metric_expression": "3.629+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.629, "friedman_reference": {"analytical_or_proved": false, "case": "squincir@34", "credit": "Maurizio Morandi", "date": "December 2010", "family_updated": "", "metric_expression": "3.629+", "metric_symbol": "r", "metric_value": 3.629, "reference_text": "r = 3.629+ | Found by Maurizio Morandi | in December 2010.", "source_image": "34.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/34.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = 3.629+ | Found by Maurizio Morandi | in December 2010.", "friedman_source_image": "34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "9214fdf4034b8f9a", "improvement": -0.005142931467513545, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0151594769247652e-06, "max_pair_overlap_depth": -1.71712946250846e-07, "metric_expression": "3.63414", "metric_symbol": "r", "metric_value": 3.6341429314675135, "n": 34, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.629, "setup": "squincir", "side": 7.268285862935027, "solution_path": "data/solutions/generated/squincir_34_9214fdf4034b8f9a.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T14:34:18+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/9214fdf4034b8f9a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squincir@35", "density": 0.8405414638481816, "friedman_analytical_or_proved": true, "friedman_credit": "David Eppstein", "friedman_date": "March 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squincir/35.gif", "friedman_family_updated": "4/13/26", "friedman_metric_expression": "\u221a53 / 2 = 3.640+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.64, "friedman_reference": {"analytical_or_proved": true, "case": "squincir@35", "credit": "David Eppstein", "date": "March 2002", "family_updated": "", "metric_expression": "\u221a53 / 2 = 3.640+", "metric_symbol": "r", "metric_value": 3.64, "reference_text": "r = \u221a53 / 2 = 3.640+ | Found by David Eppstein | in March 2002.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/squincir/35.gif", "source_page": "https://erich-friedman.github.io/packing/squincir/", "title": "Squares in Circles"}, "friedman_reference_text": "r = \u221a53 / 2 = 3.640+ | Found by David Eppstein | in March 2002.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squincir/", "friedman_title": "Squares in Circles", "frontend_seed": true, "id": "f2d91d98b4c2ee39", "improvement": -0.0006549082553228835, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.121617296348745e-07, "max_pair_overlap_depth": -2.1344301219083928e-05, "metric_expression": "3.64065", "metric_symbol": "r", "metric_value": 3.640654908255323, "n": 35, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.64, "setup": "squincir", "side": 7.281309816510646, "solution_path": "data/solutions/generated/squincir_035_f2d91d98b4c2ee39.json", "source_url": "https://erich-friedman.github.io/packing/squincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David Eppstein", "svg_path": "data/svg/f2d91d98b4c2ee39.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@1", "density": 0.499990679660235, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/2.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "1-2. | | s = 1 | Trivial. |", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/2.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "1-2. | | s = 1 | Trivial. |", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "557f8de873a17a8e", "improvement": -9.320470070184328e-06, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7525252651882113e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00001", "metric_symbol": "s", "metric_value": 1.0000093204700702, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.0, "setup": "squindom", "side": 2.0000186409401404, "solution_path": "data/solutions/generated/squindom_001_557f8de873a17a8e.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/557f8de873a17a8e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@2", "density": 0.9999882998008334, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/2.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "1-2. | | s = 1 | Trivial. |", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/2.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "1-2. | | s = 1 | Trivial. |", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "7684110dce2e5290", "improvement": -5.850150919473052e-06, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.463373909884382e-07, "max_pair_overlap_depth": -9.41881114351023e-06, "metric_expression": "1.00001", "metric_symbol": "s", "metric_value": 1.0000058501509195, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.0, "setup": "squindom", "side": 2.000011700301839, "solution_path": "data/solutions/generated/squindom_002_7684110dce2e5290.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7684110dce2e5290.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@3", "density": 0.6666417193387711, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/3.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3/2 = 1.5", "friedman_metric_symbol": "s", "friedman_metric_value": 1.5, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3/2 = 1.5", "metric_symbol": "s", "metric_value": 1.5, "reference_text": "3. | | s = 3/2 = 1.5 | Trivial. |", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/3.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "3. | | s = 3/2 = 1.5 | Trivial. |", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "3047ca63de76a853", "improvement": -2.8066531593262667e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0617527398680835e-06, "max_pair_overlap_depth": -1.9554412996725112e-05, "metric_expression": "1.50003", "metric_symbol": "s", "metric_value": 1.5000280665315933, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.5, "setup": "squindom", "side": 3.0000561330631865, "solution_path": "data/solutions/generated/squindom_003_3047ca63de76a853.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3047ca63de76a853.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@4", "density": 0.5367260778952853, "friedman_analytical_or_proved": false, "friedman_credit": "Evert Stenlund", "friedman_date": "June 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/4.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "1.930+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.93, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@4", "credit": "Evert Stenlund", "date": "June 2010", "family_updated": "", "metric_expression": "1.930+", "metric_symbol": "s", "metric_value": 1.93, "reference_text": "4. | | s = 1.930+ | Found by Evert Stenlund | in June 2010.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/4.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "4. | | s = 1.930+ | Found by Evert Stenlund | in June 2010.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "8d067d60c7138964", "improvement": -0.00036150970269521743, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.561542731343124e-06, "max_pair_overlap_depth": -3.757775137364083e-05, "metric_expression": "1.93036", "metric_symbol": "s", "metric_value": 1.9303615097026952, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.93, "setup": "squindom", "side": 3.8607230194053903, "solution_path": "data/solutions/generated/squindom_004_8d067d60c7138964.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Evert Stenlund", "svg_path": "data/svg/8d067d60c7138964.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@5", "density": 0.6249853765160349, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "5-8. | | s = 2 | Trivial. |", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "5-8. | | s = 2 | Trivial. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "05f6000ecc64a077", "improvement": -2.3397984937112426e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3736510935966635e-06, "max_pair_overlap_depth": -1.2206465427119814e-05, "metric_expression": "2.00002", "metric_symbol": "s", "metric_value": 2.000023397984937, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "squindom", "side": 4.000046795969874, "solution_path": "data/solutions/generated/squindom_005_05f6000ecc64a077.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/05f6000ecc64a077.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@6", "density": 0.749983975114432, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "5-8. | | s = 2 | Trivial. |", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "5-8. | | s = 2 | Trivial. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "095d951ace7a918a", "improvement": -2.136685649301029e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.707507375492412e-07, "max_pair_overlap_depth": -1.170836956105551e-05, "metric_expression": "2.00002", "metric_symbol": "s", "metric_value": 2.000021366856493, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "squindom", "side": 4.000042733712986, "solution_path": "data/solutions/generated/squindom_006_095d951ace7a918a.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/095d951ace7a918a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@7", "density": 0.8749639011598754, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "5-8. | | s = 2 | Trivial. |", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "5-8. | | s = 2 | Trivial. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "50aa0817a0117c40", "improvement": -4.1257093861268856e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4800520663781356e-06, "max_pair_overlap_depth": -1.885899157594295e-05, "metric_expression": "2.00004", "metric_symbol": "s", "metric_value": 2.0000412570938613, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "squindom", "side": 4.0000825141877225, "solution_path": "data/solutions/generated/squindom_007_50aa0817a0117c40.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/50aa0817a0117c40.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@8", "density": 0.9999722279382355, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@8", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "5-8. | | s = 2 | Trivial. |", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/8.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "5-8. | | s = 2 | Trivial. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "0b83841ee591689c", "improvement": -2.7772640243561852e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0495464614356251e-07, "max_pair_overlap_depth": -1.777420024382249e-07, "metric_expression": "2.00003", "metric_symbol": "s", "metric_value": 2.0000277726402436, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "squindom", "side": 4.000055545280487, "solution_path": "data/solutions/generated/squindom_008_0b83841ee591689c.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/0b83841ee591689c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@9", "density": 0.7367411181657051, "friedman_analytical_or_proved": true, "friedman_credit": "Evert Stenlund", "friedman_date": "June 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/9.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2 + \u221a2/3 = 2.471+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.471, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@9", "credit": "Evert Stenlund", "date": "June 2010", "family_updated": "", "metric_expression": "2 + \u221a2/3 = 2.471+", "metric_symbol": "s", "metric_value": 2.471, "reference_text": "9. | | s = 2 + \u221a2/3 = 2.471+ | Found by Evert Stenlund | in June 2010.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/9.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "9. | | s = 2 + \u221a2/3 = 2.471+ | Found by Evert Stenlund | in June 2010.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "2963ae62200cf2c5", "improvement": -0.000432778719658522, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8492494266174617e-06, "max_pair_overlap_depth": -7.2416104319739105e-06, "metric_expression": "2.47143", "metric_symbol": "s", "metric_value": 2.4714327787196586, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.471, "setup": "squindom", "side": 4.942865557439317, "solution_path": "data/solutions/generated/squindom_009_2963ae62200cf2c5.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Evert Stenlund", "svg_path": "data/svg/2963ae62200cf2c5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@10", "density": 0.7999865137938963, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/10.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "5/2 = 2.5", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@10", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5/2 = 2.5", "metric_symbol": "s", "metric_value": 2.5, "reference_text": "10. | | s = 5/2 = 2.5 | Trivial. |", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/10.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "10. | | s = 5/2 = 2.5 | Trivial. |", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "7d3cb25d7577069f", "improvement": -2.1072463463500668e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5655128748903735e-06, "max_pair_overlap_depth": -9.554543500822632e-07, "metric_expression": "2.50002", "metric_symbol": "s", "metric_value": 2.5000210724634635, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.5, "setup": "squindom", "side": 5.000042144926927, "solution_path": "data/solutions/generated/squindom_010_7d3cb25d7577069f.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7d3cb25d7577069f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@11", "density": 0.7109112928838592, "friedman_analytical_or_proved": false, "friedman_credit": "Evert Stenlund", "friedman_date": "June 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/11.gif", "friedman_metric_expression": "2.779+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.779, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@11", "credit": "Evert Stenlund", "date": "June 2010", "family_updated": "", "metric_expression": "2.779+", "metric_symbol": "s", "metric_value": 2.779, "reference_text": "11. | | s = 2.779+ | Found by Evert Stenlund | in June 2010. |", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/11.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "11. | | s = 2.779+ | Found by Evert Stenlund | in June 2010. |", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "eadb11fb534aa6a1", "improvement": -0.0024652481713771657, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.886034921991467e-06, "max_pair_overlap_depth": -3.432035104045106e-05, "metric_expression": "2.78147", "metric_symbol": "s", "metric_value": 2.781465248171377, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.779, "setup": "squindom", "side": 5.562930496342754, "solution_path": "data/solutions/generated/squindom_11_eadb11fb534aa6a1.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "Evert Stenlund", "svg_path": "data/svg/eadb11fb534aa6a1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@12", "density": 0.7147860825866693, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/12.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "6 (1 + \u221a2) / 5 = 2.897+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.897, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@12", "credit": "Maurizio Morandi", "date": "June 2010", "family_updated": "", "metric_expression": "6 (1 + \u221a2) / 5 = 2.897+", "metric_symbol": "s", "metric_value": 2.897, "reference_text": "12. | | s = 6 (1 + \u221a2) / 5 = 2.897+ | Found by Maurizio Morandi | in June 2010. |", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/12.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "12. | | s = 6 (1 + \u221a2) / 5 = 2.897+ | Found by Maurizio Morandi | in June 2010. |", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "4c82fd3f736a139f", "improvement": -0.0002607386911455123, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.382389093311346e-06, "max_pair_overlap_depth": -2.837543626332817e-05, "metric_expression": "2.89726", "metric_symbol": "s", "metric_value": 2.8972607386911453, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.897, "setup": "squindom", "side": 5.794521477382291, "solution_path": "data/solutions/generated/squindom_012_4c82fd3f736a139f.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/4c82fd3f736a139f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@13", "density": 0.7330751854461901, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/13.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "2.976+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.976, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@13", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "2.976+", "metric_symbol": "s", "metric_value": 2.976, "reference_text": "13. | | s = 2.976+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in June 2010.)", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/13.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "13. | | s = 2.976+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in June 2010.)", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d738dfc349a8c940", "improvement": -0.0017101333567786092, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.7694174018241284e-06, "max_pair_overlap_depth": -2.048132261660285e-05, "metric_expression": "2.97771", "metric_symbol": "s", "metric_value": 2.9777101333567786, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.976, "setup": "squindom", "side": 5.955420266713557, "solution_path": "data/solutions/generated/squindom_013_d738dfc349a8c940.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d738dfc349a8c940.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@14", "density": 0.7777474504876777, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@14", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "14-18. | | s = 3 | Trivial. |", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "167b47d5e39e076d", "improvement": -5.849005569213972e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0369016045785884e-06, "max_pair_overlap_depth": -1.5707310774493166e-05, "metric_expression": "3.00006", "metric_symbol": "s", "metric_value": 3.000058490055692, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.000116980111384, "solution_path": "data/solutions/generated/squindom_014_167b47d5e39e076d.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/167b47d5e39e076d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@15", "density": 0.8333077568349537, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@15", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "14-18. | | s = 3 | Trivial. |", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "6d686963e9ef0ece", "improvement": -4.603875684550118e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0382270816400307e-06, "max_pair_overlap_depth": -3.605764309855264e-06, "metric_expression": "3.00005", "metric_symbol": "s", "metric_value": 3.0000460387568455, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.000092077513691, "solution_path": "data/solutions/generated/squindom_015_6d686963e9ef0ece.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6d686963e9ef0ece.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@16", "density": 0.888839182217615, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@16", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "14-18. | | s = 3 | Trivial. |", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "0f551306e71d2ff0", "improvement": -8.388352586674586e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4593433725073055e-06, "max_pair_overlap_depth": -5.7555123290642385e-06, "metric_expression": "3.00008", "metric_symbol": "s", "metric_value": 3.0000838835258667, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.0001677670517335, "solution_path": "data/solutions/generated/squindom_016_0f551306e71d2ff0.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/0f551306e71d2ff0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@17", "density": 0.9444002665800334, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@17", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "14-18. | | s = 3 | Trivial. |", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "52c5d2070c49c79c", "improvement": -7.016730512532376e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.650126292496168e-06, "max_pair_overlap_depth": -3.2807804517531736e-07, "metric_expression": "3.00007", "metric_symbol": "s", "metric_value": 3.0000701673051253, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.000140334610251, "solution_path": "data/solutions/generated/squindom_017_52c5d2070c49c79c.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/52c5d2070c49c79c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@18", "density": 0.9999599238999353, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@18", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "14-18. | | s = 3 | Trivial. |", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/18.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "14-18. | | s = 3 | Trivial. |", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "a6900e1fd3f925c6", "improvement": -6.011595701327366e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.408701424045262e-07, "max_pair_overlap_depth": -1.079909316636929e-06, "metric_expression": "3.00006", "metric_symbol": "s", "metric_value": 3.0000601159570133, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.0, "setup": "squindom", "side": 6.0001202319140265, "solution_path": "data/solutions/generated/squindom_018_a6900e1fd3f925c6.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a6900e1fd3f925c6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@19", "density": 0.7988013748197259, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/19.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3 + 1/\u221a5 = 3.447+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.447, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@19", "credit": "Maurizio Morandi", "date": "June 2010", "family_updated": "", "metric_expression": "3 + 1/\u221a5 = 3.447+", "metric_symbol": "s", "metric_value": 3.447, "reference_text": "19. | | s = 3 + 1/\u221a5 = 3.447+ | Found by Maurizio Morandi | in June 2010.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/19.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "19. | | s = 3 + 1/\u221a5 = 3.447+ | Found by Maurizio Morandi | in June 2010.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "cddf920bff527727", "improvement": -0.0015966406761447338, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.98512899937964e-06, "max_pair_overlap_depth": -1.3782300862574814e-05, "metric_expression": "3.44860", "metric_symbol": "s", "metric_value": 3.448596640676145, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.447, "setup": "squindom", "side": 6.89719328135229, "solution_path": "data/solutions/generated/squindom_019_cddf920bff527727.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/cddf920bff527727.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@20", "density": 0.8162860196976518, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/21.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "7/2 = 3.5", "friedman_metric_symbol": "s", "friedman_metric_value": 3.5, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@20", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7/2 = 3.5", "metric_symbol": "s", "metric_value": 3.5, "reference_text": "20-21. | | s = 7/2 = 3.5 | Trivial. |", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/21.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "20-21. | | s = 7/2 = 3.5 | Trivial. |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "4a48c58d60a1f36b", "improvement": -8.6848505622239e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.863306484192975e-06, "max_pair_overlap_depth": -2.3696014737728532e-05, "metric_expression": "3.50009", "metric_symbol": "s", "metric_value": 3.5000868485056222, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.5, "setup": "squindom", "side": 7.0001736970112445, "solution_path": "data/solutions/generated/squindom_020_4a48c58d60a1f36b.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4a48c58d60a1f36b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@21", "density": 0.8571139620178546, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/21.gif", "friedman_metric_expression": "7/2 = 3.5", "friedman_metric_symbol": "s", "friedman_metric_value": 3.5, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@21", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7/2 = 3.5", "metric_symbol": "s", "metric_value": 3.5, "reference_text": "20-21. | | s = 7/2 = 3.5 | Trivial. |", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/21.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "20-21. | | s = 7/2 = 3.5 | Trivial. |", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "f91a28177536f6d3", "improvement": -5.899570515355279e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8534056853702905e-06, "max_pair_overlap_depth": -1.3464050134803784e-06, "metric_expression": "3.50006", "metric_symbol": "s", "metric_value": 3.5000589957051536, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.5, "setup": "squindom", "side": 7.000117991410307, "solution_path": "data/solutions/generated/squindom_021_f91a28177536f6d3.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/f91a28177536f6d3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@22", "density": 0.783535190077589, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/22.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3.745+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.745, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@22", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "3.745+", "metric_symbol": "s", "metric_value": 3.745, "reference_text": "22. | | s = 3.745+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in June 2010.)", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/22.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "22. | | s = 3.745+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in June 2010.)", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d2d5380917cfe7ce", "improvement": -0.0018567500999222553, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.72695736733975e-06, "max_pair_overlap_depth": -9.08031145729904e-06, "metric_expression": "3.74686", "metric_symbol": "s", "metric_value": 3.7468567500999224, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.745, "setup": "squindom", "side": 7.493713500199845, "solution_path": "data/solutions/generated/squindom_022_d2d5380917cfe7ce.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d2d5380917cfe7ce.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@23", "density": 0.7750857674093591, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2010", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/23.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3.849+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.849, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@23", "credit": "Maurizio Morandi", "date": "June 2010", "family_updated": "", "metric_expression": "3.849+", "metric_symbol": "s", "metric_value": 3.849, "reference_text": "23. | | s = 3.849+ | Found by Maurizio Morandi | in June 2010. |", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/23.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "23. | | s = 3.849+ | Found by Maurizio Morandi | in June 2010. |", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d12d3a9a87ba3b11", "improvement": -0.002891443742406352, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.707858897694649e-06, "max_pair_overlap_depth": -2.3445235071584136e-05, "metric_expression": "3.85189", "metric_symbol": "s", "metric_value": 3.8518914437424066, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.849, "setup": "squindom", "side": 7.703782887484813, "solution_path": "data/solutions/generated/squindom_023_d12d3a9a87ba3b11.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/d12d3a9a87ba3b11.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@24", "density": 0.7961447064751783, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell after Maurizio Morandi", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/24.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3.882+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.882, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@24", "credit": "David W. Cantrell after Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "3.882+", "metric_symbol": "s", "metric_value": 3.882, "reference_text": "24. | | s = 3.882+ | Found by DC after MM | in August 2012. |", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/24.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "24. | | s = 3.882+ | Found by DC after MM | in August 2012. |", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "f954e5304808181f", "improvement": -0.0003493916461954072, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7445287290307476e-06, "max_pair_overlap_depth": -1.9183513864096113e-05, "metric_expression": "3.88235", "metric_symbol": "s", "metric_value": 3.8823493916461955, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.882, "setup": "squindom", "side": 7.764698783292391, "solution_path": "data/solutions/generated/squindom_024_f954e5304808181f.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell after Maurizio Morandi", "svg_path": "data/svg/f954e5304808181f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@25", "density": 0.8043920132931294, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell after Maurizio Morandi", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/25.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3.939+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.939, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@25", "credit": "David W. Cantrell after Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "3.939+", "metric_symbol": "s", "metric_value": 3.939, "reference_text": "25. | | s = 3.939+ | Found by DC after MM | in August 2012. |", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/25.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "25. | | s = 3.939+ | Found by DC after MM | in August 2012. |", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "26273cb13b41bf72", "improvement": -0.003040951197706754, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.269696070480222e-06, "max_pair_overlap_depth": -2.5070959039568547e-05, "metric_expression": "3.94204", "metric_symbol": "s", "metric_value": 3.942040951197707, "n": 25, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.939, "setup": "squindom", "side": 7.884081902395414, "solution_path": "data/solutions/generated/squindom_025_26273cb13b41bf72.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell after Maurizio Morandi", "svg_path": "data/svg/26273cb13b41bf72.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@26", "density": 0.8201428647660397, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/26.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3.981+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.981, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@26", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "3.981+", "metric_symbol": "s", "metric_value": 3.981, "reference_text": "26. | | s = 3.981+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in July 2012.)", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/26.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "26. | | s = 3.981+ | Found by David W. Cantrell | in July 2012 (after Maurizio | Morandi in July 2012.)", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "a46204eea3d7f9fe", "improvement": -0.00031848875831208545, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.947984543621445e-06, "max_pair_overlap_depth": -8.516034391559124e-06, "metric_expression": "3.98132", "metric_symbol": "s", "metric_value": 3.981318488758312, "n": 26, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.981, "setup": "squindom", "side": 7.962636977516624, "solution_path": "data/solutions/generated/squindom_026_a46204eea3d7f9fe.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/a46204eea3d7f9fe.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@27", "density": 0.8436994977206457, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@27", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "27-32. | | s = 4 | Trivial. | |", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "c68e03cc8f5005db", "improvement": -0.00011971448073744284, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.175631390130775e-07, "max_pair_overlap_depth": -1.4466911184563003e-06, "metric_expression": "4.00012", "metric_symbol": "s", "metric_value": 4.000119714480737, "n": 27, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.000239428961475, "solution_path": "data/solutions/generated/squindom_027_c68e03cc8f5005db.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/c68e03cc8f5005db.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@28", "density": 0.8749068629130922, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@28", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "27-32. | | s = 4 | Trivial. | |", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "2c8c79126d60cdac", "improvement": -0.00021290176655508475, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.243746303256302e-06, "max_pair_overlap_depth": -1.6386799395987463e-05, "metric_expression": "4.00021", "metric_symbol": "s", "metric_value": 4.000212901766555, "n": 28, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.00042580353311, "solution_path": "data/solutions/generated/squindom_028_2c8c79126d60cdac.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2c8c79126d60cdac.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@29", "density": 0.9062096331295707, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@29", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "27-32. | | s = 4 | Trivial. | |", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d2548471aff7d9f5", "improvement": -8.908848335043018e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8156140130519702e-06, "max_pair_overlap_depth": -2.026030487822794e-06, "metric_expression": "4.00009", "metric_symbol": "s", "metric_value": 4.00008908848335, "n": 29, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.0001781769667, "solution_path": "data/solutions/generated/squindom_029_d2548471aff7d9f5.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d2548471aff7d9f5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@30", "density": 0.9374597146255952, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@30", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "27-32. | | s = 4 | Trivial. | |", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d67a491f7b4498c0", "improvement": -8.594490193214455e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.188605266186585e-06, "max_pair_overlap_depth": -1.644068809536492e-06, "metric_expression": "4.00009", "metric_symbol": "s", "metric_value": 4.000085944901932, "n": 30, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.000171889803864, "solution_path": "data/solutions/generated/squindom_030_d67a491f7b4498c0.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d67a491f7b4498c0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@31", "density": 0.9683879031627185, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@31", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "27-32. | | s = 4 | Trivial. | |", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "86ec622f54cecf7c", "improvement": -0.000747764390431449, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.872732694063188e-07, "max_pair_overlap_depth": -2.9208615389531944e-06, "metric_expression": "4.00075", "metric_symbol": "s", "metric_value": 4.0007477643904314, "n": 31, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.001495528780863, "solution_path": "data/solutions/generated/squindom_031_86ec622f54cecf7c.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/86ec622f54cecf7c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@32", "density": 0.9989694769831047, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@32", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "27-32. | | s = 4 | Trivial. | |", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/32.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "27-32. | | s = 4 | Trivial. | |", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "2873ac78de186b97", "improvement": -0.0020626403695489515, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0386280700511747e-06, "max_pair_overlap_depth": -2.476940408779882e-06, "metric_expression": "4.00206", "metric_symbol": "s", "metric_value": 4.002062640369549, "n": 32, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.0, "setup": "squindom", "side": 8.004125280739098, "solution_path": "data/solutions/generated/squindom_032_2873ac78de186b97.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2873ac78de186b97.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@33", "density": 0.8455857799388354, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/33.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "3 + \u221a2 = 4.414+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.414, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@33", "credit": "Maurizio Morandi", "date": "July 2012", "family_updated": "", "metric_expression": "3 + \u221a2 = 4.414+", "metric_symbol": "s", "metric_value": 4.414, "reference_text": "33. | | s = 3 + \u221a2 = 4.414+ | Found by Maurizio Morandi | in July 2012.", "source_image": "33.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/33.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "33. | | s = 3 + \u221a2 = 4.414+ | Found by Maurizio Morandi | in July 2012.", "friedman_source_image": "33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "167e5e2a52e9b5fe", "improvement": -0.00336348277204479, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.856211927670387e-06, "max_pair_overlap_depth": -6.591285420576298e-06, "metric_expression": "4.41736", "metric_symbol": "s", "metric_value": 4.4173634827720445, "n": 33, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.414, "setup": "squindom", "side": 8.834726965544089, "solution_path": "data/solutions/generated/squindom_033_167e5e2a52e9b5fe.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/167e5e2a52e9b5fe.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@34", "density": 0.8499062715939519, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/34.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4 + \u221a2/3 = 4.471+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.471, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@34", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "4 + \u221a2/3 = 4.471+", "metric_symbol": "s", "metric_value": 4.471, "reference_text": "34. | | s = 4 + \u221a2/3 = 4.471+ | Found by David W. Cantrell | in July 2012.", "source_image": "34.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/34.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "34. | | s = 4 + \u221a2/3 = 4.471+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "b631e5a58c1517e1", "improvement": -0.0013825437310881483, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.603876101403557e-07, "max_pair_overlap_depth": -3.616786977911346e-06, "metric_expression": "4.47238", "metric_symbol": "s", "metric_value": 4.472382543731088, "n": 34, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.471, "setup": "squindom", "side": 8.944765087462176, "solution_path": "data/solutions/generated/squindom_034_b631e5a58c1517e1.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/b631e5a58c1517e1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@35", "density": 0.8625827503428114, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/36.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "9/2 = 4.5", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@35", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "9/2 = 4.5", "metric_symbol": "s", "metric_value": 4.5, "reference_text": "35-36. | | s = 9/2 = 4.5 | Trivial. |", "source_image": "36.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/36.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "35-36. | | s = 9/2 = 4.5 | Trivial. |", "friedman_source_image": "36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "dcd4cfae3ff101a2", "improvement": -0.004210097373901434, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9606355010637344e-06, "max_pair_overlap_depth": -1.1537592481902781e-05, "metric_expression": "4.50421", "metric_symbol": "s", "metric_value": 4.504210097373901, "n": 35, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.5, "setup": "squindom", "side": 9.008420194747803, "solution_path": "data/solutions/generated/squindom_035_dcd4cfae3ff101a2.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/dcd4cfae3ff101a2.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@36", "density": 0.8873457711540725, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/36.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "9/2 = 4.5", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@36", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "9/2 = 4.5", "metric_symbol": "s", "metric_value": 4.5, "reference_text": "35-36. | | s = 9/2 = 4.5 | Trivial. |", "source_image": "36.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/36.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "35-36. | | s = 9/2 = 4.5 | Trivial. |", "friedman_source_image": "36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "e0fc61bd01777ea3", "improvement": -0.003911109790387002, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.310647293148122e-07, "max_pair_overlap_depth": -3.943598063660403e-06, "metric_expression": "4.50391", "metric_symbol": "s", "metric_value": 4.503911109790387, "n": 36, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.5, "setup": "squindom", "side": 9.007822219580774, "solution_path": "data/solutions/generated/squindom_036_e0fc61bd01777ea3.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/e0fc61bd01777ea3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@37", "density": 0.8199742411033116, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/37.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "10 / 3 + \u221a2 = 4.747+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.747, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@37", "credit": "Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "10 / 3 + \u221a2 = 4.747+", "metric_symbol": "s", "metric_value": 4.747, "reference_text": "37. | | s = 10 / 3 + \u221a2 = 4.747+ | Found by Maurizio Morandi | in August 2012.", "source_image": "37.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/37.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "37. | | s = 10 / 3 + \u221a2 = 4.747+ | Found by Maurizio Morandi | in August 2012.", "friedman_source_image": "37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "09a846134d0a3602", "improvement": -0.0029141409565269782, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.702594714522945e-06, "max_pair_overlap_depth": -1.6080659807293785e-05, "metric_expression": "4.74991", "metric_symbol": "s", "metric_value": 4.749914140956527, "n": 37, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.747, "setup": "squindom", "side": 9.499828281913054, "solution_path": "data/solutions/generated/squindom_037_09a846134d0a3602.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/09a846134d0a3602.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@38", "density": 0.8220103806251009, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/38.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "(13 + \u221a2) / 3 = 4.804+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.804, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@38", "credit": "Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "(13 + \u221a2) / 3 = 4.804+", "metric_symbol": "s", "metric_value": 4.804, "reference_text": "38. | | s = (13 + \u221a2) / 3 = 4.804+ | Found by Maurizio Morandi | in August 2012.", "source_image": "38.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/38.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "38. | | s = (13 + \u221a2) / 3 = 4.804+ | Found by Maurizio Morandi | in August 2012.", "friedman_source_image": "38.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d4d1cf6ab169d8aa", "improvement": -0.0037087406838747228, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.518216488058613e-06, "max_pair_overlap_depth": -2.290026661011524e-05, "metric_expression": "4.80771", "metric_symbol": "s", "metric_value": 4.807708740683875, "n": 38, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.804, "setup": "squindom", "side": 9.61541748136775, "solution_path": "data/solutions/generated/squindom_038_d4d1cf6ab169d8aa.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/d4d1cf6ab169d8aa.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@39", "density": 0.8326335013079162, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/39.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4.839+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.839, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@39", "credit": "David W. Cantrell", "date": "August 2012", "family_updated": "", "metric_expression": "4.839+", "metric_symbol": "s", "metric_value": 4.839, "reference_text": "39. | | s = 4.839+ | Found by David W. Cantrell | in August 2012.", "source_image": "39.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/39.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "39. | | s = 4.839+ | Found by David W. Cantrell | in August 2012.", "friedman_source_image": "39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "a606e4f52500fa01", "improvement": -0.00038713064593576263, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.790990827618913e-06, "max_pair_overlap_depth": -1.236762303413741e-05, "metric_expression": "4.83939", "metric_symbol": "s", "metric_value": 4.839387130645936, "n": 39, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.839, "setup": "squindom", "side": 9.678774261291872, "solution_path": "data/solutions/generated/squindom_039_a606e4f52500fa01.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/a606e4f52500fa01.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@40", "density": 0.8345719865483486, "friedman_analytical_or_proved": false, "friedman_credit": "Sigvart Brendberg", "friedman_date": "June 2023", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/40.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4.895+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.895, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@40", "credit": "Sigvart Brendberg", "date": "June 2023", "family_updated": "", "metric_expression": "4.895+", "metric_symbol": "s", "metric_value": 4.895, "reference_text": "40. | | s = 4.895+ | Found by Sigvart Brendberg | in June 2023.", "source_image": "40.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/40.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "40. | | s = 4.895+ | Found by Sigvart Brendberg | in June 2023.", "friedman_source_image": "40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "bb2811be04efb5ee", "improvement": -0.0003426573425642232, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.37233266706005e-06, "max_pair_overlap_depth": -3.9677918853286975e-06, "metric_expression": "4.89534", "metric_symbol": "s", "metric_value": 4.895342657342564, "n": 40, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.895, "setup": "squindom", "side": 9.790685314685128, "solution_path": "data/solutions/generated/squindom_040_bb2811be04efb5ee.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Sigvart Brendberg", "svg_path": "data/svg/bb2811be04efb5ee.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@41", "density": 0.8199798328031639, "friedman_analytical_or_proved": false, "friedman_credit": "Sigvart Brendberg", "friedman_date": "June 2023", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/41.gif", "friedman_metric_expression": "4.927+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.927, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@41", "credit": "Sigvart Brendberg", "date": "June 2023", "family_updated": "", "metric_expression": "4.927+", "metric_symbol": "s", "metric_value": 4.927, "reference_text": "41. | | s = 4.927+ | Found by Sigvart Brendberg | in June 2023.", "source_image": "41.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/41.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "41. | | s = 4.927+ | Found by Sigvart Brendberg | in June 2023.", "friedman_source_image": "41.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "cb1462995bd3b2c0", "improvement": -0.07306148649036626, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.465020711057036e-07, "max_pair_overlap_depth": -6.131910444850952e-06, "metric_expression": "5.00006", "metric_symbol": "s", "metric_value": 5.000061486490366, "n": 41, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.927, "setup": "squindom", "side": 10.000122972980732, "solution_path": "data/solutions/generated/squindom_41_cb1462995bd3b2c0.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "Sigvart Brendberg", "svg_path": "data/svg/cb1462995bd3b2c0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@42", "density": 0.8472590848618945, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/42.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "4.989+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.989, "friedman_reference": {"analytical_or_proved": false, "case": "squindom@42", "credit": "David W. Cantrell", "date": "August 2012", "family_updated": "", "metric_expression": "4.989+", "metric_symbol": "s", "metric_value": 4.989, "reference_text": "42. | | s = 4.989+ | Found by David W. Cantrell | in August 2012.", "source_image": "42.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/42.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "42. | | s = 4.989+ | Found by David W. Cantrell | in August 2012.", "friedman_source_image": "42.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": false, "id": "7faea8ae450b67a1", "improvement": 0.010465394774865189, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.54667148005916e-06, "max_pair_overlap_depth": -1.5216861767419765e-05, "metric_expression": "4.97853", "metric_symbol": "s", "metric_value": 4.978534605225135, "n": 42, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 4.989, "setup": "squindom", "side": 9.95706921045027, "solution_path": "data/solutions/generated/squindom_042_7faea8ae450b67a1.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/7faea8ae450b67a1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@43", "density": 0.859888117316887, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@43", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "7f9527ccddc9d648", "improvement": -0.00032527209571497906, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.1461540806485566e-08, "max_pair_overlap_depth": -2.005039219099558e-06, "metric_expression": "5.00033", "metric_symbol": "s", "metric_value": 5.000325272095715, "n": 43, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.00065054419143, "solution_path": "data/solutions/generated/squindom_043_7f9527ccddc9d648.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7f9527ccddc9d648.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@44", "density": 0.8799631966546829, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@44", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "1ffd758f1f058de5", "improvement": -0.00010455823792376862, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.245467354557775e-06, "max_pair_overlap_depth": -3.962242465904708e-06, "metric_expression": "5.00010", "metric_symbol": "s", "metric_value": 5.000104558237924, "n": 44, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000209116475848, "solution_path": "data/solutions/generated/squindom_044_1ffd758f1f058de5.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/1ffd758f1f058de5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@45", "density": 0.8999501217119913, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@45", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "d085ed9682ca2968", "improvement": -0.00013855655918781906, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0485935568738114e-06, "max_pair_overlap_depth": -1.361329106774889e-06, "metric_expression": "5.00014", "metric_symbol": "s", "metric_value": 5.000138556559188, "n": 45, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000277113118376, "solution_path": "data/solutions/generated/squindom_045_d085ed9682ca2968.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d085ed9682ca2968.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@46", "density": 0.9198953665582859, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@46", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "3b408e84cd206068", "improvement": -0.00028435426002282327, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1757865436834436e-06, "max_pair_overlap_depth": -7.423155479191479e-07, "metric_expression": "5.00028", "metric_symbol": "s", "metric_value": 5.000284354260023, "n": 46, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000568708520046, "solution_path": "data/solutions/generated/squindom_046_3b408e84cd206068.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3b408e84cd206068.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@47", "density": 0.9398436614252309, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@47", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "2525e331a392c6b9", "improvement": -0.00041584595443833194, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3105818880431173e-06, "max_pair_overlap_depth": -3.7244704524930228e-06, "metric_expression": "5.00042", "metric_symbol": "s", "metric_value": 5.000415845954438, "n": 47, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000831691908877, "solution_path": "data/solutions/generated/squindom_047_2525e331a392c6b9.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2525e331a392c6b9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@48", "density": 0.9599518570950978, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@48", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "0433c6294c4f3fe7", "improvement": -0.00012537686383318203, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0555756748686917e-06, "max_pair_overlap_depth": -4.3576353303542703e-07, "metric_expression": "5.00013", "metric_symbol": "s", "metric_value": 5.000125376863833, "n": 48, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000250753727666, "solution_path": "data/solutions/generated/squindom_048_0433c6294c4f3fe7.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/0433c6294c4f3fe7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@49", "density": 0.9798599360041962, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@49", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "c84a987ae840ebe3", "improvement": -0.0003573444166047324, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.316744310450815e-06, "max_pair_overlap_depth": -1.1644813818767474e-06, "metric_expression": "5.00036", "metric_symbol": "s", "metric_value": 5.000357344416605, "n": 49, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.00071468883321, "solution_path": "data/solutions/generated/squindom_049_c84a987ae840ebe3.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/c84a987ae840ebe3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squindom@50", "density": 0.9999308512263965, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "friedman_family_updated": "8/11/12", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "squindom@50", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "43-50. | | s = 5 | Trivial. |", "source_image": "50.gif", "source_image_url": "https://erich-friedman.github.io/packing/squindom/50.gif", "source_page": "https://erich-friedman.github.io/packing/squindom/", "title": "Squares in Dominoes"}, "friedman_reference_text": "43-50. | | s = 5 | Trivial. |", "friedman_source_image": "50.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squindom/", "friedman_title": "Squares in Dominoes", "frontend_seed": true, "id": "df0259815cc5320a", "improvement": -0.00017288089993794387, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.912590304139712e-09, "max_pair_overlap_depth": -7.234911549747203e-06, "metric_expression": "5.00017", "metric_symbol": "s", "metric_value": 5.000172880899938, "n": 50, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.0, "setup": "squindom", "side": 10.000345761799876, "solution_path": "data/solutions/generated/squindom_050_df0259815cc5320a.json", "source_url": "https://erich-friedman.github.io/packing/squindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/df0259815cc5320a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@1", "density": 0.6187919623349352, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/1.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(3+\u221a3)/6 = 0.778+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.778, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(3+\u221a3)/6 = 0.778+", "metric_symbol": "s", "metric_value": 0.778, "reference_text": "s = (3+\u221a3)/6 = 0.778+ | Trivial. |", "source_image": "1.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/1.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = (3+\u221a3)/6 = 0.778+ | Trivial. |", "friedman_source_image": "1.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "7a4ae27d520be7d9", "improvement": -0.01068162910289272, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.500249687614274e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.78868", "metric_symbol": "s", "metric_value": 0.7886816291028927, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.778, "setup": "squinhex", "side": 0.7886816291028927, "solution_path": "data/solutions/generated/squinhex_001_7a4ae27d520be7d9.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T03:07:58+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7a4ae27d520be7d9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@2", "density": 0.5773349997096187, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/2.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2/\u221a3 = 1.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.154, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2/\u221a3 = 1.154+", "metric_symbol": "s", "metric_value": 1.154, "reference_text": "s = 2/\u221a3 = 1.154+ | Trivial. |", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/2.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2/\u221a3 = 1.154+ | Trivial. |", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "21fb35b9b51e59c5", "improvement": -0.0007158081621452261, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.964730208767781e-07, "max_pair_overlap_depth": -4.154668371614267e-06, "metric_expression": "1.15472", "metric_symbol": "s", "metric_value": 1.1547158081621451, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.154, "setup": "squinhex", "side": 1.1547158081621451, "solution_path": "data/solutions/generated/squinhex_002_21fb35b9b51e59c5.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/21fb35b9b51e59c5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@3", "density": 0.6187647598988664, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/3.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(1+\u221a3)/2 = 1.366+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.366, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@3", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "(1+\u221a3)/2 = 1.366+", "metric_symbol": "s", "metric_value": 1.366, "reference_text": "s = (1+\u221a3)/2 = 1.366+ | Found by Maurizio Morandi | in March 2026.", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/3.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = (1+\u221a3)/2 = 1.366+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "a5c3dc7083e49634", "improvement": -6.667945486871218e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5592677317674368e-06, "max_pair_overlap_depth": -3.548280330942233e-05, "metric_expression": "1.36607", "metric_symbol": "s", "metric_value": 1.3660666794548688, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.366, "setup": "squinhex", "side": 1.3660666794548688, "solution_path": "data/solutions/generated/squinhex_003_a5c3dc7083e49634.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/a5c3dc7083e49634.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@4", "density": 0.6666229789487077, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/4.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2/3 1/4 = 1.519+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.519, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@4", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "2/3 1/4 = 1.519+", "metric_symbol": "s", "metric_value": 1.519, "reference_text": "s = 2/3 1/4 = 1.519+ | Found by Maurizio Morandi | in March 2026.", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/4.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2/3 1/4 = 1.519+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "49c10fcc6144044b", "improvement": -0.0007211669812854371, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.944520361455586e-06, "max_pair_overlap_depth": -3.91225282254104e-05, "metric_expression": "1.51972", "metric_symbol": "s", "metric_value": 1.5197211669812853, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.519, "setup": "squinhex", "side": 1.5197211669812853, "solution_path": "data/solutions/generated/squinhex_004_49c10fcc6144044b.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/49c10fcc6144044b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@5", "density": 0.6414868196410884, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/5.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a3 = 1.732+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.732, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3 = 1.732+", "metric_symbol": "s", "metric_value": 1.732, "reference_text": "s = \u221a3 = 1.732+ | Trivial. |", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/5.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = \u221a3 = 1.732+ | Trivial. |", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "1d1f44767245ecd6", "improvement": -6.900512462770791e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1790980134561805e-06, "max_pair_overlap_depth": -6.972508607594463e-06, "metric_expression": "1.73207", "metric_symbol": "s", "metric_value": 1.7320690051246277, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.732, "setup": "squinhex", "side": 1.7320690051246277, "solution_path": "data/solutions/generated/squinhex_005_1d1f44767245ecd6.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/1d1f44767245ecd6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@6", "density": 0.6953530967749227, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/6.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.821+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.821, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@6", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "1.821+", "metric_symbol": "s", "metric_value": 1.821, "reference_text": "s = 1.821+ | Found by Maurizio Morandi | in March 2026.", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/6.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 1.821+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "974baf9dad527270", "improvement": -0.001413758154348077, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6930873179275063e-06, "max_pair_overlap_depth": -8.425702388847878e-06, "metric_expression": "1.82241", "metric_symbol": "s", "metric_value": 1.822413758154348, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.821, "setup": "squinhex", "side": 1.822413758154348, "solution_path": "data/solutions/generated/squinhex_006_974baf9dad527270.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/974baf9dad527270.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@7", "density": 0.777727236784812, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/7.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(\u221a2)3 1/4 = 1.861+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.861, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@7", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "(\u221a2)3 1/4 = 1.861+", "metric_symbol": "s", "metric_value": 1.861, "reference_text": "s = (\u221a2)3 1/4 = 1.861+ | Found by Maurizio Morandi | in March 2026.", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/7.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = (\u221a2)3 1/4 = 1.861+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "3877067eb2c9921c", "improvement": -0.0002701930433344213, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.58302978193184e-06, "max_pair_overlap_depth": -6.096494394869367e-07, "metric_expression": "1.86127", "metric_symbol": "s", "metric_value": 1.8612701930433344, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.861, "setup": "squinhex", "side": 1.8612701930433344, "solution_path": "data/solutions/generated/squinhex_007_3877067eb2c9921c.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/3877067eb2c9921c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@8", "density": 0.6631734535885272, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/8.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.15378+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.15378, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@8", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.15378+", "metric_symbol": "s", "metric_value": 2.15378, "reference_text": "s = 2.15378+ | Found by Ian Watson | in April 2026.", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/8.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.15378+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "15b61ced9a55fd6e", "improvement": -0.0010126406302717328, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9874417895385996e-06, "max_pair_overlap_depth": -2.4862858205326743e-05, "metric_expression": "2.15479", "metric_symbol": "s", "metric_value": 2.1547926406302715, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.15378, "setup": "squinhex", "side": 2.1547926406302715, "solution_path": "data/solutions/generated/squinhex_008_15b61ced9a55fd6e.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/15b61ced9a55fd6e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@9", "density": 0.6952132004138262, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/9.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1/2+\u221a3 = 2.23205+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.23205, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@9", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1/2+\u221a3 = 2.23205+", "metric_symbol": "s", "metric_value": 2.23205, "reference_text": "s = 1/2+\u221a3 = 2.23205+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/9.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 1/2+\u221a3 = 2.23205+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "55005bb44a4d74d4", "improvement": -0.00016646218885707498, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6345465208580115e-06, "max_pair_overlap_depth": -3.4334606535568746e-05, "metric_expression": "2.23222", "metric_symbol": "s", "metric_value": 2.232216462188857, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.23205, "setup": "squinhex", "side": 2.232216462188857, "solution_path": "data/solutions/generated/squinhex_009_55005bb44a4d74d4.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/55005bb44a4d74d4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@10", "density": 0.721657360923786, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/10.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "4/\u221a3 = 2.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.309, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@10", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4/\u221a3 = 2.309+", "metric_symbol": "s", "metric_value": 2.309, "reference_text": "s = 4/\u221a3 = 2.309+ | Trivial. |", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/10.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 4/\u221a3 = 2.309+ | Trivial. |", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "1dc9bb5dff8e4212", "improvement": -0.0004498392040641619, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.82433796662779e-07, "max_pair_overlap_depth": -1.056245311698012e-05, "metric_expression": "2.30945", "metric_symbol": "s", "metric_value": 2.3094498392040643, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.309, "setup": "squinhex", "side": 2.3094498392040643, "solution_path": "data/solutions/generated/squinhex_010_1dc9bb5dff8e4212.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/1dc9bb5dff8e4212.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@11", "density": 0.7291333157228544, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/11.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.40944+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.40944, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.40944+", "metric_symbol": "s", "metric_value": 2.40944, "reference_text": "s = 2.40944+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/11.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.40944+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "705689a83f76dc66", "improvement": -0.00028191921915876605, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.56887695052102e-06, "max_pair_overlap_depth": -2.8090377436340574e-05, "metric_expression": "2.40972", "metric_symbol": "s", "metric_value": 2.409721919219159, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.40944, "setup": "squinhex", "side": 2.409721919219159, "solution_path": "data/solutions/generated/squinhex_011_705689a83f76dc66.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/705689a83f76dc66.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@12", "density": 0.7736310958293992, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/12.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1+5\u221a3/6 = 2.443+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.443, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@12", "credit": "Maurizio Morandi", "date": "March 2026", "family_updated": "", "metric_expression": "1+5\u221a3/6 = 2.443+", "metric_symbol": "s", "metric_value": 2.443, "reference_text": "s = 1+5\u221a3/6 = 2.443+ | Found by Maurizio Morandi | in March 2026.", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/12.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 1+5\u221a3/6 = 2.443+ | Found by Maurizio Morandi | in March 2026.", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "2605d29267e364f4", "improvement": -0.00041773030028169345, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4756288282540256e-06, "max_pair_overlap_depth": -2.0709990555811686e-06, "metric_expression": "2.44342", "metric_symbol": "s", "metric_value": 2.4434177303002818, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.443, "setup": "squinhex", "side": 2.4434177303002818, "solution_path": "data/solutions/generated/squinhex_012_2605d29267e364f4.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/2605d29267e364f4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@13", "density": 0.7555572001975633, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/13.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.572+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.572, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@13", "credit": "Maurizio Morandi", "date": "April 2026", "family_updated": "", "metric_expression": "2.572+", "metric_symbol": "s", "metric_value": 2.572, "reference_text": "s = 2.572+ | Found by Maurizio Morandi | in April 2026.", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/13.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.572+ | Found by Maurizio Morandi | in April 2026.", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "051502d315bb277c", "improvement": -0.001428217365099016, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2709109889167893e-06, "max_pair_overlap_depth": -2.7796124429491087e-05, "metric_expression": "2.57343", "metric_symbol": "s", "metric_value": 2.573428217365099, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.572, "setup": "squinhex", "side": 2.573428217365099, "solution_path": "data/solutions/generated/squinhex_013_051502d315bb277c.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/051502d315bb277c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@14", "density": 0.7839123498532168, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/14.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.62126+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.62126, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@14", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.62126+", "metric_symbol": "s", "metric_value": 2.62126, "reference_text": "s = 2.62126+ | Found by Jonathan Viquerat | in April 2026.", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/14.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.62126+ | Found by Jonathan Viquerat | in April 2026.", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "525e7cc585bd027c", "improvement": -0.000568749699766613, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5893267782285044e-06, "max_pair_overlap_depth": -1.4910539989809735e-05, "metric_expression": "2.62183", "metric_symbol": "s", "metric_value": 2.6218287496997665, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.62126, "setup": "squinhex", "side": 2.6218287496997665, "solution_path": "data/solutions/generated/squinhex_014_525e7cc585bd027c.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/525e7cc585bd027c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@15", "density": 0.7612787773180276, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/15.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.75365+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.75365, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@15", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.75365+", "metric_symbol": "s", "metric_value": 2.75365, "reference_text": "s = 2.75365+ | Found by Ian Watson | in April 2026.", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/15.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.75365+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "b68c71af3475ffed", "improvement": -0.00024783676108830477, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.513954837939443e-06, "max_pair_overlap_depth": -3.0296502803062175e-05, "metric_expression": "2.75390", "metric_symbol": "s", "metric_value": 2.7538978367610882, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.75365, "setup": "squinhex", "side": 2.7538978367610882, "solution_path": "data/solutions/generated/squinhex_015_b68c71af3475ffed.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/b68c71af3475ffed.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@16", "density": 0.7742958402345232, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/16.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.82004+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.82004, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@16", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.82004+", "metric_symbol": "s", "metric_value": 2.82004, "reference_text": "s = 2.82004+ | Found by Ian Watson | in April 2026.", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/16.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.82004+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "89a68dd125601a79", "improvement": -0.0001643950752789891, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.5133109066385543e-06, "max_pair_overlap_depth": -7.772490123292286e-06, "metric_expression": "2.82020", "metric_symbol": "s", "metric_value": 2.820204395075279, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.82004, "setup": "squinhex", "side": 2.820204395075279, "solution_path": "data/solutions/generated/squinhex_016_89a68dd125601a79.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/89a68dd125601a79.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@17", "density": 0.7850485918652831, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/17.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "5/\u221a3 = 2.88675+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.88675, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@17", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5/\u221a3 = 2.88675+", "metric_symbol": "s", "metric_value": 2.88675, "reference_text": "s = 5/\u221a3 = 2.88675+ | Trivial. |", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/17.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 5/\u221a3 = 2.88675+ | Trivial. |", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "68ba30b81c3844a7", "improvement": -0.0002730281066796181, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9250177449939015e-07, "max_pair_overlap_depth": -5.409854801341485e-05, "metric_expression": "2.88702", "metric_symbol": "s", "metric_value": 2.8870230281066798, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.88675, "setup": "squinhex", "side": 2.8870230281066798, "solution_path": "data/solutions/generated/squinhex_017_68ba30b81c3844a7.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/68ba30b81c3844a7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@18", "density": 0.8107158050212577, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/18.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.92111+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.92111, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@18", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.92111+", "metric_symbol": "s", "metric_value": 2.92111, "reference_text": "s = 2.92111+ | Found by Ian Watson | in April 2026.", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/18.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.92111+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "e136eeec3de765c3", "improvement": -0.0022075205933282938, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.264526989622803e-06, "max_pair_overlap_depth": -7.64695502597501e-06, "metric_expression": "2.92332", "metric_symbol": "s", "metric_value": 2.9233175205933284, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.92111, "setup": "squinhex", "side": 2.9233175205933284, "solution_path": "data/solutions/generated/squinhex_018_e136eeec3de765c3.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/e136eeec3de765c3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@19", "density": 0.8472470384096148, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/19.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.93511+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.93511, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@19", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.93511+", "metric_symbol": "s", "metric_value": 2.93511, "reference_text": "s = 2.93511+ | Found by Ian Watson | in April 2026.", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/19.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2.93511+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "ae3850bdb0ac1880", "improvement": -0.0028496545666087947, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.965076151420116e-06, "max_pair_overlap_depth": -7.201359447650546e-06, "metric_expression": "2.93796", "metric_symbol": "s", "metric_value": 2.9379596545666087, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.93511, "setup": "squinhex", "side": 2.9379596545666087, "solution_path": "data/solutions/generated/squinhex_019_ae3850bdb0ac1880.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/ae3850bdb0ac1880.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@20", "density": 0.7690325218237841, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/20.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.16123+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.16123, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@20", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.16123+", "metric_symbol": "s", "metric_value": 3.16123, "reference_text": "s = 3.16123+ | Found by Ian Watson | in April 2026.", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/20.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.16123+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "79028917ac70723d", "improvement": -0.0026259473298471114, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.697008230249216e-06, "max_pair_overlap_depth": -2.836652900617409e-05, "metric_expression": "3.16386", "metric_symbol": "s", "metric_value": 3.1638559473298473, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.16123, "setup": "squinhex", "side": 3.1638559473298473, "solution_path": "data/solutions/generated/squinhex_020_79028917ac70723d.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/79028917ac70723d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@21", "density": 0.7707944086673769, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/21.png", "friedman_metric_expression": "3.23496+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.23496, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@21", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.23496+", "metric_symbol": "s", "metric_value": 3.23496, "reference_text": "s = 3.23496+ | Found by Ian Watson | in April 2026.", "source_image": "21.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/21.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.23496+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "21.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "1d2ec517035658cd", "improvement": -0.0033202143530659534, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4830527974218215e-06, "max_pair_overlap_depth": -2.662964935497389e-05, "metric_expression": "3.23828", "metric_symbol": "s", "metric_value": 3.238280214353066, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.23496, "setup": "squinhex", "side": 3.238280214353066, "solution_path": "data/solutions/generated/squinhex_21_1d2ec517035658cd.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/1d2ec517035658cd.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@22", "density": 0.770525380181529, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/22.png", "friedman_metric_expression": "3.31573+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.31573, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@22", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.31573+", "metric_symbol": "s", "metric_value": 3.31573, "reference_text": "s = 3.31573+ | Found by Ian Watson | in April 2026.", "source_image": "22.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/22.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.31573+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "22.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "a1e7eaf12003fa2e", "improvement": 0.0006659560348909466, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.169977234601665e-06, "max_pair_overlap_depth": -1.9814540655582746e-05, "metric_expression": "3.31506", "metric_symbol": "s", "metric_value": 3.315064043965109, "n": 22, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.31573, "setup": "squinhex", "side": 3.315064043965109, "solution_path": "data/solutions/generated/squinhex_022_a1e7eaf12003fa2e.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/a1e7eaf12003fa2e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@23", "density": 0.7808954728618008, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/23.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.36437+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.36437, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@23", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.36437+", "metric_symbol": "s", "metric_value": 3.36437, "reference_text": "s = 3.36437+ | Found by Ian Watson | in April 2026.", "source_image": "23.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/23.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.36437+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "23.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "de78b89055181fd4", "improvement": -0.00261764110148599, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.240294795076849e-06, "max_pair_overlap_depth": -2.4390771091731267e-07, "metric_expression": "3.36699", "metric_symbol": "s", "metric_value": 3.366987641101486, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.36437, "setup": "squinhex", "side": 3.366987641101486, "solution_path": "data/solutions/generated/squinhex_023_de78b89055181fd4.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/de78b89055181fd4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@24", "density": 0.7904120022571139, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/24.png", "friedman_metric_expression": "3.41524+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.41524, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@24", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.41524+", "metric_symbol": "s", "metric_value": 3.41524, "reference_text": "s = 3.41524+ | Found by Ian Watson | in April 2026.", "source_image": "24.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/24.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.41524+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "24.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "73f1ceeaee97d32f", "improvement": -0.0033964158861361504, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0115328209217065e-06, "max_pair_overlap_depth": -2.5222031625055097e-05, "metric_expression": "3.41864", "metric_symbol": "s", "metric_value": 3.418636415886136, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.41524, "setup": "squinhex", "side": 3.418636415886136, "solution_path": "data/solutions/generated/squinhex_24_73f1ceeaee97d32f.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T14:34:18+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/73f1ceeaee97d32f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@25", "density": 0.8017157641998289, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/25.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2\u221a3 = 3.46410+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.4641, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@25", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2\u221a3 = 3.46410+", "metric_symbol": "s", "metric_value": 3.4641, "reference_text": "s = 2\u221a3 = 3.46410+ | Trivial. |", "source_image": "25.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/25.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2\u221a3 = 3.46410+ | Trivial. |", "friedman_source_image": "25.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "803edc96e272ee7a", "improvement": -0.00034642351026814566, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4684956309553432e-06, "max_pair_overlap_depth": -2.379294205911986e-05, "metric_expression": "3.46445", "metric_symbol": "s", "metric_value": 3.4644464235102683, "n": 25, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.4641, "setup": "squinhex", "side": 3.4644464235102683, "solution_path": "data/solutions/generated/squinhex_025_803edc96e272ee7a.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/803edc96e272ee7a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@26", "density": 0.8183272598135501, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/26.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.4939927+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.4939927, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@26", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.4939927+", "metric_symbol": "s", "metric_value": 3.4939927, "reference_text": "s = 3.4939927+ | Found by Ian Watson | in April 2026.", "source_image": "26.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/26.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.4939927+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "26.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "5fb7804364b5caaf", "improvement": -0.003020096739570377, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.392756549893556e-06, "max_pair_overlap_depth": -2.620566831379456e-05, "metric_expression": "3.49701", "metric_symbol": "s", "metric_value": 3.4970127967395706, "n": 26, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.4939927, "setup": "squinhex", "side": 3.4970127967395706, "solution_path": "data/solutions/generated/squinhex_026_5fb7804364b5caaf.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/5fb7804364b5caaf.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@27", "density": 0.8382703933092833, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/27.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(9+7\u221a3)/6 = 3.52072+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.52072, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@27", "credit": "Erich Friedman", "date": "April 2026", "family_updated": "", "metric_expression": "(9+7\u221a3)/6 = 3.52072+", "metric_symbol": "s", "metric_value": 3.52072, "reference_text": "s = (9+7\u221a3)/6 = 3.52072+ | Found by Erich Friedman | in April 2026.", "source_image": "27.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/27.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = (9+7\u221a3)/6 = 3.52072+ | Found by Erich Friedman | in April 2026.", "friedman_source_image": "27.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "9cebbb5918ba8e81", "improvement": -0.00026256883024800004, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.315485793993667e-06, "max_pair_overlap_depth": -2.3466684225592616e-05, "metric_expression": "3.52098", "metric_symbol": "s", "metric_value": 3.520982568830248, "n": 27, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.52072, "setup": "squinhex", "side": 3.520982568830248, "solution_path": "data/solutions/generated/squinhex_027_9cebbb5918ba8e81.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/9cebbb5918ba8e81.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@28", "density": 0.8183465665432688, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/28.png", "friedman_metric_expression": "3.62565+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.62565, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@28", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.62565+", "metric_symbol": "s", "metric_value": 3.62565, "reference_text": "s = 3.62565+ | Found by Ian Watson | in April 2026.", "source_image": "28.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/28.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.62565+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "28.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "184baa4d16ce7f9e", "improvement": -0.00332887546169669, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.700161000518847e-07, "max_pair_overlap_depth": -3.0973746406592895e-06, "metric_expression": "3.62898", "metric_symbol": "s", "metric_value": 3.6289788754616965, "n": 28, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.62565, "setup": "squinhex", "side": 3.6289788754616965, "solution_path": "data/solutions/generated/squinhex_28_184baa4d16ce7f9e.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T14:34:18+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/184baa4d16ce7f9e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@29", "density": 0.829741145219895, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/29.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.66437+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.66437, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@29", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.66437+", "metric_symbol": "s", "metric_value": 3.66437, "reference_text": "s = 3.66437+ | Found by Ian Watson | in April 2026.", "source_image": "29.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/29.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.66437+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "29.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "3b329ff6f7babb15", "improvement": -0.0033970322125265184, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.61123003800209e-07, "max_pair_overlap_depth": -5.962934573305878e-06, "metric_expression": "3.66777", "metric_symbol": "s", "metric_value": 3.6677670322125264, "n": 29, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.66437, "setup": "squinhex", "side": 3.6677670322125264, "solution_path": "data/solutions/generated/squinhex_029_3b329ff6f7babb15.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/3b329ff6f7babb15.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@30", "density": 0.8440944212011869, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/30.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.69554+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.69554, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@30", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.69554+", "metric_symbol": "s", "metric_value": 3.69554, "reference_text": "s = 3.69554+ | Found by Ian Watson | in April 2026.", "source_image": "30.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/30.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.69554+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "30.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "0b645f27b41cbed7", "improvement": -0.0030753622873653796, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7116676711139033e-06, "max_pair_overlap_depth": -4.452216716410717e-06, "metric_expression": "3.69862", "metric_symbol": "s", "metric_value": 3.698615362287365, "n": 30, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.69554, "setup": "squinhex", "side": 3.698615362287365, "solution_path": "data/solutions/generated/squinhex_030_0b645f27b41cbed7.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/0b645f27b41cbed7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@31", "density": 0.83334567619728, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/31.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.78410+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.7841, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.78410+", "metric_symbol": "s", "metric_value": 3.7841, "reference_text": "s = 3.78410+ | Found by Emerson Connelly | in April 2026.", "source_image": "31.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/31.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.78410+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "31.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "83b47696242e2320", "improvement": 0.00017682363310500548, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0307084382837186e-06, "max_pair_overlap_depth": -2.9858800666679453e-05, "metric_expression": "3.78392", "metric_symbol": "s", "metric_value": 3.783923176366895, "n": 31, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.7841, "setup": "squinhex", "side": 3.783923176366895, "solution_path": "data/solutions/generated/squinhex_031_83b47696242e2320.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/83b47696242e2320.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@32", "density": 0.8302114212681493, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/32.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.84795+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.84795, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@32", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.84795+", "metric_symbol": "s", "metric_value": 3.84795, "reference_text": "s = 3.84795+ | Found by Emerson Connelly | in April 2026.", "source_image": "32.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/32.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.84795+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "32.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "b233eac9bd8cc0d8", "improvement": -0.003769870972095468, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1351534734416697e-07, "max_pair_overlap_depth": -1.2013295066060081e-05, "metric_expression": "3.85172", "metric_symbol": "s", "metric_value": 3.8517198709720954, "n": 32, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.84795, "setup": "squinhex", "side": 3.8517198709720954, "solution_path": "data/solutions/generated/squinhex_032_b233eac9bd8cc0d8.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/b233eac9bd8cc0d8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@33", "density": 0.8381951703957873, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/33.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.88897+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.88897, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@33", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.88897+", "metric_symbol": "s", "metric_value": 3.88897, "reference_text": "s = 3.88897+ | Found by Emerson Connelly | in April 2026.", "source_image": "33.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/33.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.88897+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "33.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "b14228e93eb45276", "improvement": -0.003797355726243623, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.797125554432215e-06, "max_pair_overlap_depth": -8.18730119578337e-06, "metric_expression": "3.89277", "metric_symbol": "s", "metric_value": 3.8927673557262437, "n": 33, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.88897, "setup": "squinhex", "side": 3.8927673557262437, "solution_path": "data/solutions/generated/squinhex_033_b14228e93eb45276.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/b14228e93eb45276.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@34", "density": 0.8486832973249016, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/34.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.92704+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.92704, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@34", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.92704+", "metric_symbol": "s", "metric_value": 3.92704, "reference_text": "s = 3.92704+ | Found by Emerson Connelly | in April 2026.", "source_image": "34.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/34.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.92704+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "34.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "fc2f6f3b6cabd5ed", "improvement": 0.0002227711135822652, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.30656742709778e-06, "max_pair_overlap_depth": -1.5057714820798651e-05, "metric_expression": "3.92682", "metric_symbol": "s", "metric_value": 3.9268172288864176, "n": 34, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.92704, "setup": "squinhex", "side": 3.9268172288864176, "solution_path": "data/solutions/generated/squinhex_034_fc2f6f3b6cabd5ed.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/fc2f6f3b6cabd5ed.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@35", "density": 0.8429987640989463, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/35.png", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.99833+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.99833, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@35", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.99833+", "metric_symbol": "s", "metric_value": 3.99833, "reference_text": "s = 3.99833+ | Found by Emerson Connelly | in April 2026.", "source_image": "35.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/35.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 3.99833+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "35.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "1ea058e93fb408a4", "improvement": 0.0007735056288105113, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.597269939847931e-06, "max_pair_overlap_depth": -1.4652234902046146e-05, "metric_expression": "3.99756", "metric_symbol": "s", "metric_value": 3.9975564943711897, "n": 35, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.99833, "setup": "squinhex", "side": 3.9975564943711897, "solution_path": "data/solutions/generated/squinhex_035_1ea058e93fb408a4.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/1ea058e93fb408a4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@36", "density": 0.8531956067904081, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/36.png", "friedman_metric_expression": "2+7\u221a3/6 = 4.02072+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.02072, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@36", "credit": "Maurizio Morandi", "date": "May 2026", "family_updated": "", "metric_expression": "2+7\u221a3/6 = 4.02072+", "metric_symbol": "s", "metric_value": 4.02072, "reference_text": "s = 2+7\u221a3/6 = 4.02072+ | Found by Maurizio Morandi | in May 2026.", "source_image": "36.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/36.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 2+7\u221a3/6 = 4.02072+ | Found by Maurizio Morandi | in May 2026.", "friedman_source_image": "36.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "b7083884dfb200c3", "improvement": -0.009242471992498835, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.100230231840266e-07, "max_pair_overlap_depth": -2.2076071326537772e-07, "metric_expression": "4.02996", "metric_symbol": "s", "metric_value": 4.029962471992499, "n": 36, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.02072, "setup": "squinhex", "side": 4.029962471992499, "solution_path": "data/solutions/record_hunt/squinhex_36_b7083884dfb200c3.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/b7083884dfb200c3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@37", "density": 0.8718839158050435, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/37.png", "friedman_metric_expression": "7/\u221a3 = 4.04145+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.04145, "friedman_reference": {"analytical_or_proved": true, "case": "squinhex@37", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "7/\u221a3 = 4.04145+", "metric_symbol": "s", "metric_value": 4.04145, "reference_text": "s = 7/\u221a3 = 4.04145+ | Trivial. |", "source_image": "37.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/37.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 7/\u221a3 = 4.04145+ | Trivial. |", "friedman_source_image": "37.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "a455ecb05be301ad", "improvement": -7.79426845127773e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1637259667017477e-06, "max_pair_overlap_depth": -1.0162747484221413e-05, "metric_expression": "4.04153", "metric_symbol": "s", "metric_value": 4.041527942684513, "n": 37, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.04145, "setup": "squinhex", "side": 4.041527942684513, "solution_path": "data/solutions/generated/squinhex_037_a455ecb05be301ad.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a455ecb05be301ad.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@38", "density": 0.8354467372021385, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/38.png", "friedman_metric_expression": "4.18476+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.18476, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@38", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.18476+", "metric_symbol": "s", "metric_value": 4.18476, "reference_text": "s = 4.18476+ | Found by Emerson Connelly | in April 2026.", "source_image": "38.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/38.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 4.18476+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "38.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "657f5bc54656101d", "improvement": 0.0006174563357541629, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.3358638004937594e-07, "max_pair_overlap_depth": -2.4459335521420655e-06, "metric_expression": "4.18414", "metric_symbol": "s", "metric_value": 4.184142543664246, "n": 38, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.18476, "setup": "squinhex", "side": 4.184142543664246, "solution_path": "data/solutions/record_hunt/squinhex_38_657f5bc54656101d.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/657f5bc54656101d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@39", "density": 0.8275155837189719, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/39.png", "friedman_metric_expression": "4.25744+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.25744, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@39", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.25744+", "metric_symbol": "s", "metric_value": 4.25744, "reference_text": "s = 4.25744+ | Found by Emerson Connelly | in April 2026.", "source_image": "39.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/39.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 4.25744+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "39.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "85e7e9eb3b3b7f5f", "improvement": -0.001664243334674076, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.204496689819836e-06, "max_pair_overlap_depth": -1.602842749781619e-05, "metric_expression": "4.25910", "metric_symbol": "s", "metric_value": 4.259104243334674, "n": 39, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.25744, "setup": "squinhex", "side": 4.259104243334674, "solution_path": "data/solutions/generated/squinhex_039_85e7e9eb3b3b7f5f.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/85e7e9eb3b3b7f5f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinhex@40", "density": 0.8288203735668404, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinhex/40.png", "friedman_metric_expression": "4.31157+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.31157, "friedman_reference": {"analytical_or_proved": false, "case": "squinhex@40", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.31157+", "metric_symbol": "s", "metric_value": 4.31157, "reference_text": "s = 4.31157+ | Found by Emerson Connelly | in April 2026.", "source_image": "40.png", "source_image_url": "https://erich-friedman.github.io/packing/squinhex/40.png", "source_page": "https://erich-friedman.github.io/packing/squinhex/", "title": "Squares in Hexagons"}, "friedman_reference_text": "s = 4.31157+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "40.png", "friedman_source_url": "https://erich-friedman.github.io/packing/squinhex/", "friedman_title": "Squares in Hexagons", "frontend_seed": true, "id": "9859865bba74b499", "improvement": 0.001604007461533108, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.921136271031391e-07, "max_pair_overlap_depth": -1.8467799072219293e-06, "metric_expression": "4.30997", "metric_symbol": "s", "metric_value": 4.309965992538467, "n": 40, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.31157, "setup": "squinhex", "side": 4.309965992538467, "solution_path": "data/solutions/record_hunt/squinhex_40_9859865bba74b499.json", "source_url": "https://erich-friedman.github.io/packing/squinhex/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/9859865bba74b499.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@1", "density": 0.7071036297921394, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/1.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "1/\u221a(2+\u221a2) = .54119+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.54119, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1/\u221a(2+\u221a2) = .54119+", "metric_symbol": "s", "metric_value": 0.54119, "reference_text": "s = 1/\u221a(2+\u221a2) = .54119+ | Trivial. |", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/1.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 1/\u221a(2+\u221a2) = .54119+ | Trivial. |", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "b63c90f60717ba08", "improvement": -7.30613665711477e-06, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1705346114441018e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.54120", "metric_symbol": "s", "metric_value": 0.5411973061366571, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.54119, "setup": "squinoct", "side": 0.5411973061366571, "solution_path": "data/solutions/generated/squinoct_001_b63c90f60717ba08.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b63c90f60717ba08.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@2", "density": 0.5389047922692529, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "December 2013", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/2.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": ".87667+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.87667, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@2", "credit": "Maurizio Morandi", "date": "December 2013", "family_updated": "", "metric_expression": ".87667+", "metric_symbol": "s", "metric_value": 0.87667, "reference_text": "s = .87667+ | Found by Maurizio Morandi | in December 2013.", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/2.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = .87667+ | Found by Maurizio Morandi | in December 2013.", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "7194ac58cba12fa7", "improvement": -4.0351762315649076e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.607981571160174e-06, "max_pair_overlap_depth": -1.1896285956325414e-06, "metric_expression": "0.87671", "metric_symbol": "s", "metric_value": 0.8767103517623156, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.87667, "setup": "squinoct", "side": 0.8767103517623156, "solution_path": "data/solutions/generated/squinoct_002_7194ac58cba12fa7.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/7194ac58cba12fa7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@3", "density": 0.5862650280603752, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "December 2013", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/3.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "18 \u2013 12\u221a2 = 1.02940+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0294, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@3", "credit": "Erich Friedman", "date": "December 2013", "family_updated": "", "metric_expression": "18 \u2013 12\u221a2 = 1.02940+", "metric_symbol": "s", "metric_value": 1.0294, "reference_text": "s = 18 \u2013 12\u221a2 = 1.02940+ | Found by Erich Friedman | in December 2013.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/3.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 18 \u2013 12\u221a2 = 1.02940+ | Found by Erich Friedman | in December 2013.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "3d85991afac60053", "improvement": -6.311956912941064e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.835100684478856e-06, "max_pair_overlap_depth": -3.4409461626638826e-05, "metric_expression": "1.02946", "metric_symbol": "s", "metric_value": 1.0294631195691295, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.0294, "setup": "squinoct", "side": 1.0294631195691295, "solution_path": "data/solutions/generated/squinoct_003_3d85991afac60053.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3d85991afac60053.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@4", "density": 0.7070860385519216, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/4.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "\u221a(4\u20132\u221a2) = 1.08239+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.08239, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a(4\u20132\u221a2) = 1.08239+", "metric_symbol": "s", "metric_value": 1.08239, "reference_text": "s = \u221a(4\u20132\u221a2) = 1.08239+ | Trivial. |", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/4.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = \u221a(4\u20132\u221a2) = 1.08239+ | Trivial. |", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "30bfa7c46cf6dd87", "improvement": -1.8076366912644914e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.1007450963540665e-06, "max_pair_overlap_depth": -1.5610612916431253e-05, "metric_expression": "1.08241", "metric_symbol": "s", "metric_value": 1.0824080763669126, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.08239, "setup": "squinoct", "side": 1.0824080763669126, "solution_path": "data/solutions/generated/squinoct_004_30bfa7c46cf6dd87.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/30bfa7c46cf6dd87.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@5", "density": 0.6705984373456575, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/5.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "3\u221a2 \u2013 3 = 1.24261+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.24261, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3\u221a2 \u2013 3 = 1.24261+", "metric_symbol": "s", "metric_value": 1.24261, "reference_text": "s = 3\u221a2 \u2013 3 = 1.24261+ | Trivial. |", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/5.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 3\u221a2 \u2013 3 = 1.24261+ | Trivial. |", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "3bf97518bf42702c", "improvement": -4.591996404545995e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2899385621933342e-06, "max_pair_overlap_depth": -9.75715015771872e-06, "metric_expression": "1.24266", "metric_symbol": "s", "metric_value": 1.2426559199640455, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.24261, "setup": "squinoct", "side": 1.2426559199640455, "solution_path": "data/solutions/generated/squinoct_005_3bf97518bf42702c.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3bf97518bf42702c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@6", "density": 0.6887710971991018, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "December 2013", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/6.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "7 \u2013 4\u221a2 = 1.34314+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.34314, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@6", "credit": "Maurizio Morandi", "date": "December 2013", "family_updated": "", "metric_expression": "7 \u2013 4\u221a2 = 1.34314+", "metric_symbol": "s", "metric_value": 1.34314, "reference_text": "s = 7 \u2013 4\u221a2 = 1.34314+ | Found by Maurizio Morandi | in December 2013.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/6.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 7 \u2013 4\u221a2 = 1.34314+ | Found by Maurizio Morandi | in December 2013.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "6368fdc8880da7e0", "improvement": -4.34357115446371e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.645757686457117e-06, "max_pair_overlap_depth": -2.457675057909725e-05, "metric_expression": "1.34318", "metric_symbol": "s", "metric_value": 1.3431834357115446, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.34314, "setup": "squinoct", "side": 1.3431834357115446, "solution_path": "data/solutions/generated/squinoct_006_6368fdc8880da7e0.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/6368fdc8880da7e0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@7", "density": 0.702115846733366, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "December 2013", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/7.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "1.43691+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.43691, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@7", "credit": "Maurizio Morandi", "date": "December 2013", "family_updated": "", "metric_expression": "1.43691+", "metric_symbol": "s", "metric_value": 1.43691, "reference_text": "s = 1.43691+ | Found by Maurizio Morandi | in December 2013.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/7.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 1.43691+ | Found by Maurizio Morandi | in December 2013.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "35d0c06f5c122826", "improvement": -4.0447358323758564e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.691262438170272e-07, "max_pair_overlap_depth": -1.2495412028745712e-05, "metric_expression": "1.43695", "metric_symbol": "s", "metric_value": 1.4369504473583237, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.43691, "setup": "squinoct", "side": 1.4369504473583237, "solution_path": "data/solutions/generated/squinoct_007_35d0c06f5c122826.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/35d0c06f5c122826.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@8", "density": 0.6722160358662614, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/8.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "1.56967+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.56967, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@8", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.56967+", "metric_symbol": "s", "metric_value": 1.56967, "reference_text": "s = 1.56967+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/8.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 1.56967+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "ecaef16600354c9f", "improvement": -0.00028685978676801227, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.463017743294653e-07, "max_pair_overlap_depth": -1.4183990756477804e-05, "metric_expression": "1.56996", "metric_symbol": "s", "metric_value": 1.569956859786768, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.56967, "setup": "squinoct", "side": 1.569956859786768, "solution_path": "data/solutions/generated/squinoct_008_ecaef16600354c9f.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/ecaef16600354c9f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@9", "density": 0.707064805820287, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/9.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "3/\u221a(2+\u221a2) = 1.62358+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.62358, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@9", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3/\u221a(2+\u221a2) = 1.62358+", "metric_symbol": "s", "metric_value": 1.62358, "reference_text": "s = 3/\u221a(2+\u221a2) = 1.62358+ | Trivial. |", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/9.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 3/\u221a(2+\u221a2) = 1.62358+ | Trivial. |", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "703059104a2b70f8", "improvement": -5.649241589167353e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6691071523245226e-06, "max_pair_overlap_depth": -7.928509179855325e-06, "metric_expression": "1.62364", "metric_symbol": "s", "metric_value": 1.6236364924158917, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.62358, "setup": "squinoct", "side": 1.6236364924158917, "solution_path": "data/solutions/generated/squinoct_009_703059104a2b70f8.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/703059104a2b70f8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@10", "density": 0.732083417533705, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "December 2013", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/10.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2^(3/4) = 1.68179+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.68179, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@10", "credit": "Maurizio Morandi", "date": "December 2013", "family_updated": "", "metric_expression": "2^(3/4) = 1.68179+", "metric_symbol": "s", "metric_value": 1.68179, "reference_text": "s = 2^(3/4) = 1.68179+ | Found by Maurizio Morandi | in December 2013.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/10.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2^(3/4) = 1.68179+ | Found by Maurizio Morandi | in December 2013.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "03facc71923f1148", "improvement": -0.00017469136174708133, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.2978224497762506e-06, "max_pair_overlap_depth": -2.3560442145575955e-05, "metric_expression": "1.68196", "metric_symbol": "s", "metric_value": 1.681964691361747, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.68179, "setup": "squinoct", "side": 1.681964691361747, "solution_path": "data/solutions/generated/squinoct_010_03facc71923f1148.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/03facc71923f1148.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@11", "density": 0.7597329550134904, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/11.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "1.73180+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.7318, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@11", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.73180+", "metric_symbol": "s", "metric_value": 1.7318, "reference_text": "s = 1.73180+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/11.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 1.73180+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "78c32afdb2263b2d", "improvement": 0.00013842799014240903, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.409684401929808e-09, "max_pair_overlap_depth": -1.1021961062729346e-05, "metric_expression": "1.73166", "metric_symbol": "s", "metric_value": 1.7316615720098576, "n": 11, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.7318, "setup": "squinoct", "side": 1.7316615720098576, "solution_path": "data/solutions/generated/squinoct_011_78c32afdb2263b2d.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/78c32afdb2263b2d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@12", "density": 0.8083084838512736, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "December 2013", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/12.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "1.7533+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.7533, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@12", "credit": "Maurizio Morandi", "date": "December 2013", "family_updated": "", "metric_expression": "1.7533+", "metric_symbol": "s", "metric_value": 1.7533, "reference_text": "s = 1.7533+ | Found by Maurizio Morandi | in December 2013.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/12.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 1.7533+ | Found by Maurizio Morandi | in December 2013.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "7a9451cae59dbb40", "improvement": -0.00017352882969978722, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.088823188597466e-06, "max_pair_overlap_depth": -2.0797652260817756e-05, "metric_expression": "1.75347", "metric_symbol": "s", "metric_value": 1.7534735288296999, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.7533, "setup": "squinoct", "side": 1.7534735288296999, "solution_path": "data/solutions/generated/squinoct_012_7a9451cae59dbb40.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/7a9451cae59dbb40.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@13", "density": 0.7475846505942733, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/13.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "1.89760+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.8976, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@13", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.89760+", "metric_symbol": "s", "metric_value": 1.8976, "reference_text": "s = 1.89760+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/13.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 1.89760+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "c5bdcacf094e428e", "improvement": -0.00014846020367564883, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.439964464558682e-06, "max_pair_overlap_depth": -2.7883136850093138e-05, "metric_expression": "1.89775", "metric_symbol": "s", "metric_value": 1.8977484602036756, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.8976, "setup": "squinoct", "side": 1.8977484602036756, "solution_path": "data/solutions/generated/squinoct_013_c5bdcacf094e428e.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/c5bdcacf094e428e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@14", "density": 0.7369251630765782, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/14.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "1.98350+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.9835, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@14", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.98350+", "metric_symbol": "s", "metric_value": 1.9835, "reference_text": "s = 1.98350+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/14.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 1.98350+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "994e0dd57a80d4a3", "improvement": -7.891975572338339e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.49261632156563e-06, "max_pair_overlap_depth": -2.894591689805992e-05, "metric_expression": "1.98358", "metric_symbol": "s", "metric_value": 1.9835789197557234, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.9835, "setup": "squinoct", "side": 1.9835789197557234, "solution_path": "data/solutions/generated/squinoct_014_994e0dd57a80d4a3.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/994e0dd57a80d4a3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@15", "density": 0.7515649396279209, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/15.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.03259+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.03259, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@15", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.03259+", "metric_symbol": "s", "metric_value": 2.03259, "reference_text": "s = 2.03259+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/15.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.03259+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "c898d8b3cd2b55db", "improvement": -0.0005137104569299389, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7983161874182644e-07, "max_pair_overlap_depth": -1.9704810269915995e-05, "metric_expression": "2.03310", "metric_symbol": "s", "metric_value": 2.03310371045693, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.03259, "setup": "squinoct", "side": 2.03310371045693, "solution_path": "data/solutions/generated/squinoct_015_c898d8b3cd2b55db.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/c898d8b3cd2b55db.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@16", "density": 0.7725175260502497, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/16.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "5\u221a2 \u2013 5 = 2.07106+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.07106, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@16", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5\u221a2 \u2013 5 = 2.07106+", "metric_symbol": "s", "metric_value": 2.07106, "reference_text": "s = 5\u221a2 \u2013 5 = 2.07106+ | Trivial. |", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/16.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 5\u221a2 \u2013 5 = 2.07106+ | Trivial. |", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "3a2e6d99239355fd", "improvement": -4.911646819039106e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5220970331887e-06, "max_pair_overlap_depth": -8.060079695892952e-07, "metric_expression": "2.07111", "metric_symbol": "s", "metric_value": 2.0711091164681905, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.07106, "setup": "squinoct", "side": 2.0711091164681905, "solution_path": "data/solutions/generated/squinoct_016_3a2e6d99239355fd.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/3a2e6d99239355fd.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@17", "density": 0.8208079603816107, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/17.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "5\u221a2 \u2013 5 = 2.07106+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.07106, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@17", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5\u221a2 \u2013 5 = 2.07106+", "metric_symbol": "s", "metric_value": 2.07106, "reference_text": "s = 5\u221a2 \u2013 5 = 2.07106+ | Trivial. |", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/17.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 5\u221a2 \u2013 5 = 2.07106+ | Trivial. |", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "ab660f0510a3dd1c", "improvement": -3.891119068244464e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.368661750365744e-09, "max_pair_overlap_depth": -1.0480566122961221e-05, "metric_expression": "2.07110", "metric_symbol": "s", "metric_value": 2.0710989111906826, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.07106, "setup": "squinoct", "side": 2.0710989111906826, "solution_path": "data/solutions/generated/squinoct_017_ab660f0510a3dd1c.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/ab660f0510a3dd1c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@18", "density": 0.7934296897658486, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/18.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.16778+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.16778, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@18", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.16778+", "metric_symbol": "s", "metric_value": 2.16778, "reference_text": "s = 2.16778+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/18.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.16778+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "a7275e2ab7024187", "improvement": 0.00017977070139485463, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9711774532196686e-06, "max_pair_overlap_depth": -1.4522556675777842e-05, "metric_expression": "2.16760", "metric_symbol": "s", "metric_value": 2.167600229298605, "n": 18, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.16778, "setup": "squinoct", "side": 2.167600229298605, "solution_path": "data/solutions/generated/squinoct_018_a7275e2ab7024187.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/a7275e2ab7024187.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@19", "density": 0.793469906795261, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/19.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.22823+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.22823, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@19", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.22823+", "metric_symbol": "s", "metric_value": 2.22823, "reference_text": "s = 2.22823+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/19.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.22823+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "330906f1db929cdb", "improvement": 0.0012889040224068893, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0543459428412958e-06, "max_pair_overlap_depth": -1.4850274468458657e-05, "metric_expression": "2.22694", "metric_symbol": "s", "metric_value": 2.226941095977593, "n": 19, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.22823, "setup": "squinoct", "side": 2.226941095977593, "solution_path": "data/solutions/generated/squinoct_019_330906f1db929cdb.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/330906f1db929cdb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@20", "density": 0.7912106699597153, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/20.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.28730+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.2873, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@20", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.28730+", "metric_symbol": "s", "metric_value": 2.2873, "reference_text": "s = 2.28730+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "20.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/20.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.28730+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "d1597d2fdf3beee1", "improvement": -0.0007530500807275686, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.367182536359887e-05, "max_pair_overlap_depth": -8.661317408398261e-06, "metric_expression": "2.28805", "metric_symbol": "s", "metric_value": 2.2880530500807277, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.2873, "setup": "squinoct", "side": 2.2880530500807277, "solution_path": "data/solutions/generated/squinoct_020_d1597d2fdf3beee1.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/d1597d2fdf3beee1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@21", "density": 0.8091484006744551, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/21.gif", "friedman_metric_expression": "2.32167+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.32167, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@21", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.32167+", "metric_symbol": "s", "metric_value": 2.32167, "reference_text": "s = 2.32167+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/21.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.32167+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "a3759fa0325bb184", "improvement": 0.0032467851108277657, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.288437924896875e-06, "max_pair_overlap_depth": -7.589979277911052e-06, "metric_expression": "2.31842", "metric_symbol": "s", "metric_value": 2.3184232148891724, "n": 21, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.32167, "setup": "squinoct", "side": 2.3184232148891724, "solution_path": "data/solutions/generated/squinoct_021_a3759fa0325bb184.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/a3759fa0325bb184.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@22", "density": 0.779088370401765, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/22.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.41811+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.41811, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@22", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.41811+", "metric_symbol": "s", "metric_value": 2.41811, "reference_text": "s = 2.41811+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/22.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.41811+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "c01cc63fb6a654c8", "improvement": -0.00021763715445111131, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4586625716053672e-06, "max_pair_overlap_depth": -2.804598842454098e-05, "metric_expression": "2.41833", "metric_symbol": "s", "metric_value": 2.418327637154451, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.41811, "setup": "squinoct", "side": 2.418327637154451, "solution_path": "data/solutions/generated/squinoct_022_c01cc63fb6a654c8.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/c01cc63fb6a654c8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@23", "density": 0.7777209980912639, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/23.gif", "friedman_metric_expression": "2.47329+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.47329, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@23", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.47329+", "metric_symbol": "s", "metric_value": 2.47329, "reference_text": "s = 2.47329+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/23.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.47329+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "004d1c5d4d7bca47", "improvement": -0.0015616154620987643, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.985478608046375e-06, "max_pair_overlap_depth": -1.4996046389881279e-05, "metric_expression": "2.47485", "metric_symbol": "s", "metric_value": 2.4748516154620988, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.47329, "setup": "squinoct", "side": 2.4748516154620988, "solution_path": "data/solutions/generated/squinoct_023_004d1c5d4d7bca47.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/004d1c5d4d7bca47.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@24", "density": 0.804709078453695, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/24.gif", "friedman_metric_expression": "6\u221a2 \u2013 6 = 2.48528+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.48528, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@24", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6\u221a2 \u2013 6 = 2.48528+", "metric_symbol": "s", "metric_value": 2.48528, "reference_text": "s = 6\u221a2 \u2013 6 = 2.48528+ | Trivial.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/24.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 6\u221a2 \u2013 6 = 2.48528+ | Trivial.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "4f255616aca5e547", "improvement": -4.580955124477981e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.67146332394231e-06, "max_pair_overlap_depth": -9.95881463161652e-06, "metric_expression": "2.48533", "metric_symbol": "s", "metric_value": 2.4853258095512447, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.48528, "setup": "squinoct", "side": 2.4853258095512447, "solution_path": "data/solutions/generated/squinoct_024_4f255616aca5e547.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4f255616aca5e547.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@25", "density": 0.8352526418920933, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/25.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "11/(3+\u221a2) = 2.49195+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.49195, "friedman_reference": {"analytical_or_proved": true, "case": "squinoct@25", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "11/(3+\u221a2) = 2.49195+", "metric_symbol": "s", "metric_value": 2.49195, "reference_text": "s = 11/(3+\u221a2) = 2.49195+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/25.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 11/(3+\u221a2) = 2.49195+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "0d3ca23da8f19da7", "improvement": 0.0021857033362460143, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.3164106628379786e-06, "max_pair_overlap_depth": -1.12459370678053e-05, "metric_expression": "2.48976", "metric_symbol": "s", "metric_value": 2.489764296663754, "n": 25, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.49195, "setup": "squinoct", "side": 2.489764296663754, "solution_path": "data/solutions/generated/squinoct_025_0d3ca23da8f19da7.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/0d3ca23da8f19da7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@26", "density": 0.8421766228224283, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/26.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.52841+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.52841, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@26", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.52841+", "metric_symbol": "s", "metric_value": 2.52841, "reference_text": "s = 2.52841+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/26.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.52841+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "6c0cffa2ebc39455", "improvement": -0.00020227836257324938, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3125873649857306e-07, "max_pair_overlap_depth": -6.684300461579085e-06, "metric_expression": "2.52861", "metric_symbol": "s", "metric_value": 2.5286122783625733, "n": 26, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.52841, "setup": "squinoct", "side": 2.5286122783625733, "solution_path": "data/solutions/generated/squinoct_026_6c0cffa2ebc39455.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/6c0cffa2ebc39455.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@27", "density": 0.8354347334144809, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/27.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.58695+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.58695, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@27", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.58695+", "metric_symbol": "s", "metric_value": 2.58695, "reference_text": "s = 2.58695+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/27.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.58695+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "9243c0031663b10e", "improvement": -0.0002069616913620287, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1228507689492215e-06, "max_pair_overlap_depth": -1.654834709174935e-05, "metric_expression": "2.58716", "metric_symbol": "s", "metric_value": 2.587156961691362, "n": 27, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.58695, "setup": "squinoct", "side": 2.587156961691362, "solution_path": "data/solutions/generated/squinoct_027_9243c0031663b10e.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/9243c0031663b10e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@28", "density": 0.8375235613860185, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/28.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.63116+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.63116, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@28", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.63116+", "metric_symbol": "s", "metric_value": 2.63116, "reference_text": "s = 2.63116+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/28.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.63116+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "f74e0430c683aaa0", "improvement": -0.0001841851467743183, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.686366302485504e-06, "max_pair_overlap_depth": -8.053108887406957e-06, "metric_expression": "2.63134", "metric_symbol": "s", "metric_value": 2.6313441851467743, "n": 28, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.63116, "setup": "squinoct", "side": 2.6313441851467743, "solution_path": "data/solutions/generated/squinoct_028_f74e0430c683aaa0.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/f74e0430c683aaa0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@29", "density": 0.8192713469622411, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/29.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.70811+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.70811, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@29", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.70811+", "metric_symbol": "s", "metric_value": 2.70811, "reference_text": "s = 2.70811+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/29.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.70811+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "7d4fefbdad04f318", "improvement": 0.0005239033117416803, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.8746413323634285e-06, "max_pair_overlap_depth": -4.299633667326219e-06, "metric_expression": "2.70759", "metric_symbol": "s", "metric_value": 2.7075860966882583, "n": 29, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.70811, "setup": "squinoct", "side": 2.7075860966882583, "solution_path": "data/solutions/generated/squinoct_029_7d4fefbdad04f318.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/7d4fefbdad04f318.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinoct@30", "density": 0.8256466842229233, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinoct/30.gif", "friedman_family_updated": "3/17/26", "friedman_metric_expression": "2.74220+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.7422, "friedman_reference": {"analytical_or_proved": false, "case": "squinoct@30", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.74220+", "metric_symbol": "s", "metric_value": 2.7422, "reference_text": "s = 2.74220+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinoct/30.gif", "source_page": "https://erich-friedman.github.io/packing/squinoct/", "title": "Squares in Octagons"}, "friedman_reference_text": "s = 2.74220+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinoct/", "friedman_title": "Squares in Octagons", "frontend_seed": true, "id": "e6809975fe4af696", "improvement": -0.0010201760226595802, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.275653162265926e-06, "max_pair_overlap_depth": -1.2928136431611392e-05, "metric_expression": "2.74322", "metric_symbol": "s", "metric_value": 2.7432201760226596, "n": 30, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.7422, "setup": "squinoct", "side": 2.7432201760226596, "solution_path": "data/solutions/generated/squinoct_030_e6809975fe4af696.json", "source_url": "https://erich-friedman.github.io/packing/squinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/e6809975fe4af696.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@1", "density": 0.6621920830904189, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/1.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": ".936+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.936, "friedman_reference": {"analytical_or_proved": true, "case": "squinpen@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": ".936+", "metric_symbol": "s", "metric_value": 0.936, "reference_text": "s = cos(27\u00b0) / cos(18\u00b0) | s = .936+ | Trivial. |", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/1.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = cos(27\u00b0) / cos(18\u00b0) | s = .936+ | Trivial. |", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "7c4deaf46828e45f", "improvement": -0.0008790374068264883, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1116494686738636e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.93688", "metric_symbol": "s", "metric_value": 0.9368790374068265, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.936, "setup": "squinpen", "side": 0.9368790374068265, "solution_path": "data/solutions/generated/squinpen_001_7c4deaf46828e45f.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/7c4deaf46828e45f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@2", "density": 0.49287502021209156, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "March 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/2.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "1.535+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.535, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@2", "credit": "Erich Friedman", "date": "March 2012", "family_updated": "", "metric_expression": "1.535+", "metric_symbol": "s", "metric_value": 1.535, "reference_text": "s = 4\u221a(1\u20132/\u221a5)+\u221a5\u20132 | s = 1.535+ | Found by Erich Friedman | in March 2012.", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/2.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4\u221a(1\u20132/\u221a5)+\u221a5\u20132 | s = 1.535+ | Found by Erich Friedman | in March 2012.", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "e0d70422bfa035f2", "improvement": -0.0007556004426736429, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.603512178169518e-07, "max_pair_overlap_depth": -4.1992473865937185e-06, "metric_expression": "1.53576", "metric_symbol": "s", "metric_value": 1.5357556004426736, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.535, "setup": "squinpen", "side": 1.5357556004426736, "solution_path": "data/solutions/generated/squinpen_002_e0d70422bfa035f2.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/e0d70422bfa035f2.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@3", "density": 0.6064545372240481, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/3.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "1.695+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.695, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@3", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "1.695+", "metric_symbol": "s", "metric_value": 1.695, "reference_text": "s = 1.695+ | Found by David W. Cantrell | in July 2012. |", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/3.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 1.695+ | Found by David W. Cantrell | in July 2012. |", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "20345bd1acea07ee", "improvement": -0.0006531320538840646, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.096669049020619e-07, "max_pair_overlap_depth": -2.3507012493845236e-05, "metric_expression": "1.69565", "metric_symbol": "s", "metric_value": 1.6956531320538841, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.695, "setup": "squinpen", "side": 1.6956531320538841, "solution_path": "data/solutions/generated/squinpen_003_20345bd1acea07ee.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/20345bd1acea07ee.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@4", "density": 0.6707731008924028, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/4.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "1.861+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.861, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@4", "credit": "Maurizio Morandi", "date": "July 2012", "family_updated": "", "metric_expression": "1.861+", "metric_symbol": "s", "metric_value": 1.861, "reference_text": "s = 1.861+ | Found by Maurizio Morandi | in July 2012. |", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/4.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 1.861+ | Found by Maurizio Morandi | in July 2012. |", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "3b3728a72d1e463a", "improvement": -0.0007342580262259535, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8617284829398191e-06, "max_pair_overlap_depth": -1.61527471932571e-05, "metric_expression": "1.86173", "metric_symbol": "s", "metric_value": 1.861734258026226, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.861, "setup": "squinpen", "side": 1.861734258026226, "solution_path": "data/solutions/generated/squinpen_004_3b3728a72d1e463a.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/3b3728a72d1e463a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@5", "density": 0.7633756933560932, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "March 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/5.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "1.951+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.951, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@5", "credit": "Erich Friedman", "date": "March 2012", "family_updated": "", "metric_expression": "1.951+", "metric_symbol": "s", "metric_value": 1.951, "reference_text": "s = 1 + cos(18\u00b0) | s = 1.951+ | Found by Erich Friedman | in March 2012.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/5.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 1 + cos(18\u00b0) | s = 1.951+ | Found by Erich Friedman | in March 2012.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "dfdcf10f7cfe5f16", "improvement": -0.00015311806809270578, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.72748390891526e-06, "max_pair_overlap_depth": -2.251588894827683e-05, "metric_expression": "1.95115", "metric_symbol": "s", "metric_value": 1.9511531180680928, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.951, "setup": "squinpen", "side": 1.9511531180680928, "solution_path": "data/solutions/generated/squinpen_005_dfdcf10f7cfe5f16.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/dfdcf10f7cfe5f16.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@6", "density": 0.7300382566069571, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "March 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/6.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "2.185+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.185, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@6", "credit": "Erich Friedman", "date": "March 2012", "family_updated": "", "metric_expression": "2.185+", "metric_symbol": "s", "metric_value": 2.185, "reference_text": "s = 6\u221a(1\u20132/\u221a5)+\u221a5\u20132 | s = 2.185+ | Found by Erich Friedman | in March 2012.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/6.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 6\u221a(1\u20132/\u221a5)+\u221a5\u20132 | s = 2.185+ | Found by Erich Friedman | in March 2012.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "fbc8fa063fa8154f", "improvement": -0.0006384860733987807, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.046305607470139e-06, "max_pair_overlap_depth": -1.535481943848449e-05, "metric_expression": "2.18564", "metric_symbol": "s", "metric_value": 2.185638486073399, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.185, "setup": "squinpen", "side": 2.185638486073399, "solution_path": "data/solutions/generated/squinpen_006_fbc8fa063fa8154f.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/fbc8fa063fa8154f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@7", "density": 0.6937337333431398, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/7.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "2.421+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.421, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@7", "credit": "Erich Friedman", "date": "July 2012", "family_updated": "", "metric_expression": "2.421+", "metric_symbol": "s", "metric_value": 2.421, "reference_text": "s = 6\u221a(1\u20132/\u221a5)+2\u221a5\u20134 | s = 2.421+ | Found by Erich Friedman | in July 2012.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/7.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 6\u221a(1\u20132/\u221a5)+2\u221a5\u20134 | s = 2.421+ | Found by Erich Friedman | in July 2012.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "a9079c85f43f1608", "improvement": -0.0007434143519042813, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.2537276094194668e-06, "max_pair_overlap_depth": -4.1098373515402464e-05, "metric_expression": "2.42174", "metric_symbol": "s", "metric_value": 2.421743414351904, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.421, "setup": "squinpen", "side": 2.421743414351904, "solution_path": "data/solutions/generated/squinpen_007_a9079c85f43f1608.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a9079c85f43f1608.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@8", "density": 0.7064703560745587, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/8.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "2.565+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.565, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@8", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "2.565+", "metric_symbol": "s", "metric_value": 2.565, "reference_text": "s = 2.565+ | Found by David W. Cantrell | in July 2012. |", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/8.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 2.565+ | Found by David W. Cantrell | in July 2012. |", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "eedfd38786ce3d06", "improvement": -0.0005089596009386632, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2839977085297321e-06, "max_pair_overlap_depth": -2.052513031336467e-05, "metric_expression": "2.56551", "metric_symbol": "s", "metric_value": 2.5655089596009386, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.565, "setup": "squinpen", "side": 2.5655089596009386, "solution_path": "data/solutions/generated/squinpen_008_eedfd38786ce3d06.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/eedfd38786ce3d06.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@9", "density": 0.7204851744660768, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/9.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "2.693+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.693, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@9", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "2.693+", "metric_symbol": "s", "metric_value": 2.693, "reference_text": "s = 2.693+ | Found by Timo Berthold et al | in January 2026. |", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/9.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 2.693+ | Found by Timo Berthold et al | in January 2026. |", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "b050939c544c8091", "improvement": -0.0015375753912136858, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.502705162025631e-06, "max_pair_overlap_depth": -2.4857642315079254e-05, "metric_expression": "2.69454", "metric_symbol": "s", "metric_value": 2.6945375753912137, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.693, "setup": "squinpen", "side": 2.6945375753912137, "solution_path": "data/solutions/generated/squinpen_009_b050939c544c8091.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/b050939c544c8091.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@10", "density": 0.7436124164686567, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/10.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "2.793+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.793, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@10", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "2.793+", "metric_symbol": "s", "metric_value": 2.793, "reference_text": "s = 2.793+ | Found by Timo Berthold et al | in January 2026.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/10.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 2.793+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "5e3ff0fc92953694", "improvement": -0.0027748736770187854, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.695299675143831e-07, "max_pair_overlap_depth": -2.06179882833446e-05, "metric_expression": "2.79577", "metric_symbol": "s", "metric_value": 2.795774873677019, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.793, "setup": "squinpen", "side": 2.795774873677019, "solution_path": "data/solutions/generated/squinpen_010_5e3ff0fc92953694.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/5e3ff0fc92953694.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@11", "density": 0.7644165493663143, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/11.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "2.890+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.89, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@11", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "2.890+", "metric_symbol": "s", "metric_value": 2.89, "reference_text": "s = 2.890+ | Found by Timo Berthold et al | in January 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/11.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 2.890+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "eed6c273a2c34445", "improvement": -0.002056785153238039, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.157725543754771e-07, "max_pair_overlap_depth": -2.6396452143861993e-05, "metric_expression": "2.89206", "metric_symbol": "s", "metric_value": 2.892056785153238, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.89, "setup": "squinpen", "side": 2.892056785153238, "solution_path": "data/solutions/generated/squinpen_011_eed6c273a2c34445.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/eed6c273a2c34445.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@12", "density": 0.7899142461050866, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/12.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "2.969+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.969, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@12", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "2.969+", "metric_symbol": "s", "metric_value": 2.969, "reference_text": "s = 2.969+ | Found by Timo Berthold et al | in January 2026.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/12.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 2.969+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "0dc0d2750d6b1986", "improvement": -0.0025029358297610216, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6525891073399466e-06, "max_pair_overlap_depth": -2.2708088333933674e-05, "metric_expression": "2.97150", "metric_symbol": "s", "metric_value": 2.971502935829761, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.969, "setup": "squinpen", "side": 2.971502935829761, "solution_path": "data/solutions/generated/squinpen_012_0dc0d2750d6b1986.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/0dc0d2750d6b1986.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@13", "density": 0.7748259330672266, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/13.gif", "friedman_metric_expression": "3.121+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.121, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@13", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "3.121+", "metric_symbol": "s", "metric_value": 3.121, "reference_text": "s = 2\u221a(1\u20132/\u221a5)+2\u221a5\u20132 | s = 3.121+ | Found by David W. Cantrell | in July 2012.", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/13.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 2\u221a(1\u20132/\u221a5)+2\u221a5\u20132 | s = 3.121+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "13449338c659bacc", "improvement": -0.0018067975436526496, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.882913072523962e-06, "max_pair_overlap_depth": -3.6524180263897144e-05, "metric_expression": "3.12281", "metric_symbol": "s", "metric_value": 3.1228067975436526, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.121, "setup": "squinpen", "side": 3.1228067975436526, "solution_path": "data/solutions/generated/squinpen_13_13449338c659bacc.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/13449338c659bacc.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@14", "density": 0.8005863693145643, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/14.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.187+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.187, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@14", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "3.187+", "metric_symbol": "s", "metric_value": 3.187, "reference_text": "s = 3.187+ | Found by David W. Cantrell | in July 2012. |", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/14.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.187+ | Found by David W. Cantrell | in July 2012. |", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "c910e2e69c5e952c", "improvement": -0.0011256975399334124, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.345127969056754e-07, "max_pair_overlap_depth": -6.672013015962541e-06, "metric_expression": "3.18813", "metric_symbol": "s", "metric_value": 3.1881256975399332, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.187, "setup": "squinpen", "side": 3.1881256975399332, "solution_path": "data/solutions/generated/squinpen_014_c910e2e69c5e952c.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/c910e2e69c5e952c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@15", "density": 0.8079241717093747, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/15.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.284+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.284, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@15", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "3.284+", "metric_symbol": "s", "metric_value": 3.284, "reference_text": "s = (\u221a(125\u201338\u221a5)+8\u20132\u221a5)/3 | s = 3.284+ | Found by David W. Cantrell | in July 2012.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/15.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = (\u221a(125\u201338\u221a5)+8\u20132\u221a5)/3 | s = 3.284+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "d5cf3c3cc0980b70", "improvement": -0.001003544834322856, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.349360852435288e-06, "max_pair_overlap_depth": -2.4505079251779804e-05, "metric_expression": "3.28500", "metric_symbol": "s", "metric_value": 3.2850035448343227, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.284, "setup": "squinpen", "side": 3.2850035448343227, "solution_path": "data/solutions/generated/squinpen_015_d5cf3c3cc0980b70.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d5cf3c3cc0980b70.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@16", "density": 0.7674551673762713, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/16.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.480+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.48, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@16", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3.480+", "metric_symbol": "s", "metric_value": 3.48, "reference_text": "s = 3.480+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/16.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.480+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "9ca0c9eddf5257a6", "improvement": -0.0010398730905230025, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.610061847707955e-09, "max_pair_overlap_depth": -2.0020049716018562e-05, "metric_expression": "3.48104", "metric_symbol": "s", "metric_value": 3.481039873090523, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.48, "setup": "squinpen", "side": 3.481039873090523, "solution_path": "data/solutions/generated/squinpen_016_9ca0c9eddf5257a6.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/9ca0c9eddf5257a6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@17", "density": 0.7871473019367167, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/17.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.541+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.541, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@17", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3.541+", "metric_symbol": "s", "metric_value": 3.541, "reference_text": "s = 3.541+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/17.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.541+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "c1324401d6fef4ff", "improvement": -0.0020066620107224153, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3581595293550302e-06, "max_pair_overlap_depth": -1.9942519651539037e-05, "metric_expression": "3.54301", "metric_symbol": "s", "metric_value": 3.5430066620107223, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.541, "setup": "squinpen", "side": 3.5430066620107223, "solution_path": "data/solutions/generated/squinpen_017_c1324401d6fef4ff.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/c1324401d6fef4ff.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@18", "density": 0.7949963635829483, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/18.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.626+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.626, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@18", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3.626+", "metric_symbol": "s", "metric_value": 3.626, "reference_text": "s = 3.626+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/18.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.626+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "521e6ed1020d53e6", "improvement": -0.001681876520355896, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.132718950610418e-06, "max_pair_overlap_depth": -5.952134619235938e-06, "metric_expression": "3.62768", "metric_symbol": "s", "metric_value": 3.627681876520356, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.626, "setup": "squinpen", "side": 3.627681876520356, "solution_path": "data/solutions/generated/squinpen_018_521e6ed1020d53e6.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/521e6ed1020d53e6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@19", "density": 0.8016091392666894, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/19.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.710+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.71, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@19", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3.710+", "metric_symbol": "s", "metric_value": 3.71, "reference_text": "s = 3.710+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/19.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.710+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "66aeca1af6b9bf65", "improvement": -0.001683910599032501, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.240508232322902e-06, "max_pair_overlap_depth": -2.033383687716217e-05, "metric_expression": "3.71168", "metric_symbol": "s", "metric_value": 3.7116839105990325, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.71, "setup": "squinpen", "side": 3.7116839105990325, "solution_path": "data/solutions/generated/squinpen_019_66aeca1af6b9bf65.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/66aeca1af6b9bf65.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@20", "density": 0.8030009751844291, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/20.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.802+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.802, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@20", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "3.802+", "metric_symbol": "s", "metric_value": 3.802, "reference_text": "s = 3.802+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "20.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/20.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.802+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "8424f972564f51a8", "improvement": -0.00280562795003636, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.488842199203112e-06, "max_pair_overlap_depth": -3.2279665458823104e-05, "metric_expression": "3.80481", "metric_symbol": "s", "metric_value": 3.8048056279500364, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.802, "setup": "squinpen", "side": 3.8048056279500364, "solution_path": "data/solutions/generated/squinpen_020_8424f972564f51a8.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/8424f972564f51a8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@21", "density": 0.8168349282229689, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/21.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "3.864+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.864, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@21", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "3.864+", "metric_symbol": "s", "metric_value": 3.864, "reference_text": "s = 3.864+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/21.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.864+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "7f1bf6bbe100ca6c", "improvement": -0.0016097781764203845, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0975793470973527e-06, "max_pair_overlap_depth": -2.933142426009283e-05, "metric_expression": "3.86561", "metric_symbol": "s", "metric_value": 3.8656097781764203, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.864, "setup": "squinpen", "side": 3.8656097781764203, "solution_path": "data/solutions/generated/squinpen_021_7f1bf6bbe100ca6c.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/7f1bf6bbe100ca6c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@22", "density": 0.8236901899623935, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/22.gif", "friedman_metric_expression": "3.937+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.937, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@22", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "3.937+", "metric_symbol": "s", "metric_value": 3.937, "reference_text": "s = 3.937+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/22.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 3.937+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "33b95c59890c6cd7", "improvement": -0.0030787983666398233, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.1912141987053604e-06, "max_pair_overlap_depth": -2.2837560550437175e-06, "metric_expression": "3.94008", "metric_symbol": "s", "metric_value": 3.9400787983666397, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.937, "setup": "squinpen", "side": 3.9400787983666397, "solution_path": "data/solutions/generated/squinpen_022_33b95c59890c6cd7.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/33b95c59890c6cd7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@23", "density": 0.8342850114016098, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/23.gif", "friedman_metric_expression": "4.002+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.002, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@23", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.002+", "metric_symbol": "s", "metric_value": 4.002, "reference_text": "s = 4.002+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/23.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.002+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "125d15e1f42d4cc7", "improvement": -0.000968856434470311, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.142206802972531e-06, "max_pair_overlap_depth": -1.3381668518996914e-05, "metric_expression": "4.00297", "metric_symbol": "s", "metric_value": 4.00296885643447, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.002, "setup": "squinpen", "side": 4.00296885643447, "solution_path": "data/solutions/generated/squinpen_023_125d15e1f42d4cc7.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/125d15e1f42d4cc7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@24", "density": 0.8254992867774645, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/24.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.107+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.107, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@24", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.107+", "metric_symbol": "s", "metric_value": 4.107, "reference_text": "s = 4.107+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/24.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.107+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "7b6bed1452ec8f75", "improvement": -0.003766262034448964, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.187233974203672e-07, "max_pair_overlap_depth": -1.4055675707780324e-06, "metric_expression": "4.11077", "metric_symbol": "s", "metric_value": 4.110766262034449, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.107, "setup": "squinpen", "side": 4.110766262034449, "solution_path": "data/solutions/generated/squinpen_024_7b6bed1452ec8f75.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/7b6bed1452ec8f75.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@25", "density": 0.8346264431391957, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/25.gif", "friedman_metric_expression": "4.169+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.169, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@25", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.169+", "metric_symbol": "s", "metric_value": 4.169, "reference_text": "s = 4.169+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/25.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.169+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "965f543dd3a1a5b2", "improvement": -0.003529809873594303, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.8760801669609005e-06, "max_pair_overlap_depth": -9.22783607931743e-06, "metric_expression": "4.17253", "metric_symbol": "s", "metric_value": 4.172529809873594, "n": 25, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.169, "setup": "squinpen", "side": 4.172529809873594, "solution_path": "data/solutions/generated/squinpen_025_965f543dd3a1a5b2.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/965f543dd3a1a5b2.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@26", "density": 0.846720772454187, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/26.gif", "friedman_metric_expression": "4.223+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.223, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@26", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.223+", "metric_symbol": "s", "metric_value": 4.223, "reference_text": "s = 4.223+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/26.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.223+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "12ea954c92ca25d9", "improvement": -0.0016630925283189768, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.329734777233707e-07, "max_pair_overlap_depth": -5.080948892621784e-06, "metric_expression": "4.22466", "metric_symbol": "s", "metric_value": 4.224663092528319, "n": 26, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.223, "setup": "squinpen", "side": 4.224663092528319, "solution_path": "data/solutions/generated/squinpen_026_12ea954c92ca25d9.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/12ea954c92ca25d9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@27", "density": 0.8275684076653715, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/27.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.350+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.35, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@27", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.350+", "metric_symbol": "s", "metric_value": 4.35, "reference_text": "s = 4.350+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/27.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.350+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "fc7fa66fe1928440", "improvement": -0.0046719542874926745, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1174449550978842e-06, "max_pair_overlap_depth": -1.176344423625686e-05, "metric_expression": "4.35467", "metric_symbol": "s", "metric_value": 4.354671954287492, "n": 27, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.35, "setup": "squinpen", "side": 4.354671954287492, "solution_path": "data/solutions/generated/squinpen_027_fc7fa66fe1928440.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/fc7fa66fe1928440.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@28", "density": 0.8327551046015365, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/28.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.421+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.421, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@28", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.421+", "metric_symbol": "s", "metric_value": 4.421, "reference_text": "s = 4.421+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/28.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.421+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "81b95a9d0915b9b0", "improvement": 0.0002507922234444848, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3068707244667255e-06, "max_pair_overlap_depth": -1.2232567722092469e-05, "metric_expression": "4.42075", "metric_symbol": "s", "metric_value": 4.420749207776556, "n": 28, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.421, "setup": "squinpen", "side": 4.420749207776556, "solution_path": "data/solutions/generated/squinpen_028_81b95a9d0915b9b0.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/81b95a9d0915b9b0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@29", "density": 0.8326213695314967, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/29.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.499+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.499, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@29", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.499+", "metric_symbol": "s", "metric_value": 4.499, "reference_text": "s = 4.499+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/29.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.499+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "7cc5c6e51420a8c4", "improvement": -0.0003599298979271026, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.900416840645271e-06, "max_pair_overlap_depth": -1.3857736228028994e-05, "metric_expression": "4.49936", "metric_symbol": "s", "metric_value": 4.499359929897927, "n": 29, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.499, "setup": "squinpen", "side": 4.499359929897927, "solution_path": "data/solutions/generated/squinpen_029_7cc5c6e51420a8c4.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/7cc5c6e51420a8c4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@30", "density": 0.8380230132738162, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/30.gif", "friedman_metric_expression": "4.550+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.55, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@30", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "4.550+", "metric_symbol": "s", "metric_value": 4.55, "reference_text": "s = 4.550+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/30.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.550+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "45dff6d0a270c6c5", "improvement": -0.011505140806147729, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.226407328203014e-06, "max_pair_overlap_depth": -5.655700288098586e-07, "metric_expression": "4.56151", "metric_symbol": "s", "metric_value": 4.5615051408061476, "n": 30, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.55, "setup": "squinpen", "side": 4.5615051408061476, "solution_path": "data/solutions/generated/squinpen_30_45dff6d0a270c6c5.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T14:34:18+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/45dff6d0a270c6c5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@31", "density": 0.8407016075258085, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/31.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.63041+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.63041, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.63041+", "metric_symbol": "s", "metric_value": 4.63041, "reference_text": "s = 4.63041+ | Found by Emerson Connelly | in April 2026.", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/31.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.63041+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "8b17df7ad9586809", "improvement": 0.0008957860403464579, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9662880539345523e-06, "max_pair_overlap_depth": -1.825266620247845e-05, "metric_expression": "4.62951", "metric_symbol": "s", "metric_value": 4.629514213959654, "n": 31, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.63041, "setup": "squinpen", "side": 4.629514213959654, "solution_path": "data/solutions/generated/squinpen_031_8b17df7ad9586809.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/8b17df7ad9586809.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@32", "density": 0.839043760254319, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/32.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.70774+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.70774, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@32", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.70774+", "metric_symbol": "s", "metric_value": 4.70774, "reference_text": "s = 4.70774+ | Found by Emerson Connelly | in April 2026.", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/32.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.70774+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "ded24abc59c82646", "improvement": -0.000495710626569057, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.910425347385825e-07, "max_pair_overlap_depth": -1.0586072088480947e-05, "metric_expression": "4.70824", "metric_symbol": "s", "metric_value": 4.708235710626569, "n": 32, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.70774, "setup": "squinpen", "side": 4.708235710626569, "solution_path": "data/solutions/generated/squinpen_032_ded24abc59c82646.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/ded24abc59c82646.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@33", "density": 0.8454321018371573, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/33.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.75774+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.75774, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@33", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.75774+", "metric_symbol": "s", "metric_value": 4.75774, "reference_text": "s = 4.75774+ | Found by Emerson Connelly | in April 2026.", "source_image": "33.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/33.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.75774+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "a9622b0aeea26f74", "improvement": -0.00539747702354898, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5906514265382727e-06, "max_pair_overlap_depth": -5.445984948337923e-06, "metric_expression": "4.76314", "metric_symbol": "s", "metric_value": 4.763137477023549, "n": 33, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.75774, "setup": "squinpen", "side": 4.763137477023549, "solution_path": "data/solutions/generated/squinpen_033_a9622b0aeea26f74.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T03:19:00+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/a9622b0aeea26f74.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@34", "density": 0.8467800838831615, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/34.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.81817+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.81817, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@34", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.81817+", "metric_symbol": "s", "metric_value": 4.81817, "reference_text": "s = 4.81817+ | Found by Emerson Connelly | in April 2026.", "source_image": "34.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/34.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.81817+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "d7af1c3ed90603fd", "improvement": -0.012747878392191048, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.414710391441419e-07, "max_pair_overlap_depth": -6.11976491438071e-06, "metric_expression": "4.83092", "metric_symbol": "s", "metric_value": 4.830917878392191, "n": 34, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.81817, "setup": "squinpen", "side": 4.830917878392191, "solution_path": "data/solutions/generated/squinpen_034_d7af1c3ed90603fd.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/d7af1c3ed90603fd.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@35", "density": 0.8578692979479161, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/35.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.86994+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.86994, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@35", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.86994+", "metric_symbol": "s", "metric_value": 4.86994, "reference_text": "s = 4.86994+ | Found by Emerson Connelly | in April 2026.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/35.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.86994+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "d0212c88c498f951", "improvement": 0.0002762594618035408, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.74898699366526e-06, "max_pair_overlap_depth": -8.959642868955342e-06, "metric_expression": "4.86966", "metric_symbol": "s", "metric_value": 4.869663740538196, "n": 35, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.86994, "setup": "squinpen", "side": 4.869663740538196, "solution_path": "data/solutions/generated/squinpen_035_d0212c88c498f951.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/d0212c88c498f951.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@36", "density": 0.8582852742974874, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/36.gif", "friedman_metric_expression": "4.93902+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.93902, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@36", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.93902+", "metric_symbol": "s", "metric_value": 4.93902, "reference_text": "s = 4.93902+ | Found by Emerson Connelly | in April 2026.", "source_image": "36.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/36.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.93902+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "e43d96c031faf38c", "improvement": 0.0014765141690524786, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.258589189166597e-07, "max_pair_overlap_depth": -1.4614402910062552e-07, "metric_expression": "4.93754", "metric_symbol": "s", "metric_value": 4.937543485830948, "n": 36, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.93902, "setup": "squinpen", "side": 4.937543485830948, "solution_path": "data/solutions/record_hunt/squinpen_36_e43d96c031faf38c.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/e43d96c031faf38c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@37", "density": 0.8661967376016183, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/37.gif", "friedman_family_updated": "4/8/26", "friedman_metric_expression": "4.98234+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.98234, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@37", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.98234+", "metric_symbol": "s", "metric_value": 4.98234, "reference_text": "s = 4.98234+ | Found by Emerson Connelly | in April 2026.", "source_image": "37.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/37.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 4.98234+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "dba281681536eecb", "improvement": -0.00039859816128817727, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.161377353190805e-07, "max_pair_overlap_depth": -4.181534561720213e-06, "metric_expression": "4.98274", "metric_symbol": "s", "metric_value": 4.982738598161288, "n": 37, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.98234, "setup": "squinpen", "side": 4.982738598161288, "solution_path": "data/solutions/generated/squinpen_037_dba281681536eecb.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/dba281681536eecb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@38", "density": 0.8656682257376033, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/38.gif", "friedman_metric_expression": "5.05233+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.05233, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@38", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "5.05233+", "metric_symbol": "s", "metric_value": 5.05233, "reference_text": "s = 5.05233+ | Found by Emerson Connelly | in April 2026.", "source_image": "38.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/38.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 5.05233+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "38.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "00496638a2011285", "improvement": 0.001164786809802365, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.776103493630765e-07, "max_pair_overlap_depth": -5.692587072836375e-07, "metric_expression": "5.05117", "metric_symbol": "s", "metric_value": 5.051165213190198, "n": 38, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.05233, "setup": "squinpen", "side": 5.051165213190198, "solution_path": "data/solutions/record_hunt/squinpen_38_00496638a2011285.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/00496638a2011285.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@39", "density": 0.8585952949364365, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/39.gif", "friedman_metric_expression": "5.13846+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.13846, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@39", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "5.13846+", "metric_symbol": "s", "metric_value": 5.13846, "reference_text": "s = 5.13846+ | Found by Nathan Roll | in April 2026.", "source_image": "39.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/39.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 5.13846+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "b01313a4744f0eda", "improvement": 0.00022970687679979562, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2716746597467209e-07, "max_pair_overlap_depth": -7.26967224418118e-07, "metric_expression": "5.13823", "metric_symbol": "s", "metric_value": 5.1382302931232005, "n": 39, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.13846, "setup": "squinpen", "side": 5.1382302931232005, "solution_path": "data/solutions/record_hunt/squinpen_39_b01313a4744f0eda.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/b01313a4744f0eda.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squinpen@40", "density": 0.8691683896670805, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squinpen/40.gif", "friedman_metric_expression": "5.17250+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.1725, "friedman_reference": {"analytical_or_proved": false, "case": "squinpen@40", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "5.17250+", "metric_symbol": "s", "metric_value": 5.1725, "reference_text": "s = 5.17250+ | Found by Emerson Connelly | in April 2026.", "source_image": "40.gif", "source_image_url": "https://erich-friedman.github.io/packing/squinpen/40.gif", "source_page": "https://erich-friedman.github.io/packing/squinpen/", "title": "Squares in Pentagons"}, "friedman_reference_text": "s = 5.17250+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squinpen/", "friedman_title": "Squares in Pentagons", "frontend_seed": true, "id": "153dbd88227e7083", "improvement": 0.0005591675539955432, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.137577186347585e-07, "max_pair_overlap_depth": -4.818340240131391e-07, "metric_expression": "5.17194", "metric_symbol": "s", "metric_value": 5.171940832446005, "n": 40, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.1725, "setup": "squinpen", "side": 5.171940832446005, "solution_path": "data/solutions/record_hunt/squinpen_40_153dbd88227e7083.json", "source_url": "https://erich-friedman.github.io/packing/squinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/153dbd88227e7083.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@1", "density": 0.49741240626661626, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts1.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "1 + 2 / \u221a3 = 2.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1 + 2 / \u221a3 = 2.154+", "metric_symbol": "s", "metric_value": 2.154, "reference_text": "s = 1 + 2 / \u221a3 = 2.154+ | Trivial.", "source_image": "ts1.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts1.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 1 + 2 / \u221a3 = 2.154+ | Trivial.", "friedman_source_image": "ts1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "bc8215883b0d29fa", "improvement": -0.0007226428063047408, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.189638991319711e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "2.15472", "metric_symbol": "s", "metric_value": 2.1547226428063047, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.154, "setup": "squintri", "side": 2.1547226428063047, "solution_path": "data/solutions/generated/squintri_001_bc8215883b0d29fa.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/bc8215883b0d29fa.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@2", "density": 0.4640775964340414, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts2.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2 + 2 / \u221a3 = 3.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2 + 2 / \u221a3 = 3.154+", "metric_symbol": "s", "metric_value": 3.154, "reference_text": "s = 2 + 2 / \u221a3 = 3.154+ | Trivial.", "source_image": "ts2.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts2.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 2 + 2 / \u221a3 = 3.154+ | Trivial.", "friedman_source_image": "ts2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "9fcb7fc91bdb6f8d", "improvement": -0.000782174335970165, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.2952441585539525e-06, "max_pair_overlap_depth": -3.4919698730116744e-05, "metric_expression": "3.15478", "metric_symbol": "s", "metric_value": 3.15478217433597, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.154, "setup": "squintri", "side": 3.15478217433597, "solution_path": "data/solutions/generated/squintri_002_9fcb7fc91bdb6f8d.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9fcb7fc91bdb6f8d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@3", "density": 0.6632023535454387, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts3.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3 / 2 + \u221a3 = 3.232+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.232, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@3", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 / 2 + \u221a3 = 3.232+", "metric_symbol": "s", "metric_value": 3.232, "reference_text": "s = 3 / 2 + \u221a3 = 3.232+ | Found by Erich Friedman | in 1997.", "source_image": "ts3.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts3.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 / 2 + \u221a3 = 3.232+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "6a0fd09e2fb037d5", "improvement": -0.00011853663309757678, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0862631378148764e-06, "max_pair_overlap_depth": -4.0896047962125925e-05, "metric_expression": "3.23212", "metric_symbol": "s", "metric_value": 3.232118536633098, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.232, "setup": "squintri", "side": 3.232118536633098, "solution_path": "data/solutions/generated/squintri_003_6a0fd09e2fb037d5.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/6a0fd09e2fb037d5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@4", "density": 0.5351297741951115, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts4.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3 + 2 / \u221a3 = 4.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@4", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 2 / \u221a3 = 4.154+", "metric_symbol": "s", "metric_value": 4.154, "reference_text": "s = 3 + 2 / \u221a3 = 4.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts4.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts4.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 2 / \u221a3 = 4.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "a14dd3a45e182f53", "improvement": -0.000799971071848482, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3541096500101422e-06, "max_pair_overlap_depth": -2.3979848717270436e-05, "metric_expression": "4.15480", "metric_symbol": "s", "metric_value": 4.154799971071848, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.154, "setup": "squintri", "side": 4.154799971071848, "solution_path": "data/solutions/generated/squintri_004_a14dd3a45e182f53.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a14dd3a45e182f53.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@5", "density": 0.6217402820027407, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts6.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2 + 4 / \u221a3 = 4.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.309, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@5", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "2 + 4 / \u221a3 = 4.309+", "metric_symbol": "s", "metric_value": 4.309, "reference_text": "s = 2 + 4 / \u221a3 = 4.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts6.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts6.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 2 + 4 / \u221a3 = 4.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "b7ec125fb5317da3", "improvement": -0.0005327081561983249, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.818148398440643e-06, "max_pair_overlap_depth": -2.4242157156018074e-05, "metric_expression": "4.30953", "metric_symbol": "s", "metric_value": 4.3095327081561985, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.309, "setup": "squintri", "side": 4.3095327081561985, "solution_path": "data/solutions/generated/squintri_005_b7ec125fb5317da3.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/b7ec125fb5317da3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@6", "density": 0.7460914321932061, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts6.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2 + 4 / \u221a3 = 4.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.309, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@6", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "2 + 4 / \u221a3 = 4.309+", "metric_symbol": "s", "metric_value": 4.309, "reference_text": "s = 2 + 4 / \u221a3 = 4.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts6.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts6.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 2 + 4 / \u221a3 = 4.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "cafd65072dcc4c6f", "improvement": -0.0005237730564999765, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6885485560193324e-06, "max_pair_overlap_depth": -3.2625800104213454e-05, "metric_expression": "4.30952", "metric_symbol": "s", "metric_value": 4.3095237730565, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.309, "setup": "squintri", "side": 4.3095237730565, "solution_path": "data/solutions/generated/squintri_006_cafd65072dcc4c6f.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/cafd65072dcc4c6f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@7", "density": 0.6083690864087284, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts7.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 2 / \u221a3 = 5.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@7", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 2 / \u221a3 = 5.154+", "metric_symbol": "s", "metric_value": 5.154, "reference_text": "s = 4 + 2 / \u221a3 = 5.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts7.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts7.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 2 / \u221a3 = 5.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "b90420d34e1bd6e3", "improvement": -0.0008393265287383045, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.7967605582714725e-06, "max_pair_overlap_depth": -3.132565940933141e-05, "metric_expression": "5.15484", "metric_symbol": "s", "metric_value": 5.154839326528738, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.154, "setup": "squintri", "side": 5.154839326528738, "solution_path": "data/solutions/generated/squintri_007_b90420d34e1bd6e3.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/b90420d34e1bd6e3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@8", "density": 0.6568042680930309, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts8.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5.301+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.301, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@8", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "5.301+", "metric_symbol": "s", "metric_value": 5.301, "reference_text": "s = 5.301+ | Found by David W. Cantrell | in July 2002.", "source_image": "ts8.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts8.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 5.301+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "ts8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "9523798d0eb30de3", "improvement": -0.0026724224763796656, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4466353188141736e-06, "max_pair_overlap_depth": -2.591508822799571e-05, "metric_expression": "5.30367", "metric_symbol": "s", "metric_value": 5.30367242247638, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.301, "setup": "squintri", "side": 5.30367242247638, "solution_path": "data/solutions/generated/squintri_008_9523798d0eb30de3.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/9523798d0eb30de3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@9", "density": 0.7370953908179042, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts9.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3 + 4 / \u221a3 = 5.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.309, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@9", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 4 / \u221a3 = 5.309+", "metric_symbol": "s", "metric_value": 5.309, "reference_text": "s = 3 + 4 / \u221a3 = 5.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts9.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts9.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 4 / \u221a3 = 5.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "221ec438e9e78bec", "improvement": -0.0011781204346883456, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.476651564783165e-07, "max_pair_overlap_depth": -2.048243915964143e-05, "metric_expression": "5.31018", "metric_symbol": "s", "metric_value": 5.3101781204346885, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.309, "setup": "squintri", "side": 5.3101781204346885, "solution_path": "data/solutions/generated/squintri_009_221ec438e9e78bec.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/221ec438e9e78bec.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@10", "density": 0.773471405329893, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts10.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "2 + 2 \u221a3 = 5.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.464, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@10", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "2 + 2 \u221a3 = 5.464+", "metric_symbol": "s", "metric_value": 5.464, "reference_text": "s = 2 + 2 \u221a3 = 5.464+ | Found by Erich Friedman | in 1997.", "source_image": "ts10.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts10.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 2 + 2 \u221a3 = 5.464+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "4b92d4f5e6704540", "improvement": -0.00021212423215644094, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1415169103720102e-06, "max_pair_overlap_depth": -1.0706995126841257e-05, "metric_expression": "5.46421", "metric_symbol": "s", "metric_value": 5.464212124232157, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.464, "setup": "squintri", "side": 5.464212124232157, "solution_path": "data/solutions/generated/squintri_010_4b92d4f5e6704540.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/4b92d4f5e6704540.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@11", "density": 0.6705894339586221, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts11.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5 + 2 / \u221a3 = 6.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@11", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 + 2 / \u221a3 = 6.154+", "metric_symbol": "s", "metric_value": 6.154, "reference_text": "s = 5 + 2 / \u221a3 = 6.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts11.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts11.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 5 + 2 / \u221a3 = 6.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "c18f0f7b3b1e0227", "improvement": -0.0008528095099347155, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.062945956811758e-06, "max_pair_overlap_depth": -2.0039414804218225e-05, "metric_expression": "6.15485", "metric_symbol": "s", "metric_value": 6.154852809509935, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.154, "setup": "squintri", "side": 6.154852809509935, "solution_path": "data/solutions/generated/squintri_011_c18f0f7b3b1e0227.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/c18f0f7b3b1e0227.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@12", "density": 0.6970880090801083, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts12.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6.301+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.301, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@12", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "6.301+", "metric_symbol": "s", "metric_value": 6.301, "reference_text": "s = 6.301+ | Found by David W. Cantrell | in July 2002.", "source_image": "ts12.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts12.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 6.301+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "ts12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "d248ead86c97d2e6", "improvement": -0.004165620418586258, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6132876791233732e-06, "max_pair_overlap_depth": -5.9825433180948906e-05, "metric_expression": "6.30517", "metric_symbol": "s", "metric_value": 6.305165620418586, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.301, "setup": "squintri", "side": 6.305165620418586, "solution_path": "data/solutions/generated/squintri_012_d248ead86c97d2e6.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d248ead86c97d2e6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@13", "density": 0.7541442288903037, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts13.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 4 / \u221a3 = 6.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.309, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@13", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 4 / \u221a3 = 6.309+", "metric_symbol": "s", "metric_value": 6.309, "reference_text": "s = 4 + 4 / \u221a3 = 6.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts13.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts13.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 4 / \u221a3 = 6.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "3b7ab78f832490ed", "improvement": -0.0004884860946248182, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1403459653225352e-06, "max_pair_overlap_depth": -1.2003638542146255e-05, "metric_expression": "6.30949", "metric_symbol": "s", "metric_value": 6.309488486094625, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.309, "setup": "squintri", "side": 6.309488486094625, "solution_path": "data/solutions/generated/squintri_013_3b7ab78f832490ed.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3b7ab78f832490ed.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@14", "density": 0.7737221641470288, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts15.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3 + 2 \u221a3 = 6.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.464, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@14", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 2 \u221a3 = 6.464+", "metric_symbol": "s", "metric_value": 6.464, "reference_text": "s = 3 + 2 \u221a3 = 6.464+ | Found by Erich Friedman | in 1997.", "source_image": "ts15.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts15.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 2 \u221a3 = 6.464+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "ffc152ae79167d33", "improvement": -0.00029520096988377986, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.958378324017974e-07, "max_pair_overlap_depth": -1.6732970481436737e-05, "metric_expression": "6.46430", "metric_symbol": "s", "metric_value": 6.464295200969884, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.464, "setup": "squintri", "side": 6.464295200969884, "solution_path": "data/solutions/generated/squintri_014_ffc152ae79167d33.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/ffc152ae79167d33.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@15", "density": 0.8289675963035161, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts15.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3 + 2 \u221a3 = 6.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.464, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@15", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 2 \u221a3 = 6.464+", "metric_symbol": "s", "metric_value": 6.464, "reference_text": "s = 3 + 2 \u221a3 = 6.464+ | Found by Erich Friedman | in 1997.", "source_image": "ts15.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts15.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 2 \u221a3 = 6.464+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "37ace9570dd5c15b", "improvement": -0.0003748832881260711, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.039916977622108e-06, "max_pair_overlap_depth": -1.0702944550522453e-05, "metric_expression": "6.46437", "metric_symbol": "s", "metric_value": 6.4643748832881265, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.464, "setup": "squintri", "side": 6.4643748832881265, "solution_path": "data/solutions/generated/squintri_015_37ace9570dd5c15b.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/37ace9570dd5c15b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@16", "density": 0.7217958800018128, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts16.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6 + 2 / \u221a3 = 7.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@16", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "6 + 2 / \u221a3 = 7.154+", "metric_symbol": "s", "metric_value": 7.154, "reference_text": "s = 6 + 2 / \u221a3 = 7.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts16.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts16.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 6 + 2 / \u221a3 = 7.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "36afe756f11f62d6", "improvement": -0.0008819711932614993, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.5571215058068575e-06, "max_pair_overlap_depth": -2.4408532427300234e-05, "metric_expression": "7.15488", "metric_symbol": "s", "metric_value": 7.154881971193261, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.154, "setup": "squintri", "side": 7.154881971193261, "solution_path": "data/solutions/generated/squintri_016_36afe756f11f62d6.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/36afe756f11f62d6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@17", "density": 0.7357616143895036, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts17.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "7.301+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.301, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@17", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "7.301+", "metric_symbol": "s", "metric_value": 7.301, "reference_text": "s = 7.301+ | Found by David W. Cantrell | in July 2002.", "source_image": "ts17.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts17.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 7.301+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "ts17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "3a2c987432da5f1d", "improvement": -0.0037537403553287163, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.4850519352755694e-10, "max_pair_overlap_depth": -2.8059876787889237e-05, "metric_expression": "7.30475", "metric_symbol": "s", "metric_value": 7.304753740355329, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.301, "setup": "squintri", "side": 7.304753740355329, "solution_path": "data/solutions/generated/squintri_017_3a2c987432da5f1d.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/3a2c987432da5f1d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@18", "density": 0.7780278532013658, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts18.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5 + 4 / \u221a3 = 7.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.309, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@18", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 + 4 / \u221a3 = 7.309+", "metric_symbol": "s", "metric_value": 7.309, "reference_text": "s = 5 + 4 / \u221a3 = 7.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts18.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts18.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 5 + 4 / \u221a3 = 7.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "e729e13f2fcc5856", "improvement": -0.0005116409635963493, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1021248752761939e-06, "max_pair_overlap_depth": -8.579145527987109e-06, "metric_expression": "7.30951", "metric_symbol": "s", "metric_value": 7.3095116409635965, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.309, "setup": "squintri", "side": 7.3095116409635965, "solution_path": "data/solutions/generated/squintri_018_e729e13f2fcc5856.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/e729e13f2fcc5856.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@19", "density": 0.787537700321716, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts19.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 2 \u221a3 = 7.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.464, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@19", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 2 \u221a3 = 7.464+", "metric_symbol": "s", "metric_value": 7.464, "reference_text": "s = 4 + 2 \u221a3 = 7.464+ | Found by Erich Friedman | in 1997.", "source_image": "ts19.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts19.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 2 \u221a3 = 7.464+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "805d1e50c1db2842", "improvement": -0.0003295610354463463, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0609380559736765e-06, "max_pair_overlap_depth": -1.836020690809681e-05, "metric_expression": "7.46433", "metric_symbol": "s", "metric_value": 7.464329561035447, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.464, "setup": "squintri", "side": 7.464329561035447, "solution_path": "data/solutions/generated/squintri_019_805d1e50c1db2842.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/805d1e50c1db2842.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@20", "density": 0.795657127614163, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts21.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "3 + 8 / \u221a3 = 7.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.618, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@20", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 8 / \u221a3 = 7.618+", "metric_symbol": "s", "metric_value": 7.618, "reference_text": "s = 3 + 8 / \u221a3 = 7.618+ | Found by Erich Friedman | in 1997.", "source_image": "ts21.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts21.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 8 / \u221a3 = 7.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "f4d21a73470ce742", "improvement": -0.0010653922379448844, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.382631857088383e-07, "max_pair_overlap_depth": -1.1470843414018717e-05, "metric_expression": "7.61907", "metric_symbol": "s", "metric_value": 7.619065392237945, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.618, "setup": "squintri", "side": 7.619065392237945, "solution_path": "data/solutions/generated/squintri_020_f4d21a73470ce742.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f4d21a73470ce742.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@21", "density": 0.8354483980307021, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts21.gif", "friedman_metric_expression": "3 + 8 / \u221a3 = 7.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.618, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@21", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 8 / \u221a3 = 7.618+", "metric_symbol": "s", "metric_value": 7.618, "reference_text": "s = 3 + 8 / \u221a3 = 7.618+ | Found by Erich Friedman | in 1997.", "source_image": "ts21.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts21.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 8 / \u221a3 = 7.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "a2e24601c2557198", "improvement": -0.0010270252659259071, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8035391721895233e-06, "max_pair_overlap_depth": -9.7266150509423e-06, "metric_expression": "7.61903", "metric_symbol": "s", "metric_value": 7.619027025265926, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 7.618, "setup": "squintri", "side": 7.619027025265926, "solution_path": "data/solutions/generated/squintri_021_a2e24601c2557198.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a2e24601c2557198.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@22", "density": 0.7639711289147475, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts22.gif", "friedman_metric_expression": "7 + 2 / \u221a3 = 8.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@22", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "7 + 2 / \u221a3 = 8.154+", "metric_symbol": "s", "metric_value": 8.154, "reference_text": "s = 7 + 2 / \u221a3 = 8.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts22.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts22.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 7 + 2 / \u221a3 = 8.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "bbb028380d997350", "improvement": -0.00097340620692421, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4334644744109685e-07, "max_pair_overlap_depth": -2.16343036094635e-05, "metric_expression": "8.15497", "metric_symbol": "s", "metric_value": 8.154973406206924, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.154, "setup": "squintri", "side": 8.154973406206924, "solution_path": "data/solutions/generated/squintri_022_bbb028380d997350.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/bbb028380d997350.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@23", "density": 0.7697946105392801, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts23.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "8.301+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.301, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@23", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "8.301+", "metric_symbol": "s", "metric_value": 8.301, "reference_text": "s = 8.301+ | Found by David W. Cantrell | in July 2002.", "source_image": "ts23.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts23.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 8.301+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "ts23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "b1534ea73e00f283", "improvement": -0.005654877387726742, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7355116411507935e-05, "max_pair_overlap_depth": 9.193993899536679e-09, "metric_expression": "8.30665", "metric_symbol": "s", "metric_value": 8.306654877387727, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.301, "setup": "squintri", "side": 8.306654877387727, "solution_path": "data/solutions/generated/squintri_023_b1534ea73e00f283.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/b1534ea73e00f283.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@24", "density": 0.8026821430235043, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts24.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6 + 4 / \u221a3 = 8.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.309, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@24", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "6 + 4 / \u221a3 = 8.309+", "metric_symbol": "s", "metric_value": 8.309, "reference_text": "s = 6 + 4 / \u221a3 = 8.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts24.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts24.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 6 + 4 / \u221a3 = 8.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "648f7ef8ee4a197f", "improvement": -0.0006647382800331059, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.518509956206752e-06, "max_pair_overlap_depth": -1.764131142900638e-05, "metric_expression": "8.30966", "metric_symbol": "s", "metric_value": 8.309664738280032, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.309, "setup": "squintri", "side": 8.309664738280032, "solution_path": "data/solutions/generated/squintri_024_648f7ef8ee4a197f.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/648f7ef8ee4a197f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@25", "density": 0.8057999996349686, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts25.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5 + 2 \u221a3 = 8.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.464, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@25", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 + 2 \u221a3 = 8.464+", "metric_symbol": "s", "metric_value": 8.464, "reference_text": "s = 5 + 2 \u221a3 = 8.464+ | Found by Erich Friedman | in 1997.", "source_image": "ts25.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts25.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 5 + 2 \u221a3 = 8.464+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "a868d8af97a256ad", "improvement": -0.0005924926961089483, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7884356498379287e-06, "max_pair_overlap_depth": -2.4212693654490636e-06, "metric_expression": "8.46459", "metric_symbol": "s", "metric_value": 8.46459249269611, "n": 25, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.464, "setup": "squintri", "side": 8.46459249269611, "solution_path": "data/solutions/generated/squintri_025_a868d8af97a256ad.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a868d8af97a256ad.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@26", "density": 0.8076243715506172, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts27.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 8 / \u221a3 = 8.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.618, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@26", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 8 / \u221a3 = 8.618+", "metric_symbol": "s", "metric_value": 8.618, "reference_text": "s = 4 + 8 / \u221a3 = 8.618+ | Found by Erich Friedman | in 1997.", "source_image": "ts27.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts27.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 8 / \u221a3 = 8.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "249af47eefa3a74b", "improvement": -0.004469124601865104, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.4769017068652995e-06, "max_pair_overlap_depth": -2.5879119370686432e-05, "metric_expression": "8.62247", "metric_symbol": "s", "metric_value": 8.622469124601865, "n": 26, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.618, "setup": "squintri", "side": 8.622469124601865, "solution_path": "data/solutions/generated/squintri_026_249af47eefa3a74b.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/249af47eefa3a74b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@27", "density": 0.8380226358442442, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts27.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 8 / \u221a3 = 8.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.618, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@27", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 8 / \u221a3 = 8.618+", "metric_symbol": "s", "metric_value": 8.618, "reference_text": "s = 4 + 8 / \u221a3 = 8.618+ | Found by Erich Friedman | in 1997.", "source_image": "ts27.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts27.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 8 / \u221a3 = 8.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "3c62cd2126467cec", "improvement": -0.007885505517393199, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.235025861662866e-06, "max_pair_overlap_depth": -2.0721793894562524e-05, "metric_expression": "8.62589", "metric_symbol": "s", "metric_value": 8.625885505517394, "n": 27, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.618, "setup": "squintri", "side": 8.625885505517394, "solution_path": "data/solutions/generated/squintri_027_3c62cd2126467cec.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3c62cd2126467cec.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@28", "density": 0.8400113660191094, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts28.gif", "friedman_metric_expression": "3 + 10 / \u221a3 = 8.773+", "friedman_metric_symbol": "s", "friedman_metric_value": 8.773, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@28", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3 + 10 / \u221a3 = 8.773+", "metric_symbol": "s", "metric_value": 8.773, "reference_text": "s = 3 + 10 / \u221a3 = 8.773+ | Found by Erich Friedman | in 1997.", "source_image": "ts28.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts28.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 3 + 10 / \u221a3 = 8.773+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "0d20b359435ed501", "improvement": -0.0007673944032049462, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0145694937175628e-06, "max_pair_overlap_depth": -1.3941973455278145e-05, "metric_expression": "8.77377", "metric_symbol": "s", "metric_value": 8.773767394403205, "n": 28, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 8.773, "setup": "squintri", "side": 8.773767394403205, "solution_path": "data/solutions/generated/squintri_028_0d20b359435ed501.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/0d20b359435ed501.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@29", "density": 0.7990846545133872, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts29.gif", "friedman_metric_expression": "8 + 2 / \u221a3 = 9.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.154, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@29", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "8 + 2 / \u221a3 = 9.154+", "metric_symbol": "s", "metric_value": 9.154, "reference_text": "s = 8 + 2 / \u221a3 = 9.154+ | Found by Erich Friedman | in 1997.", "source_image": "ts29.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts29.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 8 + 2 / \u221a3 = 9.154+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "3947aab39c0a1a92", "improvement": -0.000872191450978832, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2628787428294703e-06, "max_pair_overlap_depth": -1.875431036468811e-05, "metric_expression": "9.15487", "metric_symbol": "s", "metric_value": 9.154872191450979, "n": 29, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.154, "setup": "squintri", "side": 9.154872191450979, "solution_path": "data/solutions/generated/squintri_029_3947aab39c0a1a92.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3947aab39c0a1a92.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@30", "density": 0.7992784153759376, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts30.gif", "friedman_metric_expression": "9.301+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.301, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@30", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "9.301+", "metric_symbol": "s", "metric_value": 9.301, "reference_text": "s = 9.301+ | Found by David W. Cantrell | in July 2002.", "source_image": "ts30.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts30.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 9.301+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "ts30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "e0b7505d8b6b1d46", "improvement": -0.009248370808917272, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5690858985090003e-06, "max_pair_overlap_depth": -2.654303096472077e-05, "metric_expression": "9.31025", "metric_symbol": "s", "metric_value": 9.310248370808917, "n": 30, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.301, "setup": "squintri", "side": 9.310248370808917, "solution_path": "data/solutions/generated/squintri_030_e0b7505d8b6b1d46.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/e0b7505d8b6b1d46.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@31", "density": 0.825993871568094, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts31.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "7 + 4 / \u221a3 = 9.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.309, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@31", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "7 + 4 / \u221a3 = 9.309+", "metric_symbol": "s", "metric_value": 9.309, "reference_text": "s = 7 + 4 / \u221a3 = 9.309+ | Found by Erich Friedman | in 1997.", "source_image": "ts31.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts31.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 7 + 4 / \u221a3 = 9.309+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "22b5f5f5707858bb", "improvement": -0.0008378379274667225, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6472234336534766e-06, "max_pair_overlap_depth": -3.090204228461957e-05, "metric_expression": "9.30984", "metric_symbol": "s", "metric_value": 9.309837837927466, "n": 31, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.309, "setup": "squintri", "side": 9.309837837927466, "solution_path": "data/solutions/generated/squintri_031_22b5f5f5707858bb.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/22b5f5f5707858bb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@32", "density": 0.8249717531277396, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts32.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "6 + 2 \u221a3 = 9.464+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.464, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@32", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "6 + 2 \u221a3 = 9.464+", "metric_symbol": "s", "metric_value": 9.464, "reference_text": "s = 6 + 2 \u221a3 = 9.464+ | Found by Erich Friedman | in 1997.", "source_image": "ts32.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts32.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 6 + 2 \u221a3 = 9.464+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "a44fbd99145b1ab3", "improvement": -0.0006624941961259623, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.1832282131171085e-07, "max_pair_overlap_depth": -1.699280632205813e-06, "metric_expression": "9.46466", "metric_symbol": "s", "metric_value": 9.464662494196126, "n": 32, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.464, "setup": "squintri", "side": 9.464662494196126, "solution_path": "data/solutions/generated/squintri_032_a44fbd99145b1ab3.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a44fbd99145b1ab3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@33", "density": 0.8228028312642216, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts34.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "5 + 8 / \u221a3 = 9.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.618, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@33", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 + 8 / \u221a3 = 9.618+", "metric_symbol": "s", "metric_value": 9.618, "reference_text": "s = 5 + 8 / \u221a3 = 9.618+ | Found by Erich Friedman | in 1997.", "source_image": "ts34.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts34.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 5 + 8 / \u221a3 = 9.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "67de1e08be2d150c", "improvement": -0.00606977957198751, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.226435414550563e-06, "max_pair_overlap_depth": -3.0944328952564604e-06, "metric_expression": "9.62407", "metric_symbol": "s", "metric_value": 9.624069779571988, "n": 33, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.618, "setup": "squintri", "side": 9.624069779571988, "solution_path": "data/solutions/generated/squintri_033_67de1e08be2d150c.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/67de1e08be2d150c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@34", "density": 0.8485900582279403, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts34.gif", "friedman_metric_expression": "5 + 8 / \u221a3 = 9.618+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.618, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@34", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "5 + 8 / \u221a3 = 9.618+", "metric_symbol": "s", "metric_value": 9.618, "reference_text": "s = 5 + 8 / \u221a3 = 9.618+ | Found by Erich Friedman | in 1997.", "source_image": "ts34.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts34.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 5 + 8 / \u221a3 = 9.618+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "3661302cdd9d0eb0", "improvement": -0.0012269381751668362, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.159334309452278e-06, "max_pair_overlap_depth": -1.741303470170763e-05, "metric_expression": "9.61923", "metric_symbol": "s", "metric_value": 9.619226938175167, "n": 34, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.618, "setup": "squintri", "side": 9.619226938175167, "solution_path": "data/solutions/generated/squintri_034_3661302cdd9d0eb0.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3661302cdd9d0eb0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@35", "density": 0.8451911560111607, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts36.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 10 / \u221a3 = 9.773+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.773, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@35", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 10 / \u221a3 = 9.773+", "metric_symbol": "s", "metric_value": 9.773, "reference_text": "s = 4 + 10 / \u221a3 = 9.773+ | Found by Erich Friedman | in 1997.", "source_image": "ts36.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts36.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 10 / \u221a3 = 9.773+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "39d52a823b632afe", "improvement": -0.006265389488104844, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.195518233524095e-06, "max_pair_overlap_depth": -2.612964804571405e-05, "metric_expression": "9.77927", "metric_symbol": "s", "metric_value": 9.779265389488105, "n": 35, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.773, "setup": "squintri", "side": 9.779265389488105, "solution_path": "data/solutions/generated/squintri_035_39d52a823b632afe.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/39d52a823b632afe.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@36", "density": 0.8698156108347095, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts36.gif", "friedman_family_updated": "4/28/26", "friedman_metric_expression": "4 + 10 / \u221a3 = 9.773+", "friedman_metric_symbol": "s", "friedman_metric_value": 9.773, "friedman_reference": {"analytical_or_proved": true, "case": "squintri@36", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "4 + 10 / \u221a3 = 9.773+", "metric_symbol": "s", "metric_value": 9.773, "reference_text": "s = 4 + 10 / \u221a3 = 9.773+ | Found by Erich Friedman | in 1997.", "source_image": "ts36.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts36.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 4 + 10 / \u221a3 = 9.773+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "ts36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "f0473aaa767ebe98", "improvement": -0.0035884436987085877, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9270562690497914e-06, "max_pair_overlap_depth": -7.45737725771356e-06, "metric_expression": "9.77659", "metric_symbol": "s", "metric_value": 9.776588443698708, "n": 36, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 9.773, "setup": "squintri", "side": 9.776588443698708, "solution_path": "data/solutions/generated/squintri_036_f0473aaa767ebe98.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f0473aaa767ebe98.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@37", "density": 0.8285699034870297, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts37.gif", "friedman_metric_expression": "10.154+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.154, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@37", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "10.154+", "metric_symbol": "s", "metric_value": 10.154, "reference_text": "s = 10.154+ | Found by Emerson Connelly | in April 2026.", "source_image": "ts37.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts37.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 10.154+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "ts37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "8530be000ac2e9ed", "improvement": -0.0011412138177444575, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3566887422690367e-06, "max_pair_overlap_depth": -1.2239987045248313e-05, "metric_expression": "10.15514", "metric_symbol": "s", "metric_value": 10.155141213817744, "n": 37, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 10.154, "setup": "squintri", "side": 10.155141213817744, "solution_path": "data/solutions/generated/squintri_037_8530be000ac2e9ed.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/8530be000ac2e9ed.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@38", "density": 0.8256293316995508, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts38.gif", "friedman_metric_expression": "10.309+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.309, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@38", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "10.309+", "metric_symbol": "s", "metric_value": 10.309, "reference_text": "s = 10.309+ | Found by Emerson Connelly | in April 2026.", "source_image": "ts38.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts38.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 10.309+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "ts38.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "896b636f9d899bf5", "improvement": -0.0007687369401132571, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.231911901047403e-06, "max_pair_overlap_depth": -1.6045122417374102e-05, "metric_expression": "10.30977", "metric_symbol": "s", "metric_value": 10.309768736940113, "n": 38, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 10.309, "setup": "squintri", "side": 10.309768736940113, "solution_path": "data/solutions/generated/squintri_038_896b636f9d899bf5.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/896b636f9d899bf5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@39", "density": 0.8473534790778084, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts39.gif", "friedman_metric_expression": "10.310+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.31, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@39", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "10.310+", "metric_symbol": "s", "metric_value": 10.31, "reference_text": "s = 10.310+ | Found by Emerson Connelly | in April 2026.", "source_image": "ts39.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts39.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 10.310+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "ts39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "c0988e1f3bf189d9", "improvement": 0.00021337573935120702, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.79079015949438e-07, "max_pair_overlap_depth": -3.1188368032619707e-06, "metric_expression": "10.30979", "metric_symbol": "s", "metric_value": 10.30978662426065, "n": 39, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.31, "setup": "squintri", "side": 10.30978662426065, "solution_path": "data/solutions/record_hunt/squintri_39_c0988e1f3bf189d9.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/c0988e1f3bf189d9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "squintri@40", "density": 0.8436138407048179, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/squintri/ts40.gif", "friedman_metric_expression": "10.46426+", "friedman_metric_symbol": "s", "friedman_metric_value": 10.46426, "friedman_reference": {"analytical_or_proved": false, "case": "squintri@40", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "10.46426+", "metric_symbol": "s", "metric_value": 10.46426, "reference_text": "s = 10.46426+ | Found by Nathan Roll | in April 2026.", "source_image": "ts40.gif", "source_image_url": "https://erich-friedman.github.io/packing/squintri/ts40.gif", "source_page": "https://erich-friedman.github.io/packing/squintri/", "title": "Squares in Triangles"}, "friedman_reference_text": "s = 10.46426+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "ts40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/squintri/", "friedman_title": "Squares in Triangles", "frontend_seed": true, "id": "13b03f25af76ce87", "improvement": 1.6681613434599285e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.839061064144293e-07, "max_pair_overlap_depth": -1.4626365829606414e-06, "metric_expression": "10.46424", "metric_symbol": "s", "metric_value": 10.464243318386565, "n": 40, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 10.46426, "setup": "squintri", "side": 10.464243318386565, "solution_path": "data/solutions/record_hunt/squintri_40_13b03f25af76ce87.json", "source_url": "https://erich-friedman.github.io/packing/squintri/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/13b03f25af76ce87.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@1", "density": 0.41349293224745, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc1.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1/\u221a3 = .577+", "friedman_metric_symbol": "r", "friedman_metric_value": 0.577, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1/\u221a3 = .577+", "metric_symbol": "r", "metric_value": 0.577, "reference_text": "r = 1/\u221a3 = .577+ | Trivial.", "source_image": "tc1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc1.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1/\u221a3 = .577+ | Trivial.", "friedman_source_image": "tc1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "49954bfbb560620d", "improvement": -0.00035287974439102143, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.775275519510515e-07, "max_pair_overlap_depth": 0.0, "metric_expression": "0.57735", "metric_symbol": "r", "metric_value": 0.577352879744391, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.577, "setup": "triincir", "side": 1.154705759488782, "solution_path": "data/solutions/generated/triincir_001_49954bfbb560620d.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/49954bfbb560620d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@2", "density": 0.3675414068402789, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc2.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "\u221a3/2 = .866+", "friedman_metric_symbol": "r", "friedman_metric_value": 0.866, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3/2 = .866+", "metric_symbol": "r", "metric_value": 0.866, "reference_text": "r = \u221a3/2 = .866+ | Trivial.", "source_image": "tc2.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc2.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = \u221a3/2 = .866+ | Trivial.", "friedman_source_image": "tc2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "92f5511b689fe041", "improvement": -3.858712037896339e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.072458078845486e-06, "max_pair_overlap_depth": -1.0890566688626091e-05, "metric_expression": "0.86604", "metric_symbol": "r", "metric_value": 0.866038587120379, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.866, "setup": "triincir", "side": 1.732077174240758, "solution_path": "data/solutions/generated/triincir_002_92f5511b689fe041.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/92f5511b689fe041.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@3", "density": 0.45476420302551357, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc3.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "3\u221a3 - 3\u221a2 = .953+", "friedman_metric_symbol": "r", "friedman_metric_value": 0.953, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@3", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "3\u221a3 - 3\u221a2 = .953+", "metric_symbol": "r", "metric_value": 0.953, "reference_text": "r = 3\u221a3 - 3\u221a2 = .953+ | Found by Erich Friedman | in 1997.", "source_image": "tc3.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc3.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 3\u221a3 - 3\u221a2 = .953+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "6ede8225b5939550", "improvement": -0.0005486885675038966, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4231654198736408e-06, "max_pair_overlap_depth": -2.0408752807413166e-05, "metric_expression": "0.95355", "metric_symbol": "r", "metric_value": 0.9535486885675039, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.953, "setup": "triincir", "side": 1.9070973771350077, "solution_path": "data/solutions/generated/triincir_003_6ede8225b5939550.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/6ede8225b5939550.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@4", "density": 0.5732690550709103, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc4.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": ".980+", "friedman_metric_symbol": "r", "friedman_metric_value": 0.98, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@4", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": ".980+", "metric_symbol": "r", "metric_value": 0.98, "reference_text": "r = .980+ | Found by Erich Friedman | in 1997.", "source_image": "tc4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc4.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = .980+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "7f9481f33de5ecda", "improvement": -0.0006773112916812485, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.642177417024662e-06, "max_pair_overlap_depth": -2.677155749965807e-05, "metric_expression": "0.98068", "metric_symbol": "r", "metric_value": 0.9806773112916812, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.98, "setup": "triincir", "side": 1.9613546225833625, "solution_path": "data/solutions/generated/triincir_004_7f9481f33de5ecda.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/7f9481f33de5ecda.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@5", "density": 0.6891272508887964, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc5.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1", "friedman_metric_symbol": "r", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "r", "metric_value": 1.0, "reference_text": "r = 1 | Trivial.", "source_image": "tc5.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc5.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1 | Trivial.", "friedman_source_image": "tc5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "cf17d061ebfe392b", "improvement": -2.4573090296353683e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.800898255106432e-06, "max_pair_overlap_depth": -9.00554050570603e-06, "metric_expression": "1.00002", "metric_symbol": "r", "metric_value": 1.0000245730902964, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.0, "setup": "triincir", "side": 2.0000491461805927, "solution_path": "data/solutions/generated/triincir_005_cf17d061ebfe392b.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/cf17d061ebfe392b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@6", "density": 0.826963635452614, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc6.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1", "friedman_metric_symbol": "r", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "r", "metric_value": 1.0, "reference_text": "r = 1 | Trivial.", "source_image": "tc6.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc6.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1 | Trivial.", "friedman_source_image": "tc6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "fe93ba87f23c6c01", "improvement": -1.7961741001881748e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.861140867618019e-08, "max_pair_overlap_depth": -9.037179045745058e-06, "metric_expression": "1.00002", "metric_symbol": "r", "metric_value": 1.0000179617410019, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.0, "setup": "triincir", "side": 2.0000359234820038, "solution_path": "data/solutions/generated/triincir_006_fe93ba87f23c6c01.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/fe93ba87f23c6c01.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@7", "density": 0.7265087780749782, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc7.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.152+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.152, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@7", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "1.152+", "metric_symbol": "r", "metric_value": 1.152, "reference_text": "r = 1.152+ | Found by Erich Friedman | in 1997.", "source_image": "tc7.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc7.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.152+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "4d32cb6092e127b9", "improvement": -0.0004019079826915739, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.469968968942808e-06, "max_pair_overlap_depth": -4.662019983989185e-06, "metric_expression": "1.15240", "metric_symbol": "r", "metric_value": 1.1524019079826915, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.152, "setup": "triincir", "side": 2.304803815965383, "solution_path": "data/solutions/generated/triincir_007_4d32cb6092e127b9.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/4d32cb6092e127b9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@8", "density": 0.6908236123299982, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc8.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.263+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.263, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@8", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "", "metric_expression": "1.263+", "metric_symbol": "r", "metric_value": 1.263, "reference_text": "r = 1.263+ | Found by Maurizio Morandi | in April 2008.", "source_image": "tc8.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc8.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.263+ | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "tc8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "33389e3c966a2171", "improvement": -0.0003881191911174753, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.08502399998045e-06, "max_pair_overlap_depth": -2.667573266001977e-06, "metric_expression": "1.26339", "metric_symbol": "r", "metric_value": 1.2633881191911174, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.263, "setup": "triincir", "side": 2.5267762383822348, "solution_path": "data/solutions/generated/triincir_008_33389e3c966a2171.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/33389e3c966a2171.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@9", "density": 0.7175995300521633, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc9.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.314+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.314, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@9", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "1.314+", "metric_symbol": "r", "metric_value": 1.314, "reference_text": "r = 1.314+ | Found by David W. Cantrell | in July 2002.", "source_image": "tc9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc9.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.314+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "tc9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "68692b6dc093b182", "improvement": -0.0007874911911363025, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.28479338054899e-06, "max_pair_overlap_depth": -4.021549232235966e-06, "metric_expression": "1.31479", "metric_symbol": "r", "metric_value": 1.3147874911911364, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.314, "setup": "triincir", "side": 2.6295749823822727, "solution_path": "data/solutions/generated/triincir_009_68692b6dc093b182.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/68692b6dc093b182.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@10", "density": 0.7187883776258066, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc10.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.385+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.385, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@10", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "1.385+", "metric_symbol": "r", "metric_value": 1.385, "reference_text": "r = 1.385+ | Found by David W. Cantrell | in July 2002.", "source_image": "tc10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc10.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.385+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "tc10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "676585e5c31a1d2f", "improvement": 0.0002388888356295027, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.405400112705848e-06, "max_pair_overlap_depth": -1.0902511275390214e-05, "metric_expression": "1.38476", "metric_symbol": "r", "metric_value": 1.3847611111643705, "n": 10, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.385, "setup": "triincir", "side": 2.769522222328741, "solution_path": "data/solutions/generated/triincir_010_676585e5c31a1d2f.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/676585e5c31a1d2f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@11", "density": 0.7044397586610418, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell after Maurizio Morandi", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc11.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.466+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.466, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@11", "credit": "David W. Cantrell after Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "1.466+", "metric_symbol": "r", "metric_value": 1.466, "reference_text": "r = 1.466+ | Found by DC after MM | in August 2012.", "source_image": "tc11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc11.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.466+ | Found by DC after MM | in August 2012.", "friedman_source_image": "tc11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "27341d9b331a210b", "improvement": -0.0010664804404059147, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5710045151440255e-06, "max_pair_overlap_depth": -1.3965597729533735e-05, "metric_expression": "1.46707", "metric_symbol": "r", "metric_value": 1.4670664804404059, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.466, "setup": "triincir", "side": 2.9341329608808118, "solution_path": "data/solutions/generated/triincir_011_27341d9b331a210b.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell after Maurizio Morandi", "svg_path": "data/svg/27341d9b331a210b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@12", "density": 0.7276318026083056, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "October 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc12.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.507+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.507, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@12", "credit": "David W. Cantrell", "date": "October 2005", "family_updated": "", "metric_expression": "1.507+", "metric_symbol": "r", "metric_value": 1.507, "reference_text": "r = 1.507+ | Found by David W. Cantrell | in October 2005.", "source_image": "tc12.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc12.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.507+ | Found by David W. Cantrell | in October 2005.", "friedman_source_image": "tc12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "0430950ce5a6ad65", "improvement": -0.0006834530254535842, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.988712507651542e-07, "max_pair_overlap_depth": -1.4096456774082e-06, "metric_expression": "1.50768", "metric_symbol": "r", "metric_value": 1.5076834530254535, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.507, "setup": "triincir", "side": 3.015366906050907, "solution_path": "data/solutions/generated/triincir_012_0430950ce5a6ad65.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/0430950ce5a6ad65.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@13", "density": 0.7678670311765523, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc13.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "\u221a(7/3) = 1.527+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.527, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@13", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a(7/3) = 1.527+", "metric_symbol": "r", "metric_value": 1.527, "reference_text": "r = \u221a(7/3) = 1.527+ | Found by Erich Friedman | in 1997.", "source_image": "tc13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc13.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = \u221a(7/3) = 1.527+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "2b014eeead965100", "improvement": -0.0005802936296193106, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.995535488687807e-06, "max_pair_overlap_depth": -6.278738689302621e-06, "metric_expression": "1.52758", "metric_symbol": "r", "metric_value": 1.5275802936296192, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.527, "setup": "triincir", "side": 3.0551605872592384, "solution_path": "data/solutions/generated/triincir_013_2b014eeead965100.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/2b014eeead965100.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@14", "density": 0.7492543391037492, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell after Maurizio Morandi", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc14.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.609+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.609, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@14", "credit": "David W. Cantrell after Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "1.609+", "metric_symbol": "r", "metric_value": 1.609, "reference_text": "r = 1.609+ | Found by DC after MM | in August 2012.", "source_image": "tc14.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc14.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.609+ | Found by DC after MM | in August 2012.", "friedman_source_image": "tc14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": false, "id": "da4114b8f6b7e383", "improvement": 0.0041857717254343285, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4204418656849072e-06, "max_pair_overlap_depth": -6.9567849071972e-06, "metric_expression": "1.60481", "metric_symbol": "r", "metric_value": 1.6048142282745657, "n": 14, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 1.609, "setup": "triincir", "side": 3.2096284565491313, "solution_path": "data/solutions/generated/triincir_014_da4114b8f6b7e383.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/da4114b8f6b7e383.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@15", "density": 0.7710002838935242, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc15.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.636+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.636, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@15", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "1.636+", "metric_symbol": "r", "metric_value": 1.636, "reference_text": "r = 1.636+ | Found by David W. Cantrell | in July 2002.", "source_image": "tc15.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc15.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.636+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "tc15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "e8f573cc9f431ed8", "improvement": -0.001546881163444036, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.392157777674768e-06, "max_pair_overlap_depth": -3.2056821285264547e-06, "metric_expression": "1.63755", "metric_symbol": "r", "metric_value": 1.637546881163444, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.636, "setup": "triincir", "side": 3.275093762326888, "solution_path": "data/solutions/generated/triincir_015_e8f573cc9f431ed8.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/e8f573cc9f431ed8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@16", "density": 0.7739404139048575, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc16.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.687+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.687, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@16", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "", "metric_expression": "1.687+", "metric_symbol": "r", "metric_value": 1.687, "reference_text": "r = 1.687+ | Found by Maurizio Morandi | in April 2008.", "source_image": "tc16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc16.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.687+ | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "tc16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "8bb5531a7df45934", "improvement": -0.0010356342518775818, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.068784430077415e-07, "max_pair_overlap_depth": -9.622687440202782e-07, "metric_expression": "1.68804", "metric_symbol": "r", "metric_value": 1.6880356342518776, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.687, "setup": "triincir", "side": 3.3760712685037553, "solution_path": "data/solutions/generated/triincir_16_8bb5531a7df45934.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/8bb5531a7df45934.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@17", "density": 0.7824875965148136, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc17.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.730+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.73, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@17", "credit": "David W. Cantrell", "date": "July 2005", "family_updated": "", "metric_expression": "1.730+", "metric_symbol": "r", "metric_value": 1.73, "reference_text": "r = 1.730+ | Found by David W. Cantrell | in July 2005.", "source_image": "tc17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc17.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.730+ | Found by David W. Cantrell | in July 2005.", "friedman_source_image": "tc17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "47c836632f7377eb", "improvement": -0.00045819251380652126, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3405989713065622e-06, "max_pair_overlap_depth": -3.2207400054717894e-06, "metric_expression": "1.73046", "metric_symbol": "r", "metric_value": 1.7304581925138065, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.73, "setup": "triincir", "side": 3.460916385027613, "solution_path": "data/solutions/generated/triincir_017_47c836632f7377eb.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/47c836632f7377eb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@18", "density": 0.763334042584198, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc18.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "\u221a13/2 = 1.802+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.802, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@18", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "\u221a13/2 = 1.802+", "metric_symbol": "r", "metric_value": 1.802, "reference_text": "r = \u221a13/2 = 1.802+ | Found by Erich Friedman | in 1997.", "source_image": "tc18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc18.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = \u221a13/2 = 1.802+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "8d495d33c78cce38", "improvement": -0.000828100052461922, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0688499290090192e-06, "max_pair_overlap_depth": -2.7547144944017887e-06, "metric_expression": "1.80283", "metric_symbol": "r", "metric_value": 1.802828100052462, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.802, "setup": "triincir", "side": 3.605656200104924, "solution_path": "data/solutions/generated/triincir_018_8d495d33c78cce38.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/8d495d33c78cce38.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@19", "density": 0.7847061820967525, "friedman_analytical_or_proved": true, "friedman_credit": "Haym Hirsh", "friedman_date": "August 2019", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc19.gif", "friedman_metric_expression": "\u221a(10/3) = 1.825+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.825, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@19", "credit": "Haym Hirsh", "date": "August 2019", "family_updated": "", "metric_expression": "\u221a(10/3) = 1.825+", "metric_symbol": "r", "metric_value": 1.825, "reference_text": "r = \u221a(10/3) = 1.825+ | Found by Haym Hirsh | in August 2019.", "source_image": "tc19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc19.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = \u221a(10/3) = 1.825+ | Found by Haym Hirsh | in August 2019.", "friedman_source_image": "tc19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "975121af758119a3", "improvement": -0.001832145989409506, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1422789436554837e-06, "max_pair_overlap_depth": -1.3515939734642934e-06, "metric_expression": "1.82683", "metric_symbol": "r", "metric_value": 1.8268321459894095, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.825, "setup": "triincir", "side": 3.653664291978819, "solution_path": "data/solutions/generated/triincir_19_975121af758119a3.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T14:34:18+00:00", "submitted_by": "Haym Hirsh", "svg_path": "data/svg/975121af758119a3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@20", "density": 0.7838286390350678, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc20.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.874+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.874, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@20", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "1.874+", "metric_symbol": "r", "metric_value": 1.874, "reference_text": "r = 1.874+ | Found by Erich Friedman | in 1997.", "source_image": "tc20.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc20.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.874+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "1217493f9010e655", "improvement": -0.0013391312177313264, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.256942872007727e-07, "max_pair_overlap_depth": -1.3401476308239069e-05, "metric_expression": "1.87534", "metric_symbol": "r", "metric_value": 1.8753391312177314, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.874, "setup": "triincir", "side": 3.750678262435463, "solution_path": "data/solutions/generated/triincir_020_1217493f9010e655.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/1217493f9010e655.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@21", "density": 0.7923995336773865, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc21.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.910+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.91, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@21", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "1.910+", "metric_symbol": "r", "metric_value": 1.91, "reference_text": "r = 1.910+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tc21.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc21.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.910+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tc21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "46b1371ff82573dd", "improvement": -0.001229866251050593, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3922552147160872e-05, "max_pair_overlap_depth": -4.454706840822942e-06, "metric_expression": "1.91123", "metric_symbol": "r", "metric_value": 1.9112298662510505, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.91, "setup": "triincir", "side": 3.822459732502101, "solution_path": "data/solutions/generated/triincir_021_46b1371ff82573dd.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/46b1371ff82573dd.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@22", "density": 0.8017796736685735, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc22.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.943+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.943, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@22", "credit": "Maurizio Morandi", "date": "June 2008", "family_updated": "", "metric_expression": "1.943+", "metric_symbol": "r", "metric_value": 1.943, "reference_text": "r = 1.943+ | Found by Maurizio Morandi | in June 2008.", "source_image": "tc22.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc22.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.943+ | Found by Maurizio Morandi | in June 2008.", "friedman_source_image": "tc22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "69faaab68dcae114", "improvement": -0.0017294951342150533, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.28347034731047e-06, "max_pair_overlap_depth": -4.770520652064292e-06, "metric_expression": "1.94473", "metric_symbol": "r", "metric_value": 1.9447294951342151, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.943, "setup": "triincir", "side": 3.8894589902684302, "solution_path": "data/solutions/generated/triincir_022_69faaab68dcae114.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/69faaab68dcae114.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@23", "density": 0.8020036651008373, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc23.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.988+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.988, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@23", "credit": "Maurizio Morandi", "date": "June 2008", "family_updated": "", "metric_expression": "1.988+", "metric_symbol": "r", "metric_value": 1.988, "reference_text": "r = 1.988+ | Found by Maurizio Morandi | in June 2008.", "source_image": "tc23.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc23.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.988+ | Found by Maurizio Morandi | in June 2008.", "friedman_source_image": "tc23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "bc03a535bc1be813", "improvement": -0.0001590443736558811, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0403434641002036e-06, "max_pair_overlap_depth": -2.2042403398092603e-06, "metric_expression": "1.98816", "metric_symbol": "r", "metric_value": 1.9881590443736559, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.988, "setup": "triincir", "side": 3.9763180887473117, "solution_path": "data/solutions/generated/triincir_023_bc03a535bc1be813.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/bc03a535bc1be813.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@24", "density": 0.82672890671807, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc24.gif", "friedman_metric_expression": "2", "friedman_metric_symbol": "r", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@24", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "r", "metric_value": 2.0, "reference_text": "r = 2 | Trivial.", "source_image": "tc24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc24.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2 | Trivial.", "friedman_source_image": "tc24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "e1de703df3a339d6", "improvement": -0.00031983310403260745, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.886123831735148e-06, "max_pair_overlap_depth": -6.330279525013083e-07, "metric_expression": "2.00032", "metric_symbol": "r", "metric_value": 2.0003198331040326, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "triincir", "side": 4.000639666208065, "solution_path": "data/solutions/generated/triincir_24_e1de703df3a339d6.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/e1de703df3a339d6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@25", "density": 0.814798554480224, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc25.gif", "friedman_metric_expression": "2.055+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.055, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@25", "credit": "David W. Cantrell", "date": "July 2005", "family_updated": "", "metric_expression": "2.055+", "metric_symbol": "r", "metric_value": 2.055, "reference_text": "r = 2.055+ | Found by David W. Cantrell | in July 2005.", "source_image": "tc25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc25.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.055+ | Found by David W. Cantrell | in July 2005.", "friedman_source_image": "tc25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "460bfc63493e0a99", "improvement": -0.0014599742843675045, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.499842621325968e-07, "max_pair_overlap_depth": -5.533591762807433e-06, "metric_expression": "2.05646", "metric_symbol": "r", "metric_value": 2.0564599742843677, "n": 25, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.055, "setup": "triincir", "side": 4.112919948568735, "solution_path": "data/solutions/generated/triincir_025_460bfc63493e0a99.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/460bfc63493e0a99.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@26", "density": 0.8055666262709128, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc26.gif", "friedman_metric_expression": "2.109+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.109, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@26", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "", "metric_expression": "2.109+", "metric_symbol": "r", "metric_value": 2.109, "reference_text": "r = 2.109+ | Found by Maurizio Morandi | in April 2008.", "source_image": "tc26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc26.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.109+ | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "tc26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "8245fa6b2a472124", "improvement": -0.00016872463985961872, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.221408735321063e-06, "max_pair_overlap_depth": -3.3631357487795555e-07, "metric_expression": "2.10917", "metric_symbol": "r", "metric_value": 2.1091687246398596, "n": 26, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.109, "setup": "triincir", "side": 4.218337449279719, "solution_path": "data/solutions/generated/triincir_26_8245fa6b2a472124.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/8245fa6b2a472124.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@27", "density": 0.8132267625810148, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc27.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "2.138+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.138, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@27", "credit": "David W. Cantrell", "date": "July 2005", "family_updated": "", "metric_expression": "2.138+", "metric_symbol": "r", "metric_value": 2.138, "reference_text": "r = 2.138+ | Found by David W. Cantrell | in July 2005.", "source_image": "tc27.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc27.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.138+ | Found by David W. Cantrell | in July 2005.", "friedman_source_image": "tc27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "6af16151fd4089a6", "improvement": -0.0012002098551513463, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.142316272914286e-07, "max_pair_overlap_depth": -2.7141719629586447e-06, "metric_expression": "2.13920", "metric_symbol": "r", "metric_value": 2.1392002098551512, "n": 27, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.138, "setup": "triincir", "side": 4.2784004197103025, "solution_path": "data/solutions/generated/triincir_027_6af16151fd4089a6.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/6af16151fd4089a6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@28", "density": 0.8029867070046219, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc28.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "2.191+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.191, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@28", "credit": "David W. Cantrell", "date": "July 2005", "family_updated": "", "metric_expression": "2.191+", "metric_symbol": "r", "metric_value": 2.191, "reference_text": "r = 2.191+ | Found by David W. Cantrell | in July 2005.", "source_image": "tc28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc28.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.191+ | Found by David W. Cantrell | in July 2005.", "friedman_source_image": "tc28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "b8a030f48c690095", "improvement": -0.0013011891131600173, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3556564240069235e-06, "max_pair_overlap_depth": -5.7499508869796045e-06, "metric_expression": "2.19230", "metric_symbol": "r", "metric_value": 2.19230118911316, "n": 28, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.191, "setup": "triincir", "side": 4.38460237822632, "solution_path": "data/solutions/generated/triincir_028_b8a030f48c690095.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/b8a030f48c690095.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@29", "density": 0.8182348688742829, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc29.gif", "friedman_metric_expression": "2.215+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.215, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@29", "credit": "Maurizio Morandi", "date": "June 2008", "family_updated": "", "metric_expression": "2.215+", "metric_symbol": "r", "metric_value": 2.215, "reference_text": "r = 2.215+ | Found by Maurizio Morandi | in June 2008.", "source_image": "tc29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc29.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.215+ | Found by Maurizio Morandi | in June 2008.", "friedman_source_image": "tc29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": false, "id": "73bcaf90b9fd2226", "improvement": 0.004780586805306797, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.1271053040349273e-06, "max_pair_overlap_depth": -1.2019034158372577e-05, "metric_expression": "2.21022", "metric_symbol": "r", "metric_value": 2.210219413194693, "n": 29, "notes": "Verified coordinate improvement; exceeds the displayed Friedman value by more than 0.15%.", "record_type": "verified", "reference_side": 2.215, "setup": "triincir", "side": 4.420438826389386, "solution_path": "data/solutions/generated/triincir_029_73bcaf90b9fd2226.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/73bcaf90b9fd2226.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@30", "density": 0.8110819400631206, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc30.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "2.258+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.258, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@30", "credit": "Maurizio Morandi", "date": "July 2008", "family_updated": "", "metric_expression": "2.258+", "metric_symbol": "r", "metric_value": 2.258, "reference_text": "r = 2.258+ | Found by Maurizio Morandi | in July 2008.", "source_image": "tc30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc30.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.258+ | Found by Maurizio Morandi | in July 2008.", "friedman_source_image": "tc30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "f720ec98817b9983", "improvement": 0.00010551194829711008, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9289733554094255e-06, "max_pair_overlap_depth": -4.657725350298847e-06, "metric_expression": "2.25789", "metric_symbol": "r", "metric_value": 2.257894488051703, "n": 30, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.258, "setup": "triincir", "side": 4.515788976103406, "solution_path": "data/solutions/generated/triincir_030_f720ec98817b9983.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/f720ec98817b9983.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triincir@31", "density": 0.8263348764897189, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc31.gif", "friedman_metric_expression": "2.272+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.272, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@31", "credit": "Maurizio Morandi", "date": "July 2008", "family_updated": "", "metric_expression": "2.272+", "metric_symbol": "r", "metric_value": 2.272, "reference_text": "r = 2.272+ | Found by Maurizio Morandi | in July 2008.", "source_image": "tc31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc31.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.272+ | Found by Maurizio Morandi | in July 2008.", "friedman_source_image": "tc31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "ed4a5072cc8d0adf", "improvement": -0.0019357369126002943, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3803713964376527e-06, "max_pair_overlap_depth": -5.497010580973738e-06, "metric_expression": "2.27394", "metric_symbol": "r", "metric_value": 2.2739357369126, "n": 31, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.272, "setup": "triincir", "side": 4.5478714738252, "solution_path": "data/solutions/generated/triincir_31_ed4a5072cc8d0adf.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/ed4a5072cc8d0adf.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@1", "density": 0.2886702903701934, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/2.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "\u221a3/2", "friedman_metric_symbol": "s", "friedman_metric_value": 0.8660254037844386, "friedman_reference": {"analytical_or_proved": true, "case": "triindom@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3/2", "metric_symbol": "s", "metric_value": 0.8660254037844386, "reference_text": "1-2. | | s = \u221a3/2 | Trivial. |", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/2.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "1-2. | | s = \u221a3/2 | Trivial. |", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "37b1dca676a779e3", "improvement": -7.266428382335377e-06, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4019537994690587e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.86603", "metric_symbol": "s", "metric_value": 0.8660326702128209, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.8660254037844386, "setup": "triindom", "side": 1.7320653404256419, "solution_path": "data/solutions/generated/triindom_001_37b1dca676a779e3.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/37b1dca676a779e3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@2", "density": 0.5773351362696808, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/2.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "\u221a3/2", "friedman_metric_symbol": "s", "friedman_metric_value": 0.8660254037844386, "friedman_reference": {"analytical_or_proved": true, "case": "triindom@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3/2", "metric_symbol": "s", "metric_value": 0.8660254037844386, "reference_text": "1-2. | | s = \u221a3/2 | Trivial. |", "source_image": "2.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/2.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "1-2. | | s = \u221a3/2 | Trivial. |", "friedman_source_image": "2.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "31518429031d2b51", "improvement": -1.1349913078717577e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.727531273176268e-07, "max_pair_overlap_depth": -0.13184728443338464, "metric_expression": "0.86604", "metric_symbol": "s", "metric_value": 0.8660367536975173, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.8660254037844386, "setup": "triindom", "side": 1.7320735073950346, "solution_path": "data/solutions/generated/triindom_002_31518429031d2b51.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/31518429031d2b51.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@3", "density": 0.6931800833949334, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/3.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": ".96796+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.96796, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@3", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": ".96796+", "metric_symbol": "s", "metric_value": 0.96796, "reference_text": "3. | | s = .96796+ | Found by Jonathan Viquerat | in April 2026", "source_image": "3.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/3.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "3. | | s = .96796+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "3.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "b10402bd10926cd1", "improvement": -3.454421484250236e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.601436348197538e-06, "max_pair_overlap_depth": -2.0206571078401625e-05, "metric_expression": "0.96799", "metric_symbol": "s", "metric_value": 0.9679945442148425, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.96796, "setup": "triindom", "side": 1.935989088429685, "solution_path": "data/solutions/generated/triindom_003_b10402bd10926cd1.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b10402bd10926cd1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@4", "density": 0.6899032945238053, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/4.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.12036+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.12036, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@4", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.12036+", "metric_symbol": "s", "metric_value": 1.12036, "reference_text": "4. | | s = 1.12036+ | Found by Jonathan Viquerat | in April 2026", "source_image": "4.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/4.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "4. | | s = 1.12036+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "4.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "daed4fc0474960d9", "improvement": -3.511453092652772e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.5154853256512695e-07, "max_pair_overlap_depth": -1.4193770285886309e-05, "metric_expression": "1.12040", "metric_symbol": "s", "metric_value": 1.1203951145309266, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.12036, "setup": "triindom", "side": 2.240790229061853, "solution_path": "data/solutions/generated/triindom_004_daed4fc0474960d9.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/daed4fc0474960d9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@5", "density": 0.6784100035927487, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/5.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.26315+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.26315, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@5", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.26315+", "metric_symbol": "s", "metric_value": 1.26315, "reference_text": "5. | | s = 1.26315+ | Found by Jonathan Viquerat | in April 2026", "source_image": "5.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/5.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "5. | | s = 1.26315+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "5.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "5f00d1fbe39c88e1", "improvement": -5.606079820497101e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.863453948036664e-06, "max_pair_overlap_depth": -2.090220821476807e-05, "metric_expression": "1.26321", "metric_symbol": "s", "metric_value": 1.263206060798205, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.26315, "setup": "triindom", "side": 2.52641212159641, "solution_path": "data/solutions/generated/triindom_005_5f00d1fbe39c88e1.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/5f00d1fbe39c88e1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@6", "density": 0.6857340379834939, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/6.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.37629+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.37629, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@6", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.37629+", "metric_symbol": "s", "metric_value": 1.37629, "reference_text": "6. | | s = 1.37629+ | Found by Jonathan Viquerat | in April 2026", "source_image": "6.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/6.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "6. | | s = 1.37629+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "6.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "cd22df1373fe40ae", "improvement": -7.332504683676966e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4791971315062398e-06, "max_pair_overlap_depth": -2.311431619983484e-05, "metric_expression": "1.37636", "metric_symbol": "s", "metric_value": 1.3763633250468368, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.37629, "setup": "triindom", "side": 2.7527266500936736, "solution_path": "data/solutions/generated/triindom_006_cd22df1373fe40ae.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/cd22df1373fe40ae.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@7", "density": 0.6816978937448389, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/7.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.49076+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.49076, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@7", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.49076+", "metric_symbol": "s", "metric_value": 1.49076, "reference_text": "7. | | s = 1.49076+ | Found by Jonathan Viquerat | in April 2026", "source_image": "7.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/7.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "7. | | s = 1.49076+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "7.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "070c4bf393378882", "improvement": -0.0002768061413715994, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6376140599128062e-06, "max_pair_overlap_depth": -1.7596834968304498e-05, "metric_expression": "1.49104", "metric_symbol": "s", "metric_value": 1.4910368061413717, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.49076, "setup": "triindom", "side": 2.9820736122827434, "solution_path": "data/solutions/generated/triindom_007_070c4bf393378882.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/070c4bf393378882.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@8", "density": 0.7089538292098625, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/8.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.56293+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.56293, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@8", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.56293+", "metric_symbol": "s", "metric_value": 1.56293, "reference_text": "8. | | s = 1.56293+ | Found by Jonathan Viquerat | in April 2026", "source_image": "8.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/8.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "8. | | s = 1.56293+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "8.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "dc89b8abdea24aaa", "improvement": -0.00011448128988833517, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.119934312653896e-07, "max_pair_overlap_depth": -2.4977335194534267e-05, "metric_expression": "1.56304", "metric_symbol": "s", "metric_value": 1.5630444812898883, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.56293, "setup": "triindom", "side": 3.1260889625797765, "solution_path": "data/solutions/generated/triindom_008_dc89b8abdea24aaa.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/dc89b8abdea24aaa.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@9", "density": 0.7166974833845204, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/9.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.64908+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.64908, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@9", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.64908+", "metric_symbol": "s", "metric_value": 1.64908, "reference_text": "9. | | s = 1.64908+ | Found by Jonathan Viquerat | in April 2026", "source_image": "9.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/9.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "9. | | s = 1.64908+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "9.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "9817bdf32d5c0df3", "improvement": 0.00020157610904458778, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.318788145301177e-06, "max_pair_overlap_depth": -1.0913014285005018e-05, "metric_expression": "1.64888", "metric_symbol": "s", "metric_value": 1.6488784238909555, "n": 9, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.64908, "setup": "triindom", "side": 3.297756847781911, "solution_path": "data/solutions/generated/triindom_009_9817bdf32d5c0df3.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/9817bdf32d5c0df3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@10", "density": 0.7634583385020529, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/10.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.68303+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.68303, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@10", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.68303+", "metric_symbol": "s", "metric_value": 1.68303, "reference_text": "10. | | s = 1.68303+ | Found by Jonathan Viquerat | in April 2026", "source_image": "10.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/10.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "10. | | s = 1.68303+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "10.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "d60b37ee7b5fe8bb", "improvement": -0.0009721776557043427, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.2406370054083306e-07, "max_pair_overlap_depth": -6.360676094274709e-06, "metric_expression": "1.68400", "metric_symbol": "s", "metric_value": 1.6840021776557044, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.68303, "setup": "triindom", "side": 3.3680043553114087, "solution_path": "data/solutions/generated/triindom_010_d60b37ee7b5fe8bb.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/d60b37ee7b5fe8bb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@11", "density": 0.7938130122561631, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/11.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.73127+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.73127, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@11", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.73127+", "metric_symbol": "s", "metric_value": 1.73127, "reference_text": "11. | | s = 1.73127+ | Found by Jonathan Viquerat | in April 2026", "source_image": "11.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/11.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "11. | | s = 1.73127+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "11.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "92f51b49b50f6469", "improvement": -0.0008283817611449695, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.752164488808219e-06, "max_pair_overlap_depth": -9.417608768792381e-06, "metric_expression": "1.73210", "metric_symbol": "s", "metric_value": 1.732098381761145, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.73127, "setup": "triindom", "side": 3.46419676352229, "solution_path": "data/solutions/generated/triindom_011_92f51b49b50f6469.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/92f51b49b50f6469.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@12", "density": 0.8578074620402373, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/12.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.74040+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.7404, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@12", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.74040+", "metric_symbol": "s", "metric_value": 1.7404, "reference_text": "12. | | s = 1.74040+ | Found by Jonathan Viquerat | in April 2026", "source_image": "12.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/12.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "12. | | s = 1.74040+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "12.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "d4d97a50d19ba7f9", "improvement": 7.229766750849187e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6179955143756928e-06, "max_pair_overlap_depth": -3.432318619178698e-05, "metric_expression": "1.74033", "metric_symbol": "s", "metric_value": 1.7403277023324915, "n": 12, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.7404, "setup": "triindom", "side": 3.480655404664983, "solution_path": "data/solutions/generated/triindom_012_d4d97a50d19ba7f9.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/d4d97a50d19ba7f9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@13", "density": 0.819430202315155, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/13.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.85319+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.85319, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@13", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.85319+", "metric_symbol": "s", "metric_value": 1.85319, "reference_text": "13. | | s = 1.85319+ | Found by Jonathan Viquerat | in April 2026", "source_image": "13.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/13.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "13. | | s = 1.85319+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "13.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "734b8b487f97f149", "improvement": -0.00013256217488510913, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3279304722789718e-06, "max_pair_overlap_depth": -3.056740874492897e-05, "metric_expression": "1.85332", "metric_symbol": "s", "metric_value": 1.853322562174885, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.85319, "setup": "triindom", "side": 3.70664512434977, "solution_path": "data/solutions/generated/triindom_013_734b8b487f97f149.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/734b8b487f97f149.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@14", "density": 0.837880420446461, "friedman_analytical_or_proved": true, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/14.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "3(3\u2013\u221a3)/2 = 1.90192+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.90192, "friedman_reference": {"analytical_or_proved": true, "case": "triindom@14", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "3(3\u2013\u221a3)/2 = 1.90192+", "metric_symbol": "s", "metric_value": 1.90192, "reference_text": "14. | | s = 3(3\u2013\u221a3)/2 = 1.90192+ | Found by Jonathan Viquerat | in April 2026", "source_image": "14.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/14.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "14. | | s = 3(3\u2013\u221a3)/2 = 1.90192+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "14.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "050249738efebc30", "improvement": -7.038452971785958e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8171942911825454e-06, "max_pair_overlap_depth": -1.6337035744662387e-05, "metric_expression": "1.90199", "metric_symbol": "s", "metric_value": 1.901990384529718, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.90192, "setup": "triindom", "side": 3.803980769059436, "solution_path": "data/solutions/generated/triindom_014_050249738efebc30.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/050249738efebc30.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@15", "density": 0.827720714535801, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/15.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "1.97950+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.9795, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@15", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "1.97950+", "metric_symbol": "s", "metric_value": 1.9795, "reference_text": "15. | | s = 1.97950+ | Found by Jonathan Viquerat | in April 2026", "source_image": "15.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/15.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "15. | | s = 1.97950+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "15.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "a012f18849dc81a9", "improvement": -0.001292749811030447, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8417244084533166e-06, "max_pair_overlap_depth": -2.1063364326057155e-05, "metric_expression": "1.98079", "metric_symbol": "s", "metric_value": 1.9807927498110305, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.9795, "setup": "triindom", "side": 3.961585499622061, "solution_path": "data/solutions/generated/triindom_015_a012f18849dc81a9.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/a012f18849dc81a9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@16", "density": 0.8658669941172707, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/16.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@16", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "16. | | s = 2 | Found by Jonathan Viquerat | in April 2026", "source_image": "16.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/16.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "16. | | s = 2 | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "16.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "97ac1b8d90281e30", "improvement": -0.0001829408254119258, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4402032156322093e-07, "max_pair_overlap_depth": -8.654468217628164e-06, "metric_expression": "2.00018", "metric_symbol": "s", "metric_value": 2.000182940825412, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "triindom", "side": 4.000365881650824, "solution_path": "data/solutions/generated/triindom_016_97ac1b8d90281e30.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/97ac1b8d90281e30.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@17", "density": 0.8312856464398565, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/17.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "2.10306+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.10306, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@17", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.10306+", "metric_symbol": "s", "metric_value": 2.10306, "reference_text": "17. | | s = 2.10306+ | Found by Jonathan Viquerat | in April 2026", "source_image": "17.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/17.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "17. | | s = 2.10306+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "17.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "3b35e95b5dec0d61", "improvement": -0.001128486819858221, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9321384165115205e-07, "max_pair_overlap_depth": -4.175356206781444e-05, "metric_expression": "2.10419", "metric_symbol": "s", "metric_value": 2.1041884868198584, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.10306, "setup": "triindom", "side": 4.208376973639717, "solution_path": "data/solutions/generated/triindom_017_3b35e95b5dec0d61.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/3b35e95b5dec0d61.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@18", "density": 0.8436836711045761, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/18.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "2.14913.+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.14913, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@18", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.14913.+", "metric_symbol": "s", "metric_value": 2.14913, "reference_text": "18. | | s = 2.14913.+ | Found by Jonathan Viquerat | in April 2026", "source_image": "18.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/18.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "18. | | s = 2.14913.+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "18.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "c7d745720c03d379", "improvement": -9.434274541320065e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.205887803834571e-06, "max_pair_overlap_depth": -1.5204184911299379e-05, "metric_expression": "2.14922", "metric_symbol": "s", "metric_value": 2.149224342745413, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.14913, "setup": "triindom", "side": 4.298448685490826, "solution_path": "data/solutions/generated/triindom_018_c7d745720c03d379.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/c7d745720c03d379.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@19", "density": 0.815605476746989, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/19.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "2.24848+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.24848, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@19", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.24848+", "metric_symbol": "s", "metric_value": 2.24848, "reference_text": "19. | | s = 2.24848+ | Found by Jonathan Viquerat | in April 2026", "source_image": "19.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/19.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "19. | | s = 2.24848+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "19.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "ccdcfac9f05e2d80", "improvement": 0.002674950513982033, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.889707633764061e-07, "max_pair_overlap_depth": -1.2330676740557323e-05, "metric_expression": "2.24581", "metric_symbol": "s", "metric_value": 2.245805049486018, "n": 19, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.24848, "setup": "triindom", "side": 4.491610098972036, "solution_path": "data/solutions/generated/triindom_019_ccdcfac9f05e2d80.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/ccdcfac9f05e2d80.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triindom@20", "density": 0.820310655560819, "friedman_analytical_or_proved": false, "friedman_credit": "Jonathan Viquerat", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triindom/20.png", "friedman_family_updated": "4/4/26", "friedman_metric_expression": "2.29730+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.2973, "friedman_reference": {"analytical_or_proved": false, "case": "triindom@20", "credit": "Jonathan Viquerat", "date": "April 2026", "family_updated": "", "metric_expression": "2.29730+", "metric_symbol": "s", "metric_value": 2.2973, "reference_text": "20. | | s = 2.29730+ | Found by Jonathan Viquerat | in April 2026", "source_image": "20.png", "source_image_url": "https://erich-friedman.github.io/packing/triindom/20.png", "source_page": "https://erich-friedman.github.io/packing/triindom/", "title": "Triangles in Dominoes"}, "friedman_reference_text": "20. | | s = 2.29730+ | Found by Jonathan Viquerat | in April 2026", "friedman_source_image": "20.png", "friedman_source_url": "https://erich-friedman.github.io/packing/triindom/", "friedman_title": "Triangles in Dominoes", "frontend_seed": true, "id": "b651b20201d5daf5", "improvement": -0.00022973895965483493, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6504306361575516e-06, "max_pair_overlap_depth": -2.6692790362048413e-05, "metric_expression": "2.29753", "metric_symbol": "s", "metric_value": 2.2975297389596547, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.2973, "setup": "triindom", "side": 4.5950594779193095, "solution_path": "data/solutions/generated/triindom_020_b651b20201d5daf5.json", "source_url": "https://erich-friedman.github.io/packing/triindom/", "submitted_at": "2026-05-01T02:49:16+00:00", "submitted_by": "Jonathan Viquerat", "svg_path": "data/svg/b651b20201d5daf5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@1", "density": 0.4999941032613389, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/1.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1/\u221a3 = .577+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.577, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1/\u221a3 = .577+", "metric_symbol": "s", "metric_value": 0.577, "reference_text": "s = 1/\u221a3 = .577+ | Trivial.", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/1.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1/\u221a3 = .577+ | Trivial.", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "e9bcdcd079c1c3b7", "improvement": -0.0003536737033924542, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.663290249487904e-07, "max_pair_overlap_depth": 0.0, "metric_expression": "0.57735", "metric_symbol": "s", "metric_value": 0.5773536737033924, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.577, "setup": "triinhex", "side": 0.5773536737033924, "solution_path": "data/solutions/generated/triinhex_001_e9bcdcd079c1c3b7.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/e9bcdcd079c1c3b7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@2", "density": 0.4444266358884555, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/2.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a3/2 = .866+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.866, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3/2 = .866+", "metric_symbol": "s", "metric_value": 0.866, "reference_text": "s = \u221a3/2 = .866+ | Trivial.", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/2.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = \u221a3/2 = .866+ | Trivial.", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "132c3ed1fad59f87", "improvement": -4.275480049964475e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.75428637420658e-06, "max_pair_overlap_depth": -4.36783093215443e-06, "metric_expression": "0.86604", "metric_symbol": "s", "metric_value": 0.8660427548004996, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.866, "setup": "triinhex", "side": 0.8660427548004996, "solution_path": "data/solutions/generated/triinhex_002_132c3ed1fad59f87.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/132c3ed1fad59f87.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@3", "density": 0.5108536175549566, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/3.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a((6+\u221a33)/12) = .989+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.989, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@3", "credit": "Erich Friedman", "date": "April 2005", "family_updated": "", "metric_expression": "\u221a((6+\u221a33)/12) = .989+", "metric_symbol": "s", "metric_value": 0.989, "reference_text": "s = \u221a((6+\u221a33)/12) = .989+ | Found by Erich Friedman | in April 2005.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/3.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = \u221a((6+\u221a33)/12) = .989+ | Found by Erich Friedman | in April 2005.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "3f19c46d9d07a54d", "improvement": -0.00031994709745319927, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.001855317125196e-07, "max_pair_overlap_depth": -5.956334362088034e-06, "metric_expression": "0.98932", "metric_symbol": "s", "metric_value": 0.9893199470974532, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.989, "setup": "triinhex", "side": 0.9893199470974532, "solution_path": "data/solutions/generated/triinhex_003_3f19c46d9d07a54d.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/3f19c46d9d07a54d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@4", "density": 0.6666347131299533, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/4.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/4.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "9a8f72cbad0da23e", "improvement": -2.396601406218224e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.5130077190296347e-08, "max_pair_overlap_depth": -1.1324383645797624e-05, "metric_expression": "1.00002", "metric_symbol": "s", "metric_value": 1.0000239660140622, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.0, "setup": "triinhex", "side": 1.0000239660140622, "solution_path": "data/solutions/generated/triinhex_004_9a8f72cbad0da23e.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9a8f72cbad0da23e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@5", "density": 0.8333022730508762, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/5.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@5", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/5.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "0d61e48787b7d5d1", "improvement": -1.863669045065741e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0710555572355552e-06, "max_pair_overlap_depth": -2.545819687190834e-06, "metric_expression": "1.00002", "metric_symbol": "s", "metric_value": 1.0000186366904507, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.0, "setup": "triinhex", "side": 1.0000186366904507, "solution_path": "data/solutions/generated/triinhex_005_0d61e48787b7d5d1.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/0d61e48787b7d5d1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@6", "density": 0.9999558832137269, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/6.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@6", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/6.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "163de15143701be8", "improvement": -2.2059123022444282e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.751347439371281e-07, "max_pair_overlap_depth": -4.053173147537176e-06, "metric_expression": "1.00002", "metric_symbol": "s", "metric_value": 1.0000220591230224, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.0, "setup": "triinhex", "side": 1.0000220591230224, "solution_path": "data/solutions/generated/triinhex_006_163de15143701be8.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/163de15143701be8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@7", "density": 0.7149966699512036, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "May 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/7.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.277+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.277, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@7", "credit": "David W. Cantrell", "date": "May 2005", "family_updated": "", "metric_expression": "1.277+", "metric_symbol": "s", "metric_value": 1.277, "reference_text": "s = 1.277+ | Found by David W. Cantrell | in May 2005.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/7.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.277+ | Found by David W. Cantrell | in May 2005.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "af96c8dfd69688ff", "improvement": -0.0003837447071250999, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8980834118002716e-06, "max_pair_overlap_depth": -1.0312174508100846e-05, "metric_expression": "1.27738", "metric_symbol": "s", "metric_value": 1.277383744707125, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.277, "setup": "triinhex", "side": 1.277383744707125, "solution_path": "data/solutions/generated/triinhex_007_af96c8dfd69688ff.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/af96c8dfd69688ff.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@8", "density": 0.724469693124155, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/8.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.356+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.356, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@8", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "1.356+", "metric_symbol": "s", "metric_value": 1.356, "reference_text": "s = 1.356+ | Found by David W. Cantrell | in July 2012.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/8.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.356+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "d557994d1935d6fb", "improvement": -0.0006232543073974028, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6958384610035216e-07, "max_pair_overlap_depth": -6.193690244168937e-06, "metric_expression": "1.35662", "metric_symbol": "s", "metric_value": 1.3566232543073975, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.356, "setup": "triinhex", "side": 1.3566232543073975, "solution_path": "data/solutions/generated/triinhex_008_d557994d1935d6fb.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d557994d1935d6fb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@9", "density": 0.7288897241647109, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "April 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/9.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.434+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.434, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@9", "credit": "Erich Friedman", "date": "April 2005", "family_updated": "", "metric_expression": "1.434+", "metric_symbol": "s", "metric_value": 1.434, "reference_text": "s = 1.434+ | Found by Erich Friedman | in April 2005.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/9.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.434+ | Found by Erich Friedman | in April 2005.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "e1592cc41ab92d4d", "improvement": -0.0005467824277936018, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.288341076546003e-06, "max_pair_overlap_depth": -3.69502244107478e-06, "metric_expression": "1.43455", "metric_symbol": "s", "metric_value": 1.4345467824277935, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.434, "setup": "triinhex", "side": 1.4345467824277935, "solution_path": "data/solutions/generated/triinhex_009_e1592cc41ab92d4d.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/e1592cc41ab92d4d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@10", "density": 0.7432082873431878, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/10.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.497+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.497, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@10", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "1.497+", "metric_symbol": "s", "metric_value": 1.497, "reference_text": "s = 1.497+ | Found by Timo Berthold et al | in January 2026.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/10.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.497+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "27f77a1826f41a7d", "improvement": -0.0005078337450310499, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4736979206377467e-06, "max_pair_overlap_depth": -5.4498855950879665e-06, "metric_expression": "1.49751", "metric_symbol": "s", "metric_value": 1.4975078337450312, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.497, "setup": "triinhex", "side": 1.4975078337450312, "solution_path": "data/solutions/generated/triinhex_010_27f77a1826f41a7d.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/27f77a1826f41a7d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@11", "density": 0.8147753094192889, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/11.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3/2 = 1.500", "friedman_metric_symbol": "s", "friedman_metric_value": 1.5, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@11", "credit": "Erich Friedman", "date": "April 2005", "family_updated": "", "metric_expression": "3/2 = 1.500", "metric_symbol": "s", "metric_value": 1.5, "reference_text": "s = 3/2 = 1.500 | Found by Erich Friedman | in April 2005.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/11.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 3/2 = 1.500 | Found by Erich Friedman | in April 2005.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "a4d7af61e2935a14", "improvement": -3.636424319730658e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0384668058064728e-06, "max_pair_overlap_depth": -9.97710385775008e-06, "metric_expression": "1.50004", "metric_symbol": "s", "metric_value": 1.5000363642431973, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.5, "setup": "triinhex", "side": 1.5000363642431973, "solution_path": "data/solutions/generated/triinhex_011_a4d7af61e2935a14.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a4d7af61e2935a14.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@12", "density": 0.7583806241339298, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/12.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.623+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.623, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@12", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "1.623+", "metric_symbol": "s", "metric_value": 1.623, "reference_text": "s = 1.623+ | Found by Timo Berthold et al | in January 2026.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/12.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.623+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "9de540041e59931a", "improvement": -0.0009452508446496921, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.928415721340796e-06, "max_pair_overlap_depth": -1.3315156502324843e-05, "metric_expression": "1.62395", "metric_symbol": "s", "metric_value": 1.6239452508446497, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.623, "setup": "triinhex", "side": 1.6239452508446497, "solution_path": "data/solutions/generated/triinhex_012_9de540041e59931a.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/9de540041e59931a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@13", "density": 0.7821155301010267, "friedman_analytical_or_proved": true, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/13.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a(587+51\u221a17)/12\u221a2 = 1.663829+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.663829, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@13", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "\u221a(587+51\u221a17)/12\u221a2 = 1.663829+", "metric_symbol": "s", "metric_value": 1.663829, "reference_text": "s = \u221a(587+51\u221a17)/12\u221a2 = 1.663829+ | Found by Ian Watson | in April 2026.", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/13.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = \u221a(587+51\u221a17)/12\u221a2 = 1.663829+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "efc43a6cbf81ae24", "improvement": -0.0005820721473515089, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.9742574580490386e-06, "max_pair_overlap_depth": -2.1993638058326326e-05, "metric_expression": "1.66441", "metric_symbol": "s", "metric_value": 1.6644110721473515, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.663829, "setup": "triinhex", "side": 1.6644110721473515, "solution_path": "data/solutions/generated/triinhex_013_efc43a6cbf81ae24.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/efc43a6cbf81ae24.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@14", "density": 0.7828804179520853, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/14.gif", "friedman_metric_expression": "1.725+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.725, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@14", "credit": "David W. Cantrell", "date": "August 2012", "family_updated": "", "metric_expression": "1.725+", "metric_symbol": "s", "metric_value": 1.725, "reference_text": "s = 1.725+ | Found by David W. Cantrell | in August 2012.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/14.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.725+ | Found by David W. Cantrell | in August 2012.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "dbfe389f1c3777d5", "improvement": -0.0013970193042536216, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.5600740703080906e-07, "max_pair_overlap_depth": -1.248505498985253e-07, "metric_expression": "1.72640", "metric_symbol": "s", "metric_value": 1.7263970193042537, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.725, "setup": "triinhex", "side": 1.7263970193042537, "solution_path": "data/solutions/generated/triinhex_14_dbfe389f1c3777d5.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/dbfe389f1c3777d5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@15", "density": 0.8162509918871499, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/15.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "7/4 = 1.750", "friedman_metric_symbol": "s", "friedman_metric_value": 1.75, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@15", "credit": "Erich Friedman", "date": "April 2005", "family_updated": "", "metric_expression": "7/4 = 1.750", "metric_symbol": "s", "metric_value": 1.75, "reference_text": "s = 7/4 = 1.750 | Found by Erich Friedman | in April 2005.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/15.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 7/4 = 1.750 | Found by Erich Friedman | in April 2005.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "20b1a0e905211ce9", "improvement": -8.097369067638915e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.758472784251539e-06, "max_pair_overlap_depth": -9.75502017092067e-06, "metric_expression": "1.75008", "metric_symbol": "s", "metric_value": 1.7500809736906764, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.75, "setup": "triinhex", "side": 1.7500809736906764, "solution_path": "data/solutions/generated/triinhex_015_20b1a0e905211ce9.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/20b1a0e905211ce9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@16", "density": 0.8229668363309378, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/16.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "9/5 = 1.800", "friedman_metric_symbol": "s", "friedman_metric_value": 1.8, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@16", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "", "metric_expression": "9/5 = 1.800", "metric_symbol": "s", "metric_value": 1.8, "reference_text": "s = 9/5 = 1.800 | Found by Maurizio Morandi | in April 2008.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/16.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 9/5 = 1.800 | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "a0b20d24de5f00ac", "improvement": -8.577060222680899e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.740972449159294e-06, "max_pair_overlap_depth": -2.3139042383268915e-06, "metric_expression": "1.80009", "metric_symbol": "s", "metric_value": 1.8000857706022269, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.8, "setup": "triinhex", "side": 1.8000857706022269, "solution_path": "data/solutions/generated/triinhex_016_a0b20d24de5f00ac.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/a0b20d24de5f00ac.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@17", "density": 0.842931104392013, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/17.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "11/6 = 1.833+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.833, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@17", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "", "metric_expression": "11/6 = 1.833+", "metric_symbol": "s", "metric_value": 1.833, "reference_text": "s = 11/6 = 1.833+ | Found by Maurizio Morandi | in April 2008.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/17.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 11/6 = 1.833+ | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "b8b9e4b4d61ec96c", "improvement": -0.00038129277582044097, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4990722513823532e-07, "max_pair_overlap_depth": -1.1854387088505725e-05, "metric_expression": "1.83338", "metric_symbol": "s", "metric_value": 1.8333812927758204, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.833, "setup": "triinhex", "side": 1.8333812927758204, "solution_path": "data/solutions/generated/triinhex_017_b8b9e4b4d61ec96c.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/b8b9e4b4d61ec96c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@18", "density": 0.8400311467465807, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/18.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "17/9 = 1.888+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.888, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@18", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "", "metric_expression": "17/9 = 1.888+", "metric_symbol": "s", "metric_value": 1.888, "reference_text": "s = 17/9 = 1.888+ | Found by Maurizio Morandi | in April 2008.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/18.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 17/9 = 1.888+ | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "7671cfd644a7af24", "improvement": -0.001787329223866152, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.484924045151217e-06, "max_pair_overlap_depth": -8.796062168636709e-06, "metric_expression": "1.88979", "metric_symbol": "s", "metric_value": 1.889787329223866, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.888, "setup": "triinhex", "side": 1.889787329223866, "solution_path": "data/solutions/generated/triinhex_018_7671cfd644a7af24.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/7671cfd644a7af24.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@19", "density": 0.8283361930438612, "friedman_analytical_or_proved": false, "friedman_credit": "Alex Adri\u00e1n Nieto", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/19.gif", "friedman_metric_expression": "1.953+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.953, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@19", "credit": "Alex Adri\u00e1n Nieto", "date": "April 2026", "family_updated": "", "metric_expression": "1.953+", "metric_symbol": "s", "metric_value": 1.953, "reference_text": "s = 1.953+ | Found by Alex Adri\u00e1n Nieto | in April 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/19.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.953+ | Found by Alex Adri\u00e1n Nieto | in April 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "451018f1291a5791", "improvement": -0.0022300191069650754, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7079807782201595e-06, "max_pair_overlap_depth": -1.3478523960663846e-05, "metric_expression": "1.95523", "metric_symbol": "s", "metric_value": 1.9552300191069651, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.953, "setup": "triinhex", "side": 1.9552300191069651, "solution_path": "data/solutions/generated/triinhex_19_451018f1291a5791.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "Alex Adri\u00e1n Nieto", "svg_path": "data/svg/451018f1291a5791.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@20", "density": 0.8524037354438128, "friedman_analytical_or_proved": false, "friedman_credit": "Aaden Abraham", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/20.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.97591+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.97591, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@20", "credit": "Aaden Abraham", "date": "April 2026", "family_updated": "", "metric_expression": "1.97591+", "metric_symbol": "s", "metric_value": 1.97591, "reference_text": "s = 1.97591+ | Found by Aaden Abraham | in April 2026.", "source_image": "20.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/20.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.97591+ | Found by Aaden Abraham | in April 2026.", "friedman_source_image": "20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "388fc73ef451e99b", "improvement": -0.0015909487133152211, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7558573966506123e-06, "max_pair_overlap_depth": -5.2993150605473716e-06, "metric_expression": "1.97750", "metric_symbol": "s", "metric_value": 1.9775009487133153, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.97591, "setup": "triinhex", "side": 1.9775009487133153, "solution_path": "data/solutions/generated/triinhex_020_388fc73ef451e99b.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Aaden Abraham", "svg_path": "data/svg/388fc73ef451e99b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@21", "density": 0.8749527320468723, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/21.gif", "friedman_metric_expression": "1.99878+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.99878, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@21", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "1.99878+", "metric_symbol": "s", "metric_value": 1.99878, "reference_text": "s = 1.99878+ | Found by Ian Watson | in April 2026.", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/21.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 1.99878+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "96f74df50573bb81", "improvement": -0.0012740227066210252, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.545179480294009e-07, "max_pair_overlap_depth": -6.765642482919887e-06, "metric_expression": "2.00005", "metric_symbol": "s", "metric_value": 2.000054022706621, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.99878, "setup": "triinhex", "side": 2.000054022706621, "solution_path": "data/solutions/generated/triinhex_021_96f74df50573bb81.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/96f74df50573bb81.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@22", "density": 0.9166150664792877, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/24.gif", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@22", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial. |", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/24.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2 | Trivial. |", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "38182a59461da74e", "improvement": -5.629349013291929e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.91184500961419e-07, "max_pair_overlap_depth": -9.942352799940934e-07, "metric_expression": "2.00006", "metric_symbol": "s", "metric_value": 2.000056293490133, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "triinhex", "side": 2.000056293490133, "solution_path": "data/solutions/generated/triinhex_022_38182a59461da74e.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/38182a59461da74e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@23", "density": 0.958278890368788, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/24.gif", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@23", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial. |", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/24.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2 | Trivial. |", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "dcd331994270a997", "improvement": -5.681247061106731e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0734329507489061e-06, "max_pair_overlap_depth": -9.41664729126046e-06, "metric_expression": "2.00006", "metric_symbol": "s", "metric_value": 2.000056812470611, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "triinhex", "side": 2.000056812470611, "solution_path": "data/solutions/generated/triinhex_023_dcd331994270a997.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/dcd331994270a997.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@24", "density": 0.9999420291406096, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/24.gif", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@24", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial. |", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/24.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2 | Trivial. |", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "13936a08bfd4d152", "improvement": -5.7973379977482864e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3635152509087512e-06, "max_pair_overlap_depth": -3.8859579855957235e-06, "metric_expression": "2.00006", "metric_symbol": "s", "metric_value": 2.0000579733799775, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "triinhex", "side": 2.0000579733799775, "solution_path": "data/solutions/generated/triinhex_024_13936a08bfd4d152.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/13936a08bfd4d152.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@25", "density": 0.8871759236838516, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/25.gif", "friedman_metric_expression": "13/6 = 2.166+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.166, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@25", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "13/6 = 2.166+", "metric_symbol": "s", "metric_value": 2.166, "reference_text": "s = 13/6 = 2.166+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/25.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 13/6 = 2.166+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "71a7801cfb4a8d31", "improvement": -0.0011526610601886311, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.780494307380179e-07, "max_pair_overlap_depth": -1.3308976067882838e-06, "metric_expression": "2.16715", "metric_symbol": "s", "metric_value": 2.1671526610601886, "n": 25, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.166, "setup": "triinhex", "side": 2.1671526610601886, "solution_path": "data/solutions/generated/triinhex_025_71a7801cfb4a8d31.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/71a7801cfb4a8d31.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@26", "density": 0.878430458154361, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/26.gif", "friedman_metric_expression": "2.21991+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.21991, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@26", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.21991+", "metric_symbol": "s", "metric_value": 2.21991, "reference_text": "s = 2.21991+ | Found by Ian Watson | in April 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/26.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.21991+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "658d7f93b604b663", "improvement": -0.0011349903788469007, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.149641376243232e-07, "max_pair_overlap_depth": -5.41804289733161e-06, "metric_expression": "2.22104", "metric_symbol": "s", "metric_value": 2.221044990378847, "n": 26, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.21991, "setup": "triinhex", "side": 2.221044990378847, "solution_path": "data/solutions/generated/triinhex_026_658d7f93b604b663.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/658d7f93b604b663.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@27", "density": 0.8525233088554102, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/27.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.29541+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.29541, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@27", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.29541+", "metric_symbol": "s", "metric_value": 2.29541, "reference_text": "s = 2.29541+ | Found by Ian Watson | in April 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/27.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.29541+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "ace1d8ac458c5bc4", "improvement": -0.0020773347367342154, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.449964623966915e-06, "max_pair_overlap_depth": -5.244956223338093e-06, "metric_expression": "2.29749", "metric_symbol": "s", "metric_value": 2.297487334736734, "n": 27, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.29541, "setup": "triinhex", "side": 2.297487334736734, "solution_path": "data/solutions/generated/triinhex_027_ace1d8ac458c5bc4.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/ace1d8ac458c5bc4.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@28", "density": 0.8636324031387004, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/28.gif", "friedman_metric_expression": "2.32289+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.32289, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@28", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.32289+", "metric_symbol": "s", "metric_value": 2.32289, "reference_text": "s = 2.32289+ | Found by Ian Watson | in April 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/28.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.32289+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "3435500023858121", "improvement": -0.0016601826518787277, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.60736131059532e-06, "max_pair_overlap_depth": -1.0038160656700157e-05, "metric_expression": "2.32455", "metric_symbol": "s", "metric_value": 2.324550182651879, "n": 28, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.32289, "setup": "triinhex", "side": 2.324550182651879, "solution_path": "data/solutions/generated/triinhex_28_3435500023858121.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/3435500023858121.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@29", "density": 0.8876189549602443, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/29.gif", "friedman_metric_expression": "7/3 = 2.333+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.333, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@29", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "7/3 = 2.333+", "metric_symbol": "s", "metric_value": 2.333, "reference_text": "s = 7/3 = 2.333+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/29.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 7/3 = 2.333+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "4339d1ebd3160f94", "improvement": -0.0005122751747741461, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.371335525980428e-06, "max_pair_overlap_depth": -4.303492490143057e-07, "metric_expression": "2.33351", "metric_symbol": "s", "metric_value": 2.3335122751747743, "n": 29, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.333, "setup": "triinhex", "side": 2.3335122751747743, "solution_path": "data/solutions/generated/triinhex_29_4339d1ebd3160f94.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/4339d1ebd3160f94.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@30", "density": 0.8862493271562808, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/30.gif", "friedman_metric_expression": "19/8 = 2.375", "friedman_metric_symbol": "s", "friedman_metric_value": 2.375, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@30", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "19/8 = 2.375", "metric_symbol": "s", "metric_value": 2.375, "reference_text": "s = 19/8 = 2.375 | Found by Ignacio Vallejo | in April 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/30.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 19/8 = 2.375 | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "7274d354e73d8050", "improvement": -0.00023750926269094919, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.946881910542413e-07, "max_pair_overlap_depth": -3.1129912760885503e-06, "metric_expression": "2.37524", "metric_symbol": "s", "metric_value": 2.375237509262691, "n": 30, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.375, "setup": "triinhex", "side": 2.375237509262691, "solution_path": "data/solutions/generated/triinhex_30_7274d354e73d8050.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/7274d354e73d8050.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@31", "density": 0.896119877899338, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/31.gif", "friedman_metric_expression": "12/5 = 2.400", "friedman_metric_symbol": "s", "friedman_metric_value": 2.4, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "12/5 = 2.400", "metric_symbol": "s", "metric_value": 2.4, "reference_text": "s = 12/5 = 2.400 | Found by Emerson Connelly | in April 2026.", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/31.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 12/5 = 2.400 | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "cae7087451040120", "improvement": -0.0011658949471424762, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9409739826535097e-07, "max_pair_overlap_depth": -2.8541100203227643e-06, "metric_expression": "2.40117", "metric_symbol": "s", "metric_value": 2.4011658949471424, "n": 31, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.4, "setup": "triinhex", "side": 2.4011658949471424, "solution_path": "data/solutions/generated/triinhex_31_cae7087451040120.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/cae7087451040120.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@32", "density": 0.8767870241121234, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/32.gif", "friedman_metric_expression": "2.465+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.465, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@32", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "2.465+", "metric_symbol": "s", "metric_value": 2.465, "reference_text": "s = 2.465+ | Found by Emerson Connelly | in April 2026.", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/32.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.465+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "8674581436e5e74d", "improvement": -0.0013363682337788063, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.925122041572899e-07, "max_pair_overlap_depth": -1.2807310418028184e-06, "metric_expression": "2.46634", "metric_symbol": "s", "metric_value": 2.4663363682337787, "n": 32, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.465, "setup": "triinhex", "side": 2.4663363682337787, "solution_path": "data/solutions/generated/triinhex_032_8674581436e5e74d.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/8674581436e5e74d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@33", "density": 0.8799443949567897, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/33.gif", "friedman_metric_expression": "2.498+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.498, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@33", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "2.498+", "metric_symbol": "s", "metric_value": 2.498, "reference_text": "s = 2.498+ | Found by Emerson Connelly | in April 2026.", "source_image": "33.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/33.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.498+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "5b836a27d2bb958c", "improvement": -0.00207898817969987, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8634496562140136e-06, "max_pair_overlap_depth": -5.829272115620565e-06, "metric_expression": "2.50008", "metric_symbol": "s", "metric_value": 2.5000789881797, "n": 33, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.498, "setup": "triinhex", "side": 2.5000789881797, "solution_path": "data/solutions/generated/triinhex_033_5b836a27d2bb958c.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/5b836a27d2bb958c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@34", "density": 0.9066102013779274, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "friedman_metric_expression": "5/2 = 2.500", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@34", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "5/2 = 2.500", "metric_symbol": "s", "metric_value": 2.5, "reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "a712744ff3b5bcc9", "improvement": -7.7851001305973e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2753913921730298e-06, "max_pair_overlap_depth": -7.960301433351802e-06, "metric_expression": "2.50008", "metric_symbol": "s", "metric_value": 2.500077851001306, "n": 34, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.5, "setup": "triinhex", "side": 2.500077851001306, "solution_path": "data/solutions/generated/triinhex_034_a712744ff3b5bcc9.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/a712744ff3b5bcc9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@35", "density": 0.93316535041203, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "friedman_metric_expression": "5/2 = 2.500", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@35", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "5/2 = 2.500", "metric_symbol": "s", "metric_value": 2.5, "reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "ec49778cf450c37b", "improvement": -0.00022500750012577342, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6207066397040256e-06, "max_pair_overlap_depth": -1.5837731757728513e-06, "metric_expression": "2.50023", "metric_symbol": "s", "metric_value": 2.5002250075001258, "n": 35, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.5, "setup": "triinhex", "side": 2.5002250075001258, "solution_path": "data/solutions/generated/triinhex_35_ec49778cf450c37b.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/ec49778cf450c37b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@36", "density": 0.8942711196026824, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/36.gif", "friedman_metric_expression": "2.59123+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.59123, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@36", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "2.59123+", "metric_symbol": "s", "metric_value": 2.59123, "reference_text": "s = 2.59123+ | Found by Nathan Roll | in April 2026.", "source_image": "36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/36.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.59123+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "d30ec5c19b8369f3", "improvement": 0.000983935961939686, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.744185746811731e-07, "max_pair_overlap_depth": -1.519675368655271e-06, "metric_expression": "2.59025", "metric_symbol": "s", "metric_value": 2.5902460640380602, "n": 36, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.59123, "setup": "triinhex", "side": 2.5902460640380602, "solution_path": "data/solutions/record_hunt/triinhex_36_d30ec5c19b8369f3.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/d30ec5c19b8369f3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@37", "density": 0.899950790089805, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/37.gif", "friedman_metric_expression": "2.61943+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.61943, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@37", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "2.61943+", "metric_symbol": "s", "metric_value": 2.61943, "reference_text": "s = 2.61943+ | Found by Nathan Roll | in April 2026.", "source_image": "37.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/37.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.61943+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "c791d76ac960cbca", "improvement": 0.0017542149405089802, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.903248309191345e-07, "max_pair_overlap_depth": -8.609976616114068e-07, "metric_expression": "2.61768", "metric_symbol": "s", "metric_value": 2.617675785059491, "n": 37, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.61943, "setup": "triinhex", "side": 2.617675785059491, "solution_path": "data/solutions/record_hunt/triinhex_37_c791d76ac960cbca.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/c791d76ac960cbca.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@38", "density": 0.8907392561961515, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", "friedman_metric_expression": "8/3 = 2.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.666, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@38", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "8/3 = 2.666+", "metric_symbol": "s", "metric_value": 2.666, "reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", "source_image": "39.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "76fd196e8c79a5c9", "improvement": -0.0004956329108489754, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.7850003642140776e-06, "max_pair_overlap_depth": -3.863933217385096e-06, "metric_expression": "2.66650", "metric_symbol": "s", "metric_value": 2.666495632910849, "n": 38, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.666, "setup": "triinhex", "side": 2.666495632910849, "solution_path": "data/solutions/generated/triinhex_038_76fd196e8c79a5c9.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/76fd196e8c79a5c9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@39", "density": 0.9136318214810123, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", "friedman_metric_expression": "8/3 = 2.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.666, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@39", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "8/3 = 2.666+", "metric_symbol": "s", "metric_value": 2.666, "reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", "source_image": "39.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "a314b4bb8467e5e8", "improvement": -0.0012951149704507614, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.891404577705932e-07, "max_pair_overlap_depth": -6.19203720853001e-07, "metric_expression": "2.66730", "metric_symbol": "s", "metric_value": 2.6672951149704507, "n": 39, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.666, "setup": "triinhex", "side": 2.6672951149704507, "solution_path": "data/solutions/generated/triinhex_39_a314b4bb8467e5e8.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/a314b4bb8467e5e8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinhex@40", "density": 0.9051419512423183, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/40.gif", "friedman_metric_expression": "2.71622+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.71622, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@40", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "2.71622+", "metric_symbol": "s", "metric_value": 2.71622, "reference_text": "s = 2.71622+ | Found by Nathan Roll | in April 2026.", "source_image": "40.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/40.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.71622+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "8b6d682b66356562", "improvement": 0.0023063615780789526, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2777984478773679e-07, "max_pair_overlap_depth": -4.470141119483628e-07, "metric_expression": "2.71391", "metric_symbol": "s", "metric_value": 2.713913638421921, "n": 40, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.71622, "setup": "triinhex", "side": 2.713913638421921, "solution_path": "data/solutions/record_hunt/triinhex_40_8b6d682b66356562.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/8b6d682b66356562.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@1", "density": 0.4152122478191623, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/1.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": ".46470+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.4647, "friedman_reference": {"analytical_or_proved": true, "case": "triinoct@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": ".46470+", "metric_symbol": "s", "metric_value": 0.4647, "reference_text": "s = .46470+ | Trivial. |", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/1.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = .46470+ | Trivial. |", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "92c66f395ff03ab9", "improvement": -4.251046545072912e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3282842257895898e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.46474", "metric_symbol": "s", "metric_value": 0.46474251046545073, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.4647, "setup": "triinoct", "side": 0.46474251046545073, "solution_path": "data/solutions/generated/triinoct_001_92c66f395ff03ab9.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/92c66f395ff03ab9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@2", "density": 0.40823825072318354, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/2.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "\u221a3/\u221a(4+2\u221a2) = .66282+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.66282, "friedman_reference": {"analytical_or_proved": true, "case": "triinoct@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a3/\u221a(4+2\u221a2) = .66282+", "metric_symbol": "s", "metric_value": 0.66282, "reference_text": "s = \u221a3/\u221a(4+2\u221a2) = .66282+ | Trivial. |", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/2.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = \u221a3/\u221a(4+2\u221a2) = .66282+ | Trivial. |", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "5706480ca6648c20", "improvement": -1.5298424264065247e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.197393779037103e-06, "max_pair_overlap_depth": -5.033792755404498e-07, "metric_expression": "0.66284", "metric_symbol": "s", "metric_value": 0.662835298424264, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.66282, "setup": "triinoct", "side": 0.662835298424264, "solution_path": "data/solutions/generated/triinoct_002_5706480ca6648c20.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/5706480ca6648c20.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@3", "density": 0.49592364137420614, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/3.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": ".73651+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.73651, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@3", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": ".73651+", "metric_symbol": "s", "metric_value": 0.73651, "reference_text": "s = .73651+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/3.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = .73651+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "f3eee07e4d3df15b", "improvement": -3.739655668555386e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.662035887956301e-06, "max_pair_overlap_depth": -2.2250265259982083e-06, "metric_expression": "0.73655", "metric_symbol": "s", "metric_value": 0.7365473965566856, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.73651, "setup": "triinoct", "side": 0.7365473965566856, "solution_path": "data/solutions/generated/triinoct_003_f3eee07e4d3df15b.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/f3eee07e4d3df15b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@4", "density": 0.593540115018531, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/4.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": ".77733+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.77733, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@4", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": ".77733+", "metric_symbol": "s", "metric_value": 0.77733, "reference_text": "s = .77733+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/4.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = .77733+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "daeed392d4fde104", "improvement": -8.413889807601382e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.025085624522854e-06, "max_pair_overlap_depth": -2.0675467336764286e-05, "metric_expression": "0.77741", "metric_symbol": "s", "metric_value": 0.777414138898076, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.77733, "setup": "triinoct", "side": 0.777414138898076, "solution_path": "data/solutions/generated/triinoct_004_daeed392d4fde104.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/daeed392d4fde104.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@5", "density": 0.6941546982548611, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/5.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": ".80367+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.80367, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@5", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": ".80367+", "metric_symbol": "s", "metric_value": 0.80367, "reference_text": "s = .80367+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/5.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = .80367+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "a0addde9772ff582", "improvement": -4.920494762683525e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.8458025218911135e-06, "max_pair_overlap_depth": -2.636296310584707e-05, "metric_expression": "0.80372", "metric_symbol": "s", "metric_value": 0.8037192049476268, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.80367, "setup": "triinoct", "side": 0.8037192049476268, "solution_path": "data/solutions/generated/triinoct_005_a0addde9772ff582.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/a0addde9772ff582.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@6", "density": 0.8085136476900721, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/6.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": ".81574+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.81574, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@6", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": ".81574+", "metric_symbol": "s", "metric_value": 0.81574, "reference_text": "s = .81574+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/6.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = .81574+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "e7ac2b490b8118b6", "improvement": -5.196804299267832e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.999019070763348e-06, "max_pair_overlap_depth": -1.523375553563397e-05, "metric_expression": "0.81579", "metric_symbol": "s", "metric_value": 0.8157919680429927, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.81574, "setup": "triinoct", "side": 0.8157919680429927, "solution_path": "data/solutions/generated/triinoct_006_e7ac2b490b8118b6.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/e7ac2b490b8118b6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@7", "density": 0.7344053269175495, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/7.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": ".92453+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.92453, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@7", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": ".92453+", "metric_symbol": "s", "metric_value": 0.92453, "reference_text": "s = .92453+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/7.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = .92453+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "f1951fc893682894", "improvement": -1.6137063615029312e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.877767128601322e-06, "max_pair_overlap_depth": -4.487804835917597e-06, "metric_expression": "0.92455", "metric_symbol": "s", "metric_value": 0.924546137063615, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.92453, "setup": "triinoct", "side": 0.924546137063615, "solution_path": "data/solutions/generated/triinoct_007_f1951fc893682894.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/f1951fc893682894.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@8", "density": 0.7351940708693757, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/8.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": ".98782+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.98782, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@8", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": ".98782+", "metric_symbol": "s", "metric_value": 0.98782, "reference_text": "s = .98782+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/8.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = .98782+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "cdfdb28bbd7c958a", "improvement": -3.106647506945226e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6027742606714177e-06, "max_pair_overlap_depth": -8.534452089259403e-06, "metric_expression": "0.98785", "metric_symbol": "s", "metric_value": 0.9878510664750695, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.98782, "setup": "triinoct", "side": 0.9878510664750695, "solution_path": "data/solutions/generated/triinoct_008_cdfdb28bbd7c958a.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/cdfdb28bbd7c958a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@9", "density": 0.7931382286148597, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/9.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.00875+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.00875, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@9", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.00875+", "metric_symbol": "s", "metric_value": 1.00875, "reference_text": "s = 1.00875+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/9.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.00875+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "0a4766beda41f84b", "improvement": -2.4953411784212065e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.992353374519455e-07, "max_pair_overlap_depth": -5.523000253249144e-06, "metric_expression": "1.00877", "metric_symbol": "s", "metric_value": 1.0087749534117842, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.00875, "setup": "triinoct", "side": 1.0087749534117842, "solution_path": "data/solutions/generated/triinoct_009_0a4766beda41f84b.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/0a4766beda41f84b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@10", "density": 0.7747683390429897, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/10.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.07565+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.07565, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@10", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.07565+", "metric_symbol": "s", "metric_value": 1.07565, "reference_text": "s = 1.07565+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/10.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.07565+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "df6976316812d041", "improvement": -0.00022432883231693346, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.3385179497460716e-06, "max_pair_overlap_depth": -1.8668924055276204e-05, "metric_expression": "1.07587", "metric_symbol": "s", "metric_value": 1.075874328832317, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.07565, "setup": "triinoct", "side": 1.075874328832317, "solution_path": "data/solutions/generated/triinoct_010_df6976316812d041.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/df6976316812d041.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@11", "density": 0.7515390994133405, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/11.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.14571+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.14571, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@11", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.14571+", "metric_symbol": "s", "metric_value": 1.14571, "reference_text": "s = 1.14571+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/11.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.14571+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "11036479a300844a", "improvement": 1.760556295749005e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.2318282708152993e-06, "max_pair_overlap_depth": -7.937162014970411e-06, "metric_expression": "1.14569", "metric_symbol": "s", "metric_value": 1.1456923944370425, "n": 11, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.14571, "setup": "triinoct", "side": 1.1456923944370425, "solution_path": "data/solutions/generated/triinoct_011_11036479a300844a.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/11036479a300844a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@12", "density": 0.74788706506276, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/12.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.19929+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.19929, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@12", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.19929+", "metric_symbol": "s", "metric_value": 1.19929, "reference_text": "s = 1.19929+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/12.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.19929+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "1fa5264e0f579a94", "improvement": -0.0002647906752053508, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2204761718366964e-06, "max_pair_overlap_depth": -1.4073169272155184e-06, "metric_expression": "1.19955", "metric_symbol": "s", "metric_value": 1.1995547906752053, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.19929, "setup": "triinoct", "side": 1.1995547906752053, "solution_path": "data/solutions/generated/triinoct_012_1fa5264e0f579a94.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/1fa5264e0f579a94.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@13", "density": 0.7604201292225432, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/13.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.23759+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.23759, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@13", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.23759+", "metric_symbol": "s", "metric_value": 1.23759, "reference_text": "s = 1.23759+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/13.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.23759+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "ebb5825b80b24749", "improvement": -0.0006144239243295324, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.763415979262496e-06, "max_pair_overlap_depth": -3.350137919300167e-06, "metric_expression": "1.23820", "metric_symbol": "s", "metric_value": 1.2382044239243295, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.23759, "setup": "triinoct", "side": 1.2382044239243295, "solution_path": "data/solutions/generated/triinoct_013_ebb5825b80b24749.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/ebb5825b80b24749.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@14", "density": 0.7704458772851622, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/14.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.27677+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.27677, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@14", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.27677+", "metric_symbol": "s", "metric_value": 1.27677, "reference_text": "s = 1.27677+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/14.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.27677+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "e00a18106d2fcfee", "improvement": 0.00021236461691676034, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.593194459024069e-06, "max_pair_overlap_depth": -5.874056281496287e-06, "metric_expression": "1.27656", "metric_symbol": "s", "metric_value": 1.2765576353830832, "n": 14, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.27677, "setup": "triinoct", "side": 1.2765576353830832, "solution_path": "data/solutions/generated/triinoct_014_e00a18106d2fcfee.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/e00a18106d2fcfee.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@15", "density": 0.7924237027261557, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/15.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.30275+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.30275, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@15", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.30275+", "metric_symbol": "s", "metric_value": 1.30275, "reference_text": "s = 1.30275+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/15.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.30275+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "e14eda962017d06d", "improvement": -0.0001598827367932376, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3032093708899356e-06, "max_pair_overlap_depth": -6.895788391969582e-06, "metric_expression": "1.30291", "metric_symbol": "s", "metric_value": 1.3029098827367933, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.30275, "setup": "triinoct", "side": 1.3029098827367933, "solution_path": "data/solutions/generated/triinoct_015_e14eda962017d06d.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/e14eda962017d06d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@16", "density": 0.7920450821484247, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/16.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.34584+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.34584, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@16", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.34584+", "metric_symbol": "s", "metric_value": 1.34584, "reference_text": "s = 1.34584+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/16.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.34584+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "640352ab60961c58", "improvement": -0.00012112963758759143, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.464975954309231e-06, "max_pair_overlap_depth": -7.38266966116008e-07, "metric_expression": "1.34596", "metric_symbol": "s", "metric_value": 1.3459611296375875, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.34584, "setup": "triinoct", "side": 1.3459611296375875, "solution_path": "data/solutions/generated/triinoct_16_640352ab60961c58.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/640352ab60961c58.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@17", "density": 0.7893502976721624, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/17.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.38964+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.38964, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@17", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.38964+", "metric_symbol": "s", "metric_value": 1.38964, "reference_text": "s = 1.38964+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/17.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.38964+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "6f00b933407c7c80", "improvement": -0.0001111742572472707, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.693112749768053e-07, "max_pair_overlap_depth": -2.384134229149737e-06, "metric_expression": "1.38975", "metric_symbol": "s", "metric_value": 1.3897511742572473, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.38964, "setup": "triinoct", "side": 1.3897511742572473, "solution_path": "data/solutions/generated/triinoct_017_6f00b933407c7c80.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/6f00b933407c7c80.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@18", "density": 0.8027011318687396, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/18.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.41836+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.41836, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@18", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.41836+", "metric_symbol": "s", "metric_value": 1.41836, "reference_text": "s = 1.41836+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/18.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.41836+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "91a694d1aa76789c", "improvement": 0.0002602110634655208, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.835525154559248e-06, "max_pair_overlap_depth": -9.960192069113916e-06, "metric_expression": "1.41810", "metric_symbol": "s", "metric_value": 1.4180997889365345, "n": 18, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.41836, "setup": "triinoct", "side": 1.4180997889365345, "solution_path": "data/solutions/generated/triinoct_018_91a694d1aa76789c.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/91a694d1aa76789c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@19", "density": 0.7911354610751659, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/19.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.46749+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.46749, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@19", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.46749+", "metric_symbol": "s", "metric_value": 1.46749, "reference_text": "s = 1.46749+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/19.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.46749+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "f92a5a5a3b54909a", "improvement": -8.00911599283527e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.983793930902579e-06, "max_pair_overlap_depth": -8.286875174401409e-06, "metric_expression": "1.46757", "metric_symbol": "s", "metric_value": 1.4675700911599283, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.46749, "setup": "triinoct", "side": 1.4675700911599283, "solution_path": "data/solutions/generated/triinoct_019_f92a5a5a3b54909a.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/f92a5a5a3b54909a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@20", "density": 0.7990175319982477, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/20.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.49832+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.49832, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@20", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.49832+", "metric_symbol": "s", "metric_value": 1.49832, "reference_text": "s = 1.49832+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "20.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/20.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.49832+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "ead92a925f9f00a7", "improvement": 6.988050923584233e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.3276003687482804e-06, "max_pair_overlap_depth": -2.2158170135400113e-06, "metric_expression": "1.49825", "metric_symbol": "s", "metric_value": 1.4982501194907643, "n": 20, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.49832, "setup": "triinoct", "side": 1.4982501194907643, "solution_path": "data/solutions/generated/triinoct_020_ead92a925f9f00a7.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/ead92a925f9f00a7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@21", "density": 0.8078137663314585, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/21.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.52615+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.52615, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@21", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.52615+", "metric_symbol": "s", "metric_value": 1.52615, "reference_text": "s = 1.52615+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/21.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.52615+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "e350f46688d32a68", "improvement": -0.0007180233722772211, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.407426670083604e-06, "max_pair_overlap_depth": -1.1097264517423433e-05, "metric_expression": "1.52687", "metric_symbol": "s", "metric_value": 1.5268680233722771, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.52615, "setup": "triinoct", "side": 1.5268680233722771, "solution_path": "data/solutions/generated/triinoct_021_e350f46688d32a68.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/e350f46688d32a68.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@22", "density": 0.8155579012833859, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/22.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.55545+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.55545, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@22", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.55545+", "metric_symbol": "s", "metric_value": 1.55545, "reference_text": "s = 1.55545+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/22.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.55545+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "c1314ed0099ebdb6", "improvement": 8.823677085589843e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5323101194107617e-06, "max_pair_overlap_depth": -6.490133103720375e-06, "metric_expression": "1.55536", "metric_symbol": "s", "metric_value": 1.555361763229144, "n": 22, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.55545, "setup": "triinoct", "side": 1.555361763229144, "solution_path": "data/solutions/generated/triinoct_022_c1314ed0099ebdb6.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/c1314ed0099ebdb6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@23", "density": 0.8210902170609379, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/23.gif", "friedman_metric_expression": "1.58458+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.58458, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@23", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.58458+", "metric_symbol": "s", "metric_value": 1.58458, "reference_text": "s = 1.58458+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/23.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.58458+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "4ca1a825a22b1f04", "improvement": -0.0003714249913302492, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.321393677649287e-06, "max_pair_overlap_depth": -1.5960535921788477e-06, "metric_expression": "1.58495", "metric_symbol": "s", "metric_value": 1.5849514249913303, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.58458, "setup": "triinoct", "side": 1.5849514249913303, "solution_path": "data/solutions/generated/triinoct_023_4ca1a825a22b1f04.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/4ca1a825a22b1f04.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@24", "density": 0.8373304168984339, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/24.gif", "friedman_metric_expression": "1.60261+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.60261, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@24", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.60261+", "metric_symbol": "s", "metric_value": 1.60261, "reference_text": "s = 1.60261+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/24.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.60261+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "e197b92ecac28558", "improvement": -0.0006526103809141315, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.085338835265612e-06, "max_pair_overlap_depth": -9.98840296584369e-06, "metric_expression": "1.60326", "metric_symbol": "s", "metric_value": 1.6032626103809142, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.60261, "setup": "triinoct", "side": 1.6032626103809142, "solution_path": "data/solutions/generated/triinoct_024_e197b92ecac28558.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/e197b92ecac28558.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@25", "density": 0.8354970198268096, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/25.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.63797+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.63797, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@25", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.63797+", "metric_symbol": "s", "metric_value": 1.63797, "reference_text": "s = 1.63797+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/25.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.63797+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "e322401fc5bafe4d", "improvement": -0.00014742221399233202, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.350806641875647e-06, "max_pair_overlap_depth": -2.3762008580785476e-06, "metric_expression": "1.63812", "metric_symbol": "s", "metric_value": 1.6381174222139923, "n": 25, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.63797, "setup": "triinoct", "side": 1.6381174222139923, "solution_path": "data/solutions/generated/triinoct_025_e322401fc5bafe4d.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/e322401fc5bafe4d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@26", "density": 0.8292613760828756, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/26.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.67666+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.67666, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@26", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.67666+", "metric_symbol": "s", "metric_value": 1.67666, "reference_text": "s = 1.67666+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/26.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.67666+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "7cb3797cd10b21d8", "improvement": -0.0001676725391090983, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6314473196666756e-05, "max_pair_overlap_depth": -5.591511314406716e-06, "metric_expression": "1.67683", "metric_symbol": "s", "metric_value": 1.6768276725391091, "n": 26, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.67666, "setup": "triinoct", "side": 1.6768276725391091, "solution_path": "data/solutions/generated/triinoct_026_7cb3797cd10b21d8.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/7cb3797cd10b21d8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@27", "density": 0.8297818493323376, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/27.gif", "friedman_metric_expression": "1.70882+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.70882, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@27", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.70882+", "metric_symbol": "s", "metric_value": 1.70882, "reference_text": "s = 1.70882+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/27.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.70882+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "c7748eb19b34ff2e", "improvement": 0.0005858720037632548, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.055699391789948e-06, "max_pair_overlap_depth": -3.877030372301249e-07, "metric_expression": "1.70823", "metric_symbol": "s", "metric_value": 1.7082341279962368, "n": 27, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.70882, "setup": "triinoct", "side": 1.7082341279962368, "solution_path": "data/solutions/generated/triinoct_27_c7748eb19b34ff2e.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/c7748eb19b34ff2e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@28", "density": 0.8299146316504411, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/28.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.73954+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.73954, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@28", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.73954+", "metric_symbol": "s", "metric_value": 1.73954, "reference_text": "s = 1.73954+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/28.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.73954+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "26e8795d82a701d8", "improvement": 9.867973407984287e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2195571711171888e-06, "max_pair_overlap_depth": -8.650951147992636e-06, "metric_expression": "1.73944", "metric_symbol": "s", "metric_value": 1.7394413202659202, "n": 28, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.73954, "setup": "triinoct", "side": 1.7394413202659202, "solution_path": "data/solutions/generated/triinoct_028_26e8795d82a701d8.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/26e8795d82a701d8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@29", "density": 0.8439313801960897, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/29.gif", "friedman_metric_expression": "1.75576+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.75576, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@29", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.75576+", "metric_symbol": "s", "metric_value": 1.75576, "reference_text": "s = 1.75576+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/29.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.75576+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "0575780ea85d68dd", "improvement": 0.00029203482208850495, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.530531415212805e-06, "max_pair_overlap_depth": -1.1528245769332202e-05, "metric_expression": "1.75547", "metric_symbol": "s", "metric_value": 1.7554679651779115, "n": 29, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.75576, "setup": "triinoct", "side": 1.7554679651779115, "solution_path": "data/solutions/generated/triinoct_29_0575780ea85d68dd.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/0575780ea85d68dd.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinoct@30", "density": 0.8466211131481989, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/30.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.78101+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.78101, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@30", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.78101+", "metric_symbol": "s", "metric_value": 1.78101, "reference_text": "s = 1.78101+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/30.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.78101+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "4c08385c68f67e77", "improvement": -0.0016296314900829945, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.472767326523421e-06, "max_pair_overlap_depth": -3.1822885246945987e-06, "metric_expression": "1.78264", "metric_symbol": "s", "metric_value": 1.782639631490083, "n": 30, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.78101, "setup": "triinoct", "side": 1.782639631490083, "solution_path": "data/solutions/generated/triinoct_030_4c08385c68f67e77.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/4c08385c68f67e77.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@1", "density": 0.4121951299103008, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/1.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": ".781+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.781, "friedman_reference": {"analytical_or_proved": true, "case": "triinpen@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": ".781+", "metric_symbol": "s", "metric_value": 0.781, "reference_text": "s = cos(42\u00b0) / cos(18\u00b0) | s = .781+ | Trivial. |", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/1.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = cos(42\u00b0) / cos(18\u00b0) | s = .781+ | Trivial. |", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "6d3af7533c544064", "improvement": -0.0004017935524769367, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.010333690326338e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.78140", "metric_symbol": "s", "metric_value": 0.781401793552477, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.781, "setup": "triinpen", "side": 0.781401793552477, "solution_path": "data/solutions/generated/triinpen_001_6d3af7533c544064.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6d3af7533c544064.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@2", "density": 0.4392461428307515, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/2.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.070+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.07, "friedman_reference": {"analytical_or_proved": true, "case": "triinpen@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1.070+", "metric_symbol": "s", "metric_value": 1.07, "reference_text": "s = \u221a(6/(3+\u221a5)) | s = 1.070+ | Trivial. |", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/2.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = \u221a(6/(3+\u221a5)) | s = 1.070+ | Trivial. |", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "eebb278fed1959e5", "improvement": -0.0005004432401414238, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.540244142797655e-06, "max_pair_overlap_depth": -4.677606255020384e-06, "metric_expression": "1.07050", "metric_symbol": "s", "metric_value": 1.0705004432401415, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.07, "setup": "triinpen", "side": 1.0705004432401415, "solution_path": "data/solutions/generated/triinpen_002_eebb278fed1959e5.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/eebb278fed1959e5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@3", "density": 0.5773621092377123, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/3.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.143+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.143, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@3", "credit": "Maurizio Morandi", "date": "August 2012", "family_updated": "", "metric_expression": "1.143+", "metric_symbol": "s", "metric_value": 1.143, "reference_text": "s = 1.143+ | Found by Maurizio Morandi | in August 2012. |", "source_image": "3.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/3.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.143+ | Found by Maurizio Morandi | in August 2012. |", "friedman_source_image": "3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "6ec3c588fb678646", "improvement": -0.0005688574140261249, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4403665141977852e-06, "max_pair_overlap_depth": -4.738354987265936e-06, "metric_expression": "1.14357", "metric_symbol": "s", "metric_value": 1.1435688574140261, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.143, "setup": "triinpen", "side": 1.1435688574140261, "solution_path": "data/solutions/generated/triinpen_003_6ec3c588fb678646.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/6ec3c588fb678646.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@4", "density": 0.6999227205319379, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/4.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.199+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.199, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@4", "credit": "Maurizio Morandi", "date": "July 2012", "family_updated": "", "metric_expression": "1.199+", "metric_symbol": "s", "metric_value": 1.199, "reference_text": "s = 1.199+ | Found by Maurizio Morandi | in July 2012. |", "source_image": "4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/4.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.199+ | Found by Maurizio Morandi | in July 2012. |", "friedman_source_image": "4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "c45dd63fd5a8806b", "improvement": -0.0003081410566185827, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9487802360351315e-06, "max_pair_overlap_depth": -2.0674441409240796e-05, "metric_expression": "1.19931", "metric_symbol": "s", "metric_value": 1.1993081410566186, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.199, "setup": "triinpen", "side": 1.1993081410566186, "solution_path": "data/solutions/generated/triinpen_004_c45dd63fd5a8806b.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/c45dd63fd5a8806b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@5", "density": 0.818292233169356, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "March 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/5.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.240+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.24, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@5", "credit": "Erich Friedman", "date": "March 2012", "family_updated": "", "metric_expression": "1.240+", "metric_symbol": "s", "metric_value": 1.24, "reference_text": "s = 1 + 2sin(12\u00b0)/\u221a3 | s = 1.240+ | Found by Erich Friedman | in March 2012.", "source_image": "5.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/5.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1 + 2sin(12\u00b0)/\u221a3 | s = 1.240+ | Found by Erich Friedman | in March 2012.", "friedman_source_image": "5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "a99f063702c9d4e8", "improvement": -9.974870359874544e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.477405190997757e-06, "max_pair_overlap_depth": -1.386320747689962e-05, "metric_expression": "1.24010", "metric_symbol": "s", "metric_value": 1.2400997487035987, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.24, "setup": "triinpen", "side": 1.2400997487035987, "solution_path": "data/solutions/generated/triinpen_005_a99f063702c9d4e8.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/a99f063702c9d4e8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@6", "density": 0.7771309707429922, "friedman_analytical_or_proved": false, "friedman_credit": "Dominik Kamp", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/6.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.393+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.393, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@6", "credit": "Dominik Kamp", "date": "April 2026", "family_updated": "", "metric_expression": "1.393+", "metric_symbol": "s", "metric_value": 1.393, "reference_text": "s = 1.393+ | Found by Dominik Kamp | in April 2026.", "source_image": "6.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/6.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.393+ | Found by Dominik Kamp | in April 2026.", "friedman_source_image": "6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "8d12939187f41554", "improvement": -0.0009729568537997224, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4992545061493154e-06, "max_pair_overlap_depth": -6.033095825291185e-06, "metric_expression": "1.39397", "metric_symbol": "s", "metric_value": 1.3939729568537997, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.393, "setup": "triinpen", "side": 1.3939729568537997, "solution_path": "data/solutions/generated/triinpen_006_8d12939187f41554.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Dominik Kamp", "svg_path": "data/svg/8d12939187f41554.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@7", "density": 0.7037922543518549, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/7.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.581+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.581, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@7", "credit": "David W. Cantrell", "date": "July 2012", "family_updated": "", "metric_expression": "1.581+", "metric_symbol": "s", "metric_value": 1.581, "reference_text": "s = 1.581+ | Found by David W. Cantrell | in July 2012.", "source_image": "7.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/7.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.581+ | Found by David W. Cantrell | in July 2012.", "friedman_source_image": "7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "d149692a95cde76c", "improvement": -0.0011680508685436575, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.5825119043122948e-06, "max_pair_overlap_depth": -1.774591125650038e-05, "metric_expression": "1.58217", "metric_symbol": "s", "metric_value": 1.5821680508685436, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.581, "setup": "triinpen", "side": 1.5821680508685436, "solution_path": "data/solutions/generated/triinpen_007_d149692a95cde76c.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/d149692a95cde76c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@8", "density": 0.7311911118691055, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/8.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.659+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.659, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@8", "credit": "Maurizio Morandi", "date": "July 2012", "family_updated": "", "metric_expression": "1.659+", "metric_symbol": "s", "metric_value": 1.659, "reference_text": "s = 1.659+ | Found by Maurizio Morandi | in July 2012.", "source_image": "8.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/8.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.659+ | Found by Maurizio Morandi | in July 2012.", "friedman_source_image": "8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "2ac4629603f9ce00", "improvement": -0.0004163833859911481, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2708160845523109e-06, "max_pair_overlap_depth": -2.147963686768839e-05, "metric_expression": "1.65942", "metric_symbol": "s", "metric_value": 1.6594163833859912, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.659, "setup": "triinpen", "side": 1.6594163833859912, "solution_path": "data/solutions/generated/triinpen_008_2ac4629603f9ce00.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/2ac4629603f9ce00.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@9", "density": 0.7240666005522262, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/9.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.767+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.767, "friedman_reference": {"analytical_or_proved": true, "case": "triinpen@9", "credit": "Maurizio Morandi", "date": "July 2012", "family_updated": "", "metric_expression": "1.767+", "metric_symbol": "s", "metric_value": 1.767, "reference_text": "s = 1.767+ | Found by Maurizio Morandi in July 2012. | Improved slightly by David W. Cantrell in August 2012.", "source_image": "9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/9.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.767+ | Found by Maurizio Morandi in July 2012. | Improved slightly by David W. Cantrell in August 2012.", "friedman_source_image": "9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "72279a3b92ae69ca", "improvement": -0.0017148783247660226, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.918216101924126e-06, "max_pair_overlap_depth": -2.2857021414468548e-08, "metric_expression": "1.76871", "metric_symbol": "s", "metric_value": 1.768714878324766, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.767, "setup": "triinpen", "side": 1.768714878324766, "solution_path": "data/solutions/generated/triinpen_009_72279a3b92ae69ca.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/72279a3b92ae69ca.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@10", "density": 0.7626299188806616, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/10.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.815+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.815, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@10", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "1.815+", "metric_symbol": "s", "metric_value": 1.815, "reference_text": "s = 1.815+ | Found by Timo Berthold et al | in January 2026.", "source_image": "10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/10.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.815+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "f05f442bda1c650b", "improvement": -0.0016401662315850274, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.176875820916777e-06, "max_pair_overlap_depth": -3.546498963225009e-06, "metric_expression": "1.81664", "metric_symbol": "s", "metric_value": 1.816640166231585, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.815, "setup": "triinpen", "side": 1.816640166231585, "solution_path": "data/solutions/generated/triinpen_010_f05f442bda1c650b.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/f05f442bda1c650b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@11", "density": 0.7898687600512876, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/11.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.871+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.871, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@11", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "1.871+", "metric_symbol": "s", "metric_value": 1.871, "reference_text": "s = 1.871+ | Found by Timo Berthold et al | in January 2026.", "source_image": "11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/11.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.871+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "ca41afc6c9b25b21", "improvement": -0.0011675168118052692, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.191218876790302e-06, "max_pair_overlap_depth": -1.6407344776342425e-05, "metric_expression": "1.87217", "metric_symbol": "s", "metric_value": 1.8721675168118053, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.871, "setup": "triinpen", "side": 1.8721675168118053, "solution_path": "data/solutions/generated/triinpen_011_ca41afc6c9b25b21.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/ca41afc6c9b25b21.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@12", "density": 0.774567014021955, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/12.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.973+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.973, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@12", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "1.973+", "metric_symbol": "s", "metric_value": 1.973, "reference_text": "s = 1.973+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "12.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/12.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 1.973+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "bf7ddcc735398246", "improvement": -0.0016355699317964767, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.087403974087953e-06, "max_pair_overlap_depth": -1.3435552785878713e-05, "metric_expression": "1.97464", "metric_symbol": "s", "metric_value": 1.9746355699317966, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.973, "setup": "triinpen", "side": 1.9746355699317966, "solution_path": "data/solutions/generated/triinpen_012_bf7ddcc735398246.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/bf7ddcc735398246.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@13", "density": 0.8029051605923488, "friedman_analytical_or_proved": false, "friedman_credit": "Timo Berthold et al", "friedman_date": "January 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/13.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.017+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.017, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@13", "credit": "Timo Berthold et al", "date": "January 2026", "family_updated": "", "metric_expression": "2.017+", "metric_symbol": "s", "metric_value": 2.017, "reference_text": "s = 2.017+ | Found by Timo Berthold et al | in January 2026.", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/13.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.017+ | Found by Timo Berthold et al | in January 2026.", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "ff07ea08219353cb", "improvement": -0.0016702531518792973, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.615779077054135e-07, "max_pair_overlap_depth": -4.3930654851198625e-05, "metric_expression": "2.01867", "metric_symbol": "s", "metric_value": 2.018670253151879, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.017, "setup": "triinpen", "side": 2.018670253151879, "solution_path": "data/solutions/generated/triinpen_013_ff07ea08219353cb.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Timo Berthold et al", "svg_path": "data/svg/ff07ea08219353cb.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@14", "density": 0.8070990172681682, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/14.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.088+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.088, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@14", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "2.088+", "metric_symbol": "s", "metric_value": 2.088, "reference_text": "s = 2.088+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/14.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.088+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "bc900c323229ee0c", "improvement": -0.0014233339778941811, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.2035527022044616e-06, "max_pair_overlap_depth": -3.9869300561434784e-05, "metric_expression": "2.08942", "metric_symbol": "s", "metric_value": 2.0894233339778943, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.088, "setup": "triinpen", "side": 2.0894233339778943, "solution_path": "data/solutions/generated/triinpen_014_bc900c323229ee0c.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/bc900c323229ee0c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@15", "density": 0.8370246386712876, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/15.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.123+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.123, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@15", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.123+", "metric_symbol": "s", "metric_value": 2.123, "reference_text": "s = 2.123+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/15.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.123+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "39b13eec51bf2ea6", "improvement": -0.000744857935933041, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.959192129393714e-06, "max_pair_overlap_depth": -3.112528873211451e-05, "metric_expression": "2.12374", "metric_symbol": "s", "metric_value": 2.1237448579359333, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.123, "setup": "triinpen", "side": 2.1237448579359333, "solution_path": "data/solutions/generated/triinpen_015_39b13eec51bf2ea6.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/39b13eec51bf2ea6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@16", "density": 0.8497807683719915, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/16.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.17598+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.17598, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@16", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.17598+", "metric_symbol": "s", "metric_value": 2.17598, "reference_text": "s = 2.17598+ | Found by Ian Watson | in April 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/16.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.17598+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "852225bc33edeebe", "improvement": -0.0008893975034314217, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.337598971955443e-06, "max_pair_overlap_depth": -3.396060024770525e-05, "metric_expression": "2.17687", "metric_symbol": "s", "metric_value": 2.1768693975034314, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.17598, "setup": "triinpen", "side": 2.1768693975034314, "solution_path": "data/solutions/generated/triinpen_016_852225bc33edeebe.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/852225bc33edeebe.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@17", "density": 0.8546812093159708, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/17.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.23658+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.23658, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@17", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.23658+", "metric_symbol": "s", "metric_value": 2.23658, "reference_text": "s = 2.23658+ | Found by Ian Watson | in April 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/17.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.23658+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "a8618eee7fdaba2a", "improvement": -0.0008435999825509022, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.39992551681681e-07, "max_pair_overlap_depth": -1.7412872881594765e-05, "metric_expression": "2.23742", "metric_symbol": "s", "metric_value": 2.237423599982551, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.23658, "setup": "triinpen", "side": 2.237423599982551, "solution_path": "data/solutions/generated/triinpen_017_a8618eee7fdaba2a.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/a8618eee7fdaba2a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@18", "density": 0.8594655455888001, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/18.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.29382+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.29382, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@18", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.29382+", "metric_symbol": "s", "metric_value": 2.29382, "reference_text": "s = 2.29382+ | Found by Ian Watson | in April 2026.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/18.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.29382+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "6bf4f9b0915d39ee", "improvement": -0.0020529370652000445, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4774387061990001e-05, "max_pair_overlap_depth": -2.9541451285197695e-08, "metric_expression": "2.29587", "metric_symbol": "s", "metric_value": 2.2958729370652002, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.29382, "setup": "triinpen", "side": 2.2958729370652002, "solution_path": "data/solutions/generated/triinpen_018_6bf4f9b0915d39ee.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/6bf4f9b0915d39ee.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@19", "density": 0.868021512388566, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/19.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.344+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.344, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@19", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.344+", "metric_symbol": "s", "metric_value": 2.344, "reference_text": "s = 2.344+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/19.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.344+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "eaa1430599d74148", "improvement": -0.003131316702724707, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9139572150349214e-06, "max_pair_overlap_depth": -1.0705368373109536e-05, "metric_expression": "2.34713", "metric_symbol": "s", "metric_value": 2.3471313167027246, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.344, "setup": "triinpen", "side": 2.3471313167027246, "solution_path": "data/solutions/generated/triinpen_19_eaa1430599d74148.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/eaa1430599d74148.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@20", "density": 0.8487102106795418, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/20.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.43350+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.4335, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@20", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.43350+", "metric_symbol": "s", "metric_value": 2.4335, "reference_text": "s = 2.43350+ | Found by Ian Watson | in April 2026.", "source_image": "20.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/20.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.43350+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "760faebc4d0b131b", "improvement": -0.001848490502489586, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.176183787084704e-07, "max_pair_overlap_depth": -8.807499014995557e-06, "metric_expression": "2.43535", "metric_symbol": "s", "metric_value": 2.4353484905024896, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.4335, "setup": "triinpen", "side": 2.4353484905024896, "solution_path": "data/solutions/generated/triinpen_020_760faebc4d0b131b.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/760faebc4d0b131b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@21", "density": 0.8589998748063701, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/21.gif", "friedman_metric_expression": "2.479+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.479, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@21", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.479+", "metric_symbol": "s", "metric_value": 2.479, "reference_text": "s = 2.479+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/21.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.479+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "115c20fad199fe9d", "improvement": -0.0014982727861765355, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.811024693767028e-06, "max_pair_overlap_depth": -2.8209410938284663e-06, "metric_expression": "2.48050", "metric_symbol": "s", "metric_value": 2.4804982727861766, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.479, "setup": "triinpen", "side": 2.4804982727861766, "solution_path": "data/solutions/generated/triinpen_021_115c20fad199fe9d.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/115c20fad199fe9d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@22", "density": 0.8389716031782235, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/22.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.56740+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5674, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@22", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.56740+", "metric_symbol": "s", "metric_value": 2.5674, "reference_text": "s = 2.56740+ | Found by Ian Watson | in April 2026.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/22.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.56740+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "929a1aedb3cfabb6", "improvement": -0.0015966714037367602, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.137059676377362e-07, "max_pair_overlap_depth": -8.172541590134585e-06, "metric_expression": "2.56900", "metric_symbol": "s", "metric_value": 2.568996671403737, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.5674, "setup": "triinpen", "side": 2.568996671403737, "solution_path": "data/solutions/generated/triinpen_022_929a1aedb3cfabb6.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/929a1aedb3cfabb6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@23", "density": 0.8432972889244729, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/23.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.61831+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.61831, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@23", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.61831+", "metric_symbol": "s", "metric_value": 2.61831, "reference_text": "s = 2.61831+ | Found by Ian Watson | in April 2026.", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/23.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.61831+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "c0d3905a26f7ef33", "improvement": -0.001678576890473682, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.992582537021139e-06, "max_pair_overlap_depth": -4.362218652298466e-06, "metric_expression": "2.61999", "metric_symbol": "s", "metric_value": 2.619988576890474, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.61831, "setup": "triinpen", "side": 2.619988576890474, "solution_path": "data/solutions/generated/triinpen_023_c0d3905a26f7ef33.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/c0d3905a26f7ef33.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@24", "density": 0.8488309418600493, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/24.gif", "friedman_metric_expression": "2.66641+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.66641, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@24", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.66641+", "metric_symbol": "s", "metric_value": 2.66641, "reference_text": "s = 2.66641+ | Found by Ian Watson | in April 2026.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/24.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.66641+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "7bc3466b163ec849", "improvement": -0.0011908775198370414, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.421732660736982e-06, "max_pair_overlap_depth": -3.3708625798434255e-05, "metric_expression": "2.66760", "metric_symbol": "s", "metric_value": 2.667600877519837, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.66641, "setup": "triinpen", "side": 2.667600877519837, "solution_path": "data/solutions/generated/triinpen_24_7bc3466b163ec849.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/7bc3466b163ec849.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@25", "density": 0.8491607934363294, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/25.gif", "friedman_metric_expression": "2.720+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.72, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@25", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.720+", "metric_symbol": "s", "metric_value": 2.72, "reference_text": "s = 2.720+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/25.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.720+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "0f20f89eba855e21", "improvement": -0.0020799025832594076, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.997044244500036e-06, "max_pair_overlap_depth": -8.121085018242269e-06, "metric_expression": "2.72208", "metric_symbol": "s", "metric_value": 2.7220799025832596, "n": 25, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.72, "setup": "triinpen", "side": 2.7220799025832596, "solution_path": "data/solutions/generated/triinpen_25_0f20f89eba855e21.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/0f20f89eba855e21.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@26", "density": 0.8520841245254238, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/26.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.771+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.771, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@26", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.771+", "metric_symbol": "s", "metric_value": 2.771, "reference_text": "s = 2.771+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/26.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.771+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "f991467944e6e634", "improvement": -0.00022168609627781777, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2986540650513047e-06, "max_pair_overlap_depth": -8.90414151222263e-06, "metric_expression": "2.77122", "metric_symbol": "s", "metric_value": 2.7712216860962777, "n": 26, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.771, "setup": "triinpen", "side": 2.7712216860962777, "solution_path": "data/solutions/generated/triinpen_026_f991467944e6e634.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/f991467944e6e634.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@27", "density": 0.8551322064876743, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/27.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.81761+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.81761, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@27", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.81761+", "metric_symbol": "s", "metric_value": 2.81761, "reference_text": "s = 2.81761+ | Found by Ian Watson | in April 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/27.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.81761+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "d6f1d3073209f163", "improvement": -0.0013640775641627911, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.642891580892597e-06, "max_pair_overlap_depth": -2.2253597504739986e-06, "metric_expression": "2.81897", "metric_symbol": "s", "metric_value": 2.818974077564163, "n": 27, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.81761, "setup": "triinpen", "side": 2.818974077564163, "solution_path": "data/solutions/generated/triinpen_027_d6f1d3073209f163.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/d6f1d3073209f163.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@28", "density": 0.8558072548880046, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/28.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.868+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.868, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@28", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "", "metric_expression": "2.868+", "metric_symbol": "s", "metric_value": 2.868, "reference_text": "s = 2.868+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/28.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.868+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "2caa877f5aa08139", "improvement": -0.0015702794780372287, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.305293369328324e-06, "max_pair_overlap_depth": -1.9072024115107666e-05, "metric_expression": "2.86957", "metric_symbol": "s", "metric_value": 2.869570279478037, "n": 28, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.868, "setup": "triinpen", "side": 2.869570279478037, "solution_path": "data/solutions/generated/triinpen_028_2caa877f5aa08139.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T03:19:00+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/2caa877f5aa08139.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@29", "density": 0.8642558401713053, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/29.gif", "friedman_metric_expression": "2.903+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.903, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@29", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "2.903+", "metric_symbol": "s", "metric_value": 2.903, "reference_text": "s = 2.903+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/29.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.903+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "9730db96941e5b52", "improvement": -0.003053928319848076, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.800701997238751e-06, "max_pair_overlap_depth": -5.0014115865471e-06, "metric_expression": "2.90605", "metric_symbol": "s", "metric_value": 2.906053928319848, "n": 29, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.903, "setup": "triinpen", "side": 2.906053928319848, "solution_path": "data/solutions/generated/triinpen_29_9730db96941e5b52.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/9730db96941e5b52.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@30", "density": 0.8662355090431414, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/30.gif", "friedman_metric_expression": "2.952+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.952, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@30", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "2.952+", "metric_symbol": "s", "metric_value": 2.952, "reference_text": "s = 2.952+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/30.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.952+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "29e4e0e684891d8a", "improvement": -0.00035425771249730786, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.990812967166742e-06, "max_pair_overlap_depth": -2.800943013125945e-05, "metric_expression": "2.95235", "metric_symbol": "s", "metric_value": 2.9523542577124973, "n": 30, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.952, "setup": "triinpen", "side": 2.9523542577124973, "solution_path": "data/solutions/generated/triinpen_30_29e4e0e684891d8a.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/29e4e0e684891d8a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@31", "density": 0.8713888671331285, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/31.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.992+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.992, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "2.992+", "metric_symbol": "s", "metric_value": 2.992, "reference_text": "s = 2.992+ | Found by Emerson Connelly | in April 2026.", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/31.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.992+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "44a0bb3f581e92e2", "improvement": -0.0002692889761504169, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7795814276238894e-06, "max_pair_overlap_depth": -1.3596177029562284e-05, "metric_expression": "2.99227", "metric_symbol": "s", "metric_value": 2.9922692889761504, "n": 31, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.992, "setup": "triinpen", "side": 2.9922692889761504, "solution_path": "data/solutions/generated/triinpen_031_44a0bb3f581e92e2.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/44a0bb3f581e92e2.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@32", "density": 0.8781198099192385, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/32.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.029+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.029, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@32", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.029+", "metric_symbol": "s", "metric_value": 3.029, "reference_text": "s = 3.029+ | Found by Emerson Connelly | in April 2026.", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/32.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.029+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "310badf403ba4fce", "improvement": 0.0005254132016050406, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.877563686504985e-07, "max_pair_overlap_depth": -2.5172545809359903e-05, "metric_expression": "3.02847", "metric_symbol": "s", "metric_value": 3.028474586798395, "n": 32, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.029, "setup": "triinpen", "side": 3.028474586798395, "solution_path": "data/solutions/generated/triinpen_032_310badf403ba4fce.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/310badf403ba4fce.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@33", "density": 0.8767362450469284, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/33.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.078+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.078, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@33", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.078+", "metric_symbol": "s", "metric_value": 3.078, "reference_text": "s = 3.078+ | Found by Emerson Connelly | in April 2026.", "source_image": "33.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/33.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.078+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "2cf1b1f38810795d", "improvement": 0.00014382895343612034, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.9579559668821387e-06, "max_pair_overlap_depth": -6.152141676318834e-06, "metric_expression": "3.07786", "metric_symbol": "s", "metric_value": 3.0778561710465637, "n": 33, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.078, "setup": "triinpen", "side": 3.0778561710465637, "solution_path": "data/solutions/generated/triinpen_033_2cf1b1f38810795d.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/2cf1b1f38810795d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@34", "density": 0.8805067280135348, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/34.gif", "friedman_metric_expression": "3.114+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.114, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@34", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.114+", "metric_symbol": "s", "metric_value": 3.114, "reference_text": "s = 3.114+ | Found by Emerson Connelly | in April 2026.", "source_image": "34.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/34.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.114+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "faff9872314d8b1a", "improvement": -0.003446083050953952, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4825467113644516e-07, "max_pair_overlap_depth": -8.773672566686486e-06, "metric_expression": "3.11745", "metric_symbol": "s", "metric_value": 3.117446083050954, "n": 34, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.114, "setup": "triinpen", "side": 3.117446083050954, "solution_path": "data/solutions/generated/triinpen_34_faff9872314d8b1a.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T14:34:18+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/faff9872314d8b1a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@35", "density": 0.8875886550862264, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/35.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.150+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.15, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@35", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.150+", "metric_symbol": "s", "metric_value": 3.15, "reference_text": "s = 3.150+ | Found by Emerson Connelly | in April 2026.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/35.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.150+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "8fe5da1df972c0b3", "improvement": -0.0003150122852528803, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4152162741179605e-06, "max_pair_overlap_depth": -9.46789992128938e-06, "metric_expression": "3.15032", "metric_symbol": "s", "metric_value": 3.150315012285253, "n": 35, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.15, "setup": "triinpen", "side": 3.150315012285253, "solution_path": "data/solutions/generated/triinpen_035_8fe5da1df972c0b3.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/8fe5da1df972c0b3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@36", "density": 0.8985572699958073, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/36.gif", "friedman_metric_expression": "3.177+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.177, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@36", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.177+", "metric_symbol": "s", "metric_value": 3.177, "reference_text": "s = 3.177+ | Found by Emerson Connelly | in April 2026.", "source_image": "36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/36.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.177+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "5183a03ffc16bcf9", "improvement": 0.0015578799748001693, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.498547309348074e-07, "max_pair_overlap_depth": -2.3029919912209706e-06, "metric_expression": "3.17544", "metric_symbol": "s", "metric_value": 3.1754421200252, "n": 36, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.177, "setup": "triinpen", "side": 3.1754421200252, "solution_path": "data/solutions/record_hunt/triinpen_36_5183a03ffc16bcf9.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/5183a03ffc16bcf9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@37", "density": 0.9016389565813171, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/37.gif", "friedman_metric_expression": "3.21529+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.21529, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@37", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "3.21529+", "metric_symbol": "s", "metric_value": 3.21529, "reference_text": "s = 3.21529+ | Found by Nathan Roll | in April 2026.", "source_image": "37.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/37.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.21529+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "08996b45ff04652c", "improvement": 0.0015527990845791706, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.641732962649716e-07, "max_pair_overlap_depth": -1.845774424413804e-07, "metric_expression": "3.21374", "metric_symbol": "s", "metric_value": 3.213737200915421, "n": 37, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.21529, "setup": "triinpen", "side": 3.213737200915421, "solution_path": "data/solutions/record_hunt/triinpen_37_08996b45ff04652c.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/08996b45ff04652c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@38", "density": 0.9010687984988149, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/38.gif", "friedman_metric_expression": "3.25877+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.25877, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@38", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "3.25877+", "metric_symbol": "s", "metric_value": 3.25877, "reference_text": "s = 3.25877+ | Found by Nathan Roll | in April 2026.", "source_image": "38.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/38.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.25877+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "38.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "cca381f54ee9b299", "improvement": 0.0008632130758914336, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.177049878621715e-07, "max_pair_overlap_depth": -8.114145322313249e-07, "metric_expression": "3.25791", "metric_symbol": "s", "metric_value": 3.2579067869241087, "n": 38, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.25877, "setup": "triinpen", "side": 3.2579067869241087, "solution_path": "data/solutions/record_hunt/triinpen_38_cca381f54ee9b299.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/cca381f54ee9b299.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@39", "density": 0.9025101556269666, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/39.gif", "friedman_metric_expression": "3.295+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.295, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@39", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.295+", "metric_symbol": "s", "metric_value": 3.295, "reference_text": "s = 3.295+ | Found by Emerson Connelly | in April 2026.", "source_image": "39.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/39.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.295+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "67a6bbf24108966a", "improvement": -0.002859023967915597, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9740347935670834e-06, "max_pair_overlap_depth": -1.0007850107629501e-05, "metric_expression": "3.29786", "metric_symbol": "s", "metric_value": 3.2978590239679155, "n": 39, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.295, "setup": "triinpen", "side": 3.2978590239679155, "solution_path": "data/solutions/generated/triinpen_039_67a6bbf24108966a.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/67a6bbf24108966a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinpen@40", "density": 0.9091859235070038, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/40.gif", "friedman_metric_expression": "3.32786+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.32786, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@40", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "3.32786+", "metric_symbol": "s", "metric_value": 3.32786, "reference_text": "s = 3.32786+ | Found by Nathan Roll | in April 2026.", "source_image": "40.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/40.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.32786+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "032ceee4e789eff6", "improvement": 0.0002725631872593226, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4037127820689358e-07, "max_pair_overlap_depth": -1.074379339049969e-06, "metric_expression": "3.32759", "metric_symbol": "s", "metric_value": 3.3275874368127405, "n": 40, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.32786, "setup": "triinpen", "side": 3.3275874368127405, "solution_path": "data/solutions/record_hunt/triinpen_40_032ceee4e789eff6.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/032ceee4e789eff6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@1", "density": 0.4640965865243202, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t1.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "(\u221a2+\u221a6)/4 = .965+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.965, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "(\u221a2+\u221a6)/4 = .965+", "metric_symbol": "s", "metric_value": 0.965, "reference_text": "s = (\u221a2+\u221a6)/4 = .965+ | Trivial.", "source_image": "t1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t1.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = (\u221a2+\u221a6)/4 = .965+ | Trivial.", "friedman_source_image": "t1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "d0da9c9aae58c37e", "improvement": -0.0009310593101359288, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.031269735994051e-07, "max_pair_overlap_depth": 0.0, "metric_expression": "0.96593", "metric_symbol": "s", "metric_value": 0.9659310593101359, "n": 1, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 0.965, "setup": "triinsqu", "side": 0.9659310593101359, "solution_path": "data/solutions/generated/triinsqu_001_d0da9c9aae58c37e.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/d0da9c9aae58c37e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@2", "density": 0.5773345801938596, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t2.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a6/2 = 1.224+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.224, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "\u221a6/2 = 1.224+", "metric_symbol": "s", "metric_value": 1.224, "reference_text": "s = \u221a6/2 = 1.224+ | Trivial.", "source_image": "t2.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t2.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = \u221a6/2 = 1.224+ | Trivial.", "friedman_source_image": "t2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "536a8bec14614050", "improvement": -0.0007615124236886039, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.5296142902332335e-06, "max_pair_overlap_depth": -8.602274652714007e-06, "metric_expression": "1.22476", "metric_symbol": "s", "metric_value": 1.2247615124236886, "n": 2, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.224, "setup": "triinsqu", "side": 1.2247615124236886, "solution_path": "data/solutions/generated/triinsqu_002_536a8bec14614050.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/536a8bec14614050.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@3", "density": 0.594304450868366, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1996", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t3.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "\u221a3/2+\u221a6/4 = 1.478+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.478, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@3", "credit": "Erich Friedman", "date": "1996", "family_updated": "", "metric_expression": "\u221a3/2+\u221a6/4 = 1.478+", "metric_symbol": "s", "metric_value": 1.478, "reference_text": "s = \u221a3/2+\u221a6/4 = 1.478+ | Found by Erich Friedman | in 1996.", "source_image": "t3.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t3.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = \u221a3/2+\u221a6/4 = 1.478+ | Found by Erich Friedman | in 1996.", "friedman_source_image": "t3.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "85d51e7a9d3c3790", "improvement": -0.00044936123470140643, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.5025974997417175e-07, "max_pair_overlap_depth": -2.9708917167997617e-05, "metric_expression": "1.47845", "metric_symbol": "s", "metric_value": 1.4784493612347014, "n": 3, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.478, "setup": "triinsqu", "side": 1.4784493612347014, "solution_path": "data/solutions/generated/triinsqu_003_85d51e7a9d3c3790.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/85d51e7a9d3c3790.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@4", "density": 0.6961073829345906, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1996", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t4.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1+1/\u221a3 = 1.577+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.577, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@4", "credit": "Erich Friedman", "date": "1996", "family_updated": "", "metric_expression": "1+1/\u221a3 = 1.577+", "metric_symbol": "s", "metric_value": 1.577, "reference_text": "s = 1+1/\u221a3 = 1.577+ | Found by Erich Friedman | in 1996.", "source_image": "t4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t4.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 1+1/\u221a3 = 1.577+ | Found by Erich Friedman | in 1996.", "friedman_source_image": "t4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "10004832454b0f3c", "improvement": -0.00040129748580719493, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.301219061208287e-07, "max_pair_overlap_depth": -2.378752457826705e-05, "metric_expression": "1.57740", "metric_symbol": "s", "metric_value": 1.5774012974858072, "n": 4, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.577, "setup": "triinsqu", "side": 1.5774012974858072, "solution_path": "data/solutions/generated/triinsqu_004_10004832454b0f3c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/10004832454b0f3c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@5", "density": 0.6658433228069607, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1996", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t5.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.803+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.803, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@5", "credit": "Erich Friedman", "date": "1996", "family_updated": "", "metric_expression": "1.803+", "metric_symbol": "s", "metric_value": 1.803, "reference_text": "s = 1.803+ | Found by Erich Friedman | in 1996.", "source_image": "t5.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t5.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 1.803+ | Found by Erich Friedman | in 1996.", "friedman_source_image": "t5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "f41eeaee150f1c15", "improvement": -0.0002224086857132157, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.6751531743821273e-06, "max_pair_overlap_depth": -2.226142234541817e-05, "metric_expression": "1.80322", "metric_symbol": "s", "metric_value": 1.8032224086857132, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.803, "setup": "triinsqu", "side": 1.8032224086857132, "solution_path": "data/solutions/generated/triinsqu_005_f41eeaee150f1c15.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f41eeaee150f1c15.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@6", "density": 0.7181966221398873, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1996", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t6.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "9/2\u20133\u221a3/2 = 1.901+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.901, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@6", "credit": "Erich Friedman", "date": "1996", "family_updated": "", "metric_expression": "9/2\u20133\u221a3/2 = 1.901+", "metric_symbol": "s", "metric_value": 1.901, "reference_text": "s = 9/2\u20133\u221a3/2 = 1.901+ | Found by Erich Friedman | in 1996.", "source_image": "t6.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t6.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 9/2\u20133\u221a3/2 = 1.901+ | Found by Erich Friedman | in 1996.", "friedman_source_image": "t6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "47de8ba8f0c9310c", "improvement": -0.0009726348176062327, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.131503356462894e-07, "max_pair_overlap_depth": -1.0226066655058541e-06, "metric_expression": "1.90197", "metric_symbol": "s", "metric_value": 1.9019726348176063, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 1.901, "setup": "triinsqu", "side": 1.9019726348176063, "solution_path": "data/solutions/generated/triinsqu_006_47de8ba8f0c9310c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/47de8ba8f0c9310c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@7", "density": 0.7576992844833931, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1996", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t7.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@7", "credit": "Erich Friedman", "date": "1996", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Found by Erich Friedman | in 1996.", "source_image": "t7.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t7.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2 | Found by Erich Friedman | in 1996.", "friedman_source_image": "t7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "d4eecc11473f7bf7", "improvement": -9.626783869354227e-05, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -7.867700488247924e-07, "max_pair_overlap_depth": -1.9950288099523666e-05, "metric_expression": "2.00010", "metric_symbol": "s", "metric_value": 2.0000962678386935, "n": 7, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.0, "setup": "triinsqu", "side": 2.0000962678386935, "solution_path": "data/solutions/generated/triinsqu_007_d4eecc11473f7bf7.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/d4eecc11473f7bf7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@8", "density": 0.7868902975296777, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1996", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t8.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3\u221a3/2\u20131/2 = 2.098+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.098, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@8", "credit": "Erich Friedman", "date": "1996", "family_updated": "", "metric_expression": "3\u221a3/2\u20131/2 = 2.098+", "metric_symbol": "s", "metric_value": 2.098, "reference_text": "s = 3\u221a3/2\u20131/2 = 2.098+ | Found by Erich Friedman | in 1996.", "source_image": "t8.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t8.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 3\u221a3/2\u20131/2 = 2.098+ | Found by Erich Friedman | in 1996.", "friedman_source_image": "t8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "bd28b88a6ba9b99d", "improvement": -0.00015813059780489525, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0169734543243664e-06, "max_pair_overlap_depth": -2.636538231393093e-05, "metric_expression": "2.09816", "metric_symbol": "s", "metric_value": 2.0981581305978048, "n": 8, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.098, "setup": "triinsqu", "side": 2.0981581305978048, "solution_path": "data/solutions/generated/triinsqu_008_bd28b88a6ba9b99d.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/bd28b88a6ba9b99d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@9", "density": 0.7446998211864622, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t9.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.287+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.287, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@9", "credit": "Maurizio Morandi", "date": "July 2008", "family_updated": "", "metric_expression": "2.287+", "metric_symbol": "s", "metric_value": 2.287, "reference_text": "s = 2.287+ | Found by Maurizio Morandi | in July 2008.", "source_image": "t9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t9.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.287+ | Found by Maurizio Morandi | in July 2008.", "friedman_source_image": "t9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "458e49fec5e5573c", "improvement": -0.0006045303821253789, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.745466993473377e-07, "max_pair_overlap_depth": -5.124657140087585e-06, "metric_expression": "2.28760", "metric_symbol": "s", "metric_value": 2.2876045303821253, "n": 9, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.287, "setup": "triinsqu", "side": 2.2876045303821253, "solution_path": "data/solutions/generated/triinsqu_009_458e49fec5e5573c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/458e49fec5e5573c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@10", "density": 0.7662234065167978, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t10.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.377+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.377, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@10", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "2.377+", "metric_symbol": "s", "metric_value": 2.377, "reference_text": "s = 2.377+ | Found by David W. Cantrell | in July 2002.", "source_image": "t10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t10.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.377+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "t10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "5b0eaf84f54fe48c", "improvement": -0.00023770927049060475, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.324926068512563e-07, "max_pair_overlap_depth": -1.1896205976635876e-05, "metric_expression": "2.37724", "metric_symbol": "s", "metric_value": 2.3772377092704904, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.377, "setup": "triinsqu", "side": 2.3772377092704904, "solution_path": "data/solutions/generated/triinsqu_10_5b0eaf84f54fe48c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/5b0eaf84f54fe48c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@11", "density": 0.7671858882860172, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t11.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.490+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.49, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@11", "credit": "Maurizio Morandi", "date": "June 2008", "family_updated": "", "metric_expression": "2.490+", "metric_symbol": "s", "metric_value": 2.49, "reference_text": "s = 2.490+ | Found by Maurizio Morandi | in June 2008.", "source_image": "t11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t11.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.490+ | Found by Maurizio Morandi | in June 2008.", "friedman_source_image": "t11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "b95fbc7e7009bb63", "improvement": -0.0017034740442234408, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.171896607131735e-07, "max_pair_overlap_depth": -3.917476745979309e-05, "metric_expression": "2.49170", "metric_symbol": "s", "metric_value": 2.4917034740442237, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.49, "setup": "triinsqu", "side": 2.4917034740442237, "solution_path": "data/solutions/generated/triinsqu_011_b95fbc7e7009bb63.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/b95fbc7e7009bb63.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@12", "density": 0.792908624011719, "friedman_analytical_or_proved": false, "friedman_credit": "Dominik Kamp", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t12.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.558+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.558, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@12", "credit": "Dominik Kamp", "date": "April 2026", "family_updated": "", "metric_expression": "2.558+", "metric_symbol": "s", "metric_value": 2.558, "reference_text": "s = 2.558+ | Found by Dominik Kamp | in April 2026.", "source_image": "t12.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t12.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.558+ | Found by Dominik Kamp | in April 2026.", "friedman_source_image": "t12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "297283972e34795d", "improvement": -0.0019375473066665094, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.5442542765183254e-06, "max_pair_overlap_depth": -1.2119755173123359e-05, "metric_expression": "2.55994", "metric_symbol": "s", "metric_value": 2.5599375473066663, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.558, "setup": "triinsqu", "side": 2.5599375473066663, "solution_path": "data/solutions/generated/triinsqu_012_297283972e34795d.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Dominik Kamp", "svg_path": "data/svg/297283972e34795d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@13", "density": 0.8355745862678572, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t13.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.595+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.595, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@13", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "2.595+", "metric_symbol": "s", "metric_value": 2.595, "reference_text": "s = 2.595+ | Found by David W. Cantrell | in July 2002.", "source_image": "t13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t13.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.595+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "t13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "7628e283de25ee68", "improvement": -0.0005499010705918295, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8264727928819919e-06, "max_pair_overlap_depth": -1.755464108432747e-05, "metric_expression": "2.59555", "metric_symbol": "s", "metric_value": 2.595549901070592, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.595, "setup": "triinsqu", "side": 2.595549901070592, "solution_path": "data/solutions/generated/triinsqu_13_7628e283de25ee68.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/7628e283de25ee68.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@14", "density": 0.8144966370845219, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t14.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.726+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.726, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@14", "credit": "David W. Cantrell", "date": "July 2002", "family_updated": "", "metric_expression": "2.726+", "metric_symbol": "s", "metric_value": 2.726, "reference_text": "s = 2.726+ | Found by David W. Cantrell | in July 2002.", "source_image": "t14.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t14.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.726+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "t14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "601dc4458719752d", "improvement": -0.0021590642579867314, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.4175513847981196e-06, "max_pair_overlap_depth": -2.112817484914431e-05, "metric_expression": "2.72816", "metric_symbol": "s", "metric_value": 2.7281590642579867, "n": 14, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.726, "setup": "triinsqu", "side": 2.7281590642579867, "solution_path": "data/solutions/generated/triinsqu_014_601dc4458719752d.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/601dc4458719752d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@15", "density": 0.8108430550585584, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t15.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.8299+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.8299, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@15", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.8299+", "metric_symbol": "s", "metric_value": 2.8299, "reference_text": "s = 2.8299+ | Found by Ian Watson | in April 2026.", "source_image": "t15.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t15.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.8299+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "430e70d954ae3759", "improvement": -0.0003679073759270679, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0258685225063818e-06, "max_pair_overlap_depth": -3.8614458098606974e-05, "metric_expression": "2.83027", "metric_symbol": "s", "metric_value": 2.830267907375927, "n": 15, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.8299, "setup": "triinsqu", "side": 2.830267907375927, "solution_path": "data/solutions/generated/triinsqu_015_430e70d954ae3759.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/430e70d954ae3759.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@16", "density": 0.8226134516626242, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t16.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.9000+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.9, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@16", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "2.9000+", "metric_symbol": "s", "metric_value": 2.9, "reference_text": "s = 2.9000+ | Found by Ian Watson | in April 2026.", "source_image": "t16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t16.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.9000+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "2350476c07d7060c", "improvement": -0.002100218284964317, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.0129376531587966e-06, "max_pair_overlap_depth": -5.176872598853383e-06, "metric_expression": "2.90210", "metric_symbol": "s", "metric_value": 2.9021002182849642, "n": 16, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.9, "setup": "triinsqu", "side": 2.9021002182849642, "solution_path": "data/solutions/generated/triinsqu_016_2350476c07d7060c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/2350476c07d7060c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@17", "density": 0.8272516069770434, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t17.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.982+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.982, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@17", "credit": "Maurizio Morandi", "date": "July 2008", "family_updated": "", "metric_expression": "2.982+", "metric_symbol": "s", "metric_value": 2.982, "reference_text": "s = 2.982+ | Found by Maurizio Morandi | in July 2008.", "source_image": "t17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t17.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.982+ | Found by Maurizio Morandi | in July 2008.", "friedman_source_image": "t17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "79a5506fa7f0974c", "improvement": -0.0010186525102788835, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8787966733926961e-06, "max_pair_overlap_depth": -9.562062775358982e-06, "metric_expression": "2.98302", "metric_symbol": "s", "metric_value": 2.983018652510279, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.982, "setup": "triinsqu", "side": 2.983018652510279, "solution_path": "data/solutions/generated/triinsqu_017_79a5506fa7f0974c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/79a5506fa7f0974c.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@18", "density": 0.8369229075763384, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t18.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.051+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.051, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@18", "credit": "Maurizio Morandi", "date": "July 2008", "family_updated": "", "metric_expression": "3.051+", "metric_symbol": "s", "metric_value": 3.051, "reference_text": "s = 3.051+ | Found by Maurizio Morandi | in July 2008.", "source_image": "t18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t18.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 3.051+ | Found by Maurizio Morandi | in July 2008.", "friedman_source_image": "t18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "0d24ad1cca3bbb64", "improvement": -0.0007140908076439523, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6577344713731995e-06, "max_pair_overlap_depth": -7.637757528100586e-06, "metric_expression": "3.05171", "metric_symbol": "s", "metric_value": 3.051714090807644, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.051, "setup": "triinsqu", "side": 3.051714090807644, "solution_path": "data/solutions/generated/triinsqu_018_0d24ad1cca3bbb64.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/0d24ad1cca3bbb64.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@19", "density": 0.8400046005998116, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t19.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.12930+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.1293, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@19", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.12930+", "metric_symbol": "s", "metric_value": 3.1293, "reference_text": "s = 3.12930+ | Found by Ian Watson | in April 2026.", "source_image": "t19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t19.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 3.12930+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "def5af7572515998", "improvement": -0.0002816463880579434, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.93050707669818e-06, "max_pair_overlap_depth": -4.5439275512260124e-06, "metric_expression": "3.12958", "metric_symbol": "s", "metric_value": 3.129581646388058, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.1293, "setup": "triinsqu", "side": 3.129581646388058, "solution_path": "data/solutions/generated/triinsqu_019_def5af7572515998.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/def5af7572515998.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@20", "density": 0.8288697801794981, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t20.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.2305+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.2305, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@20", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.2305+", "metric_symbol": "s", "metric_value": 3.2305, "reference_text": "s = 3.2305+ | Found by Ian Watson | in April 2026.", "source_image": "t20.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t20.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 3.2305+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "43e53f43942a4488", "improvement": -0.0018781535798377824, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.815940991194715e-07, "max_pair_overlap_depth": -1.1219130352735363e-05, "metric_expression": "3.23238", "metric_symbol": "s", "metric_value": 3.232378153579838, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.2305, "setup": "triinsqu", "side": 3.232378153579838, "solution_path": "data/solutions/generated/triinsqu_020_43e53f43942a4488.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/43e53f43942a4488.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@21", "density": 0.8291397947988184, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t21.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.3111+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.3111, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@21", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.3111+", "metric_symbol": "s", "metric_value": 3.3111, "reference_text": "s = 3.3111+ | Found by Ian Watson | in April 2026.", "source_image": "t21.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t21.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 3.3111+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "986572060ddb345a", "improvement": -0.0005626159903786565, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2145623034941338e-06, "max_pair_overlap_depth": -2.9730022631513364e-06, "metric_expression": "3.31166", "metric_symbol": "s", "metric_value": 3.311662615990379, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.3111, "setup": "triinsqu", "side": 3.311662615990379, "solution_path": "data/solutions/generated/triinsqu_021_986572060ddb345a.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/986572060ddb345a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@22", "density": 0.8356286267983878, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t22.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.3757+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.3757, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@22", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.3757+", "metric_symbol": "s", "metric_value": 3.3757, "reference_text": "22. | | s = 3.3757+ | Found by Ian Watson | in April 2026.", "source_image": "t22.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t22.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "22. | | s = 3.3757+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "4d69d20a7a35e585", "improvement": -0.0007086456724314516, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2855965547198588e-06, "max_pair_overlap_depth": -1.8710922459974633e-05, "metric_expression": "3.37641", "metric_symbol": "s", "metric_value": 3.3764086456724316, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.3757, "setup": "triinsqu", "side": 3.3764086456724316, "solution_path": "data/solutions/generated/triinsqu_022_4d69d20a7a35e585.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/4d69d20a7a35e585.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@23", "density": 0.8452320475065837, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t23.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.4311+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.4311, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@23", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.4311+", "metric_symbol": "s", "metric_value": 3.4311, "reference_text": "23. | | s = 3.4311+ | Found by Ian Watson | in April 2026.", "source_image": "t23.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t23.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "23. | | s = 3.4311+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "959574dda9a01133", "improvement": -0.001524187888661821, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.8602101761622194e-08, "max_pair_overlap_depth": -2.5255431334025857e-06, "metric_expression": "3.43262", "metric_symbol": "s", "metric_value": 3.4326241878886616, "n": 23, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.4311, "setup": "triinsqu", "side": 3.4326241878886616, "solution_path": "data/solutions/generated/triinsqu_023_959574dda9a01133.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/959574dda9a01133.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@24", "density": 0.8636751606923448, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "August 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t24.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.468+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.468, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@24", "credit": "Maurizio Morandi", "date": "August 2008", "family_updated": "", "metric_expression": "3.468+", "metric_symbol": "s", "metric_value": 3.468, "reference_text": "24. | | s = 3.468+ | Found by Maurizio Morandi | in August 2008.", "source_image": "t24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t24.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "24. | | s = 3.468+ | Found by Maurizio Morandi | in August 2008.", "friedman_source_image": "t24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "d96b5ad8b1001b9a", "improvement": -0.0008116902395634007, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.1658397103314651e-06, "max_pair_overlap_depth": -6.191240536246312e-06, "metric_expression": "3.46881", "metric_symbol": "s", "metric_value": 3.4688116902395634, "n": 24, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.468, "setup": "triinsqu", "side": 3.4688116902395634, "solution_path": "data/solutions/generated/triinsqu_024_d96b5ad8b1001b9a.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/d96b5ad8b1001b9a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@25", "density": 0.8645393394550887, "friedman_analytical_or_proved": false, "friedman_credit": "David W. Cantrell", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t25.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.537+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.537, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@25", "credit": "David W. Cantrell", "date": "August 2012", "family_updated": "", "metric_expression": "3.537+", "metric_symbol": "s", "metric_value": 3.537, "reference_text": "25. | | s = 3.537+ | Found by David W. Cantrell | in August 2012.", "source_image": "t25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t25.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "25. | | s = 3.537+ | Found by David W. Cantrell | in August 2012.", "friedman_source_image": "t25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "8da962f4a2ed3f11", "improvement": -0.0015712315473748184, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -6.272129416634903e-07, "max_pair_overlap_depth": -1.1961418790784606e-05, "metric_expression": "3.53857", "metric_symbol": "s", "metric_value": 3.5385712315473747, "n": 25, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.537, "setup": "triinsqu", "side": 3.5385712315473747, "solution_path": "data/solutions/generated/triinsqu_025_8da962f4a2ed3f11.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/8da962f4a2ed3f11.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@26", "density": 0.8796434821316925, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "September 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t26.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.575+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.575, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@26", "credit": "Maurizio Morandi", "date": "September 2008", "family_updated": "", "metric_expression": "3.575+", "metric_symbol": "s", "metric_value": 3.575, "reference_text": "26. | | s = 3.575+ | Found by Maurizio Morandi | in September 2008.", "source_image": "t26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t26.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "26. | | s = 3.575+ | Found by Maurizio Morandi | in September 2008.", "friedman_source_image": "t26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "c36ce120aca7c2e7", "improvement": -0.002532991640289417, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0954385221262358e-06, "max_pair_overlap_depth": -5.476191583808632e-07, "metric_expression": "3.57753", "metric_symbol": "s", "metric_value": 3.5775329916402896, "n": 26, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.575, "setup": "triinsqu", "side": 3.5775329916402896, "solution_path": "data/solutions/generated/triinsqu_026_c36ce120aca7c2e7.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/c36ce120aca7c2e7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@27", "density": 0.8658956992776567, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t27.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.6726+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.6726, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@27", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.6726+", "metric_symbol": "s", "metric_value": 3.6726, "reference_text": "27. | | s = 3.6726+ | Found by Ian Watson | in April 2026.", "source_image": "t27.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t27.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "27. | | s = 3.6726+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "4c91417b38b8be95", "improvement": -0.0019097898938134072, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.750472262434879e-07, "max_pair_overlap_depth": -1.515528041828862e-05, "metric_expression": "3.67451", "metric_symbol": "s", "metric_value": 3.6745097898938135, "n": 27, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.6726, "setup": "triinsqu", "side": 3.6745097898938135, "solution_path": "data/solutions/generated/triinsqu_027_4c91417b38b8be95.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/4c91417b38b8be95.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@28", "density": 0.8583067572738268, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t28.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.7571+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.7571, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@28", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.7571+", "metric_symbol": "s", "metric_value": 3.7571, "reference_text": "28. | | s = 3.7571+ | Found by Ian Watson | in April 2026.", "source_image": "t28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t28.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "28. | | s = 3.7571+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "9b7285926a894eec", "improvement": -0.0013438626171571322, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.7756377540625622e-06, "max_pair_overlap_depth": -5.481622750158621e-06, "metric_expression": "3.75844", "metric_symbol": "s", "metric_value": 3.758443862617157, "n": 28, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.7571, "setup": "triinsqu", "side": 3.758443862617157, "solution_path": "data/solutions/generated/triinsqu_028_9b7285926a894eec.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/9b7285926a894eec.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@29", "density": 0.8613056568479127, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t29.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.818+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.818, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@29", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "", "metric_expression": "3.818+", "metric_symbol": "s", "metric_value": 3.818, "reference_text": "29. | | s = 3.818+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "t29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t29.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "29. | | s = 3.818+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "t29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "a02ae61c838755d7", "improvement": -0.0003054483997071422, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.731563201531232e-06, "max_pair_overlap_depth": -2.352291984759891e-06, "metric_expression": "3.81831", "metric_symbol": "s", "metric_value": 3.818305448399707, "n": 29, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.818, "setup": "triinsqu", "side": 3.818305448399707, "solution_path": "data/solutions/generated/triinsqu_029_a02ae61c838755d7.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ignacio Vallejo", "svg_path": "data/svg/a02ae61c838755d7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@30", "density": 0.868927959047152, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t30.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.8662+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.8662, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@30", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "3.8662+", "metric_symbol": "s", "metric_value": 3.8662, "reference_text": "30. | | s = 3.8662+ | Found by Ian Watson | in April 2026.", "source_image": "t30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t30.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "30. | | s = 3.8662+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "d3cf7ea5ce255135", "improvement": -0.00030930450574828683, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -4.0741990212131443e-07, "max_pair_overlap_depth": -7.811820435188466e-06, "metric_expression": "3.86651", "metric_symbol": "s", "metric_value": 3.8665093045057484, "n": 30, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.8662, "setup": "triinsqu", "side": 3.8665093045057484, "solution_path": "data/solutions/generated/triinsqu_030_d3cf7ea5ce255135.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/d3cf7ea5ce255135.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@31", "density": 0.8672905272682435, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t31.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3.936+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.936, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "3.936+", "metric_symbol": "s", "metric_value": 3.936, "reference_text": "31. | | s = 3.936+ | Found by Emerson Connelly | in April 2026.", "source_image": "t31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t31.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "31. | | s = 3.936+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "90f4b9b8010910a7", "improvement": 0.001868580153832955, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1286553127417914e-06, "max_pair_overlap_depth": -1.3206709677193729e-05, "metric_expression": "3.93413", "metric_symbol": "s", "metric_value": 3.934131419846167, "n": 31, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.936, "setup": "triinsqu", "side": 3.934131419846167, "solution_path": "data/solutions/generated/triinsqu_031_90f4b9b8010910a7.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/90f4b9b8010910a7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@32", "density": 0.8675921732131934, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t32.gif", "friedman_metric_expression": "6\u20137\u221a3/6 = 3.979+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.979, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@32", "credit": "Maurizio Morandi", "date": "May 2026", "family_updated": "", "metric_expression": "6\u20137\u221a3/6 = 3.979+", "metric_symbol": "s", "metric_value": 3.979, "reference_text": "32. | | s = 6\u20137\u221a3/6 = 3.979+ | Found by Maurizio Morandi | in May 2026.", "source_image": "t32.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t32.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "32. | | s = 6\u20137\u221a3/6 = 3.979+ | Found by Maurizio Morandi | in May 2026.", "friedman_source_image": "t32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "f42801b713b9238d", "improvement": -0.01738660313705287, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.208596980261035e-06, "max_pair_overlap_depth": -2.3341135757526388e-06, "metric_expression": "3.99639", "metric_symbol": "s", "metric_value": 3.996386603137053, "n": 32, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.979, "setup": "triinsqu", "side": 3.996386603137053, "solution_path": "data/solutions/generated/triinsqu_32_f42801b713b9238d.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/f42801b713b9238d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@33", "density": 0.8675949776849048, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t33.gif", "friedman_metric_expression": "4.058+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.058, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@33", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.058+", "metric_symbol": "s", "metric_value": 4.058, "reference_text": "33. | | s = 4.058+ | Found by Emerson Connelly | in April 2026.", "source_image": "t33.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t33.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "33. | | s = 4.058+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "a6f49f2f24228d1b", "improvement": -0.00034322121207797096, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.0747182587902822e-06, "max_pair_overlap_depth": -2.8024820216354662e-05, "metric_expression": "4.05834", "metric_symbol": "s", "metric_value": 4.058343221212078, "n": 33, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.058, "setup": "triinsqu", "side": 4.058343221212078, "solution_path": "data/solutions/generated/triinsqu_33_a6f49f2f24228d1b.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/a6f49f2f24228d1b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@34", "density": 0.8767333914896571, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t34.gif", "friedman_metric_expression": "4.098+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.098, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@34", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.098+", "metric_symbol": "s", "metric_value": 4.098, "reference_text": "34. | | s = 4.098+ | Found by Emerson Connelly | in April 2026.", "source_image": "t34.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t34.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "34. | | s = 4.098+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "61b4d5d6bf22a206", "improvement": 0.00015051348449990343, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.6262786795626285e-06, "max_pair_overlap_depth": -1.499292058504409e-05, "metric_expression": "4.09785", "metric_symbol": "s", "metric_value": 4.0978494865155, "n": 34, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.098, "setup": "triinsqu", "side": 4.0978494865155, "solution_path": "data/solutions/generated/triinsqu_34_61b4d5d6bf22a206.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/61b4d5d6bf22a206.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@35", "density": 0.8677502962014568, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t35.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "9(2\u221a3\u20133) = 4.176+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.176, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@35", "credit": "Erich Friedman", "date": "April 2026", "family_updated": "", "metric_expression": "9(2\u221a3\u20133) = 4.176+", "metric_symbol": "s", "metric_value": 4.176, "reference_text": "35. | | s = 9(2\u221a3\u20133) = 4.176+ | Found by Erich Friedman | in April 2026.", "source_image": "t35.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t35.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "35. | | s = 9(2\u221a3\u20133) = 4.176+ | Found by Erich Friedman | in April 2026.", "friedman_source_image": "t35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "ab6aa3f219e08b42", "improvement": -0.0031403345722011977, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.770482464101941e-07, "max_pair_overlap_depth": -3.3297730506198775e-06, "metric_expression": "4.17914", "metric_symbol": "s", "metric_value": 4.179140334572201, "n": 35, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.176, "setup": "triinsqu", "side": 4.179140334572201, "solution_path": "data/solutions/generated/triinsqu_035_ab6aa3f219e08b42.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/ab6aa3f219e08b42.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@36", "density": 0.8790676097280682, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t36.gif", "friedman_metric_expression": "4.214+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.214, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@36", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.214+", "metric_symbol": "s", "metric_value": 4.214, "reference_text": "36. | | s = 4.214+ | Found by Emerson Connelly | in April 2026.", "source_image": "t36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t36.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "36. | | s = 4.214+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "aae755b2130a7a83", "improvement": 0.0029496973593481357, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.160284450281608e-07, "max_pair_overlap_depth": -9.933313802612531e-07, "metric_expression": "4.21105", "metric_symbol": "s", "metric_value": 4.211050302640652, "n": 36, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.214, "setup": "triinsqu", "side": 4.211050302640652, "solution_path": "data/solutions/record_hunt/triinsqu_36_aae755b2130a7a83.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/aae755b2130a7a83.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@37", "density": 0.8782570037473258, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t37.gif", "friedman_metric_expression": "4.27233+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.27233, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@37", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "4.27233+", "metric_symbol": "s", "metric_value": 4.27233, "reference_text": "37. | | s = 4.27233+ | Found by Nathan Roll | in April 2026.", "source_image": "t37.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t37.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "37. | | s = 4.27233+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "t37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "46406dc33c9746a7", "improvement": 0.0012238102051549404, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.826527513923834e-07, "max_pair_overlap_depth": -6.105510335085995e-07, "metric_expression": "4.27111", "metric_symbol": "s", "metric_value": 4.271106189794845, "n": 37, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.27233, "setup": "triinsqu", "side": 4.271106189794845, "solution_path": "data/solutions/record_hunt/triinsqu_37_46406dc33c9746a7.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/46406dc33c9746a7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@38", "density": 0.87940001283733, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t38.gif", "friedman_metric_expression": "4.327+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.327, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@38", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.327+", "metric_symbol": "s", "metric_value": 4.327, "reference_text": "38. | | s = 4.327+ | Found by Emerson Connelly | in April 2026.", "source_image": "t38.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t38.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "38. | | s = 4.327+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t38.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "f44a4bd75616bc73", "improvement": 0.0013748413573368978, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.0707071410413964e-07, "max_pair_overlap_depth": -1.0916553592199563e-06, "metric_expression": "4.32563", "metric_symbol": "s", "metric_value": 4.325625158642663, "n": 38, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.327, "setup": "triinsqu", "side": 4.325625158642663, "solution_path": "data/solutions/record_hunt/triinsqu_38_f44a4bd75616bc73.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/f44a4bd75616bc73.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@39", "density": 0.880118042721229, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t39.gif", "friedman_metric_expression": "4.383+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.383, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@39", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.383+", "metric_symbol": "s", "metric_value": 4.383, "reference_text": "39. | | s = 4.383+ | Found by Emerson Connelly | in April 2026.", "source_image": "t39.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t39.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "39. | | s = 4.383+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "263f7876a25d297b", "improvement": 0.002616247848735931, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.607883972834501e-07, "max_pair_overlap_depth": -1.86255866918561e-06, "metric_expression": "4.38038", "metric_symbol": "s", "metric_value": 4.380383752151264, "n": 39, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.383, "setup": "triinsqu", "side": 4.380383752151264, "solution_path": "data/solutions/record_hunt/triinsqu_39_263f7876a25d297b.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/263f7876a25d297b.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triinsqu@40", "density": 0.878868813713771, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t40.gif", "friedman_metric_expression": "4.437+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.437, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@40", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "4.437+", "metric_symbol": "s", "metric_value": 4.437, "reference_text": "40. | | s = 4.437+ | Found by Emerson Connelly | in April 2026.", "source_image": "t40.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t40.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "40. | | s = 4.437+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "ccf2360cc8917adc", "improvement": -0.002338759632142029, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.520888573265893e-06, "max_pair_overlap_depth": -1.6780429565876886e-05, "metric_expression": "4.43934", "metric_symbol": "s", "metric_value": 4.439338759632142, "n": 40, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.437, "setup": "triinsqu", "side": 4.439338759632142, "solution_path": "data/solutions/generated/triinsqu_040_ccf2360cc8917adc.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/ccf2360cc8917adc.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@1", "density": 1.0, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt1.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "1", "friedman_metric_symbol": "s", "friedman_metric_value": 1.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@1", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "1", "metric_symbol": "s", "metric_value": 1.0, "reference_text": "s = 1 | Trivial.", "source_image": "tt1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt1.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 1 | Trivial.", "friedman_source_image": "tt1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "b09a5ce4a2c03ad3", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.551115123125788e-17, "max_pair_overlap_depth": 0.0, "metric_expression": "1.00000", "metric_symbol": "s", "metric_value": 1.0, "n": 1, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 1.0, "setup": "triintri", "side": 1.0, "solution_path": "data/solutions/generated/triintri_001_b09a5ce4a2c03ad3.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/b09a5ce4a2c03ad3.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@2", "density": 0.5, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@2", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "tt4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "tt4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "2d6f654db11412d8", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.6653345369377346e-16, "max_pair_overlap_depth": 0.0, "metric_expression": "2.00000", "metric_symbol": "s", "metric_value": 2.0, "n": 2, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "triintri", "side": 2.0, "solution_path": "data/solutions/generated/triintri_002_2d6f654db11412d8.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2d6f654db11412d8.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@3", "density": 0.75, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@3", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "tt4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "tt4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "76381d42f8f17f41", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.6653345369377346e-16, "max_pair_overlap_depth": 3.3306690738754696e-16, "metric_expression": "2.00000", "metric_symbol": "s", "metric_value": 2.0, "n": 3, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "triintri", "side": 2.0, "solution_path": "data/solutions/generated/triintri_003_76381d42f8f17f41.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/76381d42f8f17f41.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@4", "density": 1.0, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@4", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Trivial.", "source_image": "tt4.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt4.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 | Trivial.", "friedman_source_image": "tt4.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "be1b0b4b1802d8fa", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 1.6653345369377346e-16, "max_pair_overlap_depth": 3.3306690738754696e-16, "metric_expression": "2.00000", "metric_symbol": "s", "metric_value": 2.0, "n": 4, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 2.0, "setup": "triintri", "side": 2.0, "solution_path": "data/solutions/generated/triintri_004_be1b0b4b1802d8fa.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/be1b0b4b1802d8fa.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@5", "density": 0.66984024663198, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt5.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "1 + \u221a3 = 2.732+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.732, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@5", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "1 + \u221a3 = 2.732+", "metric_symbol": "s", "metric_value": 2.732, "reference_text": "s = 1 + \u221a3 = 2.732+ | Proved by Erich Friedman | in 1997.", "source_image": "tt5.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt5.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 1 + \u221a3 = 2.732+ | Proved by Erich Friedman | in 1997.", "friedman_source_image": "tt5.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "b4ac2c13e986485e", "improvement": -0.00011756309530053244, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.644640909821311e-06, "max_pair_overlap_depth": -6.265774057961264e-06, "metric_expression": "2.73212", "metric_symbol": "s", "metric_value": 2.7321175630953007, "n": 5, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.732, "setup": "triintri", "side": 2.7321175630953007, "solution_path": "data/solutions/generated/triintri_005_b4ac2c13e986485e.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/b4ac2c13e986485e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@6", "density": 0.6768918015337515, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "August 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt6.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "13/8 + 3\u221a13/8 = 2.977+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.977, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@6", "credit": "Maurizio Morandi", "date": "August 2008", "family_updated": "", "metric_expression": "13/8 + 3\u221a13/8 = 2.977+", "metric_symbol": "s", "metric_value": 2.977, "reference_text": "s = 13/8 + 3\u221a13/8 = 2.977+ | Found by Maurizio Morandi | in August 2008.", "source_image": "tt6.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt6.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 13/8 + 3\u221a13/8 = 2.977+ | Found by Maurizio Morandi | in August 2008.", "friedman_source_image": "tt6.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "df20f3104adbd6a1", "improvement": -0.00025475946622810497, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.3999156171376625e-07, "max_pair_overlap_depth": -1.9147124426771578e-05, "metric_expression": "2.97725", "metric_symbol": "s", "metric_value": 2.977254759466228, "n": 6, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 2.977, "setup": "triintri", "side": 2.977254759466228, "solution_path": "data/solutions/generated/triintri_006_df20f3104adbd6a1.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/df20f3104adbd6a1.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@7", "density": 0.777777777777778, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@7", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "tt9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "tt9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "355f218b5ea40614", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.9605947323337506e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "3.00000", "metric_symbol": "s", "metric_value": 3.0, "n": 7, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "triintri", "side": 3.0, "solution_path": "data/solutions/generated/triintri_007_355f218b5ea40614.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/355f218b5ea40614.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@8", "density": 0.8888888888888892, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@8", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "tt9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "tt9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "43e3200cf32d77a9", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.9605947323337506e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "3.00000", "metric_symbol": "s", "metric_value": 3.0, "n": 8, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "triintri", "side": 3.0, "solution_path": "data/solutions/generated/triintri_008_43e3200cf32d77a9.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/43e3200cf32d77a9.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@9", "density": 1.0000000000000004, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@9", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "tt9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "tt9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4242c6a66848e1ea", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.9605947323337516e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "3.00000", "metric_symbol": "s", "metric_value": 3.0, "n": 9, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 3.0, "setup": "triintri", "side": 3.0, "solution_path": "data/solutions/generated/triintri_009_4242c6a66848e1ea.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4242c6a66848e1ea.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@10", "density": 0.8162770652591925, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt10.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "7/2 = 3.500", "friedman_metric_symbol": "s", "friedman_metric_value": 3.5, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@10", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "7/2 = 3.500", "metric_symbol": "s", "metric_value": 3.5, "reference_text": "s = 7/2 = 3.500 | Found by Erich Friedman | in 1997.", "source_image": "tt10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt10.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 7/2 = 3.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "bcf7fa93627e2e22", "improvement": -0.00010604617003595607, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -8.918936789115156e-08, "max_pair_overlap_depth": -1.2701382657187565e-05, "metric_expression": "3.50011", "metric_symbol": "s", "metric_value": 3.500106046170036, "n": 10, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.5, "setup": "triintri", "side": 3.500106046170036, "solution_path": "data/solutions/generated/triintri_010_bcf7fa93627e2e22.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/bcf7fa93627e2e22.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@11", "density": 0.792568600317471, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt11.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "9/4 + 9\u221a21/28 = 3.722+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.722, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@11", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "9/4 + 9\u221a21/28 = 3.722+", "metric_symbol": "s", "metric_value": 3.722, "reference_text": "s = 9/4 + 9\u221a21/28 = 3.722+ | Found by Erich Friedman | in 1997.", "source_image": "tt11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt11.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 9/4 + 9\u221a21/28 = 3.722+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "8c37202a929dc807", "improvement": -0.0034428993959463305, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7247990718051686e-06, "max_pair_overlap_depth": -2.4227159337045223e-05, "metric_expression": "3.72544", "metric_symbol": "s", "metric_value": 3.7254428993959463, "n": 11, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.722, "setup": "triintri", "side": 3.7254428993959463, "solution_path": "data/solutions/generated/triintri_11_8c37202a929dc807.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/8c37202a929dc807.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@12", "density": 0.7960259433336867, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2007", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt12.gif", "friedman_metric_expression": "2 + 2cos(\u03c0/9) = 3.879+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.879, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@12", "credit": "David W. Cantrell", "date": "July 2007", "family_updated": "", "metric_expression": "2 + 2cos(\u03c0/9) = 3.879+", "metric_symbol": "s", "metric_value": 3.879, "reference_text": "s = 2 + 2cos(\u03c0/9) = 3.879+ | Found by David W. Cantrell | in July 2007.", "source_image": "tt12.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt12.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 + 2cos(\u03c0/9) = 3.879+ | Found by David W. Cantrell | in July 2007.", "friedman_source_image": "tt12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "7642b05605706588", "improvement": -0.0036389945275763225, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -9.358547413541538e-08, "max_pair_overlap_depth": -4.590975868801195e-06, "metric_expression": "3.88264", "metric_symbol": "s", "metric_value": 3.8826389945275763, "n": 12, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.879, "setup": "triintri", "side": 3.8826389945275763, "solution_path": "data/solutions/generated/triintri_12_7642b05605706588.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T07:13:53+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/7642b05605706588.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@13", "density": 0.8146483132163195, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt13.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3.992+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.992, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@13", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "3.992+", "metric_symbol": "s", "metric_value": 3.992, "reference_text": "s = 3.992+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt13.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3.992+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "afe3e3dc6de1bc36", "improvement": -0.0027223080586638027, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.48838412272174e-06, "max_pair_overlap_depth": -3.9197094850562664e-05, "metric_expression": "3.99472", "metric_symbol": "s", "metric_value": 3.994722308058664, "n": 13, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 3.992, "setup": "triintri", "side": 3.994722308058664, "solution_path": "data/solutions/generated/triintri_013_afe3e3dc6de1bc36.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/afe3e3dc6de1bc36.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@14", "density": 0.875, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@14", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial.", "source_image": "tt16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4 | Trivial.", "friedman_source_image": "tt16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "8676384ad0301449", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.775557561562891e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "4.00000", "metric_symbol": "s", "metric_value": 4.0, "n": 14, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "triintri", "side": 4.0, "solution_path": "data/solutions/generated/triintri_014_8676384ad0301449.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/8676384ad0301449.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@15", "density": 0.9375, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@15", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial.", "source_image": "tt16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4 | Trivial.", "friedman_source_image": "tt16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "31db3ec76cc0421f", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.775557561562891e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "4.00000", "metric_symbol": "s", "metric_value": 4.0, "n": 15, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "triintri", "side": 4.0, "solution_path": "data/solutions/generated/triintri_015_31db3ec76cc0421f.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/31db3ec76cc0421f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@16", "density": 1.0, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@16", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial.", "source_image": "tt16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4 | Trivial.", "friedman_source_image": "tt16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "2ffa329b602032d6", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 2.775557561562891e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "4.00000", "metric_symbol": "s", "metric_value": 4.0, "n": 16, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 4.0, "setup": "triintri", "side": 4.0, "solution_path": "data/solutions/generated/triintri_016_2ffa329b602032d6.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/2ffa329b602032d6.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@17", "density": 0.8524249732795084, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt17.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4.465+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.465, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@17", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "4.465+", "metric_symbol": "s", "metric_value": 4.465, "reference_text": "s = 4.465+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt17.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4.465+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "ad8aac0e4c06f7d0", "improvement": -0.0007702721912288268, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8241570152012174e-06, "max_pair_overlap_depth": -7.389272764501964e-06, "metric_expression": "4.46577", "metric_symbol": "s", "metric_value": 4.465770272191229, "n": 17, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.465, "setup": "triintri", "side": 4.465770272191229, "solution_path": "data/solutions/generated/triintri_017_ad8aac0e4c06f7d0.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/ad8aac0e4c06f7d0.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@18", "density": 0.8888203671574908, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt18.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "9/2 = 4.500", "friedman_metric_symbol": "s", "friedman_metric_value": 4.5, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@18", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "9/2 = 4.500", "metric_symbol": "s", "metric_value": 4.5, "reference_text": "s = 9/2 = 4.500 | Found by Erich Friedman | in 1997.", "source_image": "tt18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt18.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 9/2 = 4.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "b4cfb14015aca75a", "improvement": -0.00017345566104154386, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.575746594607548e-06, "max_pair_overlap_depth": -5.0550449839370515e-06, "metric_expression": "4.50017", "metric_symbol": "s", "metric_value": 4.5001734556610415, "n": 18, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.5, "setup": "triintri", "side": 4.5001734556610415, "solution_path": "data/solutions/generated/triintri_018_b4cfb14015aca75a.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/b4cfb14015aca75a.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@19", "density": 0.872322152073677, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt19.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "14/3 = 4.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.666, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@19", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "14/3 = 4.666+", "metric_symbol": "s", "metric_value": 4.666, "reference_text": "s = 14/3 = 4.666+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt19.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 14/3 = 4.666+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "bf4e89c9b4dd8ae2", "improvement": -0.0010058992763006813, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.331642222161759e-07, "max_pair_overlap_depth": -9.690072424628493e-06, "metric_expression": "4.66701", "metric_symbol": "s", "metric_value": 4.667005899276301, "n": 19, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.666, "setup": "triintri", "side": 4.667005899276301, "solution_path": "data/solutions/generated/triintri_019_bf4e89c9b4dd8ae2.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/bf4e89c9b4dd8ae2.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@20", "density": 0.845304928058783, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt20.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "33/8 + 9\u221a21/56 = 4.861+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.861, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@20", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "33/8 + 9\u221a21/56 = 4.861+", "metric_symbol": "s", "metric_value": 4.861, "reference_text": "s = 33/8 + 9\u221a21/56 = 4.861+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt20.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt20.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 33/8 + 9\u221a21/56 = 4.861+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4dff2c76e1273753", "improvement": -0.00316498806837906, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.619234998604407e-06, "max_pair_overlap_depth": -7.278337082583519e-06, "metric_expression": "4.86416", "metric_symbol": "s", "metric_value": 4.864164988068379, "n": 20, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.861, "setup": "triintri", "side": 4.864164988068379, "solution_path": "data/solutions/generated/triintri_020_4dff2c76e1273753.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/4dff2c76e1273753.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@21", "density": 0.8650098201747314, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt21.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4.923+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.923, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@21", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "4.923+", "metric_symbol": "s", "metric_value": 4.923, "reference_text": "s = 4.923+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt21.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt21.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4.923+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4dc62e9e9421c0aa", "improvement": -0.004187943781052361, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.4013824985741632e-06, "max_pair_overlap_depth": -4.2136074186321926e-06, "metric_expression": "4.92719", "metric_symbol": "s", "metric_value": 4.927187943781052, "n": 21, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.923, "setup": "triintri", "side": 4.927187943781052, "solution_path": "data/solutions/generated/triintri_021_4dc62e9e9421c0aa.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/4dc62e9e9421c0aa.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@22", "density": 0.8799245341930954, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt22.gif", "friedman_metric_expression": "4.996+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.996, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@22", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "4.996+", "metric_symbol": "s", "metric_value": 4.996, "reference_text": "s = 4.996+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt22.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt22.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4.996+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "2122d833c6fcab70", "improvement": -0.004214405286988132, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -5.380108740330726e-08, "max_pair_overlap_depth": -8.335028905959163e-07, "metric_expression": "5.00021", "metric_symbol": "s", "metric_value": 5.000214405286989, "n": 22, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 4.996, "setup": "triintri", "side": 5.000214405286989, "solution_path": "data/solutions/generated/triintri_022_2122d833c6fcab70.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/2122d833c6fcab70.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@23", "density": 0.9200000000000002, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@23", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "tt25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "tt25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4f1bdd5d6c07f589", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "5.00000", "metric_symbol": "s", "metric_value": 5.0, "n": 23, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "triintri", "side": 5.0, "solution_path": "data/solutions/generated/triintri_023_4f1bdd5d6c07f589.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4f1bdd5d6c07f589.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@24", "density": 0.9600000000000002, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@24", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "tt25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "tt25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "64d77fbf13ae0be7", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "5.00000", "metric_symbol": "s", "metric_value": 5.0, "n": 24, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "triintri", "side": 5.0, "solution_path": "data/solutions/generated/triintri_024_64d77fbf13ae0be7.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/64d77fbf13ae0be7.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@25", "density": 1.0000000000000002, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "friedman_metric_expression": "5", "friedman_metric_symbol": "s", "friedman_metric_value": 5.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@25", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "5", "metric_symbol": "s", "metric_value": 5.0, "reference_text": "s = 5 | Trivial.", "source_image": "tt25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt25.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5 | Trivial.", "friedman_source_image": "tt25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "9c4806b8b37c5d37", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 7.105427357601002e-16, "max_pair_overlap_depth": 4.440892098500626e-16, "metric_expression": "5.00000", "metric_symbol": "s", "metric_value": 5.0, "n": 25, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 5.0, "setup": "triintri", "side": 5.0, "solution_path": "data/solutions/generated/triintri_025_9c4806b8b37c5d37.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/9c4806b8b37c5d37.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@26", "density": 0.8884053356293824, "friedman_analytical_or_proved": false, "friedman_credit": "Joah Frenzley", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt26.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "5.406+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.406, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@26", "credit": "Joah Frenzley", "date": "April 2026", "family_updated": "", "metric_expression": "5.406+", "metric_symbol": "s", "metric_value": 5.406, "reference_text": "s = 5.406+ | Found by Joah Frenzley | in April 2026.", "source_image": "tt26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt26.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5.406+ | Found by Joah Frenzley | in April 2026.", "friedman_source_image": "tt26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "0af5a3765ecc4828", "improvement": -0.0037985715923722907, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.296466577480461e-06, "max_pair_overlap_depth": -2.0400569136769242e-05, "metric_expression": "5.40980", "metric_symbol": "s", "metric_value": 5.409798571592372, "n": 26, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.406, "setup": "triintri", "side": 5.409798571592372, "solution_path": "data/solutions/generated/triintri_026_0af5a3765ecc4828.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Joah Frenzley", "svg_path": "data/svg/0af5a3765ecc4828.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@27", "density": 0.8924996192164427, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "friedman_metric_expression": "11/2 = 5.500", "friedman_metric_symbol": "s", "friedman_metric_value": 5.5, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@27", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "11/2 = 5.500", "metric_symbol": "s", "metric_value": 5.5, "reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "source_image": "tt28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "f1c2be61e829f1e2", "improvement": -0.00019215549305418023, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.8365186496628e-06, "max_pair_overlap_depth": -1.8064988390742442e-06, "metric_expression": "5.50019", "metric_symbol": "s", "metric_value": 5.500192155493054, "n": 27, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.5, "setup": "triintri", "side": 5.500192155493054, "solution_path": "data/solutions/generated/triintri_027_f1c2be61e829f1e2.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/f1c2be61e829f1e2.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@28", "density": 0.9255631973649003, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "friedman_metric_expression": "11/2 = 5.500", "friedman_metric_symbol": "s", "friedman_metric_value": 5.5, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@28", "credit": "Erich Friedman", "date": "1997", "family_updated": "", "metric_expression": "11/2 = 5.500", "metric_symbol": "s", "metric_value": 5.5, "reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "source_image": "tt28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "9562d7d6e075a347", "improvement": -0.00016827626559567932, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.296352348617567e-06, "max_pair_overlap_depth": -7.147352432124165e-08, "metric_expression": "5.50017", "metric_symbol": "s", "metric_value": 5.500168276265596, "n": 28, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.5, "setup": "triintri", "side": 5.500168276265596, "solution_path": "data/solutions/generated/triintri_028_9562d7d6e075a347.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/9562d7d6e075a347.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@29", "density": 0.9023512759650716, "friedman_analytical_or_proved": true, "friedman_credit": "David W. Cantrell", "friedman_date": "July 2007", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt29.gif", "friedman_metric_expression": "17/3 = 5.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.666, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@29", "credit": "David W. Cantrell", "date": "July 2007", "family_updated": "", "metric_expression": "17/3 = 5.666+", "metric_symbol": "s", "metric_value": 5.666, "reference_text": "s = 17/3 = 5.666+ | Found by David W. Cantrell | in July 2007.", "source_image": "tt29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt29.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 17/3 = 5.666+ | Found by David W. Cantrell | in July 2007.", "friedman_source_image": "tt29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "38c7a009f9059275", "improvement": -0.0030616587354490576, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9275914294015247e-06, "max_pair_overlap_depth": -5.088073942505211e-06, "metric_expression": "5.66906", "metric_symbol": "s", "metric_value": 5.669061658735449, "n": 29, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.666, "setup": "triintri", "side": 5.669061658735449, "solution_path": "data/solutions/generated/triintri_29_38c7a009f9059275.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T13:47:12+00:00", "submitted_by": "David W. Cantrell", "svg_path": "data/svg/38c7a009f9059275.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@30", "density": 0.9060985977127352, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt30.gif", "friedman_metric_expression": "23/4 = 5.75", "friedman_metric_symbol": "s", "friedman_metric_value": 5.75, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@30", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "", "metric_expression": "23/4 = 5.75", "metric_symbol": "s", "metric_value": 5.75, "reference_text": "s = 23/4 = 5.75 | Found by Maurizio Morandi | in May 2008.", "source_image": "tt30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt30.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 23/4 = 5.75 | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "149a91e0248df04e", "improvement": -0.004040286099913359, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.2882428961650933e-06, "max_pair_overlap_depth": -6.414942533283785e-06, "metric_expression": "5.75404", "metric_symbol": "s", "metric_value": 5.754040286099913, "n": 30, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.75, "setup": "triintri", "side": 5.754040286099913, "solution_path": "data/solutions/generated/triintri_30_149a91e0248df04e.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "Maurizio Morandi", "svg_path": "data/svg/149a91e0248df04e.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@31", "density": 0.8852772627280384, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt31.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "5.912+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.912, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "", "metric_expression": "5.912+", "metric_symbol": "s", "metric_value": 5.912, "reference_text": "s = 5.912+ | Found by Emerson Connelly | in April 2026.", "source_image": "tt31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt31.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5.912+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "tt31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "c729ec2b23a32942", "improvement": -0.005539858919105356, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -1.9513696602846134e-06, "max_pair_overlap_depth": -3.7790737070775293e-06, "metric_expression": "5.91754", "metric_symbol": "s", "metric_value": 5.917539858919105, "n": 31, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.912, "setup": "triintri", "side": 5.917539858919105, "solution_path": "data/solutions/generated/triintri_031_c729ec2b23a32942.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:51:36+00:00", "submitted_by": "Emerson Connelly", "svg_path": "data/svg/c729ec2b23a32942.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@32", "density": 0.9021138463561595, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt32.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "5.951+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.951, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@32", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "5.951+", "metric_symbol": "s", "metric_value": 5.951, "reference_text": "s = 5.951+ | Found by Ian Watson | in April 2026.", "source_image": "tt32.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt32.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5.951+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "tt32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "fa640909a0850f97", "improvement": -0.004857726462081935, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7950535548187937e-06, "max_pair_overlap_depth": -4.900081183589755e-06, "metric_expression": "5.95586", "metric_symbol": "s", "metric_value": 5.955857726462082, "n": 32, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 5.951, "setup": "triintri", "side": 5.955857726462082, "solution_path": "data/solutions/generated/triintri_032_fa640909a0850f97.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:29:44+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/fa640909a0850f97.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@33", "density": 0.9166666666666669, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@33", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "34a5032aebc12aff", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.921189464667501e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_symbol": "s", "metric_value": 6.0, "n": 33, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_033_34a5032aebc12aff.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/34a5032aebc12aff.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@34", "density": 0.9444444444444448, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@34", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "a2ccd0a19cfa1b75", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.921189464667501e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_symbol": "s", "metric_value": 6.0, "n": 34, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_034_a2ccd0a19cfa1b75.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/a2ccd0a19cfa1b75.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@35", "density": 0.9722222222222225, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@35", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "6325f936448cdf94", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 5.921189464667503e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_symbol": "s", "metric_value": 6.0, "n": 35, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_035_6325f936448cdf94.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/6325f936448cdf94.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@36", "density": 1.0000000000000004, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@36", "credit": "Trivial", "date": "", "family_updated": "", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "4b90836bf07f8a65", "improvement": 0.0, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": 6.661338147750939e-16, "max_pair_overlap_depth": 6.661338147750939e-16, "metric_expression": "6.00000", "metric_symbol": "s", "metric_value": 6.0, "n": 36, "notes": "Exact triangular subdivision coordinate JSON; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.0, "solution_path": "data/solutions/generated/triintri_036_4b90836bf07f8a65.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T15:42:49+00:00", "submitted_by": "Trivial", "svg_path": "data/svg/4b90836bf07f8a65.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@37", "density": 0.9223731676430976, "friedman_analytical_or_proved": true, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt37.gif", "friedman_metric_expression": "19/3 = 6.333+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.333, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@37", "credit": "Ian Watson", "date": "April 2026", "family_updated": "", "metric_expression": "19/3 = 6.333+", "metric_symbol": "s", "metric_value": 6.333, "reference_text": "s = 19/3 = 6.333+ | Found by Ian Watson | in April 2026.", "source_image": "tt37.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt37.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 19/3 = 6.333+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "tt37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "a824891b660555f5", "improvement": -0.0005547879590457327, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.1058084789851627e-06, "max_pair_overlap_depth": -1.0503324543442716e-06, "metric_expression": "6.33355", "metric_symbol": "s", "metric_value": 6.333554787959046, "n": 37, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.333, "setup": "triintri", "side": 6.333554787959046, "solution_path": "data/solutions/generated/triintri_037_a824891b660555f5.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Ian Watson", "svg_path": "data/svg/a824891b660555f5.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@38", "density": 0.9184187453437659, "friedman_analytical_or_proved": false, "friedman_credit": "Nathan Roll", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt38.gif", "friedman_metric_expression": "6.43454+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.43454, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@38", "credit": "Nathan Roll", "date": "April 2026", "family_updated": "", "metric_expression": "6.43454+", "metric_symbol": "s", "metric_value": 6.43454, "reference_text": "s = 6.43454+ | Found by Nathan Roll | in April 2026.", "source_image": "tt38.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt38.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6.43454+ | Found by Nathan Roll | in April 2026.", "friedman_source_image": "tt38.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "b1563d97c893e28d", "improvement": 0.002163914058151306, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.8714774987883973e-07, "max_pair_overlap_depth": -1.6883301962034025e-06, "metric_expression": "6.43238", "metric_symbol": "s", "metric_value": 6.432376085941849, "n": 38, "notes": "Verified coordinate layout matches or beats the Friedman metric; credited to the Friedman-listed finder.", "record_type": "verified", "reference_side": 6.43454, "setup": "triintri", "side": 6.432376085941849, "solution_path": "data/solutions/record_hunt/triintri_38_b1563d97c893e28d.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T13:33:44+00:00", "submitted_by": "Nathan Roll", "svg_path": "data/svg/b1563d97c893e28d.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@39", "density": 0.922995893638428, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt40.gif", "friedman_metric_expression": "13/2 = 6.500", "friedman_metric_symbol": "s", "friedman_metric_value": 6.5, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@39", "credit": "Erich Friedman", "date": "April 2026", "family_updated": "", "metric_expression": "13/2 = 6.500", "metric_symbol": "s", "metric_value": 6.5, "reference_text": "s = 13/2 = 6.500 | Found by Erich Friedman | in April 2026.", "source_image": "tt40.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt40.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 13/2 = 6.500 | Found by Erich Friedman | in April 2026.", "friedman_source_image": "tt40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "5ff069bd30790b7f", "improvement": -0.0002853099319573005, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -2.7141701112233934e-06, "max_pair_overlap_depth": -2.524988351737534e-07, "metric_expression": "6.50029", "metric_symbol": "s", "metric_value": 6.500285309931957, "n": 39, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.5, "setup": "triintri", "side": 6.500285309931957, "solution_path": "data/solutions/generated/triintri_039_5ff069bd30790b7f.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/5ff069bd30790b7f.svg", "tolerance": 1e-08, "verified": true}
+{"case": "triintri@40", "density": 0.9466291268019702, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt40.gif", "friedman_metric_expression": "13/2 = 6.500", "friedman_metric_symbol": "s", "friedman_metric_value": 6.5, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@40", "credit": "Erich Friedman", "date": "April 2026", "family_updated": "", "metric_expression": "13/2 = 6.500", "metric_symbol": "s", "metric_value": 6.5, "reference_text": "s = 13/2 = 6.500 | Found by Erich Friedman | in April 2026.", "source_image": "tt40.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt40.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 13/2 = 6.500 | Found by Erich Friedman | in April 2026.", "friedman_source_image": "tt40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "7890a1da940c9143", "improvement": -0.00039973752172350885, "leaderboard_consolidated_at": "2026-05-01T15:41:45+00:00", "leaderboard_current_best": true, "max_boundary_excess": -3.4343626129297504e-07, "max_pair_overlap_depth": -3.1212060811469655e-06, "metric_expression": "6.50040", "metric_symbol": "s", "metric_value": 6.5003997375217235, "n": 40, "notes": "Best available verified coordinate reconstruction, but its evaluator metric is worse than the Friedman reference.", "record_type": "verified", "reference_side": 6.5, "setup": "triintri", "side": 6.5003997375217235, "solution_path": "data/solutions/generated/triintri_040_7890a1da940c9143.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "Erich Friedman", "svg_path": "data/svg/7890a1da940c9143.svg", "tolerance": 1e-08, "verified": true}