diff --git "a/eval.jsonl" "b/eval.jsonl"
new file mode 100644--- /dev/null
+++ "b/eval.jsonl"
@@ -0,0 +1,2241 @@
+{"id": "po-ap_sum-t0-eval-000170", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -93. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -93. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -3", "ref": "-8, -3", "family": "po_ap_sum", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00907, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 6, "B": 10, "S": -93}, "seed": 141063620200000453, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 5, "valid_answers": ["-8, -3", "-3, -5", "2, -7", "7, -9"]}, "cell": "ap_sum/0", "stage": "B"}
+{"id": "po-ap_sum-t0-eval-000176", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 74. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 74. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 9", "ref": "5, 9", "family": "po_ap_sum", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004535, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 4, "B": 10, "S": 74}, "seed": 141063620200000474, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 8, "valid_answers": ["5, 9", "8, 7"]}, "cell": "ap_sum/0", "stage": "B"}
+{"id": "po-ap_sum-t0-eval-000046", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 26. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 26. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 9", "ref": "-7, 9", "family": "po_ap_sum", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.013605, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 4, "B": 10, "S": 26}, "seed": 141063620200000066, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 3, "valid_answers": ["-7, 9", "-4, 7", "-1, 5", "2, 3", "5, 1", "8, -1"]}, "cell": "ap_sum/0", "stage": "B"}
+{"id": "po-ap_sum-t0-eval-000074", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 75. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 75. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 10", "ref": "-5, 10", "family": "po_ap_sum", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.018141, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 5, "B": 10, "S": 75}, "seed": 141063620200000113, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 5, "valid_answers": ["-5, 10", "-3, 9", "-1, 8", "1, 7", "3, 6", "5, 5", "7, 4", "9, 3"]}, "cell": "ap_sum/0", "stage": "B"}
+{"id": "po-ap_sum-t0-eval-000091", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to -90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to -90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -4", "ref": "-10, -4", "family": "po_ap_sum", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.015873, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 5, "B": 10, "S": -90}, "seed": 141063620200000157, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 6, "valid_answers": ["-10, -4", "-8, -5", "-6, -6", "-4, -7", "-2, -8", "0, -9", "2, -10"]}, "cell": "ap_sum/0", "stage": "B"}
+{"id": "po-ap_sum-t0-eval-000090", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 153. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 153. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 9", "ref": "3, 9", "family": "po_ap_sum", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004535, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 6, "B": 10, "S": 153}, "seed": 141063620200000156, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 8, "valid_answers": ["3, 9", "8, 7"]}, "cell": "ap_sum/0", "stage": "B"}
+{"id": "po-ap_sum-t0-eval-000081", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 82. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 82. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 9", "ref": "7, 9", "family": "po_ap_sum", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004535, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 4, "B": 10, "S": 82}, "seed": 141063620200000131, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 9, "valid_answers": ["7, 9", "10, 7"]}, "cell": "ap_sum/0", "stage": "B"}
+{"id": "po-ap_sum-t0-eval-000175", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to -110. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to -110. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -6", "ref": "-10, -6", "family": "po_ap_sum", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.011338, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 5, "B": 10, "S": -110}, "seed": 141063620200000467, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 8, "valid_answers": ["-10, -6", "-8, -7", "-6, -8", "-4, -9", "-2, -10"]}, "cell": "ap_sum/0", "stage": "B"}
+{"id": "po-ap_sum-t0-eval-000076", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -57. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -57. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -1", "ref": "-7, -1", "family": "po_ap_sum", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00907, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 6, "B": 10, "S": -57}, "seed": 141063620200000119, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["-7, -1", "-2, -3", "3, -5", "8, -7"]}, "cell": "ap_sum/0", "stage": "B"}
+{"id": "po-ap_sum-t0-eval-000189", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -132. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -132. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -6", "ref": "-7, -6", "family": "po_ap_sum", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006803, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 6, "B": 10, "S": -132}, "seed": 141063620200000560, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 7, "valid_answers": ["-7, -6", "-2, -8", "3, -10"]}, "cell": "ap_sum/0", "stage": "B"}
+{"id": "po-ap_sum-t0-eval-000079", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 147. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 147. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 9", "ref": "2, 9", "family": "po_ap_sum", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004535, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 6, "B": 10, "S": 147}, "seed": 141063620200000122, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 7, "valid_answers": ["2, 9", "7, 7"]}, "cell": "ap_sum/0", "stage": "B"}
+{"id": "po-ap_sum-t0-eval-000112", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -108. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -108. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -4", "ref": "-8, -4", "family": "po_ap_sum", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00907, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 6, "B": 10, "S": -108}, "seed": 141063620200000211, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 6, "valid_answers": ["-8, -4", "-3, -6", "2, -8", "7, -10"]}, "cell": "ap_sum/0", "stage": "B"}
+{"id": "po-ap_sum-t1-eval-000288", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -33. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -33. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-18, 5", "ref": "-18, 5", "family": "po_ap_sum", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.004759, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 6, "B": 20, "S": -33}, "seed": 141063620201000371, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 3, "valid_answers": ["-18, 5", "-13, 3", "-8, 1", "-3, -1", "2, -3", "7, -5", "12, -7", "17, -9"]}, "cell": "ap_sum/1", "stage": "C"}
+{"id": "po-ap_sum-t1-eval-000593", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -39. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -39. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-19, 5", "ref": "-19, 5", "family": "po_ap_sum", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.004759, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 6, "B": 20, "S": -39}, "seed": 141063620201000950, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 3, "valid_answers": ["-19, 5", "-14, 3", "-9, 1", "-4, -1", "1, -3", "6, -5", "11, -7", "16, -9"]}, "cell": "ap_sum/1", "stage": "C"}
+{"id": "po-ap_sum-t1-eval-000178", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 7 terms equal to 245. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 7 terms equal to 245. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-19, 18", "ref": "-19, 18", "family": "po_ap_sum", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.008328, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 7, "B": 20, "S": 245}, "seed": 141063620201000203, "version": "po-v0", "tier": 1, "n_raw": 14, "small": 9, "valid_answers": ["-19, 18", "-16, 17", "-13, 16", "-10, 15", "-7, 14", "-4, 13", "-1, 12", "2, 11", "5, 10", "8, 9", "11, 8", "14, 7", "17, 6", "20, 5"]}, "cell": "ap_sum/1", "stage": "C"}
+{"id": "po-ap_sum-t1-eval-000353", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 9 terms equal to 792. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 9 terms equal to 792. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8, 20", "ref": "8, 20", "family": "po_ap_sum", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00238, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 9, "B": 20, "S": 792}, "seed": 141063620201000468, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 18, "valid_answers": ["8, 20", "12, 19", "16, 18", "20, 17"]}, "cell": "ap_sum/1", "stage": "C"}
+{"id": "po-ap_sum-t1-eval-000978", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 8 terms equal to 616. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 8 terms equal to 616. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 20", "ref": "7, 20", "family": "po_ap_sum", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00119, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 8, "B": 20, "S": 616}, "seed": 141063620201003996, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 18, "valid_answers": ["7, 20", "14, 18"]}, "cell": "ap_sum/1", "stage": "C"}
+{"id": "po-ap_sum-t1-eval-000138", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 7 terms equal to 469. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 7 terms equal to 469. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 20", "ref": "7, 20", "family": "po_ap_sum", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002974, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 7, "B": 20, "S": 469}, "seed": 141063620201000154, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 17, "valid_answers": ["7, 20", "10, 19", "13, 18", "16, 17", "19, 16"]}, "cell": "ap_sum/1", "stage": "C"}
+{"id": "po-ap_sum-t1-eval-000750", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 240. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 240. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 20", "ref": "-10, 20", "family": "po_ap_sum", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.004164, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 6, "B": 20, "S": 240}, "seed": 141063620201001533, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 12, "valid_answers": ["-10, 20", "-5, 18", "0, 16", "5, 14", "10, 12", "15, 10", "20, 8"]}, "cell": "ap_sum/1", "stage": "C"}
+{"id": "po-ap_sum-t1-eval-001031", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 7 terms equal to 511. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 7 terms equal to 511. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "13, 20", "ref": "13, 20", "family": "po_ap_sum", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001785, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 7, "B": 20, "S": 511}, "seed": 141063620201009064, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 19, "valid_answers": ["13, 20", "16, 19", "19, 18"]}, "cell": "ap_sum/1", "stage": "C"}
+{"id": "po-ap_sum-t1-eval-000319", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 78. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 78. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-17, 12", "ref": "-17, 12", "family": "po_ap_sum", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.004759, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 6, "B": 20, "S": 78}, "seed": 141063620201000411, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 4, "valid_answers": ["-17, 12", "-12, 10", "-7, 8", "-2, 6", "3, 4", "8, 2", "13, 0", "18, -2"]}, "cell": "ap_sum/1", "stage": "C"}
+{"id": "po-ap_sum-t1-eval-000646", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 9 terms equal to 162. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 9 terms equal to 162. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-18, 9", "ref": "-18, 9", "family": "po_ap_sum", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.005949, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 9, "B": 20, "S": 162}, "seed": 141063620201001112, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 4, "valid_answers": ["-18, 9", "-14, 8", "-10, 7", "-6, 6", "-2, 5", "2, 4", "6, 3", "10, 2", "14, 1", "18, 0"]}, "cell": "ap_sum/1", "stage": "C"}
+{"id": "po-ap_sum-t1-eval-000912", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 7 terms equal to -539. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 7 terms equal to -539. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-20, -19", "ref": "-20, -19", "family": "po_ap_sum", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00119, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 7, "B": 20, "S": -539}, "seed": 141063620201002720, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 20, "valid_answers": ["-20, -19", "-17, -20"]}, "cell": "ap_sum/1", "stage": "C"}
+{"id": "po-ap_sum-t1-eval-000740", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 8 terms equal to 548. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -20 <= a, d <= 20 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 8 terms equal to 548. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 19", "ref": "2, 19", "family": "po_ap_sum", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001785, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 8, "B": 20, "S": 548}, "seed": 141063620201001488, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 16, "valid_answers": ["2, 19", "9, 17", "16, 15"]}, "cell": "ap_sum/1", "stage": "C"}
+{"id": "po-ap_sum-t2-eval-002230", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 12 terms equal to 1494. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 12 terms equal to 1494. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-24, 27", "ref": "-24, 27", "family": "po_ap_sum", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.001344, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 12, "B": 30, "S": 1494}, "seed": 141063620202006109, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 20, "valid_answers": ["-24, 27", "-13, 25", "-2, 23", "9, 21", "20, 19"]}, "cell": "ap_sum/2", "stage": "B"}
+{"id": "po-ap_sum-t2-eval-000349", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 8 terms equal to 784. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 8 terms equal to 784. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 30", "ref": "-7, 30", "family": "po_ap_sum", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.001612, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 8, "B": 30, "S": 784}, "seed": 141063620202000375, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 22, "valid_answers": ["-7, 30", "0, 28", "7, 26", "14, 24", "21, 22", "28, 20"]}, "cell": "ap_sum/2", "stage": "B"}
+{"id": "po-ap_sum-t2-eval-000778", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 8 terms equal to 868. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 8 terms equal to 868. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 29", "ref": "7, 29", "family": "po_ap_sum", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001075, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 8, "B": 30, "S": 868}, "seed": 141063620202000959, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 25, "valid_answers": ["7, 29", "14, 27", "21, 25", "28, 23"]}, "cell": "ap_sum/2", "stage": "B"}
+{"id": "po-ap_sum-t2-eval-000225", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 10 terms equal to -600. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 10 terms equal to -600. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-24, -8", "ref": "-24, -8", "family": "po_ap_sum", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.001881, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 10, "B": 30, "S": -600}, "seed": 141063620202000240, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 12, "valid_answers": ["-24, -8", "-15, -10", "-6, -12", "3, -14", "12, -16", "21, -18", "30, -20"]}, "cell": "ap_sum/2", "stage": "B"}
+{"id": "po-ap_sum-t2-eval-001783", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 8 terms equal to 276. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 8 terms equal to 276. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-25, 17", "ref": "-25, 17", "family": "po_ap_sum", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.00215, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 8, "B": 30, "S": 276}, "seed": 141063620202003391, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 9, "valid_answers": ["-25, 17", "-18, 15", "-11, 13", "-4, 11", "3, 9", "10, 7", "17, 5", "24, 3"]}, "cell": "ap_sum/2", "stage": "B"}
+{"id": "po-ap_sum-t2-eval-000154", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 10 terms equal to -1455. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 10 terms equal to -1455. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-24, -27", "ref": "-24, -27", "family": "po_ap_sum", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000537, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 10, "B": 30, "S": -1455}, "seed": 141063620202000163, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 27, "valid_answers": ["-24, -27", "-15, -29"]}, "cell": "ap_sum/2", "stage": "B"}
+{"id": "po-ap_sum-t2-eval-002219", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 12 terms equal to 1806. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 12 terms equal to 1806. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 29", "ref": "-9, 29", "family": "po_ap_sum", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001075, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 12, "B": 30, "S": 1806}, "seed": 141063620202005990, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 24, "valid_answers": ["-9, 29", "2, 27", "13, 25", "24, 23"]}, "cell": "ap_sum/2", "stage": "B"}
+{"id": "po-ap_sum-t2-eval-002100", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 12 terms equal to 1260. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 12 terms equal to 1260. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-27, 24", "ref": "-27, 24", "family": "po_ap_sum", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.001612, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 12, "B": 30, "S": 1260}, "seed": 141063620202005070, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 17, "valid_answers": ["-27, 24", "-16, 22", "-5, 20", "6, 18", "17, 16", "28, 14"]}, "cell": "ap_sum/2", "stage": "B"}
+{"id": "po-ap_sum-t2-eval-001673", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 12 terms equal to 1488. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 12 terms equal to 1488. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-30, 28", "ref": "-30, 28", "family": "po_ap_sum", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.001612, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 12, "B": 30, "S": 1488}, "seed": 141063620202002994, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 20, "valid_answers": ["-30, 28", "-19, 26", "-8, 24", "3, 22", "14, 20", "25, 18"]}, "cell": "ap_sum/2", "stage": "B"}
+{"id": "po-ap_sum-t2-eval-000839", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 9 terms equal to 54. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 9 terms equal to 54. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-30, 9", "ref": "-30, 9", "family": "po_ap_sum", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.0043, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 9, "B": 30, "S": 54}, "seed": 141063620202001051, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 2, "valid_answers": ["-30, 9", "-26, 8", "-22, 7", "-18, 6", "-14, 5", "-10, 4", "-6, 3", "-2, 2", "2, 1", "6, 0", "10, -1", "14, -2", "18, -3", "22, -4", "26, -5", "30, -6"]}, "cell": "ap_sum/2", "stage": "B"}
+{"id": "po-ap_sum-t2-eval-001184", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 10 terms equal to 1420. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 10 terms equal to 1420. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 30", "ref": "7, 30", "family": "po_ap_sum", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000806, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 10, "B": 30, "S": 1420}, "seed": 141063620202001693, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 26, "valid_answers": ["7, 30", "16, 28", "25, 26"]}, "cell": "ap_sum/2", "stage": "B"}
+{"id": "po-ap_sum-t2-eval-001378", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 7 terms equal to -294. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -30 <= a, d <= 30 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 7 terms equal to -294. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-30, -4", "ref": "-30, -4", "family": "po_ap_sum", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.005644, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 7, "B": 30, "S": -294}, "seed": 141063620202002168, "version": "po-v0", "tier": 2, "n_raw": 21, "small": 11, "valid_answers": ["-30, -4", "-27, -5", "-24, -6", "-21, -7", "-18, -8", "-15, -9", "-12, -10", "-9, -11", "-6, -12", "-3, -13", "0, -14", "3, -15", "6, -16", "9, -17", "12, -18", "15, -19", "18, -20", "21, -21", "24, -22", "27, -23", "30, -24"]}, "cell": "ap_sum/2", "stage": "B"}
+{"id": "po-ap_sum-t3-eval-000305", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 10 terms equal to 2350. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 10 terms equal to 2350. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 50", "ref": "10, 50", "family": "po_ap_sum", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.00049, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 10, "B": 50, "S": 2350}, "seed": 141063620203000313, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 44, "valid_answers": ["10, 50", "19, 48", "28, 46", "37, 44", "46, 42"]}, "cell": "ap_sum/3", "stage": "B"}
+{"id": "po-ap_sum-t3-eval-000968", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 13 terms equal to 520. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 13 terms equal to 520. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-50, 15", "ref": "-50, 15", "family": "po_ap_sum", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 17, "bucket": "9-20", "guess_rate": 0.001667, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 13, "B": 50, "S": 520}, "seed": 141063620203001056, "version": "po-v0", "tier": 3, "n_raw": 17, "small": 6, "valid_answers": ["-50, 15", "-44, 14", "-38, 13", "-32, 12", "-26, 11", "-20, 10", "-14, 9", "-8, 8", "-2, 7", "4, 6", "10, 5", "16, 4", "22, 3", "28, 2", "34, 1", "40, 0", "46, -1"]}, "cell": "ap_sum/3", "stage": "B"}
+{"id": "po-ap_sum-t3-eval-001118", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 15 terms equal to 150. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 15 terms equal to 150. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-46, 8", "ref": "-46, 8", "family": "po_ap_sum", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.001372, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 15, "B": 50, "S": 150}, "seed": 141063620203001230, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 3, "valid_answers": ["-46, 8", "-39, 7", "-32, 6", "-25, 5", "-18, 4", "-11, 3", "-4, 2", "3, 1", "10, 0", "17, -1", "24, -2", "31, -3", "38, -4", "45, -5"]}, "cell": "ap_sum/3", "stage": "B"}
+{"id": "po-ap_sum-t3-eval-000847", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 12 terms equal to 2214. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 12 terms equal to 2214. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-41, 41", "ref": "-41, 41", "family": "po_ap_sum", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000882, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 12, "B": 50, "S": 2214}, "seed": 141063620203000914, "version": "po-v0", "tier": 3, "n_raw": 9, "small": 29, "valid_answers": ["-41, 41", "-30, 39", "-19, 37", "-8, 35", "3, 33", "14, 31", "25, 29", "36, 27", "47, 25"]}, "cell": "ap_sum/3", "stage": "B"}
+{"id": "po-ap_sum-t3-eval-001653", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 14 terms equal to -406. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 14 terms equal to -406. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-42, 2", "ref": "-42, 2", "family": "po_ap_sum", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000784, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 14, "B": 50, "S": -406}, "seed": 141063620203001929, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 4, "valid_answers": ["-42, 2", "-29, 0", "-16, -2", "-3, -4", "10, -6", "23, -8", "36, -10", "49, -12"]}, "cell": "ap_sum/3", "stage": "B"}
+{"id": "po-ap_sum-t3-eval-001712", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 13 terms equal to -195. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 13 terms equal to -195. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-45, 5", "ref": "-45, 5", "family": "po_ap_sum", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.001568, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 13, "B": 50, "S": -195}, "seed": 141063620203002006, "version": "po-v0", "tier": 3, "n_raw": 16, "small": 3, "valid_answers": ["-45, 5", "-39, 4", "-33, 3", "-27, 2", "-21, 1", "-15, 0", "-9, -1", "-3, -2", "3, -3", "9, -4", "15, -5", "21, -6", "27, -7", "33, -8", "39, -9", "45, -10"]}, "cell": "ap_sum/3", "stage": "B"}
+{"id": "po-ap_sum-t3-eval-002345", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 14 terms equal to -4599. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 14 terms equal to -4599. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-49, -43", "ref": "-49, -43", "family": "po_ap_sum", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000392, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 14, "B": 50, "S": -4599}, "seed": 141063620203002962, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 45, "valid_answers": ["-49, -43", "-36, -45", "-23, -47", "-10, -49"]}, "cell": "ap_sum/3", "stage": "B"}
+{"id": "po-ap_sum-t3-eval-000575", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 10 terms equal to 665. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 10 terms equal to 665. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-46, 25", "ref": "-46, 25", "family": "po_ap_sum", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.001078, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 10, "B": 50, "S": 665}, "seed": 141063620203000600, "version": "po-v0", "tier": 3, "n_raw": 11, "small": 13, "valid_answers": ["-46, 25", "-37, 23", "-28, 21", "-19, 19", "-10, 17", "-1, 15", "8, 13", "17, 11", "26, 9", "35, 7", "44, 5"]}, "cell": "ap_sum/3", "stage": "B"}
+{"id": "po-ap_sum-t3-eval-002490", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 15 terms equal to 705. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 15 terms equal to 705. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-44, 13", "ref": "-44, 13", "family": "po_ap_sum", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.001372, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 15, "B": 50, "S": 705}, "seed": 141063620203003186, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 6, "valid_answers": ["-44, 13", "-37, 12", "-30, 11", "-23, 10", "-16, 9", "-9, 8", "-2, 7", "5, 6", "12, 5", "19, 4", "26, 3", "33, 2", "40, 1", "47, 0"]}, "cell": "ap_sum/3", "stage": "B"}
+{"id": "po-ap_sum-t3-eval-001471", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 10 terms equal to 730. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 10 terms equal to 730. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-44, 26", "ref": "-44, 26", "family": "po_ap_sum", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.001078, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 10, "B": 50, "S": 730}, "seed": 141063620203001691, "version": "po-v0", "tier": 3, "n_raw": 11, "small": 14, "valid_answers": ["-44, 26", "-35, 24", "-26, 22", "-17, 20", "-8, 18", "1, 16", "10, 14", "19, 12", "28, 10", "37, 8", "46, 6"]}, "cell": "ap_sum/3", "stage": "B"}
+{"id": "po-ap_sum-t3-eval-001254", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 16 terms equal to 16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 16 terms equal to 16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-44, 6", "ref": "-44, 6", "family": "po_ap_sum", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000686, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 16, "B": 50, "S": 16}, "seed": 141063620203001389, "version": "po-v0", "tier": 3, "n_raw": 7, "small": 1, "valid_answers": ["-44, 6", "-29, 4", "-14, 2", "1, 0", "16, -2", "31, -4", "46, -6"]}, "cell": "ap_sum/3", "stage": "B"}
+{"id": "po-ap_sum-t3-eval-001133", "task": "cx_po_ap_sum", "source": "cx_po_ap_sum", "split": "eval", "problem": "Find integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 15 terms equal to -1020. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, d with -50 <= a, d <= 50 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 15 terms equal to -1020. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-47, -3", "ref": "-47, -3", "family": "po_ap_sum", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.001372, "solvable": true, "holdout": false, "meta": {"family": "po_ap_sum", "rung": "witness", "params": {"n": 15, "B": 50, "S": -1020}, "seed": 141063620203001247, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 9, "valid_answers": ["-47, -3", "-40, -4", "-33, -5", "-26, -6", "-19, -7", "-12, -8", "-5, -9", "2, -10", "9, -11", "16, -12", "23, -13", "30, -14", "37, -15", "44, -16"]}, "cell": "ap_sum/3", "stage": "B"}
+{"id": "po-binary_runs-t0-eval-000004", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_8 of length 8 with b_1 = 1, exactly 3 ones, exactly 3 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_8.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_8 of length 8 with b_1 = 1, exactly 3 ones, exactly 3 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_8.\n\n<think>\n", "answer": "1, 0, 0, 0, 0, 0, 1, 1", "ref": "1, 0, 0, 0, 0, 0, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.007812, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 8, "ones": 3, "runs": 3, "pairs": null, "first": 1}, "seed": 318699003900000015, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 1, "valid_answers": ["1, 0, 0, 0, 0, 0, 1, 1", "1, 1, 0, 0, 0, 0, 0, 1"]}, "cell": "binary_runs/0", "stage": "C"}
+{"id": "po-binary_runs-t0-eval-000002", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_8 of length 8 with b_1 = 1, exactly 3 ones, exactly 5 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_8.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_8 of length 8 with b_1 = 1, exactly 3 ones, exactly 5 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_8.\n\n<think>\n", "answer": "1, 0, 0, 0, 0, 1, 0, 1", "ref": "1, 0, 0, 0, 0, 1, 0, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.015625, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 8, "ones": 3, "runs": 5, "pairs": null, "first": 1}, "seed": 318699003900000007, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 1, "valid_answers": ["1, 0, 0, 0, 0, 1, 0, 1", "1, 0, 0, 0, 1, 0, 0, 1", "1, 0, 0, 1, 0, 0, 0, 1", "1, 0, 1, 0, 0, 0, 0, 1"]}, "cell": "binary_runs/0", "stage": "C"}
+{"id": "po-binary_runs-t0-eval-000001", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_8 of length 8 with b_1 = 0, exactly 3 ones, exactly 3 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_8.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_8 of length 8 with b_1 = 0, exactly 3 ones, exactly 3 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_8.\n\n<think>\n", "answer": "0, 0, 0, 0, 1, 1, 1, 0", "ref": "0, 0, 0, 0, 1, 1, 1, 0", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.015625, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 8, "ones": 3, "runs": 3, "pairs": null, "first": 0}, "seed": 318699003900000006, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 1, "valid_answers": ["0, 0, 0, 0, 1, 1, 1, 0", "0, 0, 0, 1, 1, 1, 0, 0", "0, 0, 1, 1, 1, 0, 0, 0", "0, 1, 1, 1, 0, 0, 0, 0"]}, "cell": "binary_runs/0", "stage": "C"}
+{"id": "po-binary_runs-t1-eval-000038", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_10 of length 10 with b_1 = 1, exactly 6 ones, exactly 9 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_10.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_10 of length 10 with b_1 = 1, exactly 6 ones, exactly 9 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_10.\n\n<think>\n", "answer": "1, 0, 1, 0, 1, 0, 1, 0, 1, 1", "ref": "1, 0, 1, 0, 1, 0, 1, 0, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.004883, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 10, "ones": 6, "runs": 9, "pairs": null, "first": 1}, "seed": 318699003901000231, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 1, "valid_answers": ["1, 0, 1, 0, 1, 0, 1, 0, 1, 1", "1, 0, 1, 0, 1, 0, 1, 1, 0, 1", "1, 0, 1, 0, 1, 1, 0, 1, 0, 1", "1, 0, 1, 1, 0, 1, 0, 1, 0, 1", "1, 1, 0, 1, 0, 1, 0, 1, 0, 1"]}, "cell": "binary_runs/1", "stage": "C"}
+{"id": "po-binary_runs-t1-eval-000028", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_10 of length 10 with b_1 = 0, exactly 5 ones, exactly 8 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_10.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_10 of length 10 with b_1 = 0, exactly 5 ones, exactly 8 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_10.\n\n<think>\n", "answer": "0, 0, 1, 0, 1, 0, 1, 0, 1, 1", "ref": "0, 0, 1, 0, 1, 0, 1, 0, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.015625, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 10, "ones": 5, "runs": 8, "pairs": null, "first": 0}, "seed": 318699003901000078, "version": "po-v0", "tier": 1, "n_raw": 16, "small": 1, "valid_answers": ["0, 0, 1, 0, 1, 0, 1, 0, 1, 1", "0, 0, 1, 0, 1, 0, 1, 1, 0, 1", "0, 0, 1, 0, 1, 1, 0, 1, 0, 1", "0, 0, 1, 1, 0, 1, 0, 1, 0, 1", "0, 1, 0, 0, 1, 0, 1, 0, 1, 1", "0, 1, 0, 0, 1, 0, 1, 1, 0, 1", "0, 1, 0, 0, 1, 1, 0, 1, 0, 1", "0, 1, 0, 1, 0, 0, 1, 0, 1, 1", "0, 1, 0, 1, 0, 0, 1, 1, 0, 1", "0, 1, 0, 1, 0, 1, 0, 0, 1, 1", "0, 1, 0, 1, 0, 1, 1, 0, 0, 1", "0, 1, 0, 1, 1, 0, 0, 1, 0, 1", "0, 1, 0, 1, 1, 0, 1, 0, 0, 1", "0, 1, 1, 0, 0, 1, 0, 1, 0, 1", "0, 1, 1, 0, 1, 0, 0, 1, 0, 1", "0, 1, 1, 0, 1, 0, 1, 0, 0, 1"]}, "cell": "binary_runs/1", "stage": "C"}
+{"id": "po-binary_runs-t1-eval-000005", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_10 of length 10 with b_1 = 1, exactly 4 ones, exactly 3 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_10.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_10 of length 10 with b_1 = 1, exactly 4 ones, exactly 3 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_10.\n\n<think>\n", "answer": "1, 0, 0, 0, 0, 0, 0, 1, 1, 1", "ref": "1, 0, 0, 0, 0, 0, 0, 1, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.00293, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 10, "ones": 4, "runs": 3, "pairs": null, "first": 1}, "seed": 318699003901000016, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 1, "valid_answers": ["1, 0, 0, 0, 0, 0, 0, 1, 1, 1", "1, 1, 0, 0, 0, 0, 0, 0, 1, 1", "1, 1, 1, 0, 0, 0, 0, 0, 0, 1"]}, "cell": "binary_runs/1", "stage": "C"}
+{"id": "po-binary_runs-t1-eval-000032", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_10 of length 10 with b_1 = 1, exactly 4 ones, exactly 8 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_10.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_10 of length 10 with b_1 = 1, exactly 4 ones, exactly 8 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_10.\n\n<think>\n", "answer": "1, 0, 0, 0, 1, 0, 1, 0, 1, 0", "ref": "1, 0, 0, 0, 1, 0, 1, 0, 1, 0", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.009766, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 10, "ones": 4, "runs": 8, "pairs": null, "first": 1}, "seed": 318699003901000088, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 1, "valid_answers": ["1, 0, 0, 0, 1, 0, 1, 0, 1, 0", "1, 0, 0, 1, 0, 0, 1, 0, 1, 0", "1, 0, 0, 1, 0, 1, 0, 0, 1, 0", "1, 0, 0, 1, 0, 1, 0, 1, 0, 0", "1, 0, 1, 0, 0, 0, 1, 0, 1, 0", "1, 0, 1, 0, 0, 1, 0, 0, 1, 0", "1, 0, 1, 0, 0, 1, 0, 1, 0, 0", "1, 0, 1, 0, 1, 0, 0, 0, 1, 0", "1, 0, 1, 0, 1, 0, 0, 1, 0, 0", "1, 0, 1, 0, 1, 0, 1, 0, 0, 0"]}, "cell": "binary_runs/1", "stage": "C"}
+{"id": "po-binary_runs-t1-eval-000022", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_10 of length 10 with b_1 = 1, exactly 7 ones, exactly 7 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_10.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_10 of length 10 with b_1 = 1, exactly 7 ones, exactly 7 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_10.\n\n<think>\n", "answer": "1, 0, 1, 0, 1, 0, 1, 1, 1, 1", "ref": "1, 0, 1, 0, 1, 0, 1, 1, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.019531, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 10, "ones": 7, "runs": 7, "pairs": null, "first": 1}, "seed": 318699003901000043, "version": "po-v0", "tier": 1, "n_raw": 20, "small": 1, "valid_answers": ["1, 0, 1, 0, 1, 0, 1, 1, 1, 1", "1, 0, 1, 0, 1, 1, 0, 1, 1, 1", "1, 0, 1, 0, 1, 1, 1, 0, 1, 1", "1, 0, 1, 0, 1, 1, 1, 1, 0, 1", "1, 0, 1, 1, 0, 1, 0, 1, 1, 1", "1, 0, 1, 1, 0, 1, 1, 0, 1, 1", "1, 0, 1, 1, 0, 1, 1, 1, 0, 1", "1, 0, 1, 1, 1, 0, 1, 0, 1, 1", "1, 0, 1, 1, 1, 0, 1, 1, 0, 1", "1, 0, 1, 1, 1, 1, 0, 1, 0, 1", "1, 1, 0, 1, 0, 1, 0, 1, 1, 1", "1, 1, 0, 1, 0, 1, 1, 0, 1, 1", "1, 1, 0, 1, 0, 1, 1, 1, 0, 1", "1, 1, 0, 1, 1, 0, 1, 0, 1, 1", "1, 1, 0, 1, 1, 0, 1, 1, 0, 1", "1, 1, 0, 1, 1, 1, 0, 1, 0, 1", "1, 1, 1, 0, 1, 0, 1, 0, 1, 1", "1, 1, 1, 0, 1, 0, 1, 1, 0, 1", "1, 1, 1, 0, 1, 1, 0, 1, 0, 1", "1, 1, 1, 1, 0, 1, 0, 1, 0, 1"]}, "cell": "binary_runs/1", "stage": "C"}
+{"id": "po-binary_runs-t1-eval-000006", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_10 of length 10 with b_1 = 1, exactly 3 ones, exactly 5 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_10.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_10 of length 10 with b_1 = 1, exactly 3 ones, exactly 5 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_10.\n\n<think>\n", "answer": "1, 0, 0, 0, 0, 0, 0, 1, 0, 1", "ref": "1, 0, 0, 0, 0, 0, 0, 1, 0, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.005859, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 10, "ones": 3, "runs": 5, "pairs": null, "first": 1}, "seed": 318699003901000018, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 1, "valid_answers": ["1, 0, 0, 0, 0, 0, 0, 1, 0, 1", "1, 0, 0, 0, 0, 0, 1, 0, 0, 1", "1, 0, 0, 0, 0, 1, 0, 0, 0, 1", "1, 0, 0, 0, 1, 0, 0, 0, 0, 1", "1, 0, 0, 1, 0, 0, 0, 0, 0, 1", "1, 0, 1, 0, 0, 0, 0, 0, 0, 1"]}, "cell": "binary_runs/1", "stage": "C"}
+{"id": "po-binary_runs-t1-eval-000025", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_10 of length 10 with b_1 = 1, exactly 4 ones, exactly 5 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_10.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_10 of length 10 with b_1 = 1, exactly 4 ones, exactly 5 runs (maximal blocks of equal digits). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_10.\n\n<think>\n", "answer": "1, 0, 0, 0, 0, 0, 1, 0, 1, 1", "ref": "1, 0, 0, 0, 0, 0, 1, 0, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 0.014648, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 10, "ones": 4, "runs": 5, "pairs": null, "first": 1}, "seed": 318699003901000063, "version": "po-v0", "tier": 1, "n_raw": 15, "small": 1, "valid_answers": ["1, 0, 0, 0, 0, 0, 1, 0, 1, 1", "1, 0, 0, 0, 0, 0, 1, 1, 0, 1", "1, 0, 0, 0, 0, 1, 0, 0, 1, 1", "1, 0, 0, 0, 0, 1, 1, 0, 0, 1", "1, 0, 0, 0, 1, 0, 0, 0, 1, 1", "1, 0, 0, 0, 1, 1, 0, 0, 0, 1", "1, 0, 0, 1, 0, 0, 0, 0, 1, 1", "1, 0, 0, 1, 1, 0, 0, 0, 0, 1", "1, 0, 1, 0, 0, 0, 0, 0, 1, 1", "1, 0, 1, 1, 0, 0, 0, 0, 0, 1", "1, 1, 0, 0, 0, 0, 0, 1, 0, 1", "1, 1, 0, 0, 0, 0, 1, 0, 0, 1", "1, 1, 0, 0, 0, 1, 0, 0, 0, 1", "1, 1, 0, 0, 1, 0, 0, 0, 0, 1", "1, 1, 0, 1, 0, 0, 0, 0, 0, 1"]}, "cell": "binary_runs/1", "stage": "C"}
+{"id": "po-binary_runs-t2-eval-000052", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 0, exactly 9 ones, exactly 4 runs (maximal blocks of equal digits), exactly 7 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 0, exactly 9 ones, exactly 4 runs (maximal blocks of equal digits), exactly 7 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.\n\n<think>\n", "answer": "0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1", "ref": "0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.003906, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 12, "ones": 9, "runs": 4, "pairs": 7, "first": 0}, "seed": 318699003902000160, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 1, "valid_answers": ["0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1", "0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1", "0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1", "0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1", "0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1", "0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1", "0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1", "0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1", "0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1", "0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1", "0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1", "0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1", "0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1", "0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1", "0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1", "0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1"]}, "cell": "binary_runs/2", "stage": "C"}
+{"id": "po-binary_runs-t2-eval-000037", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 1, exactly 5 ones, exactly 9 runs (maximal blocks of equal digits), exactly 0 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 1, exactly 5 ones, exactly 9 runs (maximal blocks of equal digits), exactly 0 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.\n\n<think>\n", "answer": "1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1", "ref": "1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.004883, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 12, "ones": 5, "runs": 9, "pairs": 0, "first": 1}, "seed": 318699003902000103, "version": "po-v0", "tier": 2, "n_raw": 20, "small": 1, "valid_answers": ["1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1", "1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1", "1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1", "1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1", "1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1", "1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1", "1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1", "1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1", "1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1", "1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1", "1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1", "1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1", "1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1", "1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1", "1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1", "1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1", "1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1", "1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1", "1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1", "1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1"]}, "cell": "binary_runs/2", "stage": "C"}
+{"id": "po-binary_runs-t2-eval-000033", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 1, exactly 8 ones, exactly 8 runs (maximal blocks of equal digits), exactly 4 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 1, exactly 8 ones, exactly 8 runs (maximal blocks of equal digits), exactly 4 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.\n\n<think>\n", "answer": "1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0", "ref": "1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 35, "bucket": null, "guess_rate": 0.008545, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 12, "ones": 8, "runs": 8, "pairs": 4, "first": 1}, "seed": 318699003902000097, "version": "po-v0", "tier": 2, "n_raw": 35, "small": 1, "valid_answers": ["1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0", "1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0", "1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0", "1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0", "1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0", "1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0", "1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0", "1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0", "1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0", "1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0", "1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0", "1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0", "1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0", "1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0", "1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0", "1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0", "1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0", "1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0", "1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0", "1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0", "1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0", "1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0", "1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0", "1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0", "1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0", "1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0", "1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0", "1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0", "1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0", "1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0", "1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0", "1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0", "1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0", "1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0", "1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0"]}, "cell": "binary_runs/2", "stage": "C"}
+{"id": "po-binary_runs-t2-eval-000024", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 0, exactly 9 ones, exactly 6 runs (maximal blocks of equal digits), exactly 6 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 0, exactly 9 ones, exactly 6 runs (maximal blocks of equal digits), exactly 6 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.\n\n<think>\n", "answer": "0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1", "ref": "0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 28, "bucket": null, "guess_rate": 0.006836, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 12, "ones": 9, "runs": 6, "pairs": 6, "first": 0}, "seed": 318699003902000062, "version": "po-v0", "tier": 2, "n_raw": 28, "small": 1, "valid_answers": ["0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1", "0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1", "0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1", "0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1", "0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1", "0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1", "0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1", "0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1", "0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1", "0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1", "0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1", "0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1", "0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1", "0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1", "0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1", "0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1", "0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1", "0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1", "0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1", "0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1", "0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1", "0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1", "0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1", "0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1", "0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1", "0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1", "0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1", "0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1"]}, "cell": "binary_runs/2", "stage": "C"}
+{"id": "po-binary_runs-t2-eval-000075", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 1, exactly 4 ones, exactly 5 runs (maximal blocks of equal digits), exactly 1 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 1, exactly 4 ones, exactly 5 runs (maximal blocks of equal digits), exactly 1 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.\n\n<think>\n", "answer": "1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1", "ref": "1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.005127, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 12, "ones": 4, "runs": 5, "pairs": 1, "first": 1}, "seed": 318699003902000624, "version": "po-v0", "tier": 2, "n_raw": 21, "small": 1, "valid_answers": ["1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1", "1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1", "1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1", "1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1", "1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1", "1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1", "1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1", "1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1", "1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1", "1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1", "1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1", "1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1", "1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1", "1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1", "1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1", "1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1", "1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1", "1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1", "1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1", "1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1", "1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1"]}, "cell": "binary_runs/2", "stage": "C"}
+{"id": "po-binary_runs-t2-eval-000003", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 1, exactly 5 ones, exactly 5 runs (maximal blocks of equal digits), exactly 2 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 1, exactly 5 ones, exactly 5 runs (maximal blocks of equal digits), exactly 2 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.\n\n<think>\n", "answer": "1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1", "ref": "1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 36, "bucket": null, "guess_rate": 0.008789, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 12, "ones": 5, "runs": 5, "pairs": 2, "first": 1}, "seed": 318699003902000013, "version": "po-v0", "tier": 2, "n_raw": 36, "small": 1, "valid_answers": ["1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1", "1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1", "1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1", "1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1", "1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1", "1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1", "1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1", "1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1", "1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1", "1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1", "1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1", "1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1", "1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1", "1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1", "1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1", "1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1", "1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1", "1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1", "1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1", "1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1", "1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1", "1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1", "1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1", "1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1", "1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1", "1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1", "1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1", "1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1", "1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1", "1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1", "1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1", "1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1", "1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1", "1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1", "1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1", "1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1"]}, "cell": "binary_runs/2", "stage": "C"}
+{"id": "po-binary_runs-t2-eval-000067", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 0, exactly 5 ones, exactly 4 runs (maximal blocks of equal digits), exactly 3 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 0, exactly 5 ones, exactly 4 runs (maximal blocks of equal digits), exactly 3 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.\n\n<think>\n", "answer": "0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1", "ref": "0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 24, "bucket": null, "guess_rate": 0.005859, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 12, "ones": 5, "runs": 4, "pairs": 3, "first": 0}, "seed": 318699003902000332, "version": "po-v0", "tier": 2, "n_raw": 24, "small": 1, "valid_answers": ["0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1", "0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1", "0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1", "0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1", "0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1", "0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1", "0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1", "0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1", "0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1", "0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1", "0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1", "0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1", "0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1", "0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1", "0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1", "0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1", "0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1", "0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1", "0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1", "0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1", "0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1", "0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1", "0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1", "0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1"]}, "cell": "binary_runs/2", "stage": "C"}
+{"id": "po-binary_runs-t2-eval-000047", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 0, exactly 5 ones, exactly 9 runs (maximal blocks of equal digits), exactly 1 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 0, exactly 5 ones, exactly 9 runs (maximal blocks of equal digits), exactly 1 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.\n\n<think>\n", "answer": "0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0", "ref": "0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 60, "bucket": null, "guess_rate": 0.014648, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 12, "ones": 5, "runs": 9, "pairs": 1, "first": 0}, "seed": 318699003902000136, "version": "po-v0", "tier": 2, "n_raw": 60, "small": 1, "valid_answers": ["0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0", "0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0", "0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0", "0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0", "0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0", "0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0", "0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0", "0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0", "0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0", "0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0", "0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0", "0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0", "0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0", "0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0", "0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0", "0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0", "0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0", "0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0", "0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0", "0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0", "0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0", "0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0", "0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0", "0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0", "0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0", "0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0", "0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0", "0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0", "0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0", "0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0", "0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0", "0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0", "0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0", "0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0", "0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0", "0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0", "0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0", "0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0", "0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0", "0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0", "0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0", "0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0", "0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0", "0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0", "0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0", "0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0", "0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0", "0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0", "0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0", "0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0", "0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0", "0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0", "0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0", "0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0", "0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0", "0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0", "0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0", "0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0", "0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0", "0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0"]}, "cell": "binary_runs/2", "stage": "C"}
+{"id": "po-binary_runs-t2-eval-000087", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 1, exactly 3 ones, exactly 3 runs (maximal blocks of equal digits), exactly 1 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 1, exactly 3 ones, exactly 3 runs (maximal blocks of equal digits), exactly 1 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.\n\n<think>\n", "answer": "1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1", "ref": "1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000488, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 12, "ones": 3, "runs": 3, "pairs": 1, "first": 1}, "seed": 318699003902005591, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 1, "valid_answers": ["1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1", "1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1"]}, "cell": "binary_runs/2", "stage": "C"}
+{"id": "po-binary_runs-t2-eval-000034", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 0, exactly 6 ones, exactly 10 runs (maximal blocks of equal digits), exactly 1 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 0, exactly 6 ones, exactly 10 runs (maximal blocks of equal digits), exactly 1 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.\n\n<think>\n", "answer": "0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1", "ref": "0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 25, "bucket": null, "guess_rate": 0.006104, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 12, "ones": 6, "runs": 10, "pairs": 1, "first": 0}, "seed": 318699003902000098, "version": "po-v0", "tier": 2, "n_raw": 25, "small": 1, "valid_answers": ["0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1", "0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1", "0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1", "0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1", "0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1", "0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1", "0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1", "0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1", "0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1", "0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1", "0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1", "0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1", "0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1", "0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1", "0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1", "0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1", "0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1", "0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1", "0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1", "0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1", "0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1", "0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1", "0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1", "0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1", "0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1"]}, "cell": "binary_runs/2", "stage": "C"}
+{"id": "po-binary_runs-t2-eval-000031", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 1, exactly 3 ones, exactly 6 runs (maximal blocks of equal digits), exactly 0 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 1, exactly 3 ones, exactly 6 runs (maximal blocks of equal digits), exactly 0 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.\n\n<think>\n", "answer": "1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0", "ref": "1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 28, "bucket": null, "guess_rate": 0.006836, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 12, "ones": 3, "runs": 6, "pairs": 0, "first": 1}, "seed": 318699003902000095, "version": "po-v0", "tier": 2, "n_raw": 28, "small": 1, "valid_answers": ["1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0", "1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0", "1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0", "1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0", "1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0", "1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0", "1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0", "1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0", "1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0", "1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0", "1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0", "1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0", "1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0", "1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0", "1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0", "1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0", "1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0", "1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0", "1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0", "1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0", "1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0", "1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0", "1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0", "1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0", "1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0", "1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0", "1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0", "1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0"]}, "cell": "binary_runs/2", "stage": "C"}
+{"id": "po-binary_runs-t2-eval-000085", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 0, exactly 7 ones, exactly 3 runs (maximal blocks of equal digits), exactly 6 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_12 of length 12 with b_1 = 0, exactly 7 ones, exactly 3 runs (maximal blocks of equal digits), exactly 6 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_12.\n\n<think>\n", "answer": "0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0", "ref": "0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000977, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 12, "ones": 7, "runs": 3, "pairs": 6, "first": 0}, "seed": 318699003902001351, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 1, "valid_answers": ["0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0", "0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0", "0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0", "0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0"]}, "cell": "binary_runs/2", "stage": "C"}
+{"id": "po-binary_runs-t3-eval-000019", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 0, exactly 5 ones, exactly 11 runs (maximal blocks of equal digits), exactly 0 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 0, exactly 5 ones, exactly 11 runs (maximal blocks of equal digits), exactly 0 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.\n\n<think>\n", "answer": "0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0", "ref": "0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 56, "bucket": null, "guess_rate": 0.003418, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 14, "ones": 5, "runs": 11, "pairs": 0, "first": 0}, "seed": 318699003903000210, "version": "po-v0", "tier": 3, "n_raw": 56, "small": 1, "valid_answers": ["0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0", "0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0", "0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0", "0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0", "0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0", "0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0", "0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0", "0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0", "0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0", "0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0", "0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0", "0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0", "0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0", "0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0", "0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0", "0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0", "0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0", "0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0", "0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0", "0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0", "0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0", "0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0", "0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0", "0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0", "0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0", "0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0", "0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0", "0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0", "0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0", "0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0", "0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0", "0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0", "0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0", "0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0", "0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0", "0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0", "0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0", "0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0", "0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0", "0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0", "0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0", "0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0", "0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0", "0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0", "0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0", "0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0", "0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0", "0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0", "0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0", "0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0", "0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0", "0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0", "0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0", "0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0", "0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0", "0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0"]}, "cell": "binary_runs/3", "stage": "C"}
+{"id": "po-binary_runs-t3-eval-000071", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 1, exactly 3 ones, exactly 3 runs (maximal blocks of equal digits), exactly 1 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 1, exactly 3 ones, exactly 3 runs (maximal blocks of equal digits), exactly 1 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.\n\n<think>\n", "answer": "1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1", "ref": "1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000122, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 14, "ones": 3, "runs": 3, "pairs": 1, "first": 1}, "seed": 318699003903004751, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 1, "valid_answers": ["1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1", "1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1"]}, "cell": "binary_runs/3", "stage": "C"}
+{"id": "po-binary_runs-t3-eval-000055", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 1, exactly 2 ones, exactly 4 runs (maximal blocks of equal digits), exactly 0 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 1, exactly 2 ones, exactly 4 runs (maximal blocks of equal digits), exactly 0 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.\n\n<think>\n", "answer": "1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0", "ref": "1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.000671, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 14, "ones": 2, "runs": 4, "pairs": 0, "first": 1}, "seed": 318699003903001466, "version": "po-v0", "tier": 3, "n_raw": 11, "small": 1, "valid_answers": ["1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0", "1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0", "1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0", "1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0", "1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0", "1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0", "1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0", "1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0", "1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0", "1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0", "1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0"]}, "cell": "binary_runs/3", "stage": "C"}
+{"id": "po-binary_runs-t3-eval-000035", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 0, exactly 8 ones, exactly 11 runs (maximal blocks of equal digits), exactly 3 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 0, exactly 8 ones, exactly 11 runs (maximal blocks of equal digits), exactly 3 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.\n\n<think>\n", "answer": "0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0", "ref": "0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 35, "bucket": null, "guess_rate": 0.002136, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 14, "ones": 8, "runs": 11, "pairs": 3, "first": 0}, "seed": 318699003903000509, "version": "po-v0", "tier": 3, "n_raw": 35, "small": 1, "valid_answers": ["0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0", "0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0", "0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0", "0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0", "0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0", "0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0", "0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0", "0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0", "0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0", "0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0", "0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0", "0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0", "0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0", "0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0", "0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0", "0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0", "0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0", "0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0", "0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0", "0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0", "0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0", "0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0", "0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0", "0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0", "0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0", "0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0", "0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0", "0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0", "0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0", "0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0", "0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0", "0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0", "0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0", "0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0", "0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0"]}, "cell": "binary_runs/3", "stage": "C"}
+{"id": "po-binary_runs-t3-eval-000030", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 1, exactly 9 ones, exactly 11 runs (maximal blocks of equal digits), exactly 3 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 1, exactly 9 ones, exactly 11 runs (maximal blocks of equal digits), exactly 3 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.\n\n<think>\n", "answer": "1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1", "ref": "1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 56, "bucket": null, "guess_rate": 0.003418, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 14, "ones": 9, "runs": 11, "pairs": 3, "first": 1}, "seed": 318699003903000412, "version": "po-v0", "tier": 3, "n_raw": 56, "small": 1, "valid_answers": ["1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1", "1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1", "1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1", "1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1", "1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1", "1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1", "1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1", "1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1", "1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1", "1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1", "1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1", "1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1", "1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1", "1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1", "1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1", "1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1", "1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1", "1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1", "1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1", "1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1", "1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1", "1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1", "1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1", "1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1", "1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1", "1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1", "1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1", "1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1", "1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1", "1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1", "1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1", "1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1", "1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1", "1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1", "1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1", "1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1", "1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1", "1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1", "1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1", "1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1", "1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1", "1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1", "1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1", "1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1", "1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1", "1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1", "1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1", "1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1", "1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1", "1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1", "1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1", "1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1", "1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1", "1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1", "1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1", "1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1"]}, "cell": "binary_runs/3", "stage": "C"}
+{"id": "po-binary_runs-t3-eval-000009", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 0, exactly 12 ones, exactly 4 runs (maximal blocks of equal digits), exactly 10 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 0, exactly 12 ones, exactly 4 runs (maximal blocks of equal digits), exactly 10 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.\n\n<think>\n", "answer": "0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1", "ref": "0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.000671, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 14, "ones": 12, "runs": 4, "pairs": 10, "first": 0}, "seed": 318699003903000076, "version": "po-v0", "tier": 3, "n_raw": 11, "small": 1, "valid_answers": ["0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1", "0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1", "0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1", "0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1", "0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1", "0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1", "0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1", "0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1", "0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1", "0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1", "0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1"]}, "cell": "binary_runs/3", "stage": "C"}
+{"id": "po-binary_runs-t3-eval-000026", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 0, exactly 3 ones, exactly 3 runs (maximal blocks of equal digits), exactly 2 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 0, exactly 3 ones, exactly 3 runs (maximal blocks of equal digits), exactly 2 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.\n\n<think>\n", "answer": "0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0", "ref": "0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.00061, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 14, "ones": 3, "runs": 3, "pairs": 2, "first": 0}, "seed": 318699003903000389, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 1, "valid_answers": ["0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0", "0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0", "0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0", "0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0", "0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0", "0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0", "0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0", "0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0", "0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0", "0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0"]}, "cell": "binary_runs/3", "stage": "C"}
+{"id": "po-binary_runs-t3-eval-000064", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 0, exactly 6 ones, exactly 13 runs (maximal blocks of equal digits), exactly 0 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 0, exactly 6 ones, exactly 13 runs (maximal blocks of equal digits), exactly 0 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.\n\n<think>\n", "answer": "0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0", "ref": "0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000427, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 14, "ones": 6, "runs": 13, "pairs": 0, "first": 0}, "seed": 318699003903002291, "version": "po-v0", "tier": 3, "n_raw": 7, "small": 1, "valid_answers": ["0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0", "0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0", "0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0", "0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0", "0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0", "0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0", "0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0"]}, "cell": "binary_runs/3", "stage": "C"}
+{"id": "po-binary_runs-t3-eval-000067", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 1, exactly 6 ones, exactly 3 runs (maximal blocks of equal digits), exactly 4 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 1, exactly 6 ones, exactly 3 runs (maximal blocks of equal digits), exactly 4 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.\n\n<think>\n", "answer": "1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1", "ref": "1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000305, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 14, "ones": 6, "runs": 3, "pairs": 4, "first": 1}, "seed": 318699003903003709, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 1, "valid_answers": ["1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1", "1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1", "1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1", "1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1", "1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1"]}, "cell": "binary_runs/3", "stage": "C"}
+{"id": "po-binary_runs-t3-eval-000061", "task": "cx_po_binary_runs", "source": "cx_po_binary_runs", "split": "eval", "problem": "Find a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 1, exactly 11 ones, exactly 3 runs (maximal blocks of equal digits), exactly 9 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.", "prompt": "Problem:\nFind a binary string b_1 b_2 ... b_14 of length 14 with b_1 = 1, exactly 11 ones, exactly 3 runs (maximal blocks of equal digits), exactly 9 occurrences of 11 (counting overlaps). Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as b_1, b_2, ..., b_14.\n\n<think>\n", "answer": "1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1", "ref": "1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1", "family": "po_binary_runs", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.00061, "solvable": true, "holdout": false, "meta": {"family": "po_binary_runs", "rung": "witness", "params": {"n": 14, "ones": 11, "runs": 3, "pairs": 9, "first": 1}, "seed": 318699003903001971, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 1, "valid_answers": ["1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1", "1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1", "1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1", "1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1", "1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1", "1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1", "1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1", "1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1", "1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1", "1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1"]}, "cell": "binary_runs/3", "stage": "C"}
+{"id": "po-circle_through-t0-eval-000489", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, -2) and (2, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, -2) and (2, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 9, -2", "ref": "-6, 9, -2", "family": "po_circle_through", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.00054, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-2, -2], [2, 1]], "B": 10}, "seed": 346326373000000958, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 5, "valid_answers": ["-6, 9, -2", "-3, 5, -4", "0, 1, -6", "3, -3, -8", "6, -7, -10"]}, "cell": "circle_through/0", "stage": "B"}
+{"id": "po-circle_through-t0-eval-000325", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (1, -2) and (1, -1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (1, -2) and (1, -1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 3, 10", "ref": "-9, 3, 10", "family": "po_circle_through", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.00216, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[1, -2], [1, -1]], "B": 10}, "seed": 346326373000000441, "version": "po-v0", "tier": 0, "n_raw": 20, "small": 3, "valid_answers": ["-9, 3, 10", "-8, 3, 9", "-7, 3, 8", "-6, 3, 7", "-5, 3, 6", "-4, 3, 5", "-3, 3, 4", "-2, 3, 3", "-1, 3, 2", "0, 3, 1", "1, 3, 0", "2, 3, -1", "3, 3, -2", "4, 3, -3", "5, 3, -4", "6, 3, -5", "7, 3, -6", "8, 3, -7", "9, 3, -8", "10, 3, -9"]}, "cell": "circle_through/0", "stage": "B"}
+{"id": "po-circle_through-t0-eval-000485", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, 2) and (-1, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, 2) and (-1, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -10, 4", "ref": "-4, -10, 4", "family": "po_circle_through", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 0.00162, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-2, 2], [-1, 1]], "B": 10}, "seed": 346326373000000936, "version": "po-v0", "tier": 0, "n_raw": 15, "small": 4, "valid_answers": ["-4, -10, 4", "-3, -9, 4", "-2, -8, 4", "-1, -7, 4", "0, -6, 4", "1, -5, 4", "2, -4, 4", "3, -3, 4", "4, -2, 4", "5, -1, 4", "6, 0, 4", "7, 1, 4", "8, 2, 4", "9, 3, 4", "10, 4, 4"]}, "cell": "circle_through/0", "stage": "B"}
+{"id": "po-circle_through-t0-eval-000244", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, 0) and (0, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, 0) and (0, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -1, 0", "ref": "-10, -1, 0", "family": "po_circle_through", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.002268, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[0, 0], [0, 1]], "B": 10}, "seed": 346326373000000309, "version": "po-v0", "tier": 0, "n_raw": 21, "small": 1, "valid_answers": ["-10, -1, 0", "-9, -1, 0", "-8, -1, 0", "-7, -1, 0", "-6, -1, 0", "-5, -1, 0", "-4, -1, 0", "-3, -1, 0", "-2, -1, 0", "-1, -1, 0", "0, -1, 0", "1, -1, 0", "2, -1, 0", "3, -1, 0", "4, -1, 0", "5, -1, 0", "6, -1, 0", "7, -1, 0", "8, -1, 0", "9, -1, 0", "10, -1, 0"]}, "cell": "circle_through/0", "stage": "B"}
+{"id": "po-circle_through-t0-eval-000535", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, 1) and (0, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, 1) and (0, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -1, 0", "ref": "-10, -1, 0", "family": "po_circle_through", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.002268, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[0, 1], [0, 0]], "B": 10}, "seed": 346326373000001287, "version": "po-v0", "tier": 0, "n_raw": 21, "small": 1, "valid_answers": ["-10, -1, 0", "-9, -1, 0", "-8, -1, 0", "-7, -1, 0", "-6, -1, 0", "-5, -1, 0", "-4, -1, 0", "-3, -1, 0", "-2, -1, 0", "-1, -1, 0", "0, -1, 0", "1, -1, 0", "2, -1, 0", "3, -1, 0", "4, -1, 0", "5, -1, 0", "6, -1, 0", "7, -1, 0", "8, -1, 0", "9, -1, 0", "10, -1, 0"]}, "cell": "circle_through/0", "stage": "B"}
+{"id": "po-circle_through-t0-eval-000532", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, 0) and (-2, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, 0) and (-2, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -1, -10", "ref": "-3, -1, -10", "family": "po_circle_through", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.001188, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-2, 0], [-2, 1]], "B": 10}, "seed": 346326373000001258, "version": "po-v0", "tier": 0, "n_raw": 11, "small": 2, "valid_answers": ["-3, -1, -10", "-2, -1, -8", "-1, -1, -6", "0, -1, -4", "1, -1, -2", "2, -1, 0", "3, -1, 2", "4, -1, 4", "5, -1, 6", "6, -1, 8", "7, -1, 10"]}, "cell": "circle_through/0", "stage": "B"}
+{"id": "po-circle_through-t0-eval-000299", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (1, 0) and (-1, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (1, 0) and (-1, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -10, 7", "ref": "-8, -10, 7", "family": "po_circle_through", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 0.001944, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[1, 0], [-1, 2]], "B": 10}, "seed": 346326373000000397, "version": "po-v0", "tier": 0, "n_raw": 18, "small": 2, "valid_answers": ["-8, -10, 7", "-7, -9, 6", "-6, -8, 5", "-5, -7, 4", "-4, -6, 3", "-3, -5, 2", "-2, -4, 1", "-1, -3, 0", "0, -2, -1", "1, -1, -2", "2, 0, -3", "3, 1, -4", "4, 2, -5", "5, 3, -6", "6, 4, -7", "7, 5, -8", "8, 6, -9", "9, 7, -10"]}, "cell": "circle_through/0", "stage": "B"}
+{"id": "po-circle_through-t0-eval-000421", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (2, 2) and (1, -2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (2, 2) and (1, -2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 1, 4", "ref": "-7, 1, 4", "family": "po_circle_through", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000324, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[2, 2], [1, -2]], "B": 10}, "seed": 346326373000000707, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["-7, 1, 4", "-3, 0, -2", "1, -1, -8"]}, "cell": "circle_through/0", "stage": "B"}
+{"id": "po-circle_through-t0-eval-000117", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, -2) and (-1, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, -2) and (-1, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 1, -8", "ref": "-1, 1, -8", "family": "po_circle_through", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000324, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-2, -2], [-1, 2]], "B": 10}, "seed": 346326373000000136, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["-1, 1, -8", "3, 0, -2", "7, -1, 4"]}, "cell": "circle_through/0", "stage": "B"}
+{"id": "po-circle_through-t0-eval-000534", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (1, -2) and (-1, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (1, -2) and (-1, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -5, -5", "ref": "-10, -5, -5", "family": "po_circle_through", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.001188, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[1, -2], [-1, 2]], "B": 10}, "seed": 346326373000001272, "version": "po-v0", "tier": 0, "n_raw": 11, "small": 5, "valid_answers": ["-10, -5, -5", "-8, -4, -5", "-6, -3, -5", "-4, -2, -5", "-2, -1, -5", "0, 0, -5", "2, 1, -5", "4, 2, -5", "6, 3, -5", "8, 4, -5", "10, 5, -5"]}, "cell": "circle_through/0", "stage": "B"}
+{"id": "po-circle_through-t0-eval-000365", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (1, 1) and (2, -1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (1, 1) and (2, -1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -3, 10", "ref": "-9, -3, 10", "family": "po_circle_through", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000756, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[1, 1], [2, -1]], "B": 10}, "seed": 346326373000000529, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 2, "valid_answers": ["-9, -3, 10", "-7, -2, 7", "-5, -1, 4", "-3, 0, 1", "-1, 1, -2", "1, 2, -5", "3, 3, -8"]}, "cell": "circle_through/0", "stage": "B"}
+{"id": "po-circle_through-t0-eval-000316", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (2, -1) and (-2, -2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -10 <= D, E, F <= 10 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (2, -1) and (-2, -2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 7, 4", "ref": "-1, 7, 4", "family": "po_circle_through", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000324, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[2, -1], [-2, -2]], "B": 10}, "seed": 346326373000000420, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["-1, 7, 4", "0, 3, -2", "1, -1, -8"]}, "cell": "circle_through/0", "stage": "B"}
+{"id": "po-circle_through-t1-eval-001425", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-1, 3) and (-3, 3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-1, 3) and (-3, 3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, -7, 15", "ref": "4, -7, 15", "family": "po_circle_through", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.000369, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-1, 3], [-3, 3]], "B": 15}, "seed": 346326373001002269, "version": "po-v0", "tier": 1, "n_raw": 11, "small": 4, "valid_answers": ["4, -7, 15", "4, -6, 12", "4, -5, 9", "4, -4, 6", "4, -3, 3", "4, -2, 0", "4, -1, -3", "4, 0, -6", "4, 1, -9", "4, 2, -12", "4, 3, -15"]}, "cell": "circle_through/1", "stage": "B"}
+{"id": "po-circle_through-t1-eval-000602", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, -3) and (3, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, -3) and (3, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-11, 8, 15", "ref": "-11, 8, 15", "family": "po_circle_through", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000134, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[0, -3], [3, 1]], "B": 15}, "seed": 346326373001000717, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 3, "valid_answers": ["-11, 8, 15", "-7, 5, 6", "-3, 2, -3", "1, -1, -12"]}, "cell": "circle_through/1", "stage": "B"}
+{"id": "po-circle_through-t1-eval-000728", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (2, 1) and (1, -2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (2, 1) and (1, -2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, 4, 15", "ref": "-12, 4, 15", "family": "po_circle_through", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000235, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[2, 1], [1, -2]], "B": 15}, "seed": 346326373001000880, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 3, "valid_answers": ["-12, 4, 15", "-9, 3, 10", "-6, 2, 5", "-3, 1, 0", "0, 0, -5", "3, -1, -10", "6, -2, -15"]}, "cell": "circle_through/1", "stage": "B"}
+{"id": "po-circle_through-t1-eval-000482", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-3, 3) and (-3, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-3, 3) and (-3, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -3, -15", "ref": "-2, -3, -15", "family": "po_circle_through", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.000369, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-3, 3], [-3, 0]], "B": 15}, "seed": 346326373001000556, "version": "po-v0", "tier": 1, "n_raw": 11, "small": 3, "valid_answers": ["-2, -3, -15", "-1, -3, -12", "0, -3, -9", "1, -3, -6", "2, -3, -3", "3, -3, 0", "4, -3, 3", "5, -3, 6", "6, -3, 9", "7, -3, 12", "8, -3, 15"]}, "cell": "circle_through/1", "stage": "B"}
+{"id": "po-circle_through-t1-eval-000496", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, -2) and (0, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, -2) and (0, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-15, 1, -2", "ref": "-15, 1, -2", "family": "po_circle_through", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 31, "bucket": null, "guess_rate": 0.001041, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[0, -2], [0, 1]], "B": 15}, "seed": 346326373001000574, "version": "po-v0", "tier": 1, "n_raw": 31, "small": 2, "valid_answers": ["-15, 1, -2", "-14, 1, -2", "-13, 1, -2", "-12, 1, -2", "-11, 1, -2", "-10, 1, -2", "-9, 1, -2", "-8, 1, -2", "-7, 1, -2", "-6, 1, -2", "-5, 1, -2", "-4, 1, -2", "-3, 1, -2", "-2, 1, -2", "-1, 1, -2", "0, 1, -2", "1, 1, -2", "2, 1, -2", "3, 1, -2", "4, 1, -2", "5, 1, -2", "6, 1, -2", "7, 1, -2", "8, 1, -2", "9, 1, -2", "10, 1, -2", "11, 1, -2", "12, 1, -2", "13, 1, -2", "14, 1, -2", "15, 1, -2"]}, "cell": "circle_through/1", "stage": "B"}
+{"id": "po-circle_through-t1-eval-002046", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (3, 2) and (-2, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (3, 2) and (-2, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 8, -14", "ref": "-5, 8, -14", "family": "po_circle_through", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000168, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[3, 2], [-2, 0]], "B": 15}, "seed": 346326373001004836, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 6, "valid_answers": ["-5, 8, -14", "-3, 3, -10", "-1, -2, -6", "1, -7, -2", "3, -12, 2"]}, "cell": "circle_through/1", "stage": "B"}
+{"id": "po-circle_through-t1-eval-000485", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-3, -1) and (-1, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-3, -1) and (-1, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 3, -13", "ref": "-2, 3, -13", "family": "po_circle_through", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000168, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-3, -1], [-1, 2]], "B": 15}, "seed": 346326373001000559, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 4, "valid_answers": ["-2, 3, -13", "1, 1, -6", "4, -1, 1", "7, -3, 8", "10, -5, 15"]}, "cell": "circle_through/1", "stage": "B"}
+{"id": "po-circle_through-t1-eval-000097", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-3, -1) and (2, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-3, -1) and (2, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 11, -2", "ref": "-1, 11, -2", "family": "po_circle_through", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000201, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-3, -1], [2, 0]], "B": 15}, "seed": 346326373001000103, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 6, "valid_answers": ["-1, 11, -2", "0, 6, -4", "1, 1, -6", "2, -4, -8", "3, -9, -10", "4, -14, -12"]}, "cell": "circle_through/1", "stage": "B"}
+{"id": "po-circle_through-t1-eval-001505", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-3, 2) and (-2, -2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-3, 2) and (-2, -2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -1, -8", "ref": "1, -1, -8", "family": "po_circle_through", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000101, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-3, 2], [-2, -2]], "B": 15}, "seed": 346326373001002468, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 5, "valid_answers": ["1, -1, -8", "5, 0, 2", "9, 1, 12"]}, "cell": "circle_through/1", "stage": "B"}
+{"id": "po-circle_through-t1-eval-001611", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (1, -1) and (1, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (1, -1) and (1, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-15, 1, 14", "ref": "-15, 1, 14", "family": "po_circle_through", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 30, "bucket": null, "guess_rate": 0.001007, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[1, -1], [1, 0]], "B": 15}, "seed": 346326373001002814, "version": "po-v0", "tier": 1, "n_raw": 30, "small": 1, "valid_answers": ["-15, 1, 14", "-14, 1, 13", "-13, 1, 12", "-12, 1, 11", "-11, 1, 10", "-10, 1, 9", "-9, 1, 8", "-8, 1, 7", "-7, 1, 6", "-6, 1, 5", "-5, 1, 4", "-4, 1, 3", "-3, 1, 2", "-2, 1, 1", "-1, 1, 0", "0, 1, -1", "1, 1, -2", "2, 1, -3", "3, 1, -4", "4, 1, -5", "5, 1, -6", "6, 1, -7", "7, 1, -8", "8, 1, -9", "9, 1, -10", "10, 1, -11", "11, 1, -12", "12, 1, -13", "13, 1, -14", "14, 1, -15"]}, "cell": "circle_through/1", "stage": "B"}
+{"id": "po-circle_through-t1-eval-000641", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, -3) and (0, 3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, -3) and (0, 3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 2, -15", "ref": "-4, 2, -15", "family": "po_circle_through", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000235, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-2, -3], [0, 3]], "B": 15}, "seed": 346326373001000768, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 6, "valid_answers": ["-4, 2, -15", "-1, 1, -12", "2, 0, -9", "5, -1, -6", "8, -2, -3", "11, -3, 0", "14, -4, 3"]}, "cell": "circle_through/1", "stage": "B"}
+{"id": "po-circle_through-t1-eval-001143", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, -1) and (-2, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -15 <= D, E, F <= 15 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, -1) and (-2, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -1, -14", "ref": "-4, -1, -14", "family": "po_circle_through", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 0.000504, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-2, -1], [-2, 2]], "B": 15}, "seed": 346326373001001596, "version": "po-v0", "tier": 1, "n_raw": 15, "small": 2, "valid_answers": ["-4, -1, -14", "-3, -1, -12", "-2, -1, -10", "-1, -1, -8", "0, -1, -6", "1, -1, -4", "2, -1, -2", "3, -1, 0", "4, -1, 2", "5, -1, 4", "6, -1, 6", "7, -1, 8", "8, -1, 10", "9, -1, 12", "10, -1, 14"]}, "cell": "circle_through/1", "stage": "B"}
+{"id": "po-circle_through-t2-eval-000133", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, -3) and (3, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, -3) and (3, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 9, 18", "ref": "-9, 9, 18", "family": "po_circle_through", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000189, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[0, -3], [3, 0]], "B": 20}, "seed": 346326373002000137, "version": "po-v0", "tier": 2, "n_raw": 13, "small": 3, "valid_answers": ["-9, 9, 18", "-8, 8, 15", "-7, 7, 12", "-6, 6, 9", "-5, 5, 6", "-4, 4, 3", "-3, 3, 0", "-2, 2, -3", "-1, 1, -6", "0, 0, -9", "1, -1, -12", "2, -2, -15", "3, -3, -18"]}, "cell": "circle_through/2", "stage": "B"}
+{"id": "po-circle_through-t2-eval-000174", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (3, -4) and (-3, -3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (3, -4) and (-3, -3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 1, -18", "ref": "-1, 1, -18", "family": "po_circle_through", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[3, -4], [-3, -3]], "B": 20}, "seed": 346326373002000179, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 7, "valid_answers": ["-1, 1, -18", "0, 7, 3"]}, "cell": "circle_through/2", "stage": "B"}
+{"id": "po-circle_through-t2-eval-001661", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-4, 4) and (-3, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-4, 4) and (-3, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -9, -16", "ref": "-5, -9, -16", "family": "po_circle_through", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 7.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-4, 4], [-3, 1]], "B": 20}, "seed": 346326373002001992, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 7, "valid_answers": ["-5, -9, -16", "-2, -8, -8", "1, -7, 0", "4, -6, 8", "7, -5, 16"]}, "cell": "circle_through/2", "stage": "B"}
+{"id": "po-circle_through-t2-eval-001916", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-1, 2) and (1, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-1, 2) and (1, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -15, 19", "ref": "-6, -15, 19", "family": "po_circle_through", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.000203, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-1, 2], [1, 1]], "B": 20}, "seed": 346326373002002364, "version": "po-v0", "tier": 2, "n_raw": 14, "small": 2, "valid_answers": ["-6, -15, 19", "-5, -13, 16", "-4, -11, 13", "-3, -9, 10", "-2, -7, 7", "-1, -5, 4", "0, -3, 1", "1, -1, -2", "2, 1, -5", "3, 3, -8", "4, 5, -11", "5, 7, -14", "6, 9, -17", "7, 11, -20"]}, "cell": "circle_through/2", "stage": "B"}
+{"id": "po-circle_through-t2-eval-000322", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-1, 2) and (-1, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-1, 2) and (-1, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-19, -2, -20", "ref": "-19, -2, -20", "family": "po_circle_through", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 0.00058, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-1, 2], [-1, 0]], "B": 20}, "seed": 346326373002000335, "version": "po-v0", "tier": 2, "n_raw": 40, "small": 2, "valid_answers": ["-19, -2, -20", "-18, -2, -19", "-17, -2, -18", "-16, -2, -17", "-15, -2, -16", "-14, -2, -15", "-13, -2, -14", "-12, -2, -13", "-11, -2, -12", "-10, -2, -11", "-9, -2, -10", "-8, -2, -9", "-7, -2, -8", "-6, -2, -7", "-5, -2, -6", "-4, -2, -5", "-3, -2, -4", "-2, -2, -3", "-1, -2, -2", "0, -2, -1", "1, -2, 0", "2, -2, 1", "3, -2, 2", "4, -2, 3", "5, -2, 4", "6, -2, 5", "7, -2, 6", "8, -2, 7", "9, -2, 8", "10, -2, 9", "11, -2, 10", "12, -2, 11", "13, -2, 12", "14, -2, 13", "15, -2, 14", "16, -2, 15", "17, -2, 16", "18, -2, 17", "19, -2, 18", "20, -2, 19"]}, "cell": "circle_through/2", "stage": "B"}
+{"id": "po-circle_through-t2-eval-002301", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, -3) and (3, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, -3) and (3, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-13, 7, 12", "ref": "-13, 7, 12", "family": "po_circle_through", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 5.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[0, -3], [3, 2]], "B": 20}, "seed": 346326373002002964, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 6, "valid_answers": ["-13, 7, 12", "-8, 4, 3", "-3, 1, -6", "2, -2, -15"]}, "cell": "circle_through/2", "stage": "B"}
+{"id": "po-circle_through-t2-eval-000380", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, 2) and (0, -1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, 2) and (0, -1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-20, -1, -2", "ref": "-20, -1, -2", "family": "po_circle_through", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 41, "bucket": null, "guess_rate": 0.000595, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[0, 2], [0, -1]], "B": 20}, "seed": 346326373002000397, "version": "po-v0", "tier": 2, "n_raw": 41, "small": 2, "valid_answers": ["-20, -1, -2", "-19, -1, -2", "-18, -1, -2", "-17, -1, -2", "-16, -1, -2", "-15, -1, -2", "-14, -1, -2", "-13, -1, -2", "-12, -1, -2", "-11, -1, -2", "-10, -1, -2", "-9, -1, -2", "-8, -1, -2", "-7, -1, -2", "-6, -1, -2", "-5, -1, -2", "-4, -1, -2", "-3, -1, -2", "-2, -1, -2", "-1, -1, -2", "0, -1, -2", "1, -1, -2", "2, -1, -2", "3, -1, -2", "4, -1, -2", "5, -1, -2", "6, -1, -2", "7, -1, -2", "8, -1, -2", "9, -1, -2", "10, -1, -2", "11, -1, -2", "12, -1, -2", "13, -1, -2", "14, -1, -2", "15, -1, -2", "16, -1, -2", "17, -1, -2", "18, -1, -2", "19, -1, -2", "20, -1, -2"]}, "cell": "circle_through/2", "stage": "B"}
+{"id": "po-circle_through-t2-eval-002467", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, -4) and (0, -3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, -4) and (0, -3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-20, 7, 12", "ref": "-20, 7, 12", "family": "po_circle_through", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 41, "bucket": null, "guess_rate": 0.000595, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[0, -4], [0, -3]], "B": 20}, "seed": 346326373002003245, "version": "po-v0", "tier": 2, "n_raw": 41, "small": 12, "valid_answers": ["-20, 7, 12", "-19, 7, 12", "-18, 7, 12", "-17, 7, 12", "-16, 7, 12", "-15, 7, 12", "-14, 7, 12", "-13, 7, 12", "-12, 7, 12", "-11, 7, 12", "-10, 7, 12", "-9, 7, 12", "-8, 7, 12", "-7, 7, 12", "-6, 7, 12", "-5, 7, 12", "-4, 7, 12", "-3, 7, 12", "-2, 7, 12", "-1, 7, 12", "0, 7, 12", "1, 7, 12", "2, 7, 12", "3, 7, 12", "4, 7, 12", "5, 7, 12", "6, 7, 12", "7, 7, 12", "8, 7, 12", "9, 7, 12", "10, 7, 12", "11, 7, 12", "12, 7, 12", "13, 7, 12", "14, 7, 12", "15, 7, 12", "16, 7, 12", "17, 7, 12", "18, 7, 12", "19, 7, 12", "20, 7, 12"]}, "cell": "circle_through/2", "stage": "B"}
+{"id": "po-circle_through-t2-eval-000072", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-1, 3) and (4, 4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-1, 3) and (4, 4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -2, -8", "ref": "-4, -2, -8", "family": "po_circle_through", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-1, 3], [4, 4]], "B": 20}, "seed": 346326373002000073, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 8, "valid_answers": ["-4, -2, -8", "-3, -7, 8"]}, "cell": "circle_through/2", "stage": "B"}
+{"id": "po-circle_through-t2-eval-002145", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-3, 4) and (1, -1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-3, 4) and (1, -1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-18, -19, -3", "ref": "-18, -19, -3", "family": "po_circle_through", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000116, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-3, 4], [1, -1]], "B": 20}, "seed": 346326373002002702, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 7, "valid_answers": ["-18, -19, -3", "-13, -15, -4", "-8, -11, -5", "-3, -7, -6", "2, -3, -7", "7, 1, -8", "12, 5, -9", "17, 9, -10"]}, "cell": "circle_through/2", "stage": "B"}
+{"id": "po-circle_through-t2-eval-000163", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (1, -3) and (0, -4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (1, -3) and (0, -4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 9, 20", "ref": "-3, 9, 20", "family": "po_circle_through", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.00016, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[1, -3], [0, -4]], "B": 20}, "seed": 346326373002000168, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 4, "valid_answers": ["-3, 9, 20", "-2, 8, 16", "-1, 7, 12", "0, 6, 8", "1, 5, 4", "2, 4, 0", "3, 3, -4", "4, 2, -8", "5, 1, -12", "6, 0, -16", "7, -1, -20"]}, "cell": "circle_through/2", "stage": "B"}
+{"id": "po-circle_through-t2-eval-002347", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, 1) and (-4, 4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -20 <= D, E, F <= 20 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, 1) and (-4, 4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-15, -19, -16", "ref": "-15, -19, -16", "family": "po_circle_through", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.000145, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-2, 1], [-4, 4]], "B": 20}, "seed": 346326373002003037, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 8, "valid_answers": ["-15, -19, -16", "-12, -17, -12", "-9, -15, -8", "-6, -13, -4", "-3, -11, 0", "0, -9, 4", "3, -7, 8", "6, -5, 12", "9, -3, 16", "12, -1, 20"]}, "cell": "circle_through/2", "stage": "B"}
+{"id": "po-circle_through-t3-eval-001812", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (3, -4) and (-4, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (3, -4) and (-4, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -5, -30", "ref": "-5, -5, -30", "family": "po_circle_through", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 2.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[3, -4], [-4, 2]], "B": 30}, "seed": 346326373003002003, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 10, "valid_answers": ["-5, -5, -30", "1, 2, -20", "7, 9, -10", "13, 16, 0", "19, 23, 10", "25, 30, 20"]}, "cell": "circle_through/3", "stage": "B"}
+{"id": "po-circle_through-t3-eval-002399", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, 2) and (3, 3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, 2) and (3, 3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 13, -30", "ref": "-9, 13, -30", "family": "po_circle_through", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[0, 2], [3, 3]], "B": 30}, "seed": 346326373003002714, "version": "po-v0", "tier": 3, "n_raw": 11, "small": 4, "valid_answers": ["-9, 13, -30", "-8, 10, -24", "-7, 7, -18", "-6, 4, -12", "-5, 1, -6", "-4, -2, 0", "-3, -5, 6", "-2, -8, 12", "-1, -11, 18", "0, -14, 24", "1, -17, 30"]}, "cell": "circle_through/3", "stage": "B"}
+{"id": "po-circle_through-t3-eval-002127", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (2, -3) and (-1, -1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (2, -3) and (-1, -1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-13, -14, -29", "ref": "-13, -14, -29", "family": "po_circle_through", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 5.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[2, -3], [-1, -1]], "B": 30}, "seed": 346326373003002390, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 4, "valid_answers": ["-13, -14, -29", "-11, -11, -24", "-9, -8, -19", "-7, -5, -14", "-5, -2, -9", "-3, 1, -4", "-1, 4, 1", "1, 7, 6", "3, 10, 11", "5, 13, 16", "7, 16, 21", "9, 19, 26"]}, "cell": "circle_through/3", "stage": "B"}
+{"id": "po-circle_through-t3-eval-001354", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-5, -5) and (-5, -2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-5, -5) and (-5, -2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 7, -30", "ref": "-3, 7, -30", "family": "po_circle_through", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 5.7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-5, -5], [-5, -2]], "B": 30}, "seed": 346326373003001470, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 7, "valid_answers": ["-3, 7, -30", "-2, 7, -25", "-1, 7, -20", "0, 7, -15", "1, 7, -10", "2, 7, -5", "3, 7, 0", "4, 7, 5", "5, 7, 10", "6, 7, 15", "7, 7, 20", "8, 7, 25", "9, 7, 30"]}, "cell": "circle_through/3", "stage": "B"}
+{"id": "po-circle_through-t3-eval-001761", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, -3) and (0, 4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, -3) and (0, 4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-30, -1, -12", "ref": "-30, -1, -12", "family": "po_circle_through", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 61, "bucket": null, "guess_rate": 0.000269, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[0, -3], [0, 4]], "B": 30}, "seed": 346326373003001949, "version": "po-v0", "tier": 3, "n_raw": 61, "small": 12, "valid_answers": ["-30, -1, -12", "-29, -1, -12", "-28, -1, -12", "-27, -1, -12", "-26, -1, -12", "-25, -1, -12", "-24, -1, -12", "-23, -1, -12", "-22, -1, -12", "-21, -1, -12", "-20, -1, -12", "-19, -1, -12", "-18, -1, -12", "-17, -1, -12", "-16, -1, -12", "-15, -1, -12", "-14, -1, -12", "-13, -1, -12", "-12, -1, -12", "-11, -1, -12", "-10, -1, -12", "-9, -1, -12", "-8, -1, -12", "-7, -1, -12", "-6, -1, -12", "-5, -1, -12", "-4, -1, -12", "-3, -1, -12", "-2, -1, -12", "-1, -1, -12", "0, -1, -12", "1, -1, -12", "2, -1, -12", "3, -1, -12", "4, -1, -12", "5, -1, -12", "6, -1, -12", "7, -1, -12", "8, -1, -12", "9, -1, -12", "10, -1, -12", "11, -1, -12", "12, -1, -12", "13, -1, -12", "14, -1, -12", "15, -1, -12", "16, -1, -12", "17, -1, -12", "18, -1, -12", "19, -1, -12", "20, -1, -12", "21, -1, -12", "22, -1, -12", "23, -1, -12", "24, -1, -12", "25, -1, -12", "26, -1, -12", "27, -1, -12", "28, -1, -12", "29, -1, -12", "30, -1, -12"]}, "cell": "circle_through/3", "stage": "B"}
+{"id": "po-circle_through-t3-eval-000934", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (4, 5) and (-3, -1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (4, 5) and (-3, -1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 3, -28", "ref": "-7, 3, -28", "family": "po_circle_through", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 2.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[4, 5], [-3, -1]], "B": 30}, "seed": 346326373003000985, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 11, "valid_answers": ["-7, 3, -28", "-1, -4, -17", "5, -11, -6", "11, -18, 5", "17, -25, 16"]}, "cell": "circle_through/3", "stage": "B"}
+{"id": "po-circle_through-t3-eval-002167", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-3, -1) and (-1, 5). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-3, -1) and (-1, 5). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -1, -26", "ref": "-5, -1, -26", "family": "po_circle_through", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 3.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-3, -1], [-1, 5]], "B": 30}, "seed": 346326373003002433, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 4, "valid_answers": ["-5, -1, -26", "-2, -2, -18", "1, -3, -10", "4, -4, -2", "7, -5, 6", "10, -6, 14", "13, -7, 22", "16, -8, 30"]}, "cell": "circle_through/3", "stage": "B"}
+{"id": "po-circle_through-t3-eval-001822", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-4, 2) and (-3, -3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-4, 2) and (-3, -3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -1, -30", "ref": "-3, -1, -30", "family": "po_circle_through", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-4, 2], [-3, -3]], "B": 30}, "seed": 346326373003002016, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 7, "valid_answers": ["-3, -1, -30", "2, 0, -12", "7, 1, 6", "12, 2, 24"]}, "cell": "circle_through/3", "stage": "B"}
+{"id": "po-circle_through-t3-eval-001319", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, 0) and (2, 3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (0, 0) and (2, 3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-29, 15, 0", "ref": "-29, 15, 0", "family": "po_circle_through", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 8.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[0, 0], [2, 3]], "B": 30}, "seed": 346326373003001432, "version": "po-v0", "tier": 3, "n_raw": 20, "small": 3, "valid_answers": ["-29, 15, 0", "-26, 13, 0", "-23, 11, 0", "-20, 9, 0", "-17, 7, 0", "-14, 5, 0", "-11, 3, 0", "-8, 1, 0", "-5, -1, 0", "-2, -3, 0", "1, -5, 0", "4, -7, 0", "7, -9, 0", "10, -11, 0", "13, -13, 0", "16, -15, 0", "19, -17, 0", "22, -19, 0", "25, -21, 0", "28, -23, 0"]}, "cell": "circle_through/3", "stage": "B"}
+{"id": "po-circle_through-t3-eval-000683", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (5, -4) and (-1, -4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (5, -4) and (-1, -4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -2, -29", "ref": "-4, -2, -29", "family": "po_circle_through", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 6.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[5, -4], [-1, -4]], "B": 30}, "seed": 346326373003000715, "version": "po-v0", "tier": 3, "n_raw": 15, "small": 5, "valid_answers": ["-4, -2, -29", "-4, -1, -25", "-4, 0, -21", "-4, 1, -17", "-4, 2, -13", "-4, 3, -9", "-4, 4, -5", "-4, 5, -1", "-4, 6, 3", "-4, 7, 7", "-4, 8, 11", "-4, 9, 15", "-4, 10, 19", "-4, 11, 23", "-4, 12, 27"]}, "cell": "circle_through/3", "stage": "B"}
+{"id": "po-circle_through-t3-eval-001313", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, -2) and (-2, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-2, -2) and (-2, 1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, 1, -30", "ref": "-12, 1, -30", "family": "po_circle_through", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 31, "bucket": null, "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-2, -2], [-2, 1]], "B": 30}, "seed": 346326373003001423, "version": "po-v0", "tier": 3, "n_raw": 31, "small": 2, "valid_answers": ["-12, 1, -30", "-11, 1, -28", "-10, 1, -26", "-9, 1, -24", "-8, 1, -22", "-7, 1, -20", "-6, 1, -18", "-5, 1, -16", "-4, 1, -14", "-3, 1, -12", "-2, 1, -10", "-1, 1, -8", "0, 1, -6", "1, 1, -4", "2, 1, -2", "3, 1, 0", "4, 1, 2", "5, 1, 4", "6, 1, 6", "7, 1, 8", "8, 1, 10", "9, 1, 12", "10, 1, 14", "11, 1, 16", "12, 1, 18", "13, 1, 20", "14, 1, 22", "15, 1, 24", "16, 1, 26", "17, 1, 28", "18, 1, 30"]}, "cell": "circle_through/3", "stage": "B"}
+{"id": "po-circle_through-t3-eval-002068", "task": "cx_po_circle_through", "source": "cx_po_circle_through", "split": "eval", "problem": "Find integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-5, -5) and (-4, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers D, E, F with -30 <= D, E, F <= 30 such that x^2 + y^2 + Dx + Ey + F = 0 is a circle (of positive radius) passing through (-5, -5) and (-4, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{D = ..., E = ..., F = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 4, -20", "ref": "2, 4, -20", "family": "po_circle_through", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 9e-06, "solvable": true, "holdout": false, "meta": {"family": "po_circle_through", "rung": "witness", "params": {"pts": [[-5, -5], [-4, 2]], "B": 30}, "seed": 346326373003002321, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 10, "valid_answers": ["2, 4, -20", "9, 3, 10"]}, "cell": "circle_through/3", "stage": "B"}
+{"id": "po-consecutive_sum-t1-eval-000013", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 210. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 210. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 20", "ref": "1, 20", "family": "po_consecutive_sum", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.001667, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 210}, "seed": 342287044301000014, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 12, "valid_answers": ["1, 20", "7, 15", "12, 12", "27, 7", "40, 5", "51, 4", "69, 3"]}, "cell": "consecutive_sum/1", "stage": "B"}
+{"id": "po-consecutive_sum-t1-eval-000137", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 343. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 343. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "18, 14", "ref": "18, 14", "family": "po_consecutive_sum", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000336, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 343}, "seed": 342287044301000697, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 18, "valid_answers": ["18, 14", "46, 7", "171, 2"]}, "cell": "consecutive_sum/1", "stage": "B"}
+{"id": "po-consecutive_sum-t1-eval-000088", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 972. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 972. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "23, 27", "ref": "23, 27", "family": "po_consecutive_sum", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000117, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 972}, "seed": 342287044301000138, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 27, "valid_answers": ["23, 27", "29, 24", "104, 9", "118, 8", "323, 3"]}, "cell": "consecutive_sum/1", "stage": "B"}
+{"id": "po-consecutive_sum-t1-eval-000051", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 2376. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 2376. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "26, 48", "ref": "26, 48", "family": "po_consecutive_sum", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 4.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 2376}, "seed": 342287044301000066, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 48, "valid_answers": ["26, 48", "56, 33", "75, 27", "141, 16", "211, 11", "260, 9", "791, 3"]}, "cell": "consecutive_sum/1", "stage": "B"}
+{"id": "po-consecutive_sum-t1-eval-000133", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 216. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 216. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 16", "ref": "6, 16", "family": "po_consecutive_sum", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000694, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 216}, "seed": 342287044301000563, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 16, "valid_answers": ["6, 16", "20, 9", "71, 3"]}, "cell": "consecutive_sum/1", "stage": "B"}
+{"id": "po-consecutive_sum-t1-eval-000108", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 1782. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 1782. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "19, 44", "ref": "19, 44", "family": "po_consecutive_sum", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 8.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 1782}, "seed": 342287044301000213, "version": "po-v0", "tier": 1, "n_raw": 9, "small": 36, "valid_answers": ["19, 44", "32, 36", "38, 33", "53, 27", "143, 12", "157, 11", "194, 9", "444, 4", "593, 3"]}, "cell": "consecutive_sum/1", "stage": "B"}
+{"id": "po-consecutive_sum-t1-eval-000011", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 175. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 175. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 14", "ref": "6, 14", "family": "po_consecutive_sum", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.001587, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 175}, "seed": 342287044301000011, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 13, "valid_answers": ["6, 14", "13, 10", "22, 7", "33, 5", "87, 2"]}, "cell": "consecutive_sum/1", "stage": "B"}
+{"id": "po-consecutive_sum-t1-eval-000007", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 770. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 770. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 35", "ref": "5, 35", "family": "po_consecutive_sum", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000233, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 770}, "seed": 342287044301000007, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 28, "valid_answers": ["5, 35", "14, 28", "29, 20", "65, 11", "107, 7", "152, 5", "191, 4"]}, "cell": "consecutive_sum/1", "stage": "B"}
+{"id": "po-consecutive_sum-t1-eval-000009", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 275. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 275. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 22", "ref": "2, 22", "family": "po_consecutive_sum", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000791, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 275}, "seed": 342287044301000009, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 20, "valid_answers": ["2, 22", "20, 11", "23, 10", "53, 5", "137, 2"]}, "cell": "consecutive_sum/1", "stage": "B"}
+{"id": "po-consecutive_sum-t1-eval-000073", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 660. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 660. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 33", "ref": "4, 33", "family": "po_consecutive_sum", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000295, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 660}, "seed": 342287044301000099, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 24, "valid_answers": ["4, 33", "16, 24", "37, 15", "55, 11", "79, 8", "130, 5", "219, 3"]}, "cell": "consecutive_sum/1", "stage": "B"}
+{"id": "po-consecutive_sum-t1-eval-000096", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 63. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 63. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 9", "ref": "3, 9", "family": "po_consecutive_sum", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.007215, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 63}, "seed": 342287044301000157, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 7, "valid_answers": ["3, 9", "6, 7", "8, 6", "20, 3", "31, 2"]}, "cell": "consecutive_sum/1", "stage": "B"}
+{"id": "po-consecutive_sum-t2-eval-000221", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 88088. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 88088. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "320, 208", "ref": "320, 208", "family": "po_consecutive_sum", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 88088}, "seed": 342287044302000609, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 320, "valid_answers": ["320, 208", "413, 176", "545, 143", "668, 121", "731, 112", "923, 91", "1106, 77", "5498, 16", "6770, 13", "8003, 11", "12581, 7"]}, "cell": "consecutive_sum/2", "stage": "B"}
+{"id": "po-consecutive_sum-t2-eval-000230", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 2401. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 2401. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "25, 49", "ref": "25, 49", "family": "po_consecutive_sum", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 2401}, "seed": 342287044302000733, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 49, "valid_answers": ["25, 49", "165, 14", "340, 7", "1200, 2"]}, "cell": "consecutive_sum/2", "stage": "B"}
+{"id": "po-consecutive_sum-t2-eval-000115", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 3250. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 3250. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "18, 65", "ref": "18, 65", "family": "po_consecutive_sum", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 2.7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 3250}, "seed": 342287044302000160, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 52, "valid_answers": ["18, 65", "37, 52", "118, 25", "153, 20", "244, 13", "648, 5", "811, 4"]}, "cell": "consecutive_sum/2", "stage": "B"}
+{"id": "po-consecutive_sum-t2-eval-000078", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 15400. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 15400. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 175", "ref": "1, 175", "family": "po_consecutive_sum", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 15400}, "seed": 342287044302000097, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 112, "valid_answers": ["1, 175", "82, 112", "153, 80", "162, 77", "253, 55", "423, 35", "604, 25", "955, 16", "1395, 11", "2197, 7", "3078, 5"]}, "cell": "consecutive_sum/2", "stage": "B"}
+{"id": "po-consecutive_sum-t2-eval-000090", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 24024. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 24024. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "12, 208", "ref": "12, 208", "family": "po_consecutive_sum", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 3e-06, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 24024}, "seed": 342287044302000120, "version": "po-v0", "tier": 2, "n_raw": 15, "small": 143, "valid_answers": ["12, 208", "49, 176", "97, 143", "159, 112", "219, 91", "274, 77", "477, 48", "597, 39", "712, 33", "1134, 21", "1494, 16", "1842, 13", "2179, 11", "3429, 7", "8007, 3"]}, "cell": "consecutive_sum/2", "stage": "B"}
+{"id": "po-consecutive_sum-t2-eval-000015", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 4290. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 4290. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "34, 65", "ref": "34, 65", "family": "po_consecutive_sum", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 3.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 4290}, "seed": 342287044302000015, "version": "po-v0", "tier": 2, "n_raw": 15, "small": 55, "valid_answers": ["34, 65", "42, 60", "51, 55", "57, 52", "76, 44", "91, 39", "114, 33", "205, 20", "279, 15", "324, 13", "352, 12", "385, 11", "856, 5", "1071, 4", "1429, 3"]}, "cell": "consecutive_sum/2", "stage": "B"}
+{"id": "po-consecutive_sum-t2-eval-000094", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 20328. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 20328. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "28, 176", "ref": "28, 176", "family": "po_consecutive_sum", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 3e-06, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 20328}, "seed": 342287044302000128, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 121, "valid_answers": ["28, 176", "108, 121", "126, 112", "226, 77", "400, 48", "600, 33", "958, 21", "1263, 16", "1843, 11", "2901, 7", "6775, 3"]}, "cell": "consecutive_sum/2", "stage": "B"}
+{"id": "po-consecutive_sum-t2-eval-000058", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 2145. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 2145. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 65", "ref": "1, 65", "family": "po_consecutive_sum", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 0.000108, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 2145}, "seed": 342287044302000070, "version": "po-v0", "tier": 2, "n_raw": 15, "small": 39, "valid_answers": ["1, 65", "12, 55", "36, 39", "49, 33", "57, 30", "70, 26", "87, 22", "136, 15", "159, 13", "190, 11", "210, 10", "355, 6", "427, 5", "714, 3", "1072, 2"]}, "cell": "consecutive_sum/2", "stage": "B"}
+{"id": "po-consecutive_sum-t2-eval-000089", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 11154. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 11154. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 143", "ref": "7, 143", "family": "po_consecutive_sum", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 7e-06, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 11154}, "seed": 342287044302000119, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 132, "valid_answers": ["7, 143", "19, 132", "189, 52", "232, 44", "267, 39", "322, 33", "852, 13", "924, 12", "1009, 11", "2787, 4", "3717, 3"]}, "cell": "consecutive_sum/2", "stage": "B"}
+{"id": "po-consecutive_sum-t2-eval-000127", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 10164. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 10164. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "24, 121", "ref": "24, 121", "family": "po_consecutive_sum", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 8e-06, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 10164}, "seed": 342287044302000190, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 88, "valid_answers": ["24, 121", "72, 88", "94, 77", "154, 56", "292, 33", "412, 24", "474, 21", "919, 11", "1267, 8", "1449, 7", "3387, 3"]}, "cell": "consecutive_sum/2", "stage": "B"}
+{"id": "po-consecutive_sum-t2-eval-000067", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 6050. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 6050. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 100", "ref": "11, 100", "family": "po_consecutive_sum", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 1.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 6050}, "seed": 342287044302000082, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 83, "valid_answers": ["11, 100", "83, 55", "116, 44", "230, 25", "293, 20", "545, 11", "1208, 5", "1511, 4"]}, "cell": "consecutive_sum/2", "stage": "B"}
+{"id": "po-consecutive_sum-t2-eval-000270", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 28561. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 28561. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "85, 169", "ref": "85, 169", "family": "po_consecutive_sum", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 28561}, "seed": 342287044302003238, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 169, "valid_answers": ["85, 169", "1086, 26", "2191, 13", "14280, 2"]}, "cell": "consecutive_sum/2", "stage": "B"}
+{"id": "po-consecutive_sum-t3-eval-001146", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 311542. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 311542. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "43, 748", "ref": "43, 748", "family": "po_consecutive_sum", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 17, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 311542}, "seed": 342287044303002420, "version": "po-v0", "tier": 3, "n_raw": 17, "small": 476, "valid_answers": ["43, 748", "309, 539", "417, 476", "858, 308", "934, 289", "1492, 196", "1573, 187", "2559, 119", "4008, 77", "4548, 68", "6334, 49", "7059, 44", "11113, 28", "18318, 17", "28317, 11", "44503, 7", "77884, 4"]}, "cell": "consecutive_sum/3", "stage": "B"}
+{"id": "po-consecutive_sum-t3-eval-000171", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 60515. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 60515. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "95, 266", "ref": "95, 266", "family": "po_consecutive_sum", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 23, "bucket": null, "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 60515}, "seed": 342287044303000184, "version": "po-v0", "tier": 3, "n_raw": 23, "small": 224, "valid_answers": ["95, 266", "122, 247", "125, 245", "224, 190", "242, 182", "389, 133", "401, 130", "569, 98", "590, 95", "620, 91", "830, 70", "899, 65", "1211, 49", "1574, 38", "1712, 35", "2315, 26", "3176, 19", "4316, 14", "4649, 13", "6047, 10", "8642, 7", "12101, 5", "30257, 2"]}, "cell": "consecutive_sum/3", "stage": "B"}
+{"id": "po-consecutive_sum-t3-eval-000252", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 81396. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 81396. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 399", "ref": "5, 399", "family": "po_consecutive_sum", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 23, "bucket": null, "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 81396}, "seed": 342287044303000274, "version": "po-v0", "tier": 3, "n_raw": 23, "small": 323, "valid_answers": ["5, 399", "50, 357", "91, 323", "391, 171", "401, 168", "456, 153", "460, 152", "531, 136", "546, 133", "625, 119", "1095, 72", "1261, 63", "1400, 57", "1426, 56", "1571, 51", "3380, 24", "3866, 21", "4275, 19", "4780, 17", "9040, 9", "10171, 8", "11625, 7", "27131, 3"]}, "cell": "consecutive_sum/3", "stage": "B"}
+{"id": "po-consecutive_sum-t3-eval-000344", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 135660. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 135660. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "70, 456", "ref": "70, 456", "family": "po_consecutive_sum", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 31, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 135660}, "seed": 342287044303000395, "version": "po-v0", "tier": 3, "n_raw": 31, "small": 323, "valid_answers": ["70, 456", "129, 408", "141, 399", "202, 357", "259, 323", "334, 285", "345, 280", "405, 255", "724, 168", "817, 152", "930, 136", "954, 133", "1071, 120", "1081, 119", "1240, 105", "1381, 95", "1554, 85", "2352, 57", "2395, 56", "2635, 51", "3372, 40", "3859, 35", "5641, 24", "6450, 21", "7131, 19", "7972, 17", "9037, 15", "16954, 8", "19377, 7", "27130, 5", "45219, 3"]}, "cell": "consecutive_sum/3", "stage": "B"}
+{"id": "po-consecutive_sum-t3-eval-000531", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 67881. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 67881. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 363", "ref": "6, 363", "family": "po_consecutive_sum", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 67881}, "seed": 342287044303000685, "version": "po-v0", "tier": 3, "n_raw": 15, "small": 242, "valid_answers": ["6, 363", "160, 242", "270, 187", "501, 121", "615, 102", "996, 66", "1306, 51", "1980, 34", "2041, 33", "3075, 22", "3985, 17", "6166, 11", "11311, 6", "22626, 3", "33940, 2"]}, "cell": "consecutive_sum/3", "stage": "B"}
+{"id": "po-consecutive_sum-t3-eval-000120", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 135850. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 135850. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "49, 475", "ref": "49, 475", "family": "po_consecutive_sum", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 23, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 135850}, "seed": 342287044303000126, "version": "po-v0", "tier": 3, "n_raw": 23, "small": 325, "valid_answers": ["49, 475", "168, 380", "256, 325", "357, 275", "393, 260", "427, 247", "508, 220", "546, 209", "879, 143", "1309, 100", "1383, 95", "1750, 76", "2058, 65", "2443, 55", "2587, 52", "3066, 44", "5422, 25", "6783, 20", "7141, 19", "10444, 13", "12345, 11", "27168, 5", "33961, 4"]}, "cell": "consecutive_sum/3", "stage": "B"}
+{"id": "po-consecutive_sum-t3-eval-001630", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 377245. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 377245. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "162, 722", "ref": "162, 722", "family": "po_consecutive_sum", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 377245}, "seed": 342287044303008186, "version": "po-v0", "tier": 3, "n_raw": 15, "small": 694, "valid_answers": ["162, 722", "694, 418", "865, 361", "1701, 209", "1891, 190", "3375, 110", "3924, 95", "6832, 55", "9909, 38", "17137, 22", "19846, 19", "34290, 11", "37720, 10", "75447, 5", "188622, 2"]}, "cell": "consecutive_sum/3", "stage": "B"}
+{"id": "po-consecutive_sum-t3-eval-001791", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 1982251. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 1982251. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2385, 722", "ref": "2385, 722", "family": "po_consecutive_sum", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 1982251}, "seed": 342287044303020137, "version": "po-v0", "tier": 3, "n_raw": 11, "small": 2385, "valid_answers": ["2385, 722", "2746, 646", "3141, 578", "5311, 361", "5976, 323", "6715, 289", "52146, 38", "58285, 34", "104320, 19", "116595, 17", "991125, 2"]}, "cell": "consecutive_sum/3", "stage": "B"}
+{"id": "po-consecutive_sum-t3-eval-001490", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 175692. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 175692. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "303, 363", "ref": "303, 363", "family": "po_consecutive_sum", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 175692}, "seed": 342287044303004947, "version": "po-v0", "tier": 3, "n_raw": 9, "small": 363, "valid_answers": ["303, 363", "534, 264", "1392, 121", "1953, 88", "5308, 33", "7309, 24", "15967, 11", "21958, 8", "58563, 3"]}, "cell": "consecutive_sum/3", "stage": "B"}
+{"id": "po-consecutive_sum-t3-eval-001722", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 7803. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 7803. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "26, 102", "ref": "26, 102", "family": "po_consecutive_sum", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 1.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 7803}, "seed": 342287044303012114, "version": "po-v0", "tier": 3, "n_raw": 11, "small": 102, "valid_answers": ["26, 102", "118, 54", "128, 51", "213, 34", "276, 27", "425, 18", "451, 17", "863, 9", "1298, 6", "2600, 3", "3901, 2"]}, "cell": "consecutive_sum/3", "stage": "B"}
+{"id": "po-consecutive_sum-t3-eval-000203", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 24990. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 24990. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "21, 204", "ref": "21, 204", "family": "po_consecutive_sum", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 23, "bucket": null, "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 24990}, "seed": 342287044303000224, "version": "po-v0", "tier": 3, "n_raw": 23, "small": 140, "valid_answers": ["21, 204", "30, 196", "97, 147", "109, 140", "151, 119", "186, 105", "252, 85", "256, 84", "334, 68", "387, 60", "465, 51", "486, 49", "697, 35", "879, 28", "1180, 21", "1240, 20", "1462, 17", "1659, 15", "2077, 12", "3567, 7", "4996, 5", "6246, 4", "8329, 3"]}, "cell": "consecutive_sum/3", "stage": "B"}
+{"id": "po-consecutive_sum-t3-eval-000088", "task": "cx_po_consecutive_sum", "source": "cx_po_consecutive_sum", "split": "eval", "problem": "Find positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 101400. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, n with n >= 2 such that a + (a + 1) + ... + (a + n - 1) = 101400. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "54, 400", "ref": "54, 400", "family": "po_consecutive_sum", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 17, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 101400}, "seed": 342287044303000093, "version": "po-v0", "tier": 3, "n_raw": 17, "small": 303, "valid_answers": ["54, 400", "150, 325", "303, 240", "384, 208", "423, 195", "516, 169", "1228, 80", "1315, 75", "1528, 65", "2089, 48", "2581, 39", "4044, 25", "6330, 16", "6753, 15", "7794, 13", "20278, 5", "33799, 3"]}, "cell": "consecutive_sum/3", "stage": "B"}
+{"id": "po-crt_range-t0-eval-000693", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1703 <= x <= 2524 such that x \\equiv 2 \\pmod{11}, x \\equiv 3 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1703 <= x <= 2524 such that x \\equiv 2 \\pmod{11}, x \\equiv 3 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1718", "ref": "1718", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.013382, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[2, 11], [3, 7]], "L": 1703, "U": 2524}, "seed": 401691043500000987, "version": "po-v0", "tier": 0, "n_raw": 11, "small": 1718, "valid_answers": ["1718", "1795", "1872", "1949", "2026", "2103", "2180", "2257", "2334", "2411", "2488"]}, "cell": "crt_range/0", "stage": "A"}
+{"id": "po-crt_range-t0-eval-000995", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1324 <= x <= 1718 such that x \\equiv 6 \\pmod{11}, x \\equiv 6 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1324 <= x <= 1718 such that x \\equiv 6 \\pmod{11}, x \\equiv 6 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1392", "ref": "1392", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.012658, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[6, 11], [6, 7]], "L": 1324, "U": 1718}, "seed": 401691043500001398, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 1392, "valid_answers": ["1392", "1469", "1546", "1623", "1700"]}, "cell": "crt_range/0", "stage": "A"}
+{"id": "po-crt_range-t0-eval-000654", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 984 <= x <= 1210 such that x \\equiv 8 \\pmod{11}, x \\equiv 5 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 984 <= x <= 1210 such that x \\equiv 8 \\pmod{11}, x \\equiv 5 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1031", "ref": "1031", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.008811, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[8, 11], [5, 9]], "L": 984, "U": 1210}, "seed": 401691043500000932, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 1031, "valid_answers": ["1031", "1130"]}, "cell": "crt_range/0", "stage": "A"}
+{"id": "po-crt_range-t0-eval-002039", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1654 <= x <= 2142 such that x \\equiv 3 \\pmod{7}, x \\equiv 10 \\pmod{11}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1654 <= x <= 2142 such that x \\equiv 3 \\pmod{7}, x \\equiv 10 \\pmod{11}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1704", "ref": "1704", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.01227, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[3, 7], [10, 11]], "L": 1654, "U": 2142}, "seed": 401691043500002895, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 1704, "valid_answers": ["1704", "1781", "1858", "1935", "2012", "2089"]}, "cell": "crt_range/0", "stage": "A"}
+{"id": "po-crt_range-t0-eval-000508", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 488 <= x <= 694 such that x \\equiv 9 \\pmod{11}, x \\equiv 0 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 488 <= x <= 694 such that x \\equiv 9 \\pmod{11}, x \\equiv 0 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "504", "ref": "504", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.009662, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[9, 11], [0, 9]], "L": 488, "U": 694}, "seed": 401691043500000723, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 504, "valid_answers": ["504", "603"]}, "cell": "crt_range/0", "stage": "A"}
+{"id": "po-crt_range-t0-eval-001951", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1447 <= x <= 2062 such that x \\equiv 2 \\pmod{11}, x \\equiv 3 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1447 <= x <= 2062 such that x \\equiv 2 \\pmod{11}, x \\equiv 3 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1487", "ref": "1487", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.012987, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[2, 11], [3, 7]], "L": 1447, "U": 2062}, "seed": 401691043500002767, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 1487, "valid_answers": ["1487", "1564", "1641", "1718", "1795", "1872", "1949", "2026"]}, "cell": "crt_range/0", "stage": "A"}
+{"id": "po-crt_range-t0-eval-000618", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1711 <= x <= 2114 such that x \\equiv 1 \\pmod{8}, x \\equiv 1 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1711 <= x <= 2114 such that x \\equiv 1 \\pmod{8}, x \\equiv 1 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1737", "ref": "1737", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.017327, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[1, 8], [1, 7]], "L": 1711, "U": 2114}, "seed": 401691043500000882, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 1737, "valid_answers": ["1737", "1793", "1849", "1905", "1961", "2017", "2073"]}, "cell": "crt_range/0", "stage": "A"}
+{"id": "po-crt_range-t0-eval-000241", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1753 <= x <= 2755 such that x \\equiv 3 \\pmod{11}, x \\equiv 5 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1753 <= x <= 2755 such that x \\equiv 3 \\pmod{11}, x \\equiv 5 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1796", "ref": "1796", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.00997, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[3, 11], [5, 9]], "L": 1753, "U": 2755}, "seed": 401691043500000346, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 1796, "valid_answers": ["1796", "1895", "1994", "2093", "2192", "2291", "2390", "2489", "2588", "2687"]}, "cell": "crt_range/0", "stage": "A"}
+{"id": "po-crt_range-t0-eval-000160", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 269 <= x <= 783 such that x \\equiv 3 \\pmod{9}, x \\equiv 6 \\pmod{11}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 269 <= x <= 783 such that x \\equiv 3 \\pmod{9}, x \\equiv 6 \\pmod{11}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "336", "ref": "336", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.009709, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[3, 9], [6, 11]], "L": 269, "U": 783}, "seed": 401691043500000233, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 336, "valid_answers": ["336", "435", "534", "633", "732"]}, "cell": "crt_range/0", "stage": "A"}
+{"id": "po-crt_range-t0-eval-001029", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1811 <= x <= 2053 such that x \\equiv 0 \\pmod{11}, x \\equiv 5 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1811 <= x <= 2053 such that x \\equiv 0 \\pmod{11}, x \\equiv 5 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1881", "ref": "1881", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.012346, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[0, 11], [5, 7]], "L": 1811, "U": 2053}, "seed": 401691043500001449, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 1881, "valid_answers": ["1881", "1958", "2035"]}, "cell": "crt_range/0", "stage": "A"}
+{"id": "po-crt_range-t0-eval-000981", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 637 <= x <= 1695 such that x \\equiv 7 \\pmod{8}, x \\equiv 3 \\pmod{11}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 637 <= x <= 1695 such that x \\equiv 7 \\pmod{8}, x \\equiv 3 \\pmod{11}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "663", "ref": "663", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.011331, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[7, 8], [3, 11]], "L": 637, "U": 1695}, "seed": 401691043500001374, "version": "po-v0", "tier": 0, "n_raw": 12, "small": 663, "valid_answers": ["663", "751", "839", "927", "1015", "1103", "1191", "1279", "1367", "1455", "1543", "1631"]}, "cell": "crt_range/0", "stage": "A"}
+{"id": "po-crt_range-t0-eval-000691", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 680 <= x <= 865 such that x \\equiv 0 \\pmod{7}, x \\equiv 1 \\pmod{8}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 680 <= x <= 865 such that x \\equiv 0 \\pmod{7}, x \\equiv 1 \\pmod{8}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "721", "ref": "721", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.016129, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[0, 7], [1, 8]], "L": 680, "U": 865}, "seed": 401691043500000983, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 721, "valid_answers": ["721", "777", "833"]}, "cell": "crt_range/0", "stage": "A"}
+{"id": "po-crt_range-t1-eval-001276", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 861 <= x <= 2049 such that x \\equiv 1 \\pmod{11}, x \\equiv 8 \\pmod{13}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 861 <= x <= 2049 such that x \\equiv 1 \\pmod{11}, x \\equiv 8 \\pmod{13}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "892", "ref": "892", "family": "po_crt_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.007569, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[1, 11], [8, 13]], "L": 861, "U": 2049}, "seed": 401691043501001276, "version": "po-v0", "tier": 1, "n_raw": 9, "small": 892, "valid_answers": ["892", "1035", "1178", "1321", "1464", "1607", "1750", "1893", "2036"]}, "cell": "crt_range/1", "stage": "C"}
+{"id": "po-crt_range-t1-eval-000608", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1660 <= x <= 2276 such that x \\equiv 1 \\pmod{16}, x \\equiv 10 \\pmod{11}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1660 <= x <= 2276 such that x \\equiv 1 \\pmod{16}, x \\equiv 10 \\pmod{11}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1825", "ref": "1825", "family": "po_crt_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.004862, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[1, 16], [10, 11]], "L": 1660, "U": 2276}, "seed": 401691043501000608, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 1825, "valid_answers": ["1825", "2001", "2177"]}, "cell": "crt_range/1", "stage": "C"}
+{"id": "po-crt_range-t1-eval-001951", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1971 <= x <= 4292 such that x \\equiv 14 \\pmod{16}, x \\equiv 1 \\pmod{13}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1971 <= x <= 4292 such that x \\equiv 14 \\pmod{16}, x \\equiv 1 \\pmod{13}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2094", "ref": "2094", "family": "po_crt_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.004737, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[14, 16], [1, 13]], "L": 1971, "U": 4292}, "seed": 401691043501001951, "version": "po-v0", "tier": 1, "n_raw": 11, "small": 2094, "valid_answers": ["2094", "2302", "2510", "2718", "2926", "3134", "3342", "3550", "3758", "3966", "4174"]}, "cell": "crt_range/1", "stage": "C"}
+{"id": "po-crt_range-t1-eval-002227", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1977 <= x <= 2873 such that x \\equiv 5 \\pmod{9}, x \\equiv 6 \\pmod{16}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1977 <= x <= 2873 such that x \\equiv 5 \\pmod{9}, x \\equiv 6 \\pmod{16}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2102", "ref": "2102", "family": "po_crt_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.006689, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[5, 9], [6, 16]], "L": 1977, "U": 2873}, "seed": 401691043501002227, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 2102, "valid_answers": ["2102", "2246", "2390", "2534", "2678", "2822"]}, "cell": "crt_range/1", "stage": "C"}
+{"id": "po-crt_range-t1-eval-000179", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1767 <= x <= 2625 such that x \\equiv 13 \\pmod{16}, x \\equiv 4 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1767 <= x <= 2625 such that x \\equiv 13 \\pmod{16}, x \\equiv 4 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1789", "ref": "1789", "family": "po_crt_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.009313, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[13, 16], [4, 7]], "L": 1767, "U": 2625}, "seed": 401691043501000179, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 1789, "valid_answers": ["1789", "1901", "2013", "2125", "2237", "2349", "2461", "2573"]}, "cell": "crt_range/1", "stage": "C"}
+{"id": "po-crt_range-t1-eval-001864", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 127 <= x <= 1470 such that x \\equiv 15 \\pmod{16}, x \\equiv 5 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 127 <= x <= 1470 such that x \\equiv 15 \\pmod{16}, x \\equiv 5 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "159", "ref": "159", "family": "po_crt_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.008929, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[15, 16], [5, 7]], "L": 127, "U": 1470}, "seed": 401691043501001864, "version": "po-v0", "tier": 1, "n_raw": 12, "small": 159, "valid_answers": ["159", "271", "383", "495", "607", "719", "831", "943", "1055", "1167", "1279", "1391"]}, "cell": "crt_range/1", "stage": "C"}
+{"id": "po-crt_range-t1-eval-000054", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 534 <= x <= 1002 such that x \\equiv 15 \\pmod{16}, x \\equiv 9 \\pmod{13}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 534 <= x <= 1002 such that x \\equiv 15 \\pmod{16}, x \\equiv 9 \\pmod{13}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "607", "ref": "607", "family": "po_crt_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004264, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[15, 16], [9, 13]], "L": 534, "U": 1002}, "seed": 401691043501000054, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 607, "valid_answers": ["607", "815"]}, "cell": "crt_range/1", "stage": "C"}
+{"id": "po-crt_range-t1-eval-001454", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 197 <= x <= 536 such that x \\equiv 4 \\pmod{13}, x \\equiv 5 \\pmod{11}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 197 <= x <= 536 such that x \\equiv 4 \\pmod{13}, x \\equiv 5 \\pmod{11}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "225", "ref": "225", "family": "po_crt_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.008824, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[4, 13], [5, 11]], "L": 197, "U": 536}, "seed": 401691043501001454, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 225, "valid_answers": ["225", "368", "511"]}, "cell": "crt_range/1", "stage": "C"}
+{"id": "po-crt_range-t1-eval-001155", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 290 <= x <= 1182 such that x \\equiv 7 \\pmod{11}, x \\equiv 2 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 290 <= x <= 1182 such that x \\equiv 7 \\pmod{11}, x \\equiv 2 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "359", "ref": "359", "family": "po_crt_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.012318, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[7, 11], [2, 7]], "L": 290, "U": 1182}, "seed": 401691043501001155, "version": "po-v0", "tier": 1, "n_raw": 11, "small": 359, "valid_answers": ["359", "436", "513", "590", "667", "744", "821", "898", "975", "1052", "1129"]}, "cell": "crt_range/1", "stage": "C"}
+{"id": "po-crt_range-t1-eval-001734", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 487 <= x <= 1508 such that x \\equiv 8 \\pmod{9}, x \\equiv 6 \\pmod{13}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 487 <= x <= 1508 such that x \\equiv 8 \\pmod{9}, x \\equiv 6 \\pmod{13}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "539", "ref": "539", "family": "po_crt_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.008806, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[8, 9], [6, 13]], "L": 487, "U": 1508}, "seed": 401691043501001734, "version": "po-v0", "tier": 1, "n_raw": 9, "small": 539, "valid_answers": ["539", "656", "773", "890", "1007", "1124", "1241", "1358", "1475"]}, "cell": "crt_range/1", "stage": "C"}
+{"id": "po-crt_range-t1-eval-000290", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1118 <= x <= 1815 such that x \\equiv 1 \\pmod{16}, x \\equiv 3 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1118 <= x <= 1815 such that x \\equiv 1 \\pmod{16}, x \\equiv 3 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1137", "ref": "1137", "family": "po_crt_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.007163, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[1, 16], [3, 9]], "L": 1118, "U": 1815}, "seed": 401691043501000290, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 1137, "valid_answers": ["1137", "1281", "1425", "1569", "1713"]}, "cell": "crt_range/1", "stage": "C"}
+{"id": "po-crt_range-t1-eval-002301", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1599 <= x <= 3126 such that x \\equiv 6 \\pmod{9}, x \\equiv 1 \\pmod{16}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1599 <= x <= 3126 such that x \\equiv 6 \\pmod{9}, x \\equiv 1 \\pmod{16}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1617", "ref": "1617", "family": "po_crt_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.007199, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[6, 9], [1, 16]], "L": 1599, "U": 3126}, "seed": 401691043501002301, "version": "po-v0", "tier": 1, "n_raw": 11, "small": 1617, "valid_answers": ["1617", "1761", "1905", "2049", "2193", "2337", "2481", "2625", "2769", "2913", "3057"]}, "cell": "crt_range/1", "stage": "C"}
+{"id": "po-crt_range-t2-eval-000644", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 772 <= x <= 2297 such that x \\equiv 6 \\pmod{7}, x \\equiv 0 \\pmod{9}, x \\equiv 3 \\pmod{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 772 <= x <= 2297 such that x \\equiv 6 \\pmod{7}, x \\equiv 0 \\pmod{9}, x \\equiv 3 \\pmod{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "783", "ref": "783", "family": "po_crt_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.004587, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[6, 7], [0, 9], [3, 4]], "L": 772, "U": 2297}, "seed": 401691043502000644, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 783, "valid_answers": ["783", "1035", "1287", "1539", "1791", "2043", "2295"]}, "cell": "crt_range/2", "stage": "B"}
+{"id": "po-crt_range-t2-eval-000176", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1073 <= x <= 3462 such that x \\equiv 2 \\pmod{4}, x \\equiv 3 \\pmod{7}, x \\equiv 7 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1073 <= x <= 3462 such that x \\equiv 2 \\pmod{4}, x \\equiv 3 \\pmod{7}, x \\equiv 7 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1186", "ref": "1186", "family": "po_crt_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.004184, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[2, 4], [3, 7], [7, 9]], "L": 1073, "U": 3462}, "seed": 401691043502000176, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 1186, "valid_answers": ["1186", "1438", "1690", "1942", "2194", "2446", "2698", "2950", "3202", "3454"]}, "cell": "crt_range/2", "stage": "B"}
+{"id": "po-crt_range-t2-eval-000562", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1361 <= x <= 4236 such that x \\equiv 4 \\pmod{11}, x \\equiv 0 \\pmod{4}, x \\equiv 0 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1361 <= x <= 4236 such that x \\equiv 4 \\pmod{11}, x \\equiv 0 \\pmod{4}, x \\equiv 0 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1456", "ref": "1456", "family": "po_crt_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.003477, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[4, 11], [0, 4], [0, 7]], "L": 1361, "U": 4236}, "seed": 401691043502000562, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 1456, "valid_answers": ["1456", "1764", "2072", "2380", "2688", "2996", "3304", "3612", "3920", "4228"]}, "cell": "crt_range/2", "stage": "B"}
+{"id": "po-crt_range-t2-eval-000181", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 119 <= x <= 1697 such that x \\equiv 6 \\pmod{7}, x \\equiv 1 \\pmod{4}, x \\equiv 1 \\pmod{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 119 <= x <= 1697 such that x \\equiv 6 \\pmod{7}, x \\equiv 1 \\pmod{4}, x \\equiv 1 \\pmod{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "181", "ref": "181", "family": "po_crt_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.006966, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[6, 7], [1, 4], [1, 5]], "L": 119, "U": 1697}, "seed": 401691043502000181, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 181, "valid_answers": ["181", "321", "461", "601", "741", "881", "1021", "1161", "1301", "1441", "1581"]}, "cell": "crt_range/2", "stage": "B"}
+{"id": "po-crt_range-t2-eval-001559", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1119 <= x <= 2403 such that x \\equiv 2 \\pmod{4}, x \\equiv 3 \\pmod{5}, x \\equiv 7 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1119 <= x <= 2403 such that x \\equiv 2 \\pmod{4}, x \\equiv 3 \\pmod{5}, x \\equiv 7 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1258", "ref": "1258", "family": "po_crt_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.005447, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[2, 4], [3, 5], [7, 9]], "L": 1119, "U": 2403}, "seed": 401691043502001559, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 1258, "valid_answers": ["1258", "1438", "1618", "1798", "1978", "2158", "2338"]}, "cell": "crt_range/2", "stage": "B"}
+{"id": "po-crt_range-t2-eval-000776", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1824 <= x <= 3102 such that x \\equiv 5 \\pmod{11}, x \\equiv 0 \\pmod{4}, x \\equiv 3 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1824 <= x <= 3102 such that x \\equiv 5 \\pmod{11}, x \\equiv 0 \\pmod{4}, x \\equiv 3 \\pmod{7}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2040", "ref": "2040", "family": "po_crt_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.003127, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[5, 11], [0, 4], [3, 7]], "L": 1824, "U": 3102}, "seed": 401691043502000776, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 2040, "valid_answers": ["2040", "2348", "2656", "2964"]}, "cell": "crt_range/2", "stage": "B"}
+{"id": "po-crt_range-t2-eval-000884", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1524 <= x <= 3925 such that x \\equiv 3 \\pmod{4}, x \\equiv 9 \\pmod{11}, x \\equiv 1 \\pmod{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1524 <= x <= 3925 such that x \\equiv 3 \\pmod{4}, x \\equiv 9 \\pmod{11}, x \\equiv 1 \\pmod{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1571", "ref": "1571", "family": "po_crt_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.00458, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[3, 4], [9, 11], [1, 5]], "L": 1524, "U": 3925}, "seed": 401691043502000884, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 1571, "valid_answers": ["1571", "1791", "2011", "2231", "2451", "2671", "2891", "3111", "3331", "3551", "3771"]}, "cell": "crt_range/2", "stage": "B"}
+{"id": "po-crt_range-t2-eval-001101", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1868 <= x <= 5224 such that x \\equiv 0 \\pmod{9}, x \\equiv 1 \\pmod{7}, x \\equiv 4 \\pmod{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1868 <= x <= 5224 such that x \\equiv 0 \\pmod{9}, x \\equiv 1 \\pmod{7}, x \\equiv 4 \\pmod{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1989", "ref": "1989", "family": "po_crt_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.003277, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[0, 9], [1, 7], [4, 5]], "L": 1868, "U": 5224}, "seed": 401691043502001101, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 1989, "valid_answers": ["1989", "2304", "2619", "2934", "3249", "3564", "3879", "4194", "4509", "4824", "5139"]}, "cell": "crt_range/2", "stage": "B"}
+{"id": "po-crt_range-t2-eval-000466", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1316 <= x <= 7156 such that x \\equiv 1 \\pmod{5}, x \\equiv 8 \\pmod{11}, x \\equiv 2 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1316 <= x <= 7156 such that x \\equiv 1 \\pmod{5}, x \\equiv 8 \\pmod{11}, x \\equiv 2 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1361", "ref": "1361", "family": "po_crt_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.002054, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[1, 5], [8, 11], [2, 9]], "L": 1316, "U": 7156}, "seed": 401691043502000466, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 1361, "valid_answers": ["1361", "1856", "2351", "2846", "3341", "3836", "4331", "4826", "5321", "5816", "6311", "6806"]}, "cell": "crt_range/2", "stage": "B"}
+{"id": "po-crt_range-t2-eval-000153", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1191 <= x <= 1758 such that x \\equiv 1 \\pmod{5}, x \\equiv 3 \\pmod{7}, x \\equiv 0 \\pmod{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1191 <= x <= 1758 such that x \\equiv 1 \\pmod{5}, x \\equiv 3 \\pmod{7}, x \\equiv 0 \\pmod{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1256", "ref": "1256", "family": "po_crt_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.007042, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[1, 5], [3, 7], [0, 4]], "L": 1191, "U": 1758}, "seed": 401691043502000153, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 1256, "valid_answers": ["1256", "1396", "1536", "1676"]}, "cell": "crt_range/2", "stage": "B"}
+{"id": "po-crt_range-t2-eval-001247", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 850 <= x <= 2277 such that x \\equiv 2 \\pmod{4}, x \\equiv 3 \\pmod{7}, x \\equiv 1 \\pmod{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 850 <= x <= 2277 such that x \\equiv 2 \\pmod{4}, x \\equiv 3 \\pmod{7}, x \\equiv 1 \\pmod{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "906", "ref": "906", "family": "po_crt_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.007003, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[2, 4], [3, 7], [1, 5]], "L": 850, "U": 2277}, "seed": 401691043502001247, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 906, "valid_answers": ["906", "1046", "1186", "1326", "1466", "1606", "1746", "1886", "2026", "2166"]}, "cell": "crt_range/2", "stage": "B"}
+{"id": "po-crt_range-t2-eval-001165", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1593 <= x <= 7548 such that x \\equiv 5 \\pmod{11}, x \\equiv 3 \\pmod{5}, x \\equiv 0 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1593 <= x <= 7548 such that x \\equiv 5 \\pmod{11}, x \\equiv 3 \\pmod{5}, x \\equiv 0 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1908", "ref": "1908", "family": "po_crt_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.002015, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[5, 11], [3, 5], [0, 9]], "L": 1593, "U": 7548}, "seed": 401691043502001165, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 1908, "valid_answers": ["1908", "2403", "2898", "3393", "3888", "4383", "4878", "5373", "5868", "6363", "6858", "7353"]}, "cell": "crt_range/2", "stage": "B"}
+{"id": "po-crt_range-t3-eval-000102", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 644 <= x <= 3342 such that x \\equiv 5 \\pmod{10}, x \\equiv 5 \\pmod{6}, x \\equiv 14 \\pmod{21}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 644 <= x <= 3342 such that x \\equiv 5 \\pmod{10}, x \\equiv 5 \\pmod{6}, x \\equiv 14 \\pmod{21}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "665", "ref": "665", "family": "po_crt_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.004817, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[5, 10], [5, 6], [14, 21]], "L": 644, "U": 3342}, "seed": 401691043503000108, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 665, "valid_answers": ["665", "875", "1085", "1295", "1505", "1715", "1925", "2135", "2345", "2555", "2765", "2975", "3185"]}, "cell": "crt_range/3", "stage": "B"}
+{"id": "po-crt_range-t3-eval-001720", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 590 <= x <= 1348 such that x \\equiv 7 \\pmod{15}, x \\equiv 1 \\pmod{6}, x \\equiv 4 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 590 <= x <= 1348 such that x \\equiv 7 \\pmod{15}, x \\equiv 1 \\pmod{6}, x \\equiv 4 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "607", "ref": "607", "family": "po_crt_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.011858, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[7, 15], [1, 6], [4, 9]], "L": 590, "U": 1348}, "seed": 401691043503001878, "version": "po-v0", "tier": 3, "n_raw": 9, "small": 607, "valid_answers": ["607", "697", "787", "877", "967", "1057", "1147", "1237", "1327"]}, "cell": "crt_range/3", "stage": "B"}
+{"id": "po-crt_range-t3-eval-000274", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1065 <= x <= 8966 such that x \\equiv 5 \\pmod{9}, x \\equiv 7 \\pmod{10}, x \\equiv 17 \\pmod{21}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1065 <= x <= 8966 such that x \\equiv 5 \\pmod{9}, x \\equiv 7 \\pmod{10}, x \\equiv 17 \\pmod{21}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1067", "ref": "1067", "family": "po_crt_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.001645, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[5, 9], [7, 10], [17, 21]], "L": 1065, "U": 8966}, "seed": 401691043503000296, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 1067, "valid_answers": ["1067", "1697", "2327", "2957", "3587", "4217", "4847", "5477", "6107", "6737", "7367", "7997", "8627"]}, "cell": "crt_range/3", "stage": "B"}
+{"id": "po-crt_range-t3-eval-001980", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 133 <= x <= 1292 such that x \\equiv 7 \\pmod{14}, x \\equiv 1 \\pmod{6}, x \\equiv 13 \\pmod{15}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 133 <= x <= 1292 such that x \\equiv 7 \\pmod{14}, x \\equiv 1 \\pmod{6}, x \\equiv 13 \\pmod{15}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "133", "ref": "133", "family": "po_crt_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.005172, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[7, 14], [1, 6], [13, 15]], "L": 133, "U": 1292}, "seed": 401691043503002168, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 133, "valid_answers": ["133", "343", "553", "763", "973", "1183"]}, "cell": "crt_range/3", "stage": "B"}
+{"id": "po-crt_range-t3-eval-001111", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1213 <= x <= 2586 such that x \\equiv 4 \\pmod{6}, x \\equiv 1 \\pmod{21}, x \\equiv 7 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1213 <= x <= 2586 such that x \\equiv 4 \\pmod{6}, x \\equiv 1 \\pmod{21}, x \\equiv 7 \\pmod{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1240", "ref": "1240", "family": "po_crt_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.008006, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[4, 6], [1, 21], [7, 9]], "L": 1213, "U": 2586}, "seed": 401691043503001209, "version": "po-v0", "tier": 3, "n_raw": 11, "small": 1240, "valid_answers": ["1240", "1366", "1492", "1618", "1744", "1870", "1996", "2122", "2248", "2374", "2500"]}, "cell": "crt_range/3", "stage": "B"}
+{"id": "po-crt_range-t3-eval-000554", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 443 <= x <= 6069 such that x \\equiv 1 \\pmod{10}, x \\equiv 2 \\pmod{9}, x \\equiv 17 \\pmod{21}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 443 <= x <= 6069 such that x \\equiv 1 \\pmod{10}, x \\equiv 2 \\pmod{9}, x \\equiv 17 \\pmod{21}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "731", "ref": "731", "family": "po_crt_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.001599, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[1, 10], [2, 9], [17, 21]], "L": 443, "U": 6069}, "seed": 401691043503000603, "version": "po-v0", "tier": 3, "n_raw": 9, "small": 731, "valid_answers": ["731", "1361", "1991", "2621", "3251", "3881", "4511", "5141", "5771"]}, "cell": "crt_range/3", "stage": "B"}
+{"id": "po-crt_range-t3-eval-000735", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 854 <= x <= 5332 such that x \\equiv 9 \\pmod{14}, x \\equiv 4 \\pmod{9}, x \\equiv 5 \\pmod{10}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 854 <= x <= 5332 such that x \\equiv 9 \\pmod{14}, x \\equiv 4 \\pmod{9}, x \\equiv 5 \\pmod{10}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1255", "ref": "1255", "family": "po_crt_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.001563, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[9, 14], [4, 9], [5, 10]], "L": 854, "U": 5332}, "seed": 401691043503000800, "version": "po-v0", "tier": 3, "n_raw": 7, "small": 1255, "valid_answers": ["1255", "1885", "2515", "3145", "3775", "4405", "5035"]}, "cell": "crt_range/3", "stage": "B"}
+{"id": "po-crt_range-t3-eval-001023", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1540 <= x <= 2358 such that x \\equiv 0 \\pmod{6}, x \\equiv 3 \\pmod{9}, x \\equiv 0 \\pmod{14}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1540 <= x <= 2358 such that x \\equiv 0 \\pmod{6}, x \\equiv 3 \\pmod{9}, x \\equiv 0 \\pmod{14}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1596", "ref": "1596", "family": "po_crt_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.008547, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[0, 6], [3, 9], [0, 14]], "L": 1540, "U": 2358}, "seed": 401691043503001112, "version": "po-v0", "tier": 3, "n_raw": 7, "small": 1596, "valid_answers": ["1596", "1722", "1848", "1974", "2100", "2226", "2352"]}, "cell": "crt_range/3", "stage": "B"}
+{"id": "po-crt_range-t3-eval-002495", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 994 <= x <= 1950 such that x \\equiv 7 \\pmod{10}, x \\equiv 16 \\pmod{21}, x \\equiv 9 \\pmod{14}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 994 <= x <= 1950 such that x \\equiv 7 \\pmod{10}, x \\equiv 16 \\pmod{21}, x \\equiv 9 \\pmod{14}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1087", "ref": "1087", "family": "po_crt_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.005225, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[7, 10], [16, 21], [9, 14]], "L": 994, "U": 1950}, "seed": 401691043503002762, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 1087, "valid_answers": ["1087", "1297", "1507", "1717", "1927"]}, "cell": "crt_range/3", "stage": "B"}
+{"id": "po-crt_range-t3-eval-002039", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1021 <= x <= 2533 such that x \\equiv 7 \\pmod{10}, x \\equiv 14 \\pmod{21}, x \\equiv 5 \\pmod{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1021 <= x <= 2533 such that x \\equiv 7 \\pmod{10}, x \\equiv 14 \\pmod{21}, x \\equiv 5 \\pmod{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1127", "ref": "1127", "family": "po_crt_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.004627, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[7, 10], [14, 21], [5, 6]], "L": 1021, "U": 2533}, "seed": 401691043503002236, "version": "po-v0", "tier": 3, "n_raw": 7, "small": 1127, "valid_answers": ["1127", "1337", "1547", "1757", "1967", "2177", "2387"]}, "cell": "crt_range/3", "stage": "B"}
+{"id": "po-crt_range-t3-eval-001359", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 591 <= x <= 2591 such that x \\equiv 3 \\pmod{15}, x \\equiv 3 \\pmod{14}, x \\equiv 3 \\pmod{21}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 591 <= x <= 2591 such that x \\equiv 3 \\pmod{15}, x \\equiv 3 \\pmod{14}, x \\equiv 3 \\pmod{21}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "633", "ref": "633", "family": "po_crt_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.004998, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[3, 15], [3, 14], [3, 21]], "L": 591, "U": 2591}, "seed": 401691043503001486, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 633, "valid_answers": ["633", "843", "1053", "1263", "1473", "1683", "1893", "2103", "2313", "2523"]}, "cell": "crt_range/3", "stage": "B"}
+{"id": "po-crt_range-t3-eval-002322", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 942 <= x <= 2687 such that x \\equiv 5 \\pmod{9}, x \\equiv 8 \\pmod{15}, x \\equiv 8 \\pmod{21}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 942 <= x <= 2687 such that x \\equiv 5 \\pmod{9}, x \\equiv 8 \\pmod{15}, x \\equiv 8 \\pmod{21}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1058", "ref": "1058", "family": "po_crt_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.003436, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[5, 9], [8, 15], [8, 21]], "L": 942, "U": 2687}, "seed": 401691043503002558, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 1058, "valid_answers": ["1058", "1373", "1688", "2003", "2318", "2633"]}, "cell": "crt_range/3", "stage": "B"}
+{"id": "po-diff_squares-t2-eval-000000", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 3575. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind nonnegative integers x, y such that x^2 - y^2 = 3575. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "60, 5", "ref": "60, 5", "family": "po_diff_squares", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 2e-06, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 3575}, "seed": 194317368902000000, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 60, "valid_answers": ["60, 5", "84, 59", "144, 131", "168, 157", "360, 355", "1788, 1787"]}, "cell": "diff_squares/2", "stage": "B"}
+{"id": "po-diff_squares-t2-eval-000051", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 1225. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind nonnegative integers x, y such that x^2 - y^2 = 1225. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "35, 0", "ref": "35, 0", "family": "po_diff_squares", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 1.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 1225}, "seed": 194317368902000139, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 35, "valid_answers": ["35, 0", "37, 12", "91, 84", "125, 120", "613, 612"]}, "cell": "diff_squares/2", "stage": "B"}
+{"id": "po-diff_squares-t3-eval-000125", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 196. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind nonnegative integers x, y such that x^2 - y^2 = 196. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "14, 0", "ref": "14, 0", "family": "po_diff_squares", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000204, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 196}, "seed": 194317368903000658, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 14, "valid_answers": ["14, 0", "50, 48"]}, "cell": "diff_squares/3", "stage": "B"}
+{"id": "po-diff_squares-t3-eval-000000", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 132. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind nonnegative integers x, y such that x^2 - y^2 = 132. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "14, 8", "ref": "14, 8", "family": "po_diff_squares", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000446, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 132}, "seed": 194317368903000000, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 14, "valid_answers": ["14, 8", "34, 32"]}, "cell": "diff_squares/3", "stage": "B"}
+{"id": "po-diff_squares-t3-eval-000099", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 572. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind nonnegative integers x, y such that x^2 - y^2 = 572. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "24, 2", "ref": "24, 2", "family": "po_diff_squares", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 572}, "seed": 194317368903000377, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 24, "valid_answers": ["24, 2", "144, 142"]}, "cell": "diff_squares/3", "stage": "B"}
+{"id": "po-diff_squares-t3-eval-000025", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 6171. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind nonnegative integers x, y such that x^2 - y^2 = 6171. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "86, 35", "ref": "86, 35", "family": "po_diff_squares", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 6171}, "seed": 194317368903000043, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 86, "valid_answers": ["86, 35", "110, 77", "190, 173", "286, 275", "1030, 1027", "3086, 3085"]}, "cell": "diff_squares/3", "stage": "B"}
+{"id": "po-diff_squares-t3-eval-000081", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 10115. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind nonnegative integers x, y such that x^2 - y^2 = 10115. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "102, 17", "ref": "102, 17", "family": "po_diff_squares", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 10115}, "seed": 194317368903000222, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 102, "valid_answers": ["102, 17", "162, 127", "306, 289", "726, 719", "1014, 1009", "5058, 5057"]}, "cell": "diff_squares/3", "stage": "B"}
+{"id": "po-diff_squares-t3-eval-000037", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 9163. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind nonnegative integers x, y such that x^2 - y^2 = 9163. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "98, 21", "ref": "98, 21", "family": "po_diff_squares", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 9163}, "seed": 194317368903000064, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 98, "valid_answers": ["98, 21", "118, 69", "278, 261", "422, 411", "658, 651", "4582, 4581"]}, "cell": "diff_squares/3", "stage": "B"}
+{"id": "po-diff_squares-t3-eval-000030", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 26299. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind nonnegative integers x, y such that x^2 - y^2 = 26299. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "170, 51", "ref": "170, 51", "family": "po_diff_squares", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 26299}, "seed": 194317368903000049, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 170, "valid_answers": ["170, 51", "190, 99", "782, 765", "1018, 1005", "1882, 1875", "13150, 13149"]}, "cell": "diff_squares/3", "stage": "B"}
+{"id": "po-diff_squares-t3-eval-000080", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 1225. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind nonnegative integers x, y such that x^2 - y^2 = 1225. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "35, 0", "ref": "35, 0", "family": "po_diff_squares", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 1.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 1225}, "seed": 194317368903000201, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 35, "valid_answers": ["35, 0", "37, 12", "91, 84", "125, 120", "613, 612"]}, "cell": "diff_squares/3", "stage": "B"}
+{"id": "po-diff_squares-t3-eval-000112", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 15895. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind nonnegative integers x, y such that x^2 - y^2 = 15895. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "136, 51", "ref": "136, 51", "family": "po_diff_squares", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 15895}, "seed": 194317368903000494, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 136, "valid_answers": ["136, 51", "172, 117", "476, 459", "728, 717", "1592, 1587", "7948, 7947"]}, "cell": "diff_squares/3", "stage": "B"}
+{"id": "po-diff_squares-t3-eval-000028", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 3575. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind nonnegative integers x, y such that x^2 - y^2 = 3575. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "60, 5", "ref": "60, 5", "family": "po_diff_squares", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 2e-06, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 3575}, "seed": 194317368903000046, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 60, "valid_answers": ["60, 5", "84, 59", "144, 131", "168, 157", "360, 355", "1788, 1787"]}, "cell": "diff_squares/3", "stage": "B"}
+{"id": "po-diff_squares-t3-eval-000060", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 204. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind nonnegative integers x, y such that x^2 - y^2 = 204. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "20, 14", "ref": "20, 14", "family": "po_diff_squares", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000189, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 204}, "seed": 194317368903000127, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 20, "valid_answers": ["20, 14", "52, 50"]}, "cell": "diff_squares/3", "stage": "B"}
+{"id": "po-diff_squares-t3-eval-000046", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 459. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind nonnegative integers x, y such that x^2 - y^2 = 459. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "22, 5", "ref": "22, 5", "family": "po_diff_squares", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 7.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 459}, "seed": 194317368903000082, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 22, "valid_answers": ["22, 5", "30, 21", "78, 75", "230, 229"]}, "cell": "diff_squares/3", "stage": "B"}
+{"id": "po-egyptian-t1-eval-000022", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{1}{73}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{1}{73}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "74, 5402", "ref": "74, 5402", "family": "po_egyptian", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 1, "v": 73}, "seed": 186125617101000023, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 146, "valid_answers": ["74, 5402", "146, 146"]}, "cell": "egyptian/1", "stage": "B"}
+{"id": "po-egyptian-t1-eval-000040", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{1}{50}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{1}{50}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "51, 2550", "ref": "51, 2550", "family": "po_egyptian", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 2e-06, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 1, "v": 50}, "seed": 186125617101000047, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 100, "valid_answers": ["51, 2550", "52, 1300", "54, 675", "55, 550", "60, 300", "70, 175", "75, 150", "100, 100"]}, "cell": "egyptian/1", "stage": "B"}
+{"id": "po-egyptian-t1-eval-000043", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{1}{112}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{1}{112}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "113, 12656", "ref": "113, 12656", "family": "po_egyptian", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 1, "v": 112}, "seed": 186125617101000050, "version": "po-v0", "tier": 1, "n_raw": 14, "small": 224, "valid_answers": ["113, 12656", "114, 6384", "116, 3248", "119, 1904", "120, 1680", "126, 1008", "128, 896", "140, 560", "144, 504", "161, 368", "168, 336", "176, 308", "210, 240", "224, 224"]}, "cell": "egyptian/1", "stage": "B"}
+{"id": "po-egyptian-t1-eval-000035", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{1}{38}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{1}{38}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "39, 1482", "ref": "39, 1482", "family": "po_egyptian", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 1, "v": 38}, "seed": 186125617101000040, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 76, "valid_answers": ["39, 1482", "40, 760", "42, 399", "57, 114", "76, 76"]}, "cell": "egyptian/1", "stage": "B"}
+{"id": "po-egyptian-t1-eval-000029", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{1}{64}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{1}{64}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "65, 4160", "ref": "65, 4160", "family": "po_egyptian", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 1, "v": 64}, "seed": 186125617101000034, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 128, "valid_answers": ["65, 4160", "66, 2112", "68, 1088", "72, 576", "80, 320", "96, 192", "128, 128"]}, "cell": "egyptian/1", "stage": "B"}
+{"id": "po-egyptian-t1-eval-000059", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{1}{32}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{1}{32}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "33, 1056", "ref": "33, 1056", "family": "po_egyptian", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 1.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 1, "v": 32}, "seed": 186125617101000080, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 64, "valid_answers": ["33, 1056", "34, 544", "36, 288", "40, 160", "48, 96", "64, 64"]}, "cell": "egyptian/1", "stage": "B"}
+{"id": "po-egyptian-t2-eval-000068", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{93}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{93}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "47, 4371", "ref": "47, 4371", "family": "po_egyptian", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 2, "v": 93}, "seed": 186125617102000122, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 93, "valid_answers": ["47, 4371", "48, 1488", "51, 527", "62, 186", "93, 93"]}, "cell": "egyptian/2", "stage": "B"}
+{"id": "po-egyptian-t2-eval-000049", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{145}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{145}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "73, 10585", "ref": "73, 10585", "family": "po_egyptian", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 2, "v": 145}, "seed": 186125617102000064, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 145, "valid_answers": ["73, 10585", "75, 2175", "85, 493", "87, 435", "145, 145"]}, "cell": "egyptian/2", "stage": "B"}
+{"id": "po-egyptian-t2-eval-000007", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{95}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{95}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "48, 4560", "ref": "48, 4560", "family": "po_egyptian", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 2, "v": 95}, "seed": 186125617102000007, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 95, "valid_answers": ["48, 4560", "50, 950", "57, 285", "60, 228", "95, 95"]}, "cell": "egyptian/2", "stage": "B"}
+{"id": "po-egyptian-t2-eval-000002", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{103}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{103}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "52, 5356", "ref": "52, 5356", "family": "po_egyptian", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 2, "v": 103}, "seed": 186125617102000002, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 103, "valid_answers": ["52, 5356", "103, 103"]}, "cell": "egyptian/2", "stage": "B"}
+{"id": "po-egyptian-t2-eval-000034", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{151}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{151}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "76, 11476", "ref": "76, 11476", "family": "po_egyptian", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 2, "v": 151}, "seed": 186125617102000043, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 151, "valid_answers": ["76, 11476", "151, 151"]}, "cell": "egyptian/2", "stage": "B"}
+{"id": "po-egyptian-t2-eval-000057", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{135}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{135}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "68, 9180", "ref": "68, 9180", "family": "po_egyptian", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 2, "v": 135}, "seed": 186125617102000083, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 135, "valid_answers": ["68, 9180", "69, 3105", "70, 1890", "72, 1080", "75, 675", "80, 432", "81, 405", "90, 270", "105, 189", "108, 180", "135, 135"]}, "cell": "egyptian/2", "stage": "B"}
+{"id": "po-egyptian-t2-eval-000078", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{149}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{149}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "75, 11175", "ref": "75, 11175", "family": "po_egyptian", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 2, "v": 149}, "seed": 186125617102000153, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 149, "valid_answers": ["75, 11175", "149, 149"]}, "cell": "egyptian/2", "stage": "B"}
+{"id": "po-egyptian-t2-eval-000079", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{25}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{25}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "13, 325", "ref": "13, 325", "family": "po_egyptian", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 1.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 2, "v": 25}, "seed": 186125617102000155, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 25, "valid_answers": ["13, 325", "15, 75", "25, 25"]}, "cell": "egyptian/2", "stage": "B"}
+{"id": "po-egyptian-t2-eval-000095", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{161}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{161}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "81, 13041", "ref": "81, 13041", "family": "po_egyptian", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 2, "v": 161}, "seed": 186125617102000478, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 161, "valid_answers": ["81, 13041", "84, 1932", "92, 644", "105, 345", "161, 161"]}, "cell": "egyptian/2", "stage": "B"}
+{"id": "po-egyptian-t2-eval-000091", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{63}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{63}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "32, 2016", "ref": "32, 2016", "family": "po_egyptian", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 2, "v": 63}, "seed": 186125617102000300, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 63, "valid_answers": ["32, 2016", "33, 693", "35, 315", "36, 252", "42, 126", "45, 105", "56, 72", "63, 63"]}, "cell": "egyptian/2", "stage": "B"}
+{"id": "po-egyptian-t2-eval-000006", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{127}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{2}{127}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "64, 8128", "ref": "64, 8128", "family": "po_egyptian", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 2, "v": 127}, "seed": 186125617102000006, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 127, "valid_answers": ["64, 8128", "127, 127"]}, "cell": "egyptian/2", "stage": "B"}
+{"id": "po-egyptian-t3-eval-000067", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{164}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{164}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "55, 9020", "ref": "55, 9020", "family": "po_egyptian", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 3, "v": 164}, "seed": 186125617103000122, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 164, "valid_answers": ["55, 9020", "56, 2296", "60, 615", "82, 164"]}, "cell": "egyptian/3", "stage": "B"}
+{"id": "po-egyptian-t3-eval-000077", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{34}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{34}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "12, 204", "ref": "12, 204", "family": "po_egyptian", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 3e-06, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 3, "v": 34}, "seed": 186125617103000150, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 34, "valid_answers": ["12, 204", "17, 34"]}, "cell": "egyptian/3", "stage": "B"}
+{"id": "po-egyptian-t3-eval-000121", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{140}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{140}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "47, 6580", "ref": "47, 6580", "family": "po_egyptian", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 3, "v": 140}, "seed": 186125617103000301, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 105, "valid_answers": ["47, 6580", "48, 1680", "49, 980", "50, 700", "52, 455", "55, 308", "56, 280", "60, 210", "63, 180", "70, 140", "80, 112", "84, 105"]}, "cell": "egyptian/3", "stage": "B"}
+{"id": "po-egyptian-t3-eval-000102", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{326}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{326}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "109, 35534", "ref": "109, 35534", "family": "po_egyptian", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 3, "v": 326}, "seed": 186125617103000226, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 326, "valid_answers": ["109, 35534", "110, 8965", "163, 326"]}, "cell": "egyptian/3", "stage": "B"}
+{"id": "po-egyptian-t3-eval-000081", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{203}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{203}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "68, 13804", "ref": "68, 13804", "family": "po_egyptian", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 3, "v": 203}, "seed": 186125617103000162, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 348, "valid_answers": ["68, 13804", "70, 2030", "84, 348"]}, "cell": "egyptian/3", "stage": "B"}
+{"id": "po-egyptian-t3-eval-000160", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{218}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{218}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "73, 15914", "ref": "73, 15914", "family": "po_egyptian", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 3, "v": 218}, "seed": 186125617103000727, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 218, "valid_answers": ["73, 15914", "74, 4033", "109, 218"]}, "cell": "egyptian/3", "stage": "B"}
+{"id": "po-egyptian-t3-eval-000174", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{74}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{74}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "25, 1850", "ref": "25, 1850", "family": "po_egyptian", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 3, "v": 74}, "seed": 186125617103001247, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 74, "valid_answers": ["25, 1850", "26, 481", "37, 74"]}, "cell": "egyptian/3", "stage": "B"}
+{"id": "po-egyptian-t3-eval-000127", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{188}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{188}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "63, 11844", "ref": "63, 11844", "family": "po_egyptian", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 3, "v": 188}, "seed": 186125617103000325, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 188, "valid_answers": ["63, 11844", "64, 3008", "68, 799", "94, 188"]}, "cell": "egyptian/3", "stage": "B"}
+{"id": "po-egyptian-t3-eval-000161", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{224}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{224}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "75, 16800", "ref": "75, 16800", "family": "po_egyptian", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 3, "v": 224}, "seed": 186125617103000753, "version": "po-v0", "tier": 3, "n_raw": 9, "small": 160, "valid_answers": ["75, 16800", "76, 4256", "77, 2464", "80, 1120", "84, 672", "91, 416", "96, 336", "112, 224", "140, 160"]}, "cell": "egyptian/3", "stage": "B"}
+{"id": "po-egyptian-t3-eval-000073", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{65}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{65}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "22, 1430", "ref": "22, 1430", "family": "po_egyptian", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 3, "v": 65}, "seed": 186125617103000143, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 78, "valid_answers": ["22, 1430", "26, 130", "30, 78"]}, "cell": "egyptian/3", "stage": "B"}
+{"id": "po-egyptian-t3-eval-000113", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{122}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{122}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "41, 5002", "ref": "41, 5002", "family": "po_egyptian", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 3, "v": 122}, "seed": 186125617103000281, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 122, "valid_answers": ["41, 5002", "42, 1281", "61, 122"]}, "cell": "egyptian/3", "stage": "B"}
+{"id": "po-egyptian-t3-eval-000105", "task": "cx_po_egyptian", "source": "cx_po_egyptian", "split": "eval", "problem": "Find positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{266}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b such that \\frac{1}{a} + \\frac{1}{b} = \\frac{3}{266}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "89, 23674", "ref": "89, 23674", "family": "po_egyptian", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 3, "v": 266}, "seed": 186125617103000229, "version": "po-v0", "tier": 3, "n_raw": 9, "small": 209, "valid_answers": ["89, 23674", "90, 5985", "91, 3458", "95, 1330", "98, 931", "105, 570", "114, 399", "133, 266", "154, 209"]}, "cell": "egyptian/3", "stage": "B"}
+{"id": "po-fraction_between-t0-eval-001671", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 25 such that \\frac{10}{13} < \\frac{a}{b} < \\frac{1173}{1508}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 25 such that \\frac{10}{13} < \\frac{a}{b} < \\frac{1173}{1508}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 9", "ref": "7, 9", "family": "po_fraction_between", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.012739, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "10/13", "hi": "1173/1508", "D": 25}, "seed": 79997677100025629, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 9, "valid_answers": ["7, 9", "17, 22"]}, "cell": "fraction_between/0", "stage": "C"}
+{"id": "po-fraction_between-t0-eval-000207", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 25 such that \\frac{22}{23} < \\frac{a}{b} < \\frac{2138}{2139}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 25 such that \\frac{22}{23} < \\frac{a}{b} < \\frac{2138}{2139}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "23, 24", "ref": "23, 24", "family": "po_fraction_between", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.010256, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "22/23", "hi": "2138/2139", "D": 25}, "seed": 79997677100002840, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 24, "valid_answers": ["23, 24", "24, 25"]}, "cell": "fraction_between/0", "stage": "C"}
+{"id": "po-fraction_between-t0-eval-000002", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 25 such that \\frac{15}{19} < \\frac{a}{b} < \\frac{679}{836}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 25 such that \\frac{15}{19} < \\frac{a}{b} < \\frac{679}{836}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 5", "ref": "4, 5", "family": "po_fraction_between", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.018182, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "15/19", "hi": "679/836", "D": 25}, "seed": 79997677100000018, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 5, "valid_answers": ["4, 5", "17, 21", "19, 24"]}, "cell": "fraction_between/0", "stage": "C"}
+{"id": "po-fraction_between-t0-eval-001490", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 25 such that \\frac{14}{15} < \\frac{a}{b} < \\frac{976}{1035}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 25 such that \\frac{14}{15} < \\frac{a}{b} < \\frac{976}{1035}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "15, 16", "ref": "15, 16", "family": "po_fraction_between", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.010695, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "14/15", "hi": "976/1035", "D": 25}, "seed": 79997677100022471, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 16, "valid_answers": ["15, 16", "16, 17"]}, "cell": "fraction_between/0", "stage": "C"}
+{"id": "po-fraction_between-t0-eval-000475", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 25 such that \\frac{44}{73} < \\frac{a}{b} < \\frac{10046}{16425}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 25 such that \\frac{44}{73} < \\frac{a}{b} < \\frac{10046}{16425}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 18", "ref": "11, 18", "family": "po_fraction_between", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.015748, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "44/73", "hi": "10046/16425", "D": 25}, "seed": 79997677100006741, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 18, "valid_answers": ["11, 18", "14, 23"]}, "cell": "fraction_between/0", "stage": "C"}
+{"id": "po-fraction_between-t0-eval-001817", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 25 such that \\frac{21}{22} < \\frac{a}{b} < \\frac{893}{924}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 25 such that \\frac{21}{22} < \\frac{a}{b} < \\frac{893}{924}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "22, 23", "ref": "22, 23", "family": "po_fraction_between", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.015385, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "21/22", "hi": "893/924", "D": 25}, "seed": 79997677100028200, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 23, "valid_answers": ["22, 23", "23, 24", "24, 25"]}, "cell": "fraction_between/0", "stage": "C"}
+{"id": "po-fraction_between-t0-eval-000784", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 25 such that \\frac{8}{11} < \\frac{a}{b} < \\frac{41}{55}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 25 such that \\frac{8}{11} < \\frac{a}{b} < \\frac{41}{55}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 15", "ref": "11, 15", "family": "po_fraction_between", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.02, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "8/11", "hi": "41/55", "D": 25}, "seed": 79997677100011323, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 15, "valid_answers": ["11, 15", "14, 19", "17, 23"]}, "cell": "fraction_between/0", "stage": "C"}
+{"id": "po-fraction_between-t0-eval-000403", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 25 such that \\frac{17}{28} < \\frac{a}{b} < \\frac{103}{168}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 25 such that \\frac{17}{28} < \\frac{a}{b} < \\frac{103}{168}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 18", "ref": "11, 18", "family": "po_fraction_between", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.015748, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "17/28", "hi": "103/168", "D": 25}, "seed": 79997677100005944, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 18, "valid_answers": ["11, 18", "14, 23"]}, "cell": "fraction_between/0", "stage": "C"}
+{"id": "po-fraction_between-t0-eval-000581", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 25 such that \\frac{40}{53} < \\frac{a}{b} < \\frac{4253}{5565}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 25 such that \\frac{40}{53} < \\frac{a}{b} < \\frac{4253}{5565}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "16, 21", "ref": "16, 21", "family": "po_fraction_between", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.012739, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "40/53", "hi": "4253/5565", "D": 25}, "seed": 79997677100008404, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 21, "valid_answers": ["16, 21", "19, 25"]}, "cell": "fraction_between/0", "stage": "C"}
+{"id": "po-fraction_between-t0-eval-000544", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 25 such that \\frac{58}{69} < \\frac{a}{b} < \\frac{433}{506}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 25 such that \\frac{58}{69} < \\frac{a}{b} < \\frac{433}{506}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 13", "ref": "11, 13", "family": "po_fraction_between", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017442, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "58/69", "hi": "433/506", "D": 25}, "seed": 79997677100007812, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 13, "valid_answers": ["11, 13", "16, 19", "17, 20"]}, "cell": "fraction_between/0", "stage": "C"}
+{"id": "po-fraction_between-t0-eval-001631", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 25 such that \\frac{52}{61} < \\frac{a}{b} < \\frac{4117}{4758}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 25 such that \\frac{52}{61} < \\frac{a}{b} < \\frac{4117}{4758}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 7", "ref": "6, 7", "family": "po_fraction_between", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011628, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "52/61", "hi": "4117/4758", "D": 25}, "seed": 79997677100024894, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 7, "valid_answers": ["6, 7", "19, 22"]}, "cell": "fraction_between/0", "stage": "C"}
+{"id": "po-fraction_between-t0-eval-001092", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 25 such that \\frac{23}{26} < \\frac{a}{b} < \\frac{1475}{1638}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 25 such that \\frac{23}{26} < \\frac{a}{b} < \\frac{1475}{1638}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8, 9", "ref": "8, 9", "family": "po_fraction_between", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.016667, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "23/26", "hi": "1475/1638", "D": 25}, "seed": 79997677100016345, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 9, "valid_answers": ["8, 9", "9, 10", "17, 19"]}, "cell": "fraction_between/0", "stage": "C"}
+{"id": "po-fraction_between-t1-eval-001881", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 40 such that \\frac{59}{62} < \\frac{a}{b} < \\frac{18709}{19530}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 40 such that \\frac{59}{62} < \\frac{a}{b} < \\frac{18709}{19530}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "20, 21", "ref": "20, 21", "family": "po_fraction_between", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.00625, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "59/62", "hi": "18709/19530", "D": 40}, "seed": 79997677101009122, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 21, "valid_answers": ["20, 21", "21, 22", "22, 23"]}, "cell": "fraction_between/1", "stage": "C"}
+{"id": "po-fraction_between-t1-eval-002051", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 40 such that \\frac{75}{97} < \\frac{a}{b} < \\frac{26263}{33465}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 40 such that \\frac{75}{97} < \\frac{a}{b} < \\frac{26263}{33465}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 9", "ref": "7, 9", "family": "po_fraction_between", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.015152, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "75/97", "hi": "26263/33465", "D": 40}, "seed": 79997677101009959, "version": "po-v0", "tier": 1, "n_raw": 9, "small": 9, "valid_answers": ["7, 9", "18, 23", "24, 31", "25, 32", "29, 37", "31, 40"]}, "cell": "fraction_between/1", "stage": "C"}
+{"id": "po-fraction_between-t1-eval-000167", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 40 such that \\frac{5}{8} < \\frac{a}{b} < \\frac{433}{680}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 40 such that \\frac{5}{8} < \\frac{a}{b} < \\frac{433}{680}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 11", "ref": "7, 11", "family": "po_fraction_between", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.015432, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "5/8", "hi": "433/680", "D": 40}, "seed": 79997677101000821, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 11, "valid_answers": ["7, 11", "12, 19", "17, 27", "19, 30", "22, 35"]}, "cell": "fraction_between/1", "stage": "C"}
+{"id": "po-fraction_between-t1-eval-001989", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 40 such that \\frac{14}{33} < \\frac{a}{b} < \\frac{767}{1782}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 40 such that \\frac{14}{33} < \\frac{a}{b} < \\frac{767}{1782}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 7", "ref": "3, 7", "family": "po_fraction_between", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.008772, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "14/33", "hi": "767/1782", "D": 40}, "seed": 79997677101009639, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 7, "valid_answers": ["3, 7", "17, 40"]}, "cell": "fraction_between/1", "stage": "C"}
+{"id": "po-fraction_between-t1-eval-002062", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 40 such that \\frac{76}{111} < \\frac{a}{b} < \\frac{6343}{9102}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 40 such that \\frac{76}{111} < \\frac{a}{b} < \\frac{6343}{9102}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9, 13", "ref": "9, 13", "family": "po_fraction_between", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.017241, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "76/111", "hi": "6343/9102", "D": 40}, "seed": 79997677101010001, "version": "po-v0", "tier": 1, "n_raw": 9, "small": 13, "valid_answers": ["9, 13", "11, 16", "16, 23", "20, 29", "24, 35", "25, 36"]}, "cell": "fraction_between/1", "stage": "C"}
+{"id": "po-fraction_between-t1-eval-000265", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 40 such that \\frac{31}{35} < \\frac{a}{b} < \\frac{1037}{1155}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 40 such that \\frac{31}{35} < \\frac{a}{b} < \\frac{1037}{1155}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8, 9", "ref": "8, 9", "family": "po_fraction_between", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.013514, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "31/35", "hi": "1037/1155", "D": 40}, "seed": 79997677101001337, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 9, "valid_answers": ["8, 9", "17, 19", "25, 28", "26, 29", "33, 37", "35, 39"]}, "cell": "fraction_between/1", "stage": "C"}
+{"id": "po-fraction_between-t1-eval-000880", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 40 such that \\frac{11}{26} < \\frac{a}{b} < \\frac{353}{819}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 40 such that \\frac{11}{26} < \\frac{a}{b} < \\frac{353}{819}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 7", "ref": "3, 7", "family": "po_fraction_between", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.013158, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "11/26", "hi": "353/819", "D": 40}, "seed": 79997677101004235, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 7, "valid_answers": ["3, 7", "14, 33", "17, 40"]}, "cell": "fraction_between/1", "stage": "C"}
+{"id": "po-fraction_between-t1-eval-002166", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 40 such that \\frac{7}{9} < \\frac{a}{b} < \\frac{419}{531}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 40 such that \\frac{7}{9} < \\frac{a}{b} < \\frac{419}{531}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 14", "ref": "11, 14", "family": "po_fraction_between", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.012626, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "7/9", "hi": "419/531", "D": 40}, "seed": 79997677101010478, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 14, "valid_answers": ["11, 14", "18, 23", "25, 32", "26, 33", "29, 37"]}, "cell": "fraction_between/1", "stage": "C"}
+{"id": "po-fraction_between-t1-eval-000341", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 40 such that \\frac{51}{73} < \\frac{a}{b} < \\frac{7417}{10512}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 40 such that \\frac{51}{73} < \\frac{a}{b} < \\frac{7417}{10512}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 10", "ref": "7, 10", "family": "po_fraction_between", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.008333, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "51/73", "hi": "7417/10512", "D": 40}, "seed": 79997677101001731, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 10, "valid_answers": ["7, 10", "19, 27", "26, 37"]}, "cell": "fraction_between/1", "stage": "C"}
+{"id": "po-fraction_between-t1-eval-001279", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 40 such that \\frac{45}{49} < \\frac{a}{b} < \\frac{3154}{3381}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 40 such that \\frac{45}{49} < \\frac{a}{b} < \\frac{3154}{3381}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "12, 13", "ref": "12, 13", "family": "po_fraction_between", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.017094, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "45/49", "hi": "3154/3381", "D": 40}, "seed": 79997677101006179, "version": "po-v0", "tier": 1, "n_raw": 11, "small": 13, "valid_answers": ["12, 13", "13, 14", "23, 25", "25, 27", "27, 29", "34, 37", "35, 38", "37, 40"]}, "cell": "fraction_between/1", "stage": "C"}
+{"id": "po-fraction_between-t1-eval-000854", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 40 such that \\frac{5}{6} < \\frac{a}{b} < \\frac{587}{690}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 40 such that \\frac{5}{6} < \\frac{a}{b} < \\frac{587}{690}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 13", "ref": "11, 13", "family": "po_fraction_between", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.018519, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "5/6", "hi": "587/690", "D": 40}, "seed": 79997677101004092, "version": "po-v0", "tier": 1, "n_raw": 12, "small": 13, "valid_answers": ["11, 13", "16, 19", "17, 20", "21, 25", "26, 31", "27, 32", "28, 33", "31, 37"]}, "cell": "fraction_between/1", "stage": "C"}
+{"id": "po-fraction_between-t1-eval-002187", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 40 such that \\frac{49}{60} < \\frac{a}{b} < \\frac{799}{960}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 40 such that \\frac{49}{60} < \\frac{a}{b} < \\frac{799}{960}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9, 11", "ref": "9, 11", "family": "po_fraction_between", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.019048, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "49/60", "hi": "799/960", "D": 40}, "seed": 79997677101010589, "version": "po-v0", "tier": 1, "n_raw": 11, "small": 11, "valid_answers": ["9, 11", "14, 17", "19, 23", "23, 28", "24, 29", "29, 35", "32, 39", "33, 40"]}, "cell": "fraction_between/1", "stage": "C"}
+{"id": "po-fraction_between-t2-eval-002441", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 70 such that \\frac{27}{37} < \\frac{a}{b} < \\frac{5140}{6993}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 70 such that \\frac{27}{37} < \\frac{a}{b} < \\frac{5140}{6993}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 15", "ref": "11, 15", "family": "po_fraction_between", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.007188, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "27/37", "hi": "5140/6993", "D": 70}, "seed": 79997677102006153, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 15, "valid_answers": ["11, 15", "19, 26", "30, 41", "36, 49", "41, 56", "46, 63", "47, 64", "49, 67"]}, "cell": "fraction_between/2", "stage": "C"}
+{"id": "po-fraction_between-t2-eval-000008", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 70 such that \\frac{16}{81} < \\frac{a}{b} < \\frac{329}{1620}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 70 such that \\frac{16}{81} < \\frac{a}{b} < \\frac{329}{1620}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 5", "ref": "1, 5", "family": "po_fraction_between", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.005952, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "16/81", "hi": "329/1620", "D": 70}, "seed": 79997677102000027, "version": "po-v0", "tier": 2, "n_raw": 15, "small": 5, "valid_answers": ["1, 5", "14, 69"]}, "cell": "fraction_between/2", "stage": "C"}
+{"id": "po-fraction_between-t2-eval-000200", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 70 such that \\frac{111}{118} < \\frac{a}{b} < \\frac{10493}{11092}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 70 such that \\frac{111}{118} < \\frac{a}{b} < \\frac{10493}{11092}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "16, 17", "ref": "16, 17", "family": "po_fraction_between", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.005602, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "111/118", "hi": "10493/11092", "D": 70}, "seed": 79997677102000502, "version": "po-v0", "tier": 2, "n_raw": 14, "small": 17, "valid_answers": ["16, 17", "17, 18", "33, 35", "35, 37", "49, 52", "50, 53", "52, 55", "65, 69"]}, "cell": "fraction_between/2", "stage": "C"}
+{"id": "po-fraction_between-t2-eval-002477", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 70 such that \\frac{61}{99} < \\frac{a}{b} < \\frac{8573}{13860}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 70 such that \\frac{61}{99} < \\frac{a}{b} < \\frac{8573}{13860}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "21, 34", "ref": "21, 34", "family": "po_fraction_between", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.004233, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "61/99", "hi": "8573/13860", "D": 70}, "seed": 79997677102006229, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 34, "valid_answers": ["21, 34", "29, 47", "34, 55", "37, 60"]}, "cell": "fraction_between/2", "stage": "C"}
+{"id": "po-fraction_between-t2-eval-001677", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 70 such that \\frac{10}{17} < \\frac{a}{b} < \\frac{1684}{2805}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 70 such that \\frac{10}{17} < \\frac{a}{b} < \\frac{1684}{2805}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 5", "ref": "3, 5", "family": "po_fraction_between", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 17, "bucket": "9-20", "guess_rate": 0.018398, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "10/17", "hi": "1684/2805", "D": 70}, "seed": 79997677102004262, "version": "po-v0", "tier": 2, "n_raw": 34, "small": 5, "valid_answers": ["3, 5", "13, 22", "16, 27", "19, 32", "22, 37", "23, 39", "25, 42", "28, 47", "29, 49", "31, 52", "33, 56", "34, 57", "35, 59", "36, 61", "37, 62", "40, 67", "41, 69"]}, "cell": "fraction_between/2", "stage": "C"}
+{"id": "po-fraction_between-t2-eval-001585", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 70 such that \\frac{41}{86} < \\frac{a}{b} < \\frac{8655}{17974}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 70 such that \\frac{41}{86} < \\frac{a}{b} < \\frac{8655}{17974}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 23", "ref": "11, 23", "family": "po_fraction_between", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.010884, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "41/86", "hi": "8655/17974", "D": 70}, "seed": 79997677102004039, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 23, "valid_answers": ["11, 23", "12, 25", "13, 27", "21, 44", "23, 48", "25, 52", "31, 65", "32, 67"]}, "cell": "fraction_between/2", "stage": "C"}
+{"id": "po-fraction_between-t2-eval-002357", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 70 such that \\frac{57}{64} < \\frac{a}{b} < \\frac{15005}{16704}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 70 such that \\frac{57}{64} < \\frac{a}{b} < \\frac{15005}{16704}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "17, 19", "ref": "17, 19", "family": "po_fraction_between", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.010417, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "57/64", "hi": "15005/16704", "D": 70}, "seed": 79997677102005962, "version": "po-v0", "tier": 2, "n_raw": 18, "small": 19, "valid_answers": ["17, 19", "25, 28", "26, 29", "33, 37", "35, 39", "41, 46", "42, 47", "43, 48", "44, 49", "49, 55", "58, 65", "59, 66", "60, 67", "61, 68"]}, "cell": "fraction_between/2", "stage": "C"}
+{"id": "po-fraction_between-t2-eval-000234", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 70 such that \\frac{107}{145} < \\frac{a}{b} < \\frac{8783}{11745}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 70 such that \\frac{107}{145} < \\frac{a}{b} < \\frac{8783}{11745}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "17, 23", "ref": "17, 23", "family": "po_fraction_between", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 17, "bucket": "9-20", "guess_rate": 0.014991, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "107/145", "hi": "8783/11745", "D": 70}, "seed": 79997677102000584, "version": "po-v0", "tier": 2, "n_raw": 22, "small": 23, "valid_answers": ["17, 23", "20, 27", "23, 31", "26, 35", "29, 39", "31, 42", "32, 43", "35, 47", "37, 50", "38, 51", "41, 55", "43, 58", "44, 59", "47, 63", "48, 65", "49, 66", "50, 67"]}, "cell": "fraction_between/2", "stage": "C"}
+{"id": "po-fraction_between-t2-eval-002112", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 70 such that \\frac{43}{63} < \\frac{a}{b} < \\frac{161}{234}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 70 such that \\frac{43}{63} < \\frac{a}{b} < \\frac{161}{234}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 16", "ref": "11, 16", "family": "po_fraction_between", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.007619, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "43/63", "hi": "161/234", "D": 70}, "seed": 79997677102005325, "version": "po-v0", "tier": 2, "n_raw": 14, "small": 16, "valid_answers": ["11, 16", "13, 19", "24, 35", "28, 41", "35, 51", "37, 54", "41, 60", "46, 67"]}, "cell": "fraction_between/2", "stage": "C"}
+{"id": "po-fraction_between-t2-eval-000936", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 70 such that \\frac{2}{5} < \\frac{a}{b} < \\frac{373}{920}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 70 such that \\frac{2}{5} < \\frac{a}{b} < \\frac{373}{920}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "15, 37", "ref": "15, 37", "family": "po_fraction_between", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.011111, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "2/5", "hi": "373/920", "D": 70}, "seed": 79997677102002449, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 37, "valid_answers": ["15, 37", "17, 42", "19, 47", "21, 52", "23, 57", "25, 62", "27, 67"]}, "cell": "fraction_between/2", "stage": "C"}
+{"id": "po-fraction_between-t2-eval-000445", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 70 such that \\frac{98}{107} < \\frac{a}{b} < \\frac{29534}{31779}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 70 such that \\frac{98}{107} < \\frac{a}{b} < \\frac{29534}{31779}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 12", "ref": "11, 12", "family": "po_fraction_between", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.014925, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "98/107", "hi": "29534/31779", "D": 70}, "seed": 79997677102001139, "version": "po-v0", "tier": 2, "n_raw": 35, "small": 12, "valid_answers": ["11, 12", "12, 13", "13, 14", "23, 25", "25, 27", "34, 37", "35, 38", "37, 40", "38, 41", "45, 49", "47, 51", "49, 53", "51, 55", "56, 61", "57, 62", "58, 63", "59, 64", "61, 66", "62, 67", "63, 68", "64, 69"]}, "cell": "fraction_between/2", "stage": "C"}
+{"id": "po-fraction_between-t2-eval-000669", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 70 such that \\frac{73}{157} < \\frac{a}{b} < \\frac{18991}{40506}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 70 such that \\frac{73}{157} < \\frac{a}{b} < \\frac{18991}{40506}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 15", "ref": "7, 15", "family": "po_fraction_between", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.008403, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "73/157", "hi": "18991/40506", "D": 70}, "seed": 79997677102001743, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 15, "valid_answers": ["7, 15", "15, 32", "20, 43", "22, 47", "27, 58", "29, 62"]}, "cell": "fraction_between/2", "stage": "C"}
+{"id": "po-fraction_between-t3-eval-000070", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 120 such that \\frac{37}{75} < \\frac{a}{b} < \\frac{8771}{17475}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 120 such that \\frac{37}{75} < \\frac{a}{b} < \\frac{8771}{17475}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 2", "ref": "1, 2", "family": "po_fraction_between", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 23, "bucket": null, "guess_rate": 0.010305, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "37/75", "hi": "8771/17475", "D": 120}, "seed": 79997677103000124, "version": "po-v0", "tier": 3, "n_raw": 82, "small": 2, "valid_answers": ["1, 2", "38, 77", "39, 79", "40, 81", "41, 83", "42, 85", "43, 87", "44, 89", "45, 91", "46, 93", "47, 95", "48, 97", "49, 99", "50, 101", "51, 103", "52, 105", "53, 107", "54, 109", "55, 111", "56, 113", "57, 115", "58, 117", "59, 119"]}, "cell": "fraction_between/3", "stage": "C"}
+{"id": "po-fraction_between-t3-eval-001097", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 120 such that \\frac{19}{22} < \\frac{a}{b} < \\frac{5199}{5918}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 120 such that \\frac{19}{22} < \\frac{a}{b} < \\frac{5199}{5918}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 8", "ref": "7, 8", "family": "po_fraction_between", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 64, "bucket": null, "guess_rate": 0.016615, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "19/22", "hi": "5199/5918", "D": 120}, "seed": 79997677103001810, "version": "po-v0", "tier": 3, "n_raw": 106, "small": 8, "valid_answers": ["7, 8", "13, 15", "20, 23", "27, 31", "32, 37", "33, 38", "34, 39", "36, 41", "41, 47", "43, 49", "45, 52", "46, 53", "47, 54", "48, 55", "50, 57", "51, 59", "53, 61", "55, 63", "57, 65", "58, 67", "59, 68", "61, 70", "62, 71", "64, 73", "65, 74", "67, 77", "69, 79", "70, 81", "71, 81", "71, 82", "72, 83", "73, 84", "74, 85", "75, 86", "76, 87", "77, 89", "78, 89", "79, 90", "79, 91", "83, 95", "83, 96", "84, 97", "85, 97", "85, 98", "86, 99", "87, 100", "88, 101", "89, 102", "89, 103", "90, 103", "92, 105", "93, 106", "93, 107", "94, 107", "95, 109", "97, 111", "97, 112", "98, 113", "99, 113", "101, 115", "101, 116", "103, 118", "103, 119", "104, 119"]}, "cell": "fraction_between/3", "stage": "C"}
+{"id": "po-fraction_between-t3-eval-001759", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 120 such that \\frac{73}{320} < \\frac{a}{b} < \\frac{5701}{24640}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 120 such that \\frac{73}{320} < \\frac{a}{b} < \\frac{5701}{24640}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 13", "ref": "3, 13", "family": "po_fraction_between", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.011494, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "73/320", "hi": "5701/24640", "D": 120}, "seed": 79997677103002874, "version": "po-v0", "tier": 3, "n_raw": 23, "small": 13, "valid_answers": ["3, 13", "8, 35", "11, 48", "14, 61", "17, 74", "19, 83", "20, 87", "21, 92", "23, 100", "25, 109", "26, 113", "27, 118"]}, "cell": "fraction_between/3", "stage": "C"}
+{"id": "po-fraction_between-t3-eval-001135", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 120 such that \\frac{88}{309} < \\frac{a}{b} < \\frac{23453}{81267}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 120 such that \\frac{88}{309} < \\frac{a}{b} < \\frac{23453}{81267}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 7", "ref": "2, 7", "family": "po_fraction_between", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.009259, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "88/309", "hi": "23453/81267", "D": 120}, "seed": 79997677103001870, "version": "po-v0", "tier": 3, "n_raw": 30, "small": 7, "valid_answers": ["2, 7", "15, 52", "17, 59", "19, 66", "21, 73", "23, 80", "25, 87", "27, 94", "29, 101", "31, 108", "32, 111", "33, 115"]}, "cell": "fraction_between/3", "stage": "C"}
+{"id": "po-fraction_between-t3-eval-000927", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 120 such that \\frac{259}{270} < \\frac{a}{b} < \\frac{3067}{3186}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 120 such that \\frac{259}{270} < \\frac{a}{b} < \\frac{3067}{3186}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "24, 25", "ref": "24, 25", "family": "po_fraction_between", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.003324, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "259/270", "hi": "3067/3186", "D": 120}, "seed": 79997677103001538, "version": "po-v0", "tier": 3, "n_raw": 22, "small": 25, "valid_answers": ["24, 25", "25, 26", "49, 51", "51, 53", "71, 74", "73, 76", "74, 77", "76, 79", "77, 80", "95, 99", "97, 101", "99, 103", "101, 105", "103, 107"]}, "cell": "fraction_between/3", "stage": "C"}
+{"id": "po-fraction_between-t3-eval-000023", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 120 such that \\frac{103}{106} < \\frac{a}{b} < \\frac{70973}{72822}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 120 such that \\frac{103}{106} < \\frac{a}{b} < \\frac{70973}{72822}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "35, 36", "ref": "35, 36", "family": "po_fraction_between", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.003766, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "103/106", "hi": "70973/72822", "D": 120}, "seed": 79997677103000041, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 36, "valid_answers": ["35, 36", "36, 37", "37, 38", "38, 39", "69, 71", "71, 73", "73, 75", "75, 77", "104, 107", "106, 109", "107, 110", "109, 112", "110, 113", "112, 115", "113, 116", "115, 118"]}, "cell": "fraction_between/3", "stage": "C"}
+{"id": "po-fraction_between-t3-eval-000658", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 120 such that \\frac{50}{93} < \\frac{a}{b} < \\frac{5231}{9672}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 120 such that \\frac{50}{93} < \\frac{a}{b} < \\frac{5231}{9672}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 13", "ref": "7, 13", "family": "po_fraction_between", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.005471, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "50/93", "hi": "5231/9672", "D": 120}, "seed": 79997677103001109, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 13, "valid_answers": ["7, 13", "20, 37", "27, 50", "34, 63", "41, 76", "47, 87", "48, 89", "53, 98", "55, 102", "57, 106", "61, 113", "62, 115", "64, 119"]}, "cell": "fraction_between/3", "stage": "C"}
+{"id": "po-fraction_between-t3-eval-001831", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 120 such that \\frac{251}{319} < \\frac{a}{b} < \\frac{59121}{74327}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 120 such that \\frac{251}{319} < \\frac{a}{b} < \\frac{59121}{74327}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "15, 19", "ref": "15, 19", "family": "po_fraction_between", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 39, "bucket": null, "guess_rate": 0.011168, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "251/319", "hi": "59121/74327", "D": 120}, "seed": 79997677103003012, "version": "po-v0", "tier": 3, "n_raw": 61, "small": 19, "valid_answers": ["15, 19", "19, 24", "23, 29", "26, 33", "27, 34", "31, 39", "34, 43", "37, 47", "41, 52", "42, 53", "48, 61", "49, 62", "50, 63", "53, 67", "56, 71", "58, 73", "61, 77", "63, 80", "64, 81", "65, 82", "66, 83", "67, 85", "71, 90", "72, 91", "73, 92", "77, 97", "79, 100", "80, 101", "83, 105", "85, 107", "85, 108", "86, 109", "87, 110", "88, 111", "89, 112", "89, 113", "91, 115", "93, 118", "94, 119"]}, "cell": "fraction_between/3", "stage": "C"}
+{"id": "po-fraction_between-t3-eval-001930", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 120 such that \\frac{23}{73} < \\frac{a}{b} < \\frac{24649}{77307}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 120 such that \\frac{23}{73} < \\frac{a}{b} < \\frac{24649}{77307}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 19", "ref": "6, 19", "family": "po_fraction_between", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 17, "bucket": "9-20", "guess_rate": 0.011806, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "23/73", "hi": "24649/77307", "D": 120}, "seed": 79997677103003164, "version": "po-v0", "tier": 3, "n_raw": 28, "small": 19, "valid_answers": ["6, 19", "7, 22", "13, 41", "19, 60", "20, 63", "22, 69", "25, 79", "27, 85", "29, 91", "29, 92", "31, 98", "32, 101", "33, 104", "34, 107", "35, 111", "36, 113", "37, 117"]}, "cell": "fraction_between/3", "stage": "C"}
+{"id": "po-fraction_between-t3-eval-002101", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 120 such that \\frac{62}{123} < \\frac{a}{b} < \\frac{15355}{30381}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 120 such that \\frac{62}{123} < \\frac{a}{b} < \\frac{15355}{30381}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "47, 93", "ref": "47, 93", "family": "po_fraction_between", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.006272, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "62/123", "hi": "15355/30381", "D": 120}, "seed": 79997677103003479, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 93, "valid_answers": ["47, 93", "48, 95", "49, 97", "50, 99", "51, 101", "52, 103", "53, 105", "54, 107", "55, 109", "56, 111", "57, 113", "58, 115", "59, 117", "60, 119"]}, "cell": "fraction_between/3", "stage": "C"}
+{"id": "po-fraction_between-t3-eval-001754", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 120 such that \\frac{13}{14} < \\frac{a}{b} < \\frac{475}{504}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 120 such that \\frac{13}{14} < \\frac{a}{b} < \\frac{475}{504}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "14, 15", "ref": "14, 15", "family": "po_fraction_between", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 63, "bucket": null, "guess_rate": 0.015217, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "13/14", "hi": "475/504", "D": 120}, "seed": 79997677103002863, "version": "po-v0", "tier": 3, "n_raw": 98, "small": 15, "valid_answers": ["14, 15", "15, 16", "16, 17", "27, 29", "29, 31", "31, 33", "40, 43", "41, 44", "43, 46", "44, 47", "46, 49", "47, 50", "49, 52", "53, 57", "55, 59", "57, 61", "59, 63", "61, 65", "63, 67", "65, 69", "66, 71", "67, 72", "68, 73", "69, 74", "71, 76", "72, 77", "73, 78", "74, 79", "76, 81", "77, 82", "78, 83", "79, 84", "79, 85", "81, 86", "83, 89", "85, 91", "89, 95", "91, 97", "92, 99", "93, 100", "94, 101", "95, 101", "95, 102", "96, 103", "97, 103", "97, 104", "99, 106", "100, 107", "101, 108", "102, 109", "103, 110", "104, 111", "105, 113", "106, 113", "107, 114", "107, 115", "108, 115", "109, 116", "109, 117", "110, 117", "111, 118", "111, 119", "113, 120"]}, "cell": "fraction_between/3", "stage": "C"}
+{"id": "po-fraction_between-t3-eval-001397", "task": "cx_po_fraction_between", "source": "cx_po_fraction_between", "split": "eval", "problem": "Find integers a, b with 1 <= b <= 120 such that \\frac{142}{207} < \\frac{a}{b} < \\frac{16}{23}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b with 1 <= b <= 120 such that \\frac{142}{207} < \\frac{a}{b} < \\frac{16}{23}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9, 13", "ref": "9, 13", "family": "po_fraction_between", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 43, "bucket": null, "guess_rate": 0.014052, "solvable": true, "holdout": false, "meta": {"family": "po_fraction_between", "rung": "witness", "params": {"lo": "142/207", "hi": "16/23", "D": 120}, "seed": 79997677103002267, "version": "po-v0", "tier": 3, "n_raw": 68, "small": 13, "valid_answers": ["9, 13", "11, 16", "20, 29", "25, 36", "29, 42", "31, 45", "34, 49", "35, 51", "38, 55", "41, 59", "42, 61", "43, 62", "46, 67", "47, 68", "49, 71", "51, 74", "52, 75", "53, 77", "56, 81", "57, 82", "57, 83", "59, 85", "59, 86", "61, 88", "64, 93", "65, 94", "66, 95", "67, 97", "68, 99", "69, 100", "70, 101", "71, 103", "73, 105", "73, 106", "74, 107", "75, 109", "77, 111", "78, 113", "79, 114", "79, 115", "81, 118", "82, 119", "83, 120"]}, "cell": "fraction_between/3", "stage": "C"}
+{"id": "po-gauss_factor-t1-eval-000122", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = 3 + 3i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = 3 + 3i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 3, 0", "ref": "1, 1, 3, 0", "family": "po_gauss_factor", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000988, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": 3, "v": 3}, "seed": 258960587001000228, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 3, "valid_answers": ["1, 1, 3, 0", "3, 0, 1, 1"]}, "cell": "gauss_factor/1", "stage": "B"}
+{"id": "po-gauss_factor-t1-eval-000016", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = 4 - 6i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = 4 - 6i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -1, -5", "ref": "1, 1, -1, -5", "family": "po_gauss_factor", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000278, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": 4, "v": -6}, "seed": 258960587001000020, "version": "po-v0", "tier": 1, "n_raw": 16, "small": 3, "valid_answers": ["1, 1, -1, -5", "1, 5, -1, -1", "2, 0, 2, -3", "3, 2, 0, -2"]}, "cell": "gauss_factor/1", "stage": "B"}
+{"id": "po-gauss_factor-t1-eval-000076", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -8 - 4i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -8 - 4i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -6, 2", "ref": "1, 1, -6, 2", "family": "po_gauss_factor", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000342, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -8, "v": -4}, "seed": 258960587001000109, "version": "po-v0", "tier": 1, "n_raw": 32, "small": 3, "valid_answers": ["1, 1, -6, 2", "1, 3, -2, 2", "2, 0, -4, -2", "2, 1, -4, 0", "2, 2, -3, 1", "2, 6, -1, 1", "4, 0, -2, -1", "4, 2, -2, 0"]}, "cell": "gauss_factor/1", "stage": "B"}
+{"id": "po-gauss_factor-t1-eval-000123", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -5 + 1i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -5 + 1i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -2, 3", "ref": "1, 1, -2, 3", "family": "po_gauss_factor", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000459, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -5, "v": 1}, "seed": 258960587001000229, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 3, "valid_answers": ["1, 1, -2, 3", "3, 2, -1, 1"]}, "cell": "gauss_factor/1", "stage": "B"}
+{"id": "po-gauss_factor-t1-eval-000130", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = 3 - 15i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = 3 - 15i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -6, -9", "ref": "1, 1, -6, -9", "family": "po_gauss_factor", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": 3, "v": -15}, "seed": 258960587001000248, "version": "po-v0", "tier": 1, "n_raw": 24, "small": 3, "valid_answers": ["1, 1, -6, -9", "2, 3, -3, -3", "3, 0, 1, -5", "3, 3, -2, -3", "5, 1, 0, -3", "6, 9, -1, -1"]}, "cell": "gauss_factor/1", "stage": "B"}
+{"id": "po-gauss_factor-t1-eval-000113", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -3 - 11i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -3 - 11i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -7, -4", "ref": "1, 1, -7, -4", "family": "po_gauss_factor", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 7.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -3, "v": -11}, "seed": 258960587001000191, "version": "po-v0", "tier": 1, "n_raw": 24, "small": 3, "valid_answers": ["1, 1, -7, -4", "1, 2, -5, -1", "2, 3, -3, -1", "3, 1, -2, -3", "5, 1, -1, -2", "7, 4, -1, -1"]}, "cell": "gauss_factor/1", "stage": "B"}
+{"id": "po-gauss_factor-t1-eval-000044", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = 7 - 4i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = 7 - 4i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 1, 2, -3", "ref": "2, 1, 2, -3", "family": "po_gauss_factor", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 8.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": 7, "v": -4}, "seed": 258960587001000054, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 3, "valid_answers": ["2, 1, 2, -3", "3, 2, 1, -2"]}, "cell": "gauss_factor/1", "stage": "B"}
+{"id": "po-gauss_factor-t1-eval-000111", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -8 - 6i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -8 - 6i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -7, 1", "ref": "1, 1, -7, 1", "family": "po_gauss_factor", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000131, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -8, "v": -6}, "seed": 258960587001000188, "version": "po-v0", "tier": 1, "n_raw": 28, "small": 3, "valid_answers": ["1, 1, -7, 1", "1, 2, -4, 2", "1, 7, -1, 1", "2, 0, -4, -3", "2, 4, -2, 1", "3, 1, -3, -1", "4, 3, -2, 0"]}, "cell": "gauss_factor/1", "stage": "B"}
+{"id": "po-gauss_factor-t1-eval-000120", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -11 + 3i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -11 + 3i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -4, 7", "ref": "1, 1, -4, 7", "family": "po_gauss_factor", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 7.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -11, "v": 3}, "seed": 258960587001000222, "version": "po-v0", "tier": 1, "n_raw": 24, "small": 3, "valid_answers": ["1, 1, -4, 7", "1, 2, -1, 5", "2, 3, -1, 3", "3, 1, -3, 2", "5, 1, -2, 1", "7, 4, -1, 1"]}, "cell": "gauss_factor/1", "stage": "B"}
+{"id": "po-gauss_factor-t1-eval-000007", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = 3 + 9i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = 3 + 9i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 6, 3", "ref": "1, 1, 6, 3", "family": "po_gauss_factor", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000166, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": 3, "v": 9}, "seed": 258960587001000008, "version": "po-v0", "tier": 1, "n_raw": 24, "small": 3, "valid_answers": ["1, 1, 6, 3", "1, 3, 3, 0", "2, 1, 3, 3", "3, 0, 1, 3", "3, 3, 2, 1", "6, 3, 1, 1"]}, "cell": "gauss_factor/1", "stage": "B"}
+{"id": "po-gauss_factor-t1-eval-000118", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = 0 + 13i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = 0 + 13i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 3, 3, 2", "ref": "2, 3, 3, 2", "family": "po_gauss_factor", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": 0, "v": 13}, "seed": 258960587001000217, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 3, "valid_answers": ["2, 3, 3, 2", "3, 2, 2, 3"]}, "cell": "gauss_factor/1", "stage": "B"}
+{"id": "po-gauss_factor-t1-eval-000092", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -7 + 1i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -7 + 1i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -3, 4", "ref": "1, 1, -3, 4", "family": "po_gauss_factor", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000278, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -7, "v": 1}, "seed": 258960587001000151, "version": "po-v0", "tier": 1, "n_raw": 16, "small": 3, "valid_answers": ["1, 1, -3, 4", "1, 2, -1, 3", "3, 1, -2, 1", "4, 3, -1, 1"]}, "cell": "gauss_factor/1", "stage": "B"}
+{"id": "po-gauss_factor-t2-eval-000085", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -7 + 1i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -7 + 1i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -3, 4", "ref": "1, 1, -3, 4", "family": "po_gauss_factor", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000278, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -7, "v": 1}, "seed": 258960587002000094, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 3, "valid_answers": ["1, 1, -3, 4", "1, 2, -1, 3", "3, 1, -2, 1", "4, 3, -1, 1"]}, "cell": "gauss_factor/2", "stage": "B"}
+{"id": "po-gauss_factor-t2-eval-000018", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -2 - 11i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -2 - 11i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 1, -3, -4", "ref": "2, 1, -3, -4", "family": "po_gauss_factor", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -2, "v": -11}, "seed": 258960587002000019, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 4, "valid_answers": ["2, 1, -3, -4", "3, 4, -2, -1"]}, "cell": "gauss_factor/2", "stage": "B"}
+{"id": "po-gauss_factor-t2-eval-000313", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -14 - 2i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -14 - 2i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -8, 6", "ref": "1, 1, -8, 6", "family": "po_gauss_factor", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 5.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -14, "v": -2}, "seed": 258960587002000506, "version": "po-v0", "tier": 2, "n_raw": 40, "small": 4, "valid_answers": ["1, 1, -8, 6", "1, 3, -2, 4", "2, 0, -7, -1", "2, 1, -6, 2", "2, 2, -4, 3", "2, 6, -1, 2", "3, 4, -2, 2", "4, 2, -3, 1", "6, 8, -1, 1", "7, 1, -2, 0"]}, "cell": "gauss_factor/2", "stage": "B"}
+{"id": "po-gauss_factor-t2-eval-000470", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -6 - 17i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -6 - 17i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 2, -8, -1", "ref": "1, 2, -8, -1", "family": "po_gauss_factor", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 8e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -6, "v": -17}, "seed": 258960587002001348, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 4, "valid_answers": ["1, 2, -8, -1", "3, 2, -4, -3", "4, 3, -3, -2", "8, 1, -1, -2"]}, "cell": "gauss_factor/2", "stage": "B"}
+{"id": "po-gauss_factor-t2-eval-000107", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = 3 - 12i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = 3 - 12i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 0, 1, -4", "ref": "3, 0, 1, -4", "family": "po_gauss_factor", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": 3, "v": -12}, "seed": 258960587002000122, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 4, "valid_answers": ["3, 0, 1, -4", "4, 1, 0, -3"]}, "cell": "gauss_factor/2", "stage": "B"}
+{"id": "po-gauss_factor-t2-eval-000466", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -3 + 21i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -3 + 21i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 9, 12", "ref": "1, 1, 9, 12", "family": "po_gauss_factor", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 1.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -3, "v": 21}, "seed": 258960587002001277, "version": "po-v0", "tier": 2, "n_raw": 40, "small": 4, "valid_answers": ["1, 1, 9, 12", "1, 3, 6, 3", "2, 1, 3, 9", "3, 0, -1, 7", "3, 3, 3, 4", "3, 4, 3, 3", "3, 9, 2, 1", "6, 3, 1, 3", "7, 1, 0, 3", "9, 12, 1, 1"]}, "cell": "gauss_factor/2", "stage": "B"}
+{"id": "po-gauss_factor-t2-eval-000110", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = 5 - 15i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = 5 - 15i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -5, -10", "ref": "1, 1, -5, -10", "family": "po_gauss_factor", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 4.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": 5, "v": -15}, "seed": 258960587002000126, "version": "po-v0", "tier": 2, "n_raw": 40, "small": 4, "valid_answers": ["1, 1, -5, -10", "1, 2, -5, -5", "1, 3, -4, -3", "1, 7, -2, -1", "2, 1, -1, -7", "3, 1, 0, -5", "4, 3, -1, -3", "5, 0, 1, -3", "5, 5, -1, -2", "5, 10, -1, -1"]}, "cell": "gauss_factor/2", "stage": "B"}
+{"id": "po-gauss_factor-t2-eval-000064", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -3 - 11i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -3 - 11i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -7, -4", "ref": "1, 1, -7, -4", "family": "po_gauss_factor", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 7.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -3, "v": -11}, "seed": 258960587002000069, "version": "po-v0", "tier": 2, "n_raw": 24, "small": 3, "valid_answers": ["1, 1, -7, -4", "1, 2, -5, -1", "2, 3, -3, -1", "3, 1, -2, -3", "5, 1, -1, -2", "7, 4, -1, -1"]}, "cell": "gauss_factor/2", "stage": "B"}
+{"id": "po-gauss_factor-t2-eval-000378", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -3 - 5i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -3 - 5i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -4, -1", "ref": "1, 1, -4, -1", "family": "po_gauss_factor", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000459, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -3, "v": -5}, "seed": 258960587002000700, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 4, "valid_answers": ["1, 1, -4, -1", "4, 1, -1, -1"]}, "cell": "gauss_factor/2", "stage": "B"}
+{"id": "po-gauss_factor-t2-eval-000282", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -9 + 15i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -9 + 15i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 3, 12", "ref": "1, 1, 3, 12", "family": "po_gauss_factor", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -9, "v": 15}, "seed": 258960587002000425, "version": "po-v0", "tier": 2, "n_raw": 24, "small": 4, "valid_answers": ["1, 1, 3, 12", "1, 4, 3, 3", "3, 0, -3, 5", "3, 3, 1, 4", "3, 12, 1, 1", "5, 3, 0, 3"]}, "cell": "gauss_factor/2", "stage": "B"}
+{"id": "po-gauss_factor-t2-eval-000457", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -16 - 13i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -16 - 13i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 4, -4, 3", "ref": "1, 4, -4, 3", "family": "po_gauss_factor", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -16, "v": -13}, "seed": 258960587002001173, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 4, "valid_answers": ["1, 4, -4, 3", "2, 1, -9, -2", "3, 4, -4, 1", "9, 2, -2, -1"]}, "cell": "gauss_factor/2", "stage": "B"}
+{"id": "po-gauss_factor-t2-eval-000488", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = 14 + 5i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = 14 + 5i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 4, 2, -3", "ref": "1, 4, 2, -3", "family": "po_gauss_factor", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": 14, "v": 5}, "seed": 258960587002001514, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 4, "valid_answers": ["1, 4, 2, -3", "3, 2, 4, -1"]}, "cell": "gauss_factor/2", "stage": "B"}
+{"id": "po-gauss_factor-t3-eval-001846", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -30 + 24i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -30 + 24i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -3, 27", "ref": "1, 1, -3, 27", "family": "po_gauss_factor", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -30, "v": 24}, "seed": 258960587003004173, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 6, "valid_answers": ["1, 1, -3, 27", "2, 0, -15, 12", "3, 0, -10, 8", "3, 3, -1, 9", "4, 5, 0, 6", "6, 0, -5, 4", "8, 10, 0, 3", "9, 1, -3, 3", "12, 15, 0, 2", "27, 3, -1, 1"]}, "cell": "gauss_factor/3", "stage": "B"}
+{"id": "po-gauss_factor-t3-eval-001014", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = 14 + 22i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = 14 + 22i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 18, 4", "ref": "1, 1, 18, 4", "family": "po_gauss_factor", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 7e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": 14, "v": 22}, "seed": 258960587003001467, "version": "po-v0", "tier": 3, "n_raw": 56, "small": 5, "valid_answers": ["1, 1, 18, 4", "1, 3, 8, -2", "1, 4, 6, -2", "2, 0, 7, 11", "2, 1, 10, 6", "2, 2, 9, 2", "2, 6, 4, -1", "2, 8, 3, -1", "4, 2, 5, 3", "5, 3, 4, 2", "7, 11, 2, 0", "9, 2, 2, 2", "10, 6, 2, 1", "18, 4, 1, 1"]}, "cell": "gauss_factor/3", "stage": "B"}
+{"id": "po-gauss_factor-t3-eval-000357", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -9 + 15i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -9 + 15i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 3, 12", "ref": "1, 1, 3, 12", "family": "po_gauss_factor", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -9, "v": 15}, "seed": 258960587003000399, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 4, "valid_answers": ["1, 1, 3, 12", "1, 4, 3, 3", "3, 0, -3, 5", "3, 3, 1, 4", "3, 12, 1, 1", "5, 3, 0, 3"]}, "cell": "gauss_factor/3", "stage": "B"}
+{"id": "po-gauss_factor-t3-eval-000489", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = 23 - 27i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = 23 - 27i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -2, -25", "ref": "1, 1, -2, -25", "family": "po_gauss_factor", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": 23, "v": -27}, "seed": 258960587003000578, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 6, "valid_answers": ["1, 1, -2, -25", "1, 4, -5, -7", "2, 25, -1, -1", "5, 3, 1, -6", "5, 7, -1, -4", "6, 1, 3, -5"]}, "cell": "gauss_factor/3", "stage": "B"}
+{"id": "po-gauss_factor-t3-eval-000859", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -3 + 27i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -3 + 27i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 12, 15", "ref": "1, 1, 12, 15", "family": "po_gauss_factor", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 3e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -3, "v": 27}, "seed": 258960587003001172, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 5, "valid_answers": ["1, 1, 12, 15", "3, 0, -1, 9", "3, 3, 4, 5", "4, 5, 3, 3", "9, 1, 0, 3", "12, 15, 1, 1"]}, "cell": "gauss_factor/3", "stage": "B"}
+{"id": "po-gauss_factor-t3-eval-002385", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -12 - 1i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -12 - 1i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 1, -5, 2", "ref": "2, 1, -5, 2", "family": "po_gauss_factor", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -12, "v": -1}, "seed": 258960587003018210, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 5, "valid_answers": ["2, 1, -5, 2", "2, 5, -1, 2"]}, "cell": "gauss_factor/3", "stage": "B"}
+{"id": "po-gauss_factor-t3-eval-000079", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = 17 - 17i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = 17 - 17i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 0, -17", "ref": "1, 1, 0, -17", "family": "po_gauss_factor", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": 17, "v": -17}, "seed": 258960587003000081, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 5, "valid_answers": ["1, 1, 0, -17", "1, 4, -3, -5", "3, 5, -1, -4", "4, 1, 3, -5", "5, 3, 1, -4", "17, 0, 1, -1"]}, "cell": "gauss_factor/3", "stage": "B"}
+{"id": "po-gauss_factor-t3-eval-000140", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -36 + 12i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -36 + 12i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -12, 24", "ref": "1, 1, -12, 24", "family": "po_gauss_factor", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 3e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -36, "v": 12}, "seed": 258960587003000147, "version": "po-v0", "tier": 3, "n_raw": 88, "small": 6, "valid_answers": ["1, 1, -12, 24", "1, 3, 0, 12", "2, 0, -18, 6", "2, 1, -12, 12", "2, 2, -6, 12", "2, 6, 0, 6", "3, 0, -12, 4", "3, 3, -4, 8", "3, 9, 0, 4", "4, 0, -9, 3", "4, 2, -6, 6", "4, 4, -3, 6", "4, 12, 0, 3", "6, 0, -6, 2", "6, 3, -4, 4", "6, 6, -2, 4", "6, 18, 0, 2", "8, 4, -3, 3", "12, 0, -3, 1", "12, 6, -2, 2", "12, 12, -1, 2", "24, 12, -1, 1"]}, "cell": "gauss_factor/3", "stage": "B"}
+{"id": "po-gauss_factor-t3-eval-000127", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -2 + 22i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -2 + 22i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 10, 12", "ref": "1, 1, 10, 12", "family": "po_gauss_factor", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 6e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -2, "v": 22}, "seed": 258960587003000133, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 6, "valid_answers": ["1, 1, 10, 12", "2, 0, -1, 11", "2, 2, 5, 6", "5, 6, 2, 2", "10, 12, 1, 1", "11, 1, 0, 2"]}, "cell": "gauss_factor/3", "stage": "B"}
+{"id": "po-gauss_factor-t3-eval-000661", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -8 - 19i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -8 - 19i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 1, -7, -6", "ref": "2, 1, -7, -6", "family": "po_gauss_factor", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -8, "v": -19}, "seed": 258960587003000844, "version": "po-v0", "tier": 3, "n_raw": 16, "small": 4, "valid_answers": ["2, 1, -7, -6", "3, 4, -4, -1", "4, 1, -3, -4", "7, 6, -2, -1"]}, "cell": "gauss_factor/3", "stage": "B"}
+{"id": "po-gauss_factor-t3-eval-001918", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = -6 + 27i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = -6 + 27i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 4, 6, 3", "ref": "1, 4, 6, 3", "family": "po_gauss_factor", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 3e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": -6, "v": 27}, "seed": 258960587003004689, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 6, "valid_answers": ["1, 4, 6, 3", "2, 1, 3, 12", "3, 0, -2, 9", "3, 12, 2, 1", "6, 3, 1, 4", "9, 2, 0, 3"]}, "cell": "gauss_factor/3", "stage": "B"}
+{"id": "po-gauss_factor-t3-eval-000809", "task": "cx_po_gauss_factor", "source": "cx_po_gauss_factor", "split": "eval", "problem": "Find integers a, b, c, d such that (a + bi)(c + di) = 18 - 39i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d such that (a + bi)(c + di) = 18 - 39i, where a^2 + b^2 > 1 and c^2 + d^2 > 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 2, -12, -15", "ref": "1, 2, -12, -15", "family": "po_gauss_factor", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gauss_factor", "rung": "witness", "params": {"u": 18, "v": -39}, "seed": 258960587003001071, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 6, "valid_answers": ["1, 2, -12, -15", "3, 0, 6, -13", "3, 6, -4, -5", "4, 5, -3, -6", "12, 15, -1, -2", "13, 6, 0, -3"]}, "cell": "gauss_factor/3", "stage": "B"}
+{"id": "po-gcd_lcm-t0-eval-000259", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 231. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 231. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 231", "ref": "3, 231", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 3, "L": 231, "k": 2}, "seed": 43074069100000471, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 33, "valid_answers": ["3, 231", "21, 33"]}, "cell": "gcd_lcm/0", "stage": "A"}
+{"id": "po-gcd_lcm-t0-eval-000139", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 117. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 117. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 117", "ref": "3, 117", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00029, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 3, "L": 117, "k": 2}, "seed": 43074069100000176, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 39, "valid_answers": ["3, 117", "9, 39"]}, "cell": "gcd_lcm/0", "stage": "A"}
+{"id": "po-gcd_lcm-t0-eval-000183", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 6 and whose least common multiple is 3630. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 6 and whose least common multiple is 3630. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 3630", "ref": "6, 3630", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 6, "L": 3630, "k": 2}, "seed": 43074069100000251, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 726, "valid_answers": ["6, 3630", "30, 726"]}, "cell": "gcd_lcm/0", "stage": "A"}
+{"id": "po-gcd_lcm-t0-eval-000269", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 210. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 210. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 210", "ref": "10, 210", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 10, "L": 210, "k": 2}, "seed": 43074069100000512, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 70, "valid_answers": ["10, 210", "30, 70"]}, "cell": "gcd_lcm/0", "stage": "A"}
+{"id": "po-gcd_lcm-t0-eval-000025", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 4 and whose least common multiple is 40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 4 and whose least common multiple is 40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 40", "ref": "4, 40", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002439, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 4, "L": 40, "k": 2}, "seed": 43074069100000026, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 20, "valid_answers": ["4, 40", "8, 20"]}, "cell": "gcd_lcm/0", "stage": "A"}
+{"id": "po-gcd_lcm-t0-eval-000285", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 6 and whose least common multiple is 9126. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 6 and whose least common multiple is 9126. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 9126", "ref": "6, 9126", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 6, "L": 9126, "k": 2}, "seed": 43074069100000591, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 1014, "valid_answers": ["6, 9126", "54, 1014"]}, "cell": "gcd_lcm/0", "stage": "A"}
+{"id": "po-gcd_lcm-t0-eval-000306", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 5390. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 5390. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 5390", "ref": "10, 5390", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 10, "L": 5390, "k": 2}, "seed": 43074069100000691, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 490, "valid_answers": ["10, 5390", "110, 490"]}, "cell": "gcd_lcm/0", "stage": "A"}
+{"id": "po-gcd_lcm-t0-eval-000235", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 2 and whose least common multiple is 286. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 2 and whose least common multiple is 286. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 286", "ref": "2, 286", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 4.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 2, "L": 286, "k": 2}, "seed": 43074069100000392, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 26, "valid_answers": ["2, 286", "22, 26"]}, "cell": "gcd_lcm/0", "stage": "A"}
+{"id": "po-gcd_lcm-t0-eval-000220", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 1 and whose least common multiple is 363. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 1 and whose least common multiple is 363. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 363", "ref": "1, 363", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 1, "L": 363, "k": 2}, "seed": 43074069100000345, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 121, "valid_answers": ["1, 363", "3, 121"]}, "cell": "gcd_lcm/0", "stage": "A"}
+{"id": "po-gcd_lcm-t0-eval-000205", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 4 and whose least common multiple is 1100. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 4 and whose least common multiple is 1100. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 1100", "ref": "4, 1100", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 3e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 4, "L": 1100, "k": 2}, "seed": 43074069100000298, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 100, "valid_answers": ["4, 1100", "44, 100"]}, "cell": "gcd_lcm/0", "stage": "A"}
+{"id": "po-gcd_lcm-t0-eval-000330", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 5577. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 5577. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 5577", "ref": "3, 5577", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 3, "L": 5577, "k": 2}, "seed": 43074069100000957, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 507, "valid_answers": ["3, 5577", "33, 507"]}, "cell": "gcd_lcm/0", "stage": "A"}
+{"id": "po-gcd_lcm-t0-eval-000255", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 1 and whose least common multiple is 175. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 1 and whose least common multiple is 175. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 175", "ref": "1, 175", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00013, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 1, "L": 175, "k": 2}, "seed": 43074069100000467, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 25, "valid_answers": ["1, 175", "7, 25"]}, "cell": "gcd_lcm/0", "stage": "A"}
+{"id": "po-gcd_lcm-t1-eval-000234", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 3000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 3000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 3000", "ref": "10, 3000", "family": "po_gcd_lcm", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 10, "L": 3000, "k": 2}, "seed": 43074069101000261, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 250, "valid_answers": ["10, 3000", "30, 1000", "40, 750", "120, 250"]}, "cell": "gcd_lcm/1", "stage": "B"}
+{"id": "po-gcd_lcm-t1-eval-000270", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 21780. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 21780. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 21780", "ref": "10, 21780", "family": "po_gcd_lcm", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 10, "L": 21780, "k": 2}, "seed": 43074069101000304, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 1210, "valid_answers": ["10, 21780", "20, 10890", "90, 2420", "180, 1210"]}, "cell": "gcd_lcm/1", "stage": "B"}
+{"id": "po-gcd_lcm-t1-eval-000569", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 4 and whose least common multiple is 165620. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 4 and whose least common multiple is 165620. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 165620", "ref": "4, 165620", "family": "po_gcd_lcm", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 4, "L": 165620, "k": 2}, "seed": 43074069101000842, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 980, "valid_answers": ["4, 165620", "20, 33124", "196, 3380", "676, 980"]}, "cell": "gcd_lcm/1", "stage": "B"}
+{"id": "po-gcd_lcm-t1-eval-000863", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 37180. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 37180. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 37180", "ref": "10, 37180", "family": "po_gcd_lcm", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 10, "L": 37180, "k": 2}, "seed": 43074069101002238, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 1690, "valid_answers": ["10, 37180", "20, 18590", "110, 3380", "220, 1690"]}, "cell": "gcd_lcm/1", "stage": "B"}
+{"id": "po-gcd_lcm-t1-eval-000412", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 76050. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 76050. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 76050", "ref": "10, 76050", "family": "po_gcd_lcm", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 10, "L": 76050, "k": 2}, "seed": 43074069101000534, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 1690, "valid_answers": ["10, 76050", "50, 15210", "90, 8450", "450, 1690"]}, "cell": "gcd_lcm/1", "stage": "B"}
+{"id": "po-gcd_lcm-t1-eval-000013", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 6 and whose least common multiple is 2730. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 6 and whose least common multiple is 2730. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 2730", "ref": "6, 2730", "family": "po_gcd_lcm", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 6, "L": 2730, "k": 2}, "seed": 43074069101000013, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 210, "valid_answers": ["6, 2730", "30, 546", "42, 390", "78, 210"]}, "cell": "gcd_lcm/1", "stage": "B"}
+{"id": "po-gcd_lcm-t1-eval-000167", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 2 and whose least common multiple is 5390. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 2 and whose least common multiple is 5390. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 5390", "ref": "2, 5390", "family": "po_gcd_lcm", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 2, "L": 5390, "k": 2}, "seed": 43074069101000183, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 110, "valid_answers": ["2, 5390", "10, 1078", "22, 490", "98, 110"]}, "cell": "gcd_lcm/1", "stage": "B"}
+{"id": "po-gcd_lcm-t1-eval-000197", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 110250. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 110250. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 110250", "ref": "10, 110250", "family": "po_gcd_lcm", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 10, "L": 110250, "k": 2}, "seed": 43074069101000219, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 2250, "valid_answers": ["10, 110250", "90, 12250", "250, 4410", "490, 2250"]}, "cell": "gcd_lcm/1", "stage": "B"}
+{"id": "po-gcd_lcm-t1-eval-000383", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 1 and whose least common multiple is 11025. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 1 and whose least common multiple is 11025. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 11025", "ref": "1, 11025", "family": "po_gcd_lcm", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 1, "L": 11025, "k": 2}, "seed": 43074069101000483, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 225, "valid_answers": ["1, 11025", "9, 1225", "25, 441", "49, 225"]}, "cell": "gcd_lcm/1", "stage": "B"}
+{"id": "po-gcd_lcm-t1-eval-000484", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 585. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 585. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 585", "ref": "3, 585", "family": "po_gcd_lcm", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 2.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 3, "L": 585, "k": 2}, "seed": 43074069101000667, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 45, "valid_answers": ["3, 585", "9, 195", "15, 117", "39, 45"]}, "cell": "gcd_lcm/1", "stage": "B"}
+{"id": "po-gcd_lcm-t1-eval-000494", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 6 and whose least common multiple is 4200. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 6 and whose least common multiple is 4200. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 4200", "ref": "6, 4200", "family": "po_gcd_lcm", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 6, "L": 4200, "k": 2}, "seed": 43074069101000689, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 168, "valid_answers": ["6, 4200", "24, 1050", "42, 600", "150, 168"]}, "cell": "gcd_lcm/1", "stage": "B"}
+{"id": "po-gcd_lcm-t1-eval-000535", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 17199. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 17199. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 17199", "ref": "3, 17199", "family": "po_gcd_lcm", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 3, "L": 17199, "k": 2}, "seed": 43074069101000775, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 351, "valid_answers": ["3, 17199", "27, 1911", "39, 1323", "147, 351"]}, "cell": "gcd_lcm/1", "stage": "B"}
+{"id": "po-gcd_lcm-t2-eval-000311", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 1 and whose least common multiple is 2145. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 1 and whose least common multiple is 2145. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 2145", "ref": "1, 2145", "family": "po_gcd_lcm", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 3e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 1, "L": 2145, "k": 2}, "seed": 43074069102000342, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 55, "valid_answers": ["1, 2145", "3, 715", "5, 429", "11, 195", "13, 165", "15, 143", "33, 65", "39, 55"]}, "cell": "gcd_lcm/2", "stage": "B"}
+{"id": "po-gcd_lcm-t2-eval-000961", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 159705. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 159705. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 159705", "ref": "3, 159705", "family": "po_gcd_lcm", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 3, "L": 159705, "k": 2}, "seed": 43074069102001608, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 945, "valid_answers": ["3, 159705", "15, 31941", "21, 22815", "27, 17745", "105, 4563", "135, 3549", "189, 2535", "507, 945"]}, "cell": "gcd_lcm/2", "stage": "B"}
+{"id": "po-gcd_lcm-t2-eval-000152", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 9009. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 9009. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 9009", "ref": "3, 9009", "family": "po_gcd_lcm", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 3, "L": 9009, "k": 2}, "seed": 43074069102000158, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 231, "valid_answers": ["3, 9009", "9, 3003", "21, 1287", "33, 819", "39, 693", "63, 429", "99, 273", "117, 231"]}, "cell": "gcd_lcm/2", "stage": "B"}
+{"id": "po-gcd_lcm-t2-eval-000163", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 828100. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 828100. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 828100", "ref": "10, 828100", "family": "po_gcd_lcm", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 10, "L": 828100, "k": 2}, "seed": 43074069102000169, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 3380, "valid_answers": ["10, 828100", "20, 414050", "50, 165620", "100, 82810", "490, 16900", "980, 8450", "1690, 4900", "2450, 3380"]}, "cell": "gcd_lcm/2", "stage": "B"}
+{"id": "po-gcd_lcm-t2-eval-000762", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 4 and whose least common multiple is 3312400. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 4 and whose least common multiple is 3312400. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 3312400", "ref": "4, 3312400", "family": "po_gcd_lcm", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 4, "L": 3312400, "k": 2}, "seed": 43074069102001091, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 4900, "valid_answers": ["4, 3312400", "16, 828100", "100, 132496", "196, 67600", "400, 33124", "676, 19600", "784, 16900", "2704, 4900"]}, "cell": "gcd_lcm/2", "stage": "B"}
+{"id": "po-gcd_lcm-t2-eval-000493", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 2 and whose least common multiple is 55770. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 2 and whose least common multiple is 55770. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 55770", "ref": "2, 55770", "family": "po_gcd_lcm", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 2, "L": 55770, "k": 2}, "seed": 43074069102000609, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 338, "valid_answers": ["2, 55770", "6, 18590", "10, 11154", "22, 5070", "30, 3718", "66, 1690", "110, 1014", "330, 338"]}, "cell": "gcd_lcm/2", "stage": "B"}
+{"id": "po-gcd_lcm-t2-eval-000467", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 4 and whose least common multiple is 13200. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 4 and whose least common multiple is 13200. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 13200", "ref": "4, 13200", "family": "po_gcd_lcm", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 4, "L": 13200, "k": 2}, "seed": 43074069102000568, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 300, "valid_answers": ["4, 13200", "12, 4400", "16, 3300", "44, 1200", "48, 1100", "100, 528", "132, 400", "176, 300"]}, "cell": "gcd_lcm/2", "stage": "B"}
+{"id": "po-gcd_lcm-t2-eval-000806", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 2 and whose least common multiple is 2044900. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 2 and whose least common multiple is 2044900. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 2044900", "ref": "2, 2044900", "family": "po_gcd_lcm", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 2, "L": 2044900, "k": 2}, "seed": 43074069102001203, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 6050, "valid_answers": ["2, 2044900", "4, 1022450", "50, 81796", "100, 40898", "242, 16900", "338, 12100", "484, 8450", "676, 6050"]}, "cell": "gcd_lcm/2", "stage": "B"}
+{"id": "po-gcd_lcm-t2-eval-001044", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 9900. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 10 and whose least common multiple is 9900. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 9900", "ref": "10, 9900", "family": "po_gcd_lcm", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 10, "L": 9900, "k": 2}, "seed": 43074069102001886, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 450, "valid_answers": ["10, 9900", "20, 4950", "50, 1980", "90, 1100", "100, 990", "110, 900", "180, 550", "220, 450"]}, "cell": "gcd_lcm/2", "stage": "B"}
+{"id": "po-gcd_lcm-t2-eval-001130", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 6 and whose least common multiple is 1260. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 6 and whose least common multiple is 1260. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 1260", "ref": "6, 1260", "family": "po_gcd_lcm", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 6, "L": 1260, "k": 2}, "seed": 43074069102002230, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 90, "valid_answers": ["6, 1260", "12, 630", "18, 420", "30, 252", "36, 210", "42, 180", "60, 126", "84, 90"]}, "cell": "gcd_lcm/2", "stage": "B"}
+{"id": "po-gcd_lcm-t2-eval-000426", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 525525. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 3 and whose least common multiple is 525525. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 525525", "ref": "3, 525525", "family": "po_gcd_lcm", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 3, "L": 525525, "k": 2}, "seed": 43074069102000504, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 1617, "valid_answers": ["3, 525525", "33, 47775", "39, 40425", "75, 21021", "147, 10725", "429, 3675", "825, 1911", "975, 1617"]}, "cell": "gcd_lcm/2", "stage": "B"}
+{"id": "po-gcd_lcm-t2-eval-000818", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b whose greatest common divisor is 4 and whose least common multiple is 176400. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b whose greatest common divisor is 4 and whose least common multiple is 176400. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 176400", "ref": "4, 176400", "family": "po_gcd_lcm", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 4, "L": 176400, "k": 2}, "seed": 43074069102001229, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 900, "valid_answers": ["4, 176400", "16, 44100", "36, 19600", "100, 7056", "144, 4900", "196, 3600", "400, 1764", "784, 900"]}, "cell": "gcd_lcm/2", "stage": "B"}
+{"id": "po-gcd_lcm-t3-eval-000037", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b, c whose greatest common divisor is 1 and whose least common multiple is 1001. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b, c whose greatest common divisor is 1 and whose least common multiple is 1001. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 1001", "ref": "1, 1, 1001", "family": "po_gcd_lcm", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 1, "L": 1001, "k": 3}, "seed": 43074069103000340, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 13, "valid_answers": ["1, 1, 1001", "1, 7, 143", "1, 7, 1001", "1, 11, 91", "1, 11, 1001", "1, 13, 77", "1, 13, 1001", "1, 77, 91", "1, 77, 143", "1, 77, 1001", "1, 91, 143", "1, 91, 1001", "1, 143, 1001", "1, 1001, 1001", "7, 7, 143", "7, 11, 13", "7, 11, 91", "7, 11, 143", "7, 11, 1001", "7, 13, 77", "7, 13, 143", "7, 13, 1001", "7, 77, 143", "7, 91, 143", "7, 143, 143", "7, 143, 1001", "11, 11, 91", "11, 13, 77", "11, 13, 91", "11, 13, 1001", "11, 77, 91", "11, 91, 91", "11, 91, 143", "11, 91, 1001", "13, 13, 77", "13, 77, 77", "13, 77, 91", "13, 77, 143", "13, 77, 1001", "77, 91, 143"]}, "cell": "gcd_lcm/3", "stage": "B"}
+{"id": "po-gcd_lcm-t3-eval-000105", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b, c whose greatest common divisor is 1 and whose least common multiple is 195. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b, c whose greatest common divisor is 1 and whose least common multiple is 195. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 195", "ref": "1, 1, 195", "family": "po_gcd_lcm", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 3.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 1, "L": 195, "k": 3}, "seed": 43074069103002039, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 13, "valid_answers": ["1, 1, 195", "1, 3, 65", "1, 3, 195", "1, 5, 39", "1, 5, 195", "1, 13, 15", "1, 13, 195", "1, 15, 39", "1, 15, 65", "1, 15, 195", "1, 39, 65", "1, 39, 195", "1, 65, 195", "1, 195, 195", "3, 3, 65", "3, 5, 13", "3, 5, 39", "3, 5, 65", "3, 5, 195", "3, 13, 15", "3, 13, 65", "3, 13, 195", "3, 15, 65", "3, 39, 65", "3, 65, 65", "3, 65, 195", "5, 5, 39", "5, 13, 15", "5, 13, 39", "5, 13, 195", "5, 15, 39", "5, 39, 39", "5, 39, 65", "5, 39, 195", "13, 13, 15", "13, 15, 15", "13, 15, 39", "13, 15, 65", "13, 15, 195", "15, 39, 65"]}, "cell": "gcd_lcm/3", "stage": "B"}
+{"id": "po-gcd_lcm-t3-eval-000074", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b, c whose greatest common divisor is 1 and whose least common multiple is 130. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b, c whose greatest common divisor is 1 and whose least common multiple is 130. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 130", "ref": "1, 1, 130", "family": "po_gcd_lcm", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 0.000107, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 1, "L": 130, "k": 3}, "seed": 43074069103000972, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 13, "valid_answers": ["1, 1, 130", "1, 2, 65", "1, 2, 130", "1, 5, 26", "1, 5, 130", "1, 10, 13", "1, 10, 26", "1, 10, 65", "1, 10, 130", "1, 13, 130", "1, 26, 65", "1, 26, 130", "1, 65, 130", "1, 130, 130", "2, 2, 65", "2, 5, 13", "2, 5, 26", "2, 5, 65", "2, 5, 130", "2, 10, 13", "2, 10, 65", "2, 13, 65", "2, 13, 130", "2, 26, 65", "2, 65, 65", "2, 65, 130", "5, 5, 26", "5, 10, 13", "5, 10, 26", "5, 13, 26", "5, 13, 130", "5, 26, 26", "5, 26, 65", "5, 26, 130", "10, 10, 13", "10, 13, 13", "10, 13, 26", "10, 13, 65", "10, 13, 130", "10, 26, 65"]}, "cell": "gcd_lcm/3", "stage": "B"}
+{"id": "po-gcd_lcm-t3-eval-000118", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b, c whose greatest common divisor is 2 and whose least common multiple is 210. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b, c whose greatest common divisor is 2 and whose least common multiple is 210. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 2, 210", "ref": "2, 2, 210", "family": "po_gcd_lcm", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 2.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 2, "L": 210, "k": 3}, "seed": 43074069103004668, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 14, "valid_answers": ["2, 2, 210", "2, 6, 70", "2, 6, 210", "2, 10, 42", "2, 10, 210", "2, 14, 30", "2, 14, 210", "2, 30, 42", "2, 30, 70", "2, 30, 210", "2, 42, 70", "2, 42, 210", "2, 70, 210", "2, 210, 210", "6, 6, 70", "6, 10, 14", "6, 10, 42", "6, 10, 70", "6, 10, 210", "6, 14, 30", "6, 14, 70", "6, 14, 210", "6, 30, 70", "6, 42, 70", "6, 70, 70", "6, 70, 210", "10, 10, 42", "10, 14, 30", "10, 14, 42", "10, 14, 210", "10, 30, 42", "10, 42, 42", "10, 42, 70", "10, 42, 210", "14, 14, 30", "14, 30, 30", "14, 30, 42", "14, 30, 70", "14, 30, 210", "30, 42, 70"]}, "cell": "gcd_lcm/3", "stage": "B"}
+{"id": "po-gcd_lcm-t3-eval-000006", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b, c whose greatest common divisor is 4 and whose least common multiple is 420. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b, c whose greatest common divisor is 4 and whose least common multiple is 420. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 4, 420", "ref": "4, 4, 420", "family": "po_gcd_lcm", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 3e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 4, "L": 420, "k": 3}, "seed": 43074069103000082, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 28, "valid_answers": ["4, 4, 420", "4, 12, 140", "4, 12, 420", "4, 20, 84", "4, 20, 420", "4, 28, 60", "4, 28, 420", "4, 60, 84", "4, 60, 140", "4, 60, 420", "4, 84, 140", "4, 84, 420", "4, 140, 420", "4, 420, 420", "12, 12, 140", "12, 20, 28", "12, 20, 84", "12, 20, 140", "12, 20, 420", "12, 28, 60", "12, 28, 140", "12, 28, 420", "12, 60, 140", "12, 84, 140", "12, 140, 140", "12, 140, 420", "20, 20, 84", "20, 28, 60", "20, 28, 84", "20, 28, 420", "20, 60, 84", "20, 84, 84", "20, 84, 140", "20, 84, 420", "28, 28, 60", "28, 60, 60", "28, 60, 84", "28, 60, 140", "28, 60, 420", "60, 84, 140"]}, "cell": "gcd_lcm/3", "stage": "B"}
+{"id": "po-gcd_lcm-t3-eval-000038", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b, c whose greatest common divisor is 6 and whose least common multiple is 1638. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b, c whose greatest common divisor is 6 and whose least common multiple is 1638. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 6, 1638", "ref": "6, 6, 1638", "family": "po_gcd_lcm", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 6, "L": 1638, "k": 3}, "seed": 43074069103000397, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 78, "valid_answers": ["6, 6, 1638", "6, 18, 546", "6, 18, 1638", "6, 42, 234", "6, 42, 1638", "6, 78, 126", "6, 78, 1638", "6, 126, 234", "6, 126, 546", "6, 126, 1638", "6, 234, 546", "6, 234, 1638", "6, 546, 1638", "6, 1638, 1638", "18, 18, 546", "18, 42, 78", "18, 42, 234", "18, 42, 546", "18, 42, 1638", "18, 78, 126", "18, 78, 546", "18, 78, 1638", "18, 126, 546", "18, 234, 546", "18, 546, 546", "18, 546, 1638", "42, 42, 234", "42, 78, 126", "42, 78, 234", "42, 78, 1638", "42, 126, 234", "42, 234, 234", "42, 234, 546", "42, 234, 1638", "78, 78, 126", "78, 126, 126", "78, 126, 234", "78, 126, 546", "78, 126, 1638", "126, 234, 546"]}, "cell": "gcd_lcm/3", "stage": "B"}
+{"id": "po-gcd_lcm-t3-eval-000011", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b, c whose greatest common divisor is 1 and whose least common multiple is 429. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b, c whose greatest common divisor is 1 and whose least common multiple is 429. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 429", "ref": "1, 1, 429", "family": "po_gcd_lcm", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 3e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 1, "L": 429, "k": 3}, "seed": 43074069103000109, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 13, "valid_answers": ["1, 1, 429", "1, 3, 143", "1, 3, 429", "1, 11, 39", "1, 11, 429", "1, 13, 33", "1, 13, 429", "1, 33, 39", "1, 33, 143", "1, 33, 429", "1, 39, 143", "1, 39, 429", "1, 143, 429", "1, 429, 429", "3, 3, 143", "3, 11, 13", "3, 11, 39", "3, 11, 143", "3, 11, 429", "3, 13, 33", "3, 13, 143", "3, 13, 429", "3, 33, 143", "3, 39, 143", "3, 143, 143", "3, 143, 429", "11, 11, 39", "11, 13, 33", "11, 13, 39", "11, 13, 429", "11, 33, 39", "11, 39, 39", "11, 39, 143", "11, 39, 429", "13, 13, 33", "13, 33, 33", "13, 33, 39", "13, 33, 143", "13, 33, 429", "33, 39, 143"]}, "cell": "gcd_lcm/3", "stage": "B"}
+{"id": "po-gcd_lcm-t3-eval-000059", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b, c whose greatest common divisor is 2 and whose least common multiple is 132. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b, c whose greatest common divisor is 2 and whose least common multiple is 132. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 2, 132", "ref": "2, 2, 132", "family": "po_gcd_lcm", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 0.000102, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 2, "L": 132, "k": 3}, "seed": 43074069103000711, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 22, "valid_answers": ["2, 2, 132", "2, 4, 66", "2, 4, 132", "2, 6, 44", "2, 6, 132", "2, 12, 22", "2, 12, 44", "2, 12, 66", "2, 12, 132", "2, 22, 132", "2, 44, 66", "2, 44, 132", "2, 66, 132", "2, 132, 132", "4, 4, 66", "4, 6, 22", "4, 6, 44", "4, 6, 66", "4, 6, 132", "4, 12, 22", "4, 12, 66", "4, 22, 66", "4, 22, 132", "4, 44, 66", "4, 66, 66", "4, 66, 132", "6, 6, 44", "6, 12, 22", "6, 12, 44", "6, 22, 44", "6, 22, 132", "6, 44, 44", "6, 44, 66", "6, 44, 132", "12, 12, 22", "12, 22, 22", "12, 22, 44", "12, 22, 66", "12, 22, 132", "12, 44, 66"]}, "cell": "gcd_lcm/3", "stage": "B"}
+{"id": "po-gcd_lcm-t3-eval-000008", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b, c whose greatest common divisor is 10 and whose least common multiple is 300. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b, c whose greatest common divisor is 10 and whose least common multiple is 300. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 10, 300", "ref": "10, 10, 300", "family": "po_gcd_lcm", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 9e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 10, "L": 300, "k": 3}, "seed": 43074069103000098, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 50, "valid_answers": ["10, 10, 300", "10, 20, 150", "10, 20, 300", "10, 30, 100", "10, 30, 300", "10, 50, 60", "10, 50, 300", "10, 60, 100", "10, 60, 150", "10, 60, 300", "10, 100, 150", "10, 100, 300", "10, 150, 300", "10, 300, 300", "20, 20, 150", "20, 30, 50", "20, 30, 100", "20, 30, 150", "20, 30, 300", "20, 50, 60", "20, 50, 150", "20, 50, 300", "20, 60, 150", "20, 100, 150", "20, 150, 150", "20, 150, 300", "30, 30, 100", "30, 50, 60", "30, 50, 100", "30, 50, 300", "30, 60, 100", "30, 100, 100", "30, 100, 150", "30, 100, 300", "50, 50, 60", "50, 60, 60", "50, 60, 100", "50, 60, 150", "50, 60, 300", "60, 100, 150"]}, "cell": "gcd_lcm/3", "stage": "B"}
+{"id": "po-gcd_lcm-t3-eval-000081", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b, c whose greatest common divisor is 2 and whose least common multiple is 546. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b, c whose greatest common divisor is 2 and whose least common multiple is 546. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 2, 546", "ref": "2, 2, 546", "family": "po_gcd_lcm", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 2, "L": 546, "k": 3}, "seed": 43074069103001098, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 26, "valid_answers": ["2, 2, 546", "2, 6, 182", "2, 6, 546", "2, 14, 78", "2, 14, 546", "2, 26, 42", "2, 26, 546", "2, 42, 78", "2, 42, 182", "2, 42, 546", "2, 78, 182", "2, 78, 546", "2, 182, 546", "2, 546, 546", "6, 6, 182", "6, 14, 26", "6, 14, 78", "6, 14, 182", "6, 14, 546", "6, 26, 42", "6, 26, 182", "6, 26, 546", "6, 42, 182", "6, 78, 182", "6, 182, 182", "6, 182, 546", "14, 14, 78", "14, 26, 42", "14, 26, 78", "14, 26, 546", "14, 42, 78", "14, 78, 78", "14, 78, 182", "14, 78, 546", "26, 26, 42", "26, 42, 42", "26, 42, 78", "26, 42, 182", "26, 42, 546", "42, 78, 182"]}, "cell": "gcd_lcm/3", "stage": "B"}
+{"id": "po-gcd_lcm-t3-eval-000078", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b, c whose greatest common divisor is 3 and whose least common multiple is 390. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b, c whose greatest common divisor is 3 and whose least common multiple is 390. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 3, 390", "ref": "3, 3, 390", "family": "po_gcd_lcm", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 3, "L": 390, "k": 3}, "seed": 43074069103001020, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 39, "valid_answers": ["3, 3, 390", "3, 6, 195", "3, 6, 390", "3, 15, 78", "3, 15, 390", "3, 30, 39", "3, 30, 78", "3, 30, 195", "3, 30, 390", "3, 39, 390", "3, 78, 195", "3, 78, 390", "3, 195, 390", "3, 390, 390", "6, 6, 195", "6, 15, 39", "6, 15, 78", "6, 15, 195", "6, 15, 390", "6, 30, 39", "6, 30, 195", "6, 39, 195", "6, 39, 390", "6, 78, 195", "6, 195, 195", "6, 195, 390", "15, 15, 78", "15, 30, 39", "15, 30, 78", "15, 39, 78", "15, 39, 390", "15, 78, 78", "15, 78, 195", "15, 78, 390", "30, 30, 39", "30, 39, 39", "30, 39, 78", "30, 39, 195", "30, 39, 390", "30, 78, 195"]}, "cell": "gcd_lcm/3", "stage": "B"}
+{"id": "po-gcd_lcm-t3-eval-000066", "task": "cx_po_gcd_lcm", "source": "cx_po_gcd_lcm", "split": "eval", "problem": "Find positive integers a, b, c whose greatest common divisor is 2 and whose least common multiple is 858. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a, b, c whose greatest common divisor is 2 and whose least common multiple is 858. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 2, 858", "ref": "2, 2, 858", "family": "po_gcd_lcm", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 2, "L": 858, "k": 3}, "seed": 43074069103000775, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 26, "valid_answers": ["2, 2, 858", "2, 6, 286", "2, 6, 858", "2, 22, 78", "2, 22, 858", "2, 26, 66", "2, 26, 858", "2, 66, 78", "2, 66, 286", "2, 66, 858", "2, 78, 286", "2, 78, 858", "2, 286, 858", "2, 858, 858", "6, 6, 286", "6, 22, 26", "6, 22, 78", "6, 22, 286", "6, 22, 858", "6, 26, 66", "6, 26, 286", "6, 26, 858", "6, 66, 286", "6, 78, 286", "6, 286, 286", "6, 286, 858", "22, 22, 78", "22, 26, 66", "22, 26, 78", "22, 26, 858", "22, 66, 78", "22, 78, 78", "22, 78, 286", "22, 78, 858", "26, 26, 66", "26, 66, 66", "26, 66, 78", "26, 66, 286", "26, 66, 858", "66, 78, 286"]}, "cell": "gcd_lcm/3", "stage": "B"}
+{"id": "po-gp_term-t0-eval-000070", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 91875. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 91875. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -175", "ref": "3, -175", "family": "po_gp_term", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": 91875}, "seed": 162977326500000104, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 75, "valid_answers": ["3, -175", "3, 175", "75, -35", "75, 35", "147, -25", "147, 25", "1875, -7", "1875, 7", "3675, -5", "3675, 5"]}, "cell": "gp_term/0", "stage": "B"}
+{"id": "po-gp_term-t0-eval-000089", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 900. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 900. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -30", "ref": "1, -30", "family": "po_gp_term", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.000127, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": 900}, "seed": 162977326500000174, "version": "po-v0", "tier": 0, "n_raw": 14, "small": 10, "valid_answers": ["1, -30", "1, 30", "4, -15", "4, 15", "9, -10", "9, 10", "25, -6", "25, 6", "36, -5", "36, 5", "100, -3", "100, 3", "225, -2", "225, 2"]}, "cell": "gp_term/0", "stage": "B"}
+{"id": "po-gp_term-t0-eval-000024", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -784. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -784. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-196, -2", "ref": "-196, -2", "family": "po_gp_term", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.000112, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": -784}, "seed": 162977326500000031, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 14, "valid_answers": ["-196, -2", "-196, 2", "-49, -4", "-49, 4", "-16, -7", "-16, 7", "-4, -14", "-4, 14", "-1, -28", "-1, 28"]}, "cell": "gp_term/0", "stage": "B"}
+{"id": "po-gp_term-t0-eval-000016", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -6075. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -6075. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-675, -3", "ref": "-675, -3", "family": "po_gp_term", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": -6075}, "seed": 162977326500000020, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 27, "valid_answers": ["-675, -3", "-675, 3", "-243, -5", "-243, 5", "-75, -9", "-75, 9", "-27, -15", "-27, 15", "-3, -45", "-3, 45"]}, "cell": "gp_term/0", "stage": "B"}
+{"id": "po-gp_term-t0-eval-000055", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -352947. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -352947. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7203, -7", "ref": "-7203, -7", "family": "po_gp_term", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": -352947}, "seed": 162977326500000080, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 147, "valid_answers": ["-7203, -7", "-7203, 7", "-147, -49", "-147, 49", "-3, -343", "-3, 343"]}, "cell": "gp_term/0", "stage": "B"}
+{"id": "po-gp_term-t0-eval-000049", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -144. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -144. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-36, -2", "ref": "-36, -2", "family": "po_gp_term", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.001384, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": -144}, "seed": 162977326500000069, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 6, "valid_answers": ["-36, -2", "-36, 2", "-16, -3", "-16, 3", "-9, -4", "-9, 4", "-4, -6", "-4, 6", "-1, -12", "-1, 12"]}, "cell": "gp_term/0", "stage": "B"}
+{"id": "po-gp_term-t0-eval-000085", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 11250. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 11250. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -75", "ref": "2, -75", "family": "po_gp_term", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 2e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": 11250}, "seed": 162977326500000152, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 25, "valid_answers": ["2, -75", "2, 75", "18, -25", "18, 25", "50, -15", "50, 15", "450, -5", "450, 5", "1250, -3", "1250, 3"]}, "cell": "gp_term/0", "stage": "B"}
+{"id": "po-gp_term-t0-eval-000031", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -900. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -900. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-225, -2", "ref": "-225, -2", "family": "po_gp_term", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.000127, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": -900}, "seed": 162977326500000043, "version": "po-v0", "tier": 0, "n_raw": 14, "small": 10, "valid_answers": ["-225, -2", "-225, 2", "-100, -3", "-100, 3", "-36, -5", "-36, 5", "-25, -6", "-25, 6", "-9, -10", "-9, 10", "-4, -15", "-4, 15", "-1, -30", "-1, 30"]}, "cell": "gp_term/0", "stage": "B"}
+{"id": "po-gp_term-t0-eval-000074", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -22050. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -22050. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2450, -3", "ref": "-2450, -3", "family": "po_gp_term", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": -22050}, "seed": 162977326500000115, "version": "po-v0", "tier": 0, "n_raw": 14, "small": 35, "valid_answers": ["-2450, -3", "-2450, 3", "-882, -5", "-882, 5", "-450, -7", "-450, 7", "-98, -15", "-98, 15", "-50, -21", "-50, 21", "-18, -35", "-18, 35", "-2, -105", "-2, 105"]}, "cell": "gp_term/0", "stage": "B"}
+{"id": "po-gp_term-t0-eval-000094", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 6075. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 6075. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -45", "ref": "3, -45", "family": "po_gp_term", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": 6075}, "seed": 162977326500000185, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 27, "valid_answers": ["3, -45", "3, 45", "27, -15", "27, 15", "75, -9", "75, 9", "243, -5", "243, 5", "675, -3", "675, 3"]}, "cell": "gp_term/0", "stage": "B"}
+{"id": "po-gp_term-t0-eval-000084", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -16875. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -16875. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1875, -3", "ref": "-1875, -3", "family": "po_gp_term", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": -16875}, "seed": 162977326500000150, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 27, "valid_answers": ["-1875, -3", "-1875, 3", "-675, -5", "-675, 5", "-75, -15", "-75, 15", "-27, -25", "-27, 25", "-3, -75", "-3, 75"]}, "cell": "gp_term/0", "stage": "B"}
+{"id": "po-gp_term-t0-eval-000010", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -43218. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -43218. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4802, -3", "ref": "-4802, -3", "family": "po_gp_term", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": -43218}, "seed": 162977326500000014, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 49, "valid_answers": ["-4802, -3", "-4802, 3", "-882, -7", "-882, 7", "-98, -21", "-98, 21", "-18, -49", "-18, 49", "-2, -147", "-2, 147"]}, "cell": "gp_term/0", "stage": "B"}
+{"id": "po-gp_term-t1-eval-000092", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -1536. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -1536. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-192, 2", "ref": "-192, 2", "family": "po_gp_term", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 8.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 3, "T": -1536}, "seed": 162977326501000162, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 8, "valid_answers": ["-192, 2", "-24, 4", "-3, 8", "3, -8", "24, -4", "192, -2"]}, "cell": "gp_term/1", "stage": "B"}
+{"id": "po-gp_term-t1-eval-000051", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -59049. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -59049. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2187, 3", "ref": "-2187, 3", "family": "po_gp_term", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 3, "T": -59049}, "seed": 162977326501000068, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 27, "valid_answers": ["-2187, 3", "-81, 9", "-3, 27", "3, -27", "81, -9", "2187, -3"]}, "cell": "gp_term/1", "stage": "B"}
+{"id": "po-gp_term-t1-eval-000044", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -24000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -24000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3000, 2", "ref": "-3000, 2", "family": "po_gp_term", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 3, "T": -24000}, "seed": 162977326501000056, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 20, "valid_answers": ["-3000, 2", "-375, 4", "-192, 5", "-24, 10", "-3, 20", "3, -20", "24, -10", "192, -5", "375, -4", "3000, -2"]}, "cell": "gp_term/1", "stage": "B"}
+{"id": "po-gp_term-t1-eval-000008", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -3176523. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -3176523. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-117649, 3", "ref": "-117649, 3", "family": "po_gp_term", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 3, "T": -3176523}, "seed": 162977326501000008, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 49, "valid_answers": ["-117649, 3", "-9261, 7", "-343, 21", "-27, 49", "-1, 147", "1, -147", "27, -49", "343, -21", "9261, -7", "117649, -3"]}, "cell": "gp_term/1", "stage": "B"}
+{"id": "po-gp_term-t1-eval-000021", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = 455625. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = 455625. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-16875, -3", "ref": "-16875, -3", "family": "po_gp_term", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 3, "T": 455625}, "seed": 162977326501000023, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 45, "valid_answers": ["-16875, -3", "-3645, -5", "-625, -9", "-135, -15", "-5, -45", "5, 45", "135, 15", "625, 9", "3645, 5", "16875, 3"]}, "cell": "gp_term/1", "stage": "B"}
+{"id": "po-gp_term-t1-eval-000035", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -14706125. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -14706125. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-117649, 5", "ref": "-117649, 5", "family": "po_gp_term", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 3, "T": -14706125}, "seed": 162977326501000043, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 125, "valid_answers": ["-117649, 5", "-42875, 7", "-343, 35", "-125, 49", "-1, 245", "1, -245", "125, -49", "343, -35", "42875, -7", "117649, -5"]}, "cell": "gp_term/1", "stage": "B"}
+{"id": "po-gp_term-t1-eval-000014", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = 10718750. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = 10718750. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-85750, -5", "ref": "-85750, -5", "family": "po_gp_term", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 3, "T": 10718750}, "seed": 162977326501000015, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 175, "valid_answers": ["-85750, -5", "-31250, -7", "-686, -25", "-250, -35", "-2, -175", "2, 175", "250, 35", "686, 25", "31250, 7", "85750, 5"]}, "cell": "gp_term/1", "stage": "B"}
+{"id": "po-gp_term-t1-eval-000057", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -3456. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -3456. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-432, 2", "ref": "-432, 2", "family": "po_gp_term", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 4.7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 3, "T": -3456}, "seed": 162977326501000081, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 12, "valid_answers": ["-432, 2", "-128, 3", "-54, 4", "-16, 6", "-2, 12", "2, -12", "16, -6", "54, -4", "128, -3", "432, -2"]}, "cell": "gp_term/1", "stage": "B"}
+{"id": "po-gp_term-t1-eval-000069", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -843750. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -843750. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-31250, 3", "ref": "-31250, 3", "family": "po_gp_term", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 3, "T": -843750}, "seed": 162977326501000112, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 54, "valid_answers": ["-31250, 3", "-6750, 5", "-250, 15", "-54, 25", "-2, 75", "2, -75", "54, -25", "250, -15", "6750, -5", "31250, -3"]}, "cell": "gp_term/1", "stage": "B"}
+{"id": "po-gp_term-t1-eval-000071", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -941192. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = -941192. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-117649, 2", "ref": "-117649, 2", "family": "po_gp_term", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 3, "T": -941192}, "seed": 162977326501000118, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 49, "valid_answers": ["-117649, 2", "-2744, 7", "-343, 14", "-8, 49", "-1, 98", "1, -98", "8, -49", "343, -14", "2744, -7", "117649, -2"]}, "cell": "gp_term/1", "stage": "B"}
+{"id": "po-gp_term-t1-eval-000015", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = 750141. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = 750141. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-27783, -3", "ref": "-27783, -3", "family": "po_gp_term", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 3, "T": 750141}, "seed": 162977326501000016, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 63, "valid_answers": ["-27783, -3", "-2187, -7", "-1029, -9", "-81, -21", "-3, -63", "3, 63", "81, 21", "1029, 9", "2187, 7", "27783, 3"]}, "cell": "gp_term/1", "stage": "B"}
+{"id": "po-gp_term-t1-eval-000133", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = 6353046. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^3 = 6353046. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-235298, -3", "ref": "-235298, -3", "family": "po_gp_term", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 3, "T": 6353046}, "seed": 162977326501000730, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 54, "valid_answers": ["-235298, -3", "-18522, -7", "-686, -21", "-54, -49", "-2, -147", "2, 147", "54, 49", "686, 21", "18522, 7", "235298, 3"]}, "cell": "gp_term/1", "stage": "B"}
+{"id": "po-gp_term-t2-eval-000037", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 211768200. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 211768200. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -10290", "ref": "2, -10290", "family": "po_gp_term", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 62, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": 211768200}, "seed": 162977326502000053, "version": "po-v0", "tier": 2, "n_raw": 62, "small": 686, "valid_answers": ["2, -10290", "2, 10290", "8, -5145", "8, 5145", "18, -3430", "18, 3430", "50, -2058", "50, 2058", "72, -1715", "72, 1715", "98, -1470", "98, 1470", "200, -1029", "200, 1029", "392, -735", "392, 735", "450, -686", "450, 686", "882, -490", "882, 490", "1800, -343", "1800, 343", "2450, -294", "2450, 294", "3528, -245", "3528, 245", "4802, -210", "4802, 210", "9800, -147", "9800, 147", "19208, -105", "19208, 105", "22050, -98", "22050, 98", "43218, -70", "43218, 70", "88200, -49", "88200, 49", "120050, -42", "120050, 42", "172872, -35", "172872, 35", "235298, -30", "235298, 30", "480200, -21", "480200, 21", "941192, -15", "941192, 15", "1080450, -14", "1080450, 14", "2117682, -10", "2117682, 10", "4321800, -7", "4321800, 7", "5882450, -6", "5882450, 6", "8470728, -5", "8470728, 5", "23529800, -3", "23529800, 3", "52942050, -2", "52942050, 2"]}, "cell": "gp_term/2", "stage": "B"}
+{"id": "po-gp_term-t2-eval-000046", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 61250000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 61250000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, -3500", "ref": "5, -3500", "family": "po_gp_term", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 46, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": 61250000}, "seed": 162977326502000066, "version": "po-v0", "tier": 2, "n_raw": 46, "small": 500, "valid_answers": ["5, -3500", "5, 3500", "20, -1750", "20, 1750", "80, -875", "80, 875", "125, -700", "125, 700", "245, -500", "245, 500", "500, -350", "500, 350", "980, -250", "980, 250", "2000, -175", "2000, 175", "3125, -140", "3125, 140", "3920, -125", "3920, 125", "6125, -100", "6125, 100", "12500, -70", "12500, 70", "24500, -50", "24500, 50", "50000, -35", "50000, 35", "78125, -28", "78125, 28", "98000, -25", "98000, 25", "153125, -20", "153125, 20", "312500, -14", "312500, 14", "612500, -10", "612500, 10", "1250000, -7", "1250000, 7", "2450000, -5", "2450000, 5", "3828125, -4", "3828125, 4", "15312500, -2", "15312500, 2"]}, "cell": "gp_term/2", "stage": "B"}
+{"id": "po-gp_term-t2-eval-000165", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 16074450. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 16074450. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -2835", "ref": "2, -2835", "family": "po_gp_term", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 38, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": 16074450}, "seed": 162977326502000297, "version": "po-v0", "tier": 2, "n_raw": 38, "small": 315, "valid_answers": ["2, -2835", "2, 2835", "18, -945", "18, 945", "50, -567", "50, 567", "98, -405", "98, 405", "162, -315", "162, 315", "450, -189", "450, 189", "882, -135", "882, 135", "1458, -105", "1458, 105", "2450, -81", "2450, 81", "4050, -63", "4050, 63", "7938, -45", "7938, 45", "13122, -35", "13122, 35", "22050, -27", "22050, 27", "36450, -21", "36450, 21", "71442, -15", "71442, 15", "198450, -9", "198450, 9", "328050, -7", "328050, 7", "642978, -5", "642978, 5", "1786050, -3", "1786050, 3"]}, "cell": "gp_term/2", "stage": "B"}
+{"id": "po-gp_term-t2-eval-000195", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -762048. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -762048. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-190512, -2", "ref": "-190512, -2", "family": "po_gp_term", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 46, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": -762048}, "seed": 162977326502000373, "version": "po-v0", "tier": 2, "n_raw": 46, "small": 108, "valid_answers": ["-190512, -2", "-190512, 2", "-84672, -3", "-84672, 3", "-47628, -4", "-47628, 4", "-21168, -6", "-21168, 6", "-15552, -7", "-15552, 7", "-11907, -8", "-11907, 8", "-9408, -9", "-9408, 9", "-5292, -12", "-5292, 12", "-3888, -14", "-3888, 14", "-2352, -18", "-2352, 18", "-1728, -21", "-1728, 21", "-1323, -24", "-1323, 24", "-972, -28", "-972, 28", "-588, -36", "-588, 36", "-432, -42", "-432, 42", "-243, -56", "-243, 56", "-192, -63", "-192, 63", "-147, -72", "-147, 72", "-108, -84", "-108, 84", "-48, -126", "-48, 126", "-27, -168", "-27, 168", "-12, -252", "-12, 252", "-3, -504", "-3, 504"]}, "cell": "gp_term/2", "stage": "B"}
+{"id": "po-gp_term-t2-eval-000030", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -571536. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -571536. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-142884, -2", "ref": "-142884, -2", "family": "po_gp_term", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 46, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": -571536}, "seed": 162977326502000043, "version": "po-v0", "tier": 2, "n_raw": 46, "small": 84, "valid_answers": ["-142884, -2", "-142884, 2", "-63504, -3", "-63504, 3", "-35721, -4", "-35721, 4", "-15876, -6", "-15876, 6", "-11664, -7", "-11664, 7", "-7056, -9", "-7056, 9", "-3969, -12", "-3969, 12", "-2916, -14", "-2916, 14", "-1764, -18", "-1764, 18", "-1296, -21", "-1296, 21", "-784, -27", "-784, 27", "-729, -28", "-729, 28", "-441, -36", "-441, 36", "-324, -42", "-324, 42", "-196, -54", "-196, 54", "-144, -63", "-144, 63", "-81, -84", "-81, 84", "-49, -108", "-49, 108", "-36, -126", "-36, 126", "-16, -189", "-16, 189", "-9, -252", "-9, 252", "-4, -378", "-4, 378", "-1, -756", "-1, 756"]}, "cell": "gp_term/2", "stage": "B"}
+{"id": "po-gp_term-t2-eval-000305", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -87890625. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -87890625. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9765625, -3", "ref": "-9765625, -3", "family": "po_gp_term", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": -87890625}, "seed": 162977326502000772, "version": "po-v0", "tier": 2, "n_raw": 22, "small": 625, "valid_answers": ["-9765625, -3", "-9765625, 3", "-3515625, -5", "-3515625, 5", "-390625, -15", "-390625, 15", "-140625, -25", "-140625, 25", "-15625, -75", "-15625, 75", "-5625, -125", "-5625, 125", "-625, -375", "-625, 375", "-225, -625", "-225, 625", "-25, -1875", "-25, 1875", "-9, -3125", "-9, 3125", "-1, -9375", "-1, 9375"]}, "cell": "gp_term/2", "stage": "B"}
+{"id": "po-gp_term-t2-eval-000012", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 524880. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 524880. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, -324", "ref": "5, -324", "family": "po_gp_term", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 28, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": 524880}, "seed": 162977326502000017, "version": "po-v0", "tier": 2, "n_raw": 28, "small": 81, "valid_answers": ["5, -324", "5, 324", "20, -162", "20, 162", "45, -108", "45, 108", "80, -81", "80, 81", "180, -54", "180, 54", "405, -36", "405, 36", "720, -27", "720, 27", "1620, -18", "1620, 18", "3645, -12", "3645, 12", "6480, -9", "6480, 9", "14580, -6", "14580, 6", "32805, -4", "32805, 4", "58320, -3", "58320, 3", "131220, -2", "131220, 2"]}, "cell": "gp_term/2", "stage": "B"}
+{"id": "po-gp_term-t2-eval-000076", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 172800. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 172800. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -240", "ref": "3, -240", "family": "po_gp_term", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 38, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": 172800}, "seed": 162977326502000113, "version": "po-v0", "tier": 2, "n_raw": 38, "small": 60, "valid_answers": ["3, -240", "3, 240", "12, -120", "12, 120", "27, -80", "27, 80", "48, -60", "48, 60", "75, -48", "75, 48", "108, -40", "108, 40", "192, -30", "192, 30", "300, -24", "300, 24", "432, -20", "432, 20", "675, -16", "675, 16", "768, -15", "768, 15", "1200, -12", "1200, 12", "1728, -10", "1728, 10", "2700, -8", "2700, 8", "4800, -6", "4800, 6", "6912, -5", "6912, 5", "10800, -4", "10800, 4", "19200, -3", "19200, 3", "43200, -2", "43200, 2"]}, "cell": "gp_term/2", "stage": "B"}
+{"id": "po-gp_term-t2-eval-000315", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 720000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 720000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -600", "ref": "2, -600", "family": "po_gp_term", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 46, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": 720000}, "seed": 162977326502000825, "version": "po-v0", "tier": 2, "n_raw": 46, "small": 100, "valid_answers": ["2, -600", "2, 600", "8, -300", "8, 300", "18, -200", "18, 200", "32, -150", "32, 150", "50, -120", "50, 120", "72, -100", "72, 100", "128, -75", "128, 75", "200, -60", "200, 60", "288, -50", "288, 50", "450, -40", "450, 40", "800, -30", "800, 30", "1152, -25", "1152, 25", "1250, -24", "1250, 24", "1800, -20", "1800, 20", "3200, -15", "3200, 15", "5000, -12", "5000, 12", "7200, -10", "7200, 10", "11250, -8", "11250, 8", "20000, -6", "20000, 6", "28800, -5", "28800, 5", "45000, -4", "45000, 4", "80000, -3", "80000, 3", "180000, -2", "180000, 2"]}, "cell": "gp_term/2", "stage": "B"}
+{"id": "po-gp_term-t2-eval-000426", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 27648. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 27648. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -96", "ref": "3, -96", "family": "po_gp_term", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": 27648}, "seed": 162977326502002372, "version": "po-v0", "tier": 2, "n_raw": 22, "small": 32, "valid_answers": ["3, -96", "3, 96", "12, -48", "12, 48", "27, -32", "27, 32", "48, -24", "48, 24", "108, -16", "108, 16", "192, -12", "192, 12", "432, -8", "432, 8", "768, -6", "768, 6", "1728, -4", "1728, 4", "3072, -3", "3072, 3", "6912, -2", "6912, 2"]}, "cell": "gp_term/2", "stage": "B"}
+{"id": "po-gp_term-t2-eval-000407", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -622598508. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = -622598508. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-155649627, -2", "ref": "-155649627, -2", "family": "po_gp_term", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 38, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": -622598508}, "seed": 162977326502001979, "version": "po-v0", "tier": 2, "n_raw": 38, "small": 1029, "valid_answers": ["-155649627, -2", "-155649627, 2", "-69177612, -3", "-69177612, 3", "-17294403, -6", "-17294403, 6", "-12706092, -7", "-12706092, 7", "-3176523, -14", "-3176523, 14", "-1411788, -21", "-1411788, 21", "-352947, -42", "-352947, 42", "-259308, -49", "-259308, 49", "-64827, -98", "-64827, 98", "-28812, -147", "-28812, 147", "-7203, -294", "-7203, 294", "-5292, -343", "-5292, 343", "-1323, -686", "-1323, 686", "-588, -1029", "-588, 1029", "-147, -2058", "-147, 2058", "-108, -2401", "-108, 2401", "-27, -4802", "-27, 4802", "-12, -7203", "-12, 7203", "-3, -14406", "-3, 14406"]}, "cell": "gp_term/2", "stage": "B"}
+{"id": "po-gp_term-t2-eval-000237", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 207532836. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^2 = 207532836. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -14406", "ref": "1, -14406", "family": "po_gp_term", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 38, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 2, "T": 207532836}, "seed": 162977326502000509, "version": "po-v0", "tier": 2, "n_raw": 38, "small": 686, "valid_answers": ["1, -14406", "1, 14406", "4, -7203", "4, 7203", "9, -4802", "9, 4802", "36, -2401", "36, 2401", "49, -2058", "49, 2058", "196, -1029", "196, 1029", "441, -686", "441, 686", "1764, -343", "1764, 343", "2401, -294", "2401, 294", "9604, -147", "9604, 147", "21609, -98", "21609, 98", "86436, -49", "86436, 49", "117649, -42", "117649, 42", "470596, -21", "470596, 21", "1058841, -14", "1058841, 14", "4235364, -7", "4235364, 7", "5764801, -6", "5764801, 6", "23059204, -3", "23059204, 3", "51883209, -2", "51883209, 2"]}, "cell": "gp_term/2", "stage": "B"}
+{"id": "po-gp_term-t3-eval-000154", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = 288240050000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = 288240050000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, -490", "ref": "5, -490", "family": "po_gp_term", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 4, "T": 288240050000}, "seed": 162977326503000324, "version": "po-v0", "tier": 3, "n_raw": 22, "small": 245, "valid_answers": ["5, -490", "5, 490", "80, -245", "80, 245", "3125, -98", "3125, 98", "12005, -70", "12005, 70", "50000, -49", "50000, 49", "192080, -35", "192080, 35", "7503125, -14", "7503125, 14", "28824005, -10", "28824005, 10", "120050000, -7", "120050000, 7", "461184080, -5", "461184080, 5", "18015003125, -2", "18015003125, 2"]}, "cell": "gp_term/3", "stage": "B"}
+{"id": "po-gp_term-t3-eval-000165", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = -99574272. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = -99574272. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6223392, -2", "ref": "-6223392, -2", "family": "po_gp_term", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 4, "T": -99574272}, "seed": 162977326503000374, "version": "po-v0", "tier": 3, "n_raw": 22, "small": 42, "valid_answers": ["-6223392, -2", "-6223392, 2", "-1229312, -3", "-1229312, 3", "-388962, -4", "-388962, 4", "-76832, -6", "-76832, 6", "-41472, -7", "-41472, 7", "-4802, -12", "-4802, 12", "-2592, -14", "-2592, 14", "-512, -21", "-512, 21", "-162, -28", "-162, 28", "-32, -42", "-32, 42", "-2, -84", "-2, 84"]}, "cell": "gp_term/3", "stage": "B"}
+{"id": "po-gp_term-t3-eval-000187", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = -19668992. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = -19668992. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1229312, -2", "ref": "-1229312, -2", "family": "po_gp_term", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 4, "T": -19668992}, "seed": 162977326503000590, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 32, "valid_answers": ["-1229312, -2", "-1229312, 2", "-76832, -4", "-76832, 4", "-8192, -7", "-8192, 7", "-4802, -8", "-4802, 8", "-512, -14", "-512, 14", "-32, -28", "-32, 28", "-2, -56", "-2, 56"]}, "cell": "gp_term/3", "stage": "B"}
+{"id": "po-gp_term-t3-eval-000073", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = -25920000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = -25920000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1620000, -2", "ref": "-1620000, -2", "family": "po_gp_term", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 4, "T": -25920000}, "seed": 162977326503000093, "version": "po-v0", "tier": 3, "n_raw": 22, "small": 32, "valid_answers": ["-1620000, -2", "-1620000, 2", "-320000, -3", "-320000, 3", "-101250, -4", "-101250, 4", "-41472, -5", "-41472, 5", "-20000, -6", "-20000, 6", "-2592, -10", "-2592, 10", "-1250, -12", "-1250, 12", "-512, -15", "-512, 15", "-162, -20", "-162, 20", "-32, -30", "-32, 30", "-2, -60", "-2, 60"]}, "cell": "gp_term/3", "stage": "B"}
+{"id": "po-gp_term-t3-eval-000151", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = 25952413501875. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = 25952413501875. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -1715", "ref": "3, -1715", "family": "po_gp_term", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 4, "T": 25952413501875}, "seed": 162977326503000312, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 1715, "valid_answers": ["3, -1715", "3, 1715", "1875, -343", "1875, 343", "7203, -245", "7203, 245", "4501875, -49", "4501875, 49", "17294403, -35", "17294403, 35", "10809001875, -7", "10809001875, 7", "41523861603, -5", "41523861603, 5"]}, "cell": "gp_term/3", "stage": "B"}
+{"id": "po-gp_term-t3-eval-000033", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = 115296020000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = 115296020000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -490", "ref": "2, -490", "family": "po_gp_term", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 4, "T": 115296020000}, "seed": 162977326503000038, "version": "po-v0", "tier": 3, "n_raw": 22, "small": 245, "valid_answers": ["2, -490", "2, 490", "32, -245", "32, 245", "1250, -98", "1250, 98", "4802, -70", "4802, 70", "20000, -49", "20000, 49", "76832, -35", "76832, 35", "3001250, -14", "3001250, 14", "11529602, -10", "11529602, 10", "48020000, -7", "48020000, 7", "184473632, -5", "184473632, 5", "7206001250, -2", "7206001250, 2"]}, "cell": "gp_term/3", "stage": "B"}
+{"id": "po-gp_term-t3-eval-000129", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = 5038848. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = 5038848. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -36", "ref": "3, -36", "family": "po_gp_term", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 4, "T": 5038848}, "seed": 162977326503000226, "version": "po-v0", "tier": 3, "n_raw": 16, "small": 36, "valid_answers": ["3, -36", "3, 36", "48, -18", "48, 18", "243, -12", "243, 12", "768, -9", "768, 9", "3888, -6", "3888, 6", "19683, -4", "19683, 4", "62208, -3", "62208, 3", "314928, -2", "314928, 2"]}, "cell": "gp_term/3", "stage": "B"}
+{"id": "po-gp_term-t3-eval-000078", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = 19668992. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = 19668992. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -56", "ref": "2, -56", "family": "po_gp_term", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 4, "T": 19668992}, "seed": 162977326503000099, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 32, "valid_answers": ["2, -56", "2, 56", "32, -28", "32, 28", "512, -14", "512, 14", "4802, -8", "4802, 8", "8192, -7", "8192, 7", "76832, -4", "76832, 4", "1229312, -2", "1229312, 2"]}, "cell": "gp_term/3", "stage": "B"}
+{"id": "po-gp_term-t3-eval-000088", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = -22413546288. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = -22413546288. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1400846643, -2", "ref": "-1400846643, -2", "family": "po_gp_term", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 4, "T": -22413546288}, "seed": 162977326503000116, "version": "po-v0", "tier": 3, "n_raw": 22, "small": 147, "valid_answers": ["-1400846643, -2", "-1400846643, 2", "-276710448, -3", "-276710448, 3", "-17294403, -6", "-17294403, 6", "-9335088, -7", "-9335088, 7", "-583443, -14", "-583443, 14", "-115248, -21", "-115248, 21", "-7203, -42", "-7203, 42", "-3888, -49", "-3888, 49", "-243, -98", "-243, 98", "-48, -147", "-48, 147", "-3, -294", "-3, 294"]}, "cell": "gp_term/3", "stage": "B"}
+{"id": "po-gp_term-t3-eval-000057", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = 49228003125. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = 49228003125. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, -315", "ref": "5, -315", "family": "po_gp_term", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 4, "T": 49228003125}, "seed": 162977326503000068, "version": "po-v0", "tier": 3, "n_raw": 22, "small": 315, "valid_answers": ["5, -315", "5, 315", "405, -105", "405, 105", "3125, -63", "3125, 63", "12005, -45", "12005, 45", "32805, -35", "32805, 35", "253125, -21", "253125, 21", "972405, -15", "972405, 15", "7503125, -9", "7503125, 9", "20503125, -7", "20503125, 7", "78764805, -5", "78764805, 5", "607753125, -3", "607753125, 3"]}, "cell": "gp_term/3", "stage": "B"}
+{"id": "po-gp_term-t3-eval-000220", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = -29503488. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = -29503488. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1843968, -2", "ref": "-1843968, -2", "family": "po_gp_term", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 4, "T": -29503488}, "seed": 162977326503001206, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 48, "valid_answers": ["-1843968, -2", "-1843968, 2", "-115248, -4", "-115248, 4", "-12288, -7", "-12288, 7", "-7203, -8", "-7203, 8", "-768, -14", "-768, 14", "-48, -28", "-48, 28", "-3, -56", "-3, 56"]}, "cell": "gp_term/3", "stage": "B"}
+{"id": "po-gp_term-t3-eval-000034", "task": "cx_po_gp_term", "source": "cx_po_gp_term", "split": "eval", "problem": "Find integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = -221460595216. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, r with r \\ne 0, 1, -1 such that the geometric sequence a, ar, ar^2, ... has ar^4 = -221460595216. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., r = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-13841287201, -2", "ref": "-13841287201, -2", "family": "po_gp_term", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_gp_term", "rung": "witness", "params": {"m": 4, "T": -221460595216}, "seed": 162977326503000040, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 343, "valid_answers": ["-13841287201, -2", "-13841287201, 2", "-92236816, -7", "-92236816, 7", "-5764801, -14", "-5764801, 14", "-38416, -49", "-38416, 49", "-2401, -98", "-2401, 98", "-16, -343", "-16, 343", "-1, -686", "-1, 686"]}, "cell": "gp_term/3", "stage": "B"}
+{"id": "po-int_triangle-t0-eval-000107", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 85 and c - a = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 85 and c - a = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "23, 30, 32", "ref": "23, 30, 32", "family": "po_int_triangle", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.02, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 85, "g": 9, "kind": null}, "seed": 205690953300001176, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 32, "valid_answers": ["23, 30, 32", "24, 28, 33", "25, 26, 34"]}, "cell": "int_triangle/0", "stage": "C"}
+{"id": "po-int_triangle-t0-eval-000003", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 96 and c - a = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 96 and c - a = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "29, 33, 34", "ref": "29, 33, 34", "family": "po_int_triangle", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.010417, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 96, "g": 5, "kind": null}, "seed": 205690953300000030, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 34, "valid_answers": ["29, 33, 34", "30, 31, 35"]}, "cell": "int_triangle/0", "stage": "C"}
+{"id": "po-int_triangle-t0-eval-000002", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 82 and c - a = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 82 and c - a = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "24, 29, 29", "ref": "24, 29, 29", "family": "po_int_triangle", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.014286, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 82, "g": 5, "kind": null}, "seed": 205690953300000026, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 29, "valid_answers": ["24, 29, 29", "25, 27, 30"]}, "cell": "int_triangle/0", "stage": "C"}
+{"id": "po-int_triangle-t0-eval-000123", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 91 and c - a = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 91 and c - a = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "25, 33, 33", "ref": "25, 33, 33", "family": "po_int_triangle", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017442, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 91, "g": 8, "kind": null}, "seed": 205690953300001454, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 33, "valid_answers": ["25, 33, 33", "26, 31, 34", "27, 29, 35"]}, "cell": "int_triangle/0", "stage": "C"}
+{"id": "po-int_triangle-t0-eval-000134", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 89 and c - a = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 89 and c - a = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "25, 32, 32", "ref": "25, 32, 32", "family": "po_int_triangle", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.018182, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 89, "g": 7, "kind": null}, "seed": 205690953300001994, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 32, "valid_answers": ["25, 32, 32", "26, 30, 33", "27, 28, 34"]}, "cell": "int_triangle/0", "stage": "C"}
+{"id": "po-int_triangle-t0-eval-000147", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 97 and c - a = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 97 and c - a = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "30, 33, 34", "ref": "30, 33, 34", "family": "po_int_triangle", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.010204, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 97, "g": 4, "kind": null}, "seed": 205690953300003390, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 34, "valid_answers": ["30, 33, 34", "31, 31, 35"]}, "cell": "int_triangle/0", "stage": "C"}
+{"id": "po-int_triangle-t0-eval-000000", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 74 and c - a = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 74 and c - a = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "22, 25, 27", "ref": "22, 25, 27", "family": "po_int_triangle", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.017544, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 74, "g": 5, "kind": null}, "seed": 205690953300000015, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 27, "valid_answers": ["22, 25, 27", "23, 23, 28"]}, "cell": "int_triangle/0", "stage": "C"}
+{"id": "po-int_triangle-t0-eval-000063", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 95 and c - a = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 95 and c - a = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "26, 34, 35", "ref": "26, 34, 35", "family": "po_int_triangle", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.015957, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 95, "g": 9, "kind": null}, "seed": 205690953300000559, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 35, "valid_answers": ["26, 34, 35", "27, 32, 36", "28, 30, 37"]}, "cell": "int_triangle/0", "stage": "C"}
+{"id": "po-int_triangle-t0-eval-000049", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 98 and c - a = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 98 and c - a = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "27, 35, 36", "ref": "27, 35, 36", "family": "po_int_triangle", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.015, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 98, "g": 9, "kind": null}, "seed": 205690953300000371, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 36, "valid_answers": ["27, 35, 36", "28, 33, 37", "29, 31, 38"]}, "cell": "int_triangle/0", "stage": "C"}
+{"id": "po-int_triangle-t0-eval-000104", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 87 and c - a = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 87 and c - a = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "26, 30, 31", "ref": "26, 30, 31", "family": "po_int_triangle", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.012739, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 87, "g": 5, "kind": null}, "seed": 205690953300001117, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 31, "valid_answers": ["26, 30, 31", "27, 28, 32"]}, "cell": "int_triangle/0", "stage": "C"}
+{"id": "po-int_triangle-t0-eval-000015", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 91 and c - a = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 91 and c - a = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "28, 31, 32", "ref": "28, 31, 32", "family": "po_int_triangle", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011628, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 91, "g": 4, "kind": null}, "seed": 205690953300000102, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 32, "valid_answers": ["28, 31, 32", "29, 29, 33"]}, "cell": "int_triangle/0", "stage": "C"}
+{"id": "po-int_triangle-t0-eval-000142", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 71 and c - a = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 71 and c - a = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "20, 25, 26", "ref": "20, 25, 26", "family": "po_int_triangle", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.019048, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 71, "g": 6, "kind": null}, "seed": 205690953300002537, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 26, "valid_answers": ["20, 25, 26", "21, 23, 27"]}, "cell": "int_triangle/0", "stage": "C"}
+{"id": "po-int_triangle-t1-eval-000614", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 157 and c - a = 33. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 157 and c - a = 33. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "34, 56, 67", "ref": "34, 56, 67", "family": "po_int_triangle", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.015595, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 157, "g": 33, "kind": "obtuse"}, "seed": 205690953301007431, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 67, "valid_answers": ["34, 56, 67", "35, 54, 68", "36, 52, 69", "37, 50, 70", "38, 48, 71", "39, 46, 72", "40, 44, 73", "41, 42, 74"]}, "cell": "int_triangle/1", "stage": "C"}
+{"id": "po-int_triangle-t1-eval-000504", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 141 and c - a = 23. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 141 and c - a = 23. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "36, 46, 59", "ref": "36, 46, 59", "family": "po_int_triangle", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.009662, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 141, "g": 23, "kind": "obtuse"}, "seed": 205690953301003361, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 59, "valid_answers": ["36, 46, 59", "37, 44, 60", "38, 42, 61", "39, 40, 62"]}, "cell": "int_triangle/1", "stage": "C"}
+{"id": "po-int_triangle-t1-eval-000340", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 152 and c - a = 20. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 152 and c - a = 20. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "43, 46, 63", "ref": "43, 46, 63", "family": "po_int_triangle", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004158, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 152, "g": 20, "kind": "obtuse"}, "seed": 205690953301001582, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 63, "valid_answers": ["43, 46, 63", "44, 44, 64"]}, "cell": "int_triangle/1", "stage": "C"}
+{"id": "po-int_triangle-t1-eval-000281", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 117 and c - a = 17. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 117 and c - a = 17. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "32, 36, 49", "ref": "32, 36, 49", "family": "po_int_triangle", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.007018, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 117, "g": 17, "kind": "obtuse"}, "seed": 205690953301001152, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 49, "valid_answers": ["32, 36, 49", "33, 34, 50"]}, "cell": "int_triangle/1", "stage": "C"}
+{"id": "po-int_triangle-t1-eval-000058", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 157 and c - a = 27. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 157 and c - a = 27. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "39, 52, 66", "ref": "39, 52, 66", "family": "po_int_triangle", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.009747, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 157, "g": 27, "kind": "obtuse"}, "seed": 205690953301000189, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 66, "valid_answers": ["39, 52, 66", "40, 50, 67", "41, 48, 68", "42, 46, 69", "43, 44, 70"]}, "cell": "int_triangle/1", "stage": "C"}
+{"id": "po-int_triangle-t1-eval-000289", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 158 and c - a = 22. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 158 and c - a = 22. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "44, 48, 66", "ref": "44, 48, 66", "family": "po_int_triangle", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.003846, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 158, "g": 22, "kind": "obtuse"}, "seed": 205690953301001216, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 66, "valid_answers": ["44, 48, 66", "45, 46, 67"]}, "cell": "int_triangle/1", "stage": "C"}
+{"id": "po-int_triangle-t1-eval-000178", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 150 and c - a = 26. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 150 and c - a = 26. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "37, 50, 63", "ref": "37, 50, 63", "family": "po_int_triangle", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.010684, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 150, "g": 26, "kind": "obtuse"}, "seed": 205690953301000646, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 63, "valid_answers": ["37, 50, 63", "38, 48, 64", "39, 46, 65", "40, 44, 66", "41, 42, 67"]}, "cell": "int_triangle/1", "stage": "C"}
+{"id": "po-int_triangle-t1-eval-000142", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 148 and c - a = 29. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 148 and c - a = 29. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "34, 51, 63", "ref": "34, 51, 63", "family": "po_int_triangle", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.013158, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 148, "g": 29, "kind": "obtuse"}, "seed": 205690953301000487, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 63, "valid_answers": ["34, 51, 63", "35, 49, 64", "36, 47, 65", "37, 45, 66", "38, 43, 67", "39, 41, 68"]}, "cell": "int_triangle/1", "stage": "C"}
+{"id": "po-int_triangle-t1-eval-000270", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 112 and c - a = 22. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 112 and c - a = 22. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "26, 38, 48", "ref": "26, 38, 48", "family": "po_int_triangle", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.019157, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 112, "g": 22, "kind": "obtuse"}, "seed": 205690953301001095, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 48, "valid_answers": ["26, 38, 48", "27, 36, 49", "28, 34, 50", "29, 32, 51", "30, 30, 52"]}, "cell": "int_triangle/1", "stage": "C"}
+{"id": "po-int_triangle-t1-eval-000589", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 140 and c - a = 30. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 140 and c - a = 30. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "30, 50, 60", "ref": "30, 50, 60", "family": "po_int_triangle", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.017157, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 140, "g": 30, "kind": "obtuse"}, "seed": 205690953301005675, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 60, "valid_answers": ["30, 50, 60", "31, 48, 61", "32, 46, 62", "33, 44, 63", "34, 42, 64", "35, 40, 65", "36, 38, 66"]}, "cell": "int_triangle/1", "stage": "C"}
+{"id": "po-int_triangle-t1-eval-000490", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 130 and c - a = 28. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 130 and c - a = 28. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "28, 46, 56", "ref": "28, 46, 56", "family": "po_int_triangle", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.019886, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 130, "g": 28, "kind": "obtuse"}, "seed": 205690953301003083, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 56, "valid_answers": ["28, 46, 56", "29, 44, 57", "30, 42, 58", "31, 40, 59", "32, 38, 60", "33, 36, 61", "34, 34, 62"]}, "cell": "int_triangle/1", "stage": "C"}
+{"id": "po-int_triangle-t1-eval-000473", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 123 and c - a = 20. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 123 and c - a = 20. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "32, 39, 52", "ref": "32, 39, 52", "family": "po_int_triangle", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.009524, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 123, "g": 20, "kind": "obtuse"}, "seed": 205690953301002816, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 52, "valid_answers": ["32, 39, 52", "33, 37, 53", "34, 35, 54"]}, "cell": "int_triangle/1", "stage": "C"}
+{"id": "po-int_triangle-t2-eval-000773", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 218 and c - a = 29. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 218 and c - a = 29. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "54, 81, 83", "ref": "54, 81, 83", "family": "po_int_triangle", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.008081, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 218, "g": 29, "kind": "acute"}, "seed": 205690953302000902, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 83, "valid_answers": ["54, 81, 83", "55, 79, 84", "56, 77, 85", "57, 75, 86", "58, 73, 87", "59, 71, 88", "60, 69, 89", "61, 67, 90"]}, "cell": "int_triangle/2", "stage": "C"}
+{"id": "po-int_triangle-t2-eval-000557", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 232 and c - a = 26. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 232 and c - a = 26. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "60, 86, 86", "ref": "60, 86, 86", "family": "po_int_triangle", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.008029, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 232, "g": 26, "kind": "acute"}, "seed": 205690953302000626, "version": "po-v0", "tier": 2, "n_raw": 9, "small": 86, "valid_answers": ["60, 86, 86", "61, 84, 87", "62, 82, 88", "63, 80, 89", "64, 78, 90", "65, 76, 91", "66, 74, 92", "67, 72, 93", "68, 70, 94"]}, "cell": "int_triangle/2", "stage": "C"}
+{"id": "po-int_triangle-t2-eval-002382", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 216 and c - a = 44. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 216 and c - a = 44. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "43, 86, 87", "ref": "43, 86, 87", "family": "po_int_triangle", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.005144, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 216, "g": 44, "kind": "acute"}, "seed": 205690953302004022, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 87, "valid_answers": ["43, 86, 87", "44, 84, 88", "45, 82, 89", "46, 80, 90", "47, 78, 91"]}, "cell": "int_triangle/2", "stage": "C"}
+{"id": "po-int_triangle-t2-eval-000752", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 217 and c - a = 30. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 217 and c - a = 30. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "53, 81, 83", "ref": "53, 81, 83", "family": "po_int_triangle", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.007136, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 217, "g": 30, "kind": "acute"}, "seed": 205690953302000874, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 83, "valid_answers": ["53, 81, 83", "54, 79, 84", "55, 77, 85", "56, 75, 86", "57, 73, 87", "58, 71, 88", "59, 69, 89"]}, "cell": "int_triangle/2", "stage": "C"}
+{"id": "po-int_triangle-t2-eval-001974", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 163 and c - a = 32. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 163 and c - a = 32. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "33, 65, 65", "ref": "33, 65, 65", "family": "po_int_triangle", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.007233, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 163, "g": 32, "kind": "acute"}, "seed": 205690953302002988, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 65, "valid_answers": ["33, 65, 65", "34, 63, 66", "35, 61, 67", "36, 59, 68"]}, "cell": "int_triangle/2", "stage": "C"}
+{"id": "po-int_triangle-t2-eval-002068", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 228 and c - a = 7. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 228 and c - a = 7. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "72, 77, 79", "ref": "72, 77, 79", "family": "po_int_triangle", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001847, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 228, "g": 7, "kind": "acute"}, "seed": 205690953302003206, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 79, "valid_answers": ["72, 77, 79", "73, 75, 80"]}, "cell": "int_triangle/2", "stage": "C"}
+{"id": "po-int_triangle-t2-eval-002436", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 164 and c - a = 32. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 164 and c - a = 32. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "34, 64, 66", "ref": "34, 64, 66", "family": "po_int_triangle", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.005357, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 164, "g": 32, "kind": "acute"}, "seed": 205690953302004202, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 66, "valid_answers": ["34, 64, 66", "35, 62, 67", "36, 60, 68"]}, "cell": "int_triangle/2", "stage": "C"}
+{"id": "po-int_triangle-t2-eval-001598", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 200 and c - a = 29. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 200 and c - a = 29. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "48, 75, 77", "ref": "48, 75, 77", "family": "po_int_triangle", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.007203, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 200, "g": 29, "kind": "acute"}, "seed": 205690953302002211, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 77, "valid_answers": ["48, 75, 77", "49, 73, 78", "50, 71, 79", "51, 69, 80", "52, 67, 81", "53, 65, 82"]}, "cell": "int_triangle/2", "stage": "C"}
+{"id": "po-int_triangle-t2-eval-002464", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 191 and c - a = 42. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 191 and c - a = 42. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "36, 77, 78", "ref": "36, 77, 78", "family": "po_int_triangle", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005263, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 191, "g": 42, "kind": "acute"}, "seed": 205690953302004297, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 78, "valid_answers": ["36, 77, 78", "37, 75, 79", "38, 73, 80", "39, 71, 81"]}, "cell": "int_triangle/2", "stage": "C"}
+{"id": "po-int_triangle-t2-eval-001002", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 191 and c - a = 19. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 191 and c - a = 19. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "51, 70, 70", "ref": "51, 70, 70", "family": "po_int_triangle", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.009211, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 191, "g": 19, "kind": "acute"}, "seed": 205690953302001219, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 70, "valid_answers": ["51, 70, 70", "52, 68, 71", "53, 66, 72", "54, 64, 73", "55, 62, 74", "56, 60, 75", "57, 58, 76"]}, "cell": "int_triangle/2", "stage": "C"}
+{"id": "po-int_triangle-t2-eval-001081", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 183 and c - a = 39. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 183 and c - a = 39. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "35, 74, 74", "ref": "35, 74, 74", "family": "po_int_triangle", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005739, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 183, "g": 39, "kind": "acute"}, "seed": 205690953302001333, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 74, "valid_answers": ["35, 74, 74", "36, 72, 75", "37, 70, 76", "38, 68, 77"]}, "cell": "int_triangle/2", "stage": "C"}
+{"id": "po-int_triangle-t2-eval-000071", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 217 and c - a = 47. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 217 and c - a = 47. The triangle must be acute. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "41, 88, 88", "ref": "41, 88, 88", "family": "po_int_triangle", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.004077, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 217, "g": 47, "kind": "acute"}, "seed": 205690953302000074, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 88, "valid_answers": ["41, 88, 88", "42, 86, 89", "43, 84, 90", "44, 82, 91"]}, "cell": "int_triangle/2", "stage": "C"}
+{"id": "po-int_triangle-t3-eval-001788", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 388 and c - a = 84. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 388 and c - a = 84. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "81, 142, 165", "ref": "81, 142, 165", "family": "po_int_triangle", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.006696, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 388, "g": 84, "kind": "obtuse"}, "seed": 205690953303004303, "version": "po-v0", "tier": 3, "n_raw": 21, "small": 165, "valid_answers": ["81, 142, 165", "82, 140, 166", "83, 138, 167", "84, 136, 168", "85, 134, 169", "86, 132, 170", "87, 130, 171", "88, 128, 172", "89, 126, 173", "90, 124, 174", "91, 122, 175", "92, 120, 176", "93, 118, 177", "94, 116, 178", "95, 114, 179", "96, 112, 180", "97, 110, 181", "98, 108, 182", "99, 106, 183", "100, 104, 184", "101, 102, 185"]}, "cell": "int_triangle/3", "stage": "C"}
+{"id": "po-int_triangle-t3-eval-001705", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 394 and c - a = 95. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 394 and c - a = 95. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "74, 151, 169", "ref": "74, 151, 169", "family": "po_int_triangle", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 26, "bucket": null, "guess_rate": 0.00804, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 394, "g": 95, "kind": "obtuse"}, "seed": 205690953303004064, "version": "po-v0", "tier": 3, "n_raw": 26, "small": 169, "valid_answers": ["74, 151, 169", "75, 149, 170", "76, 147, 171", "77, 145, 172", "78, 143, 173", "79, 141, 174", "80, 139, 175", "81, 137, 176", "82, 135, 177", "83, 133, 178", "84, 131, 179", "85, 129, 180", "86, 127, 181", "87, 125, 182", "88, 123, 183", "89, 121, 184", "90, 119, 185", "91, 117, 186", "92, 115, 187", "93, 113, 188", "94, 111, 189", "95, 109, 190", "96, 107, 191", "97, 105, 192", "98, 103, 193", "99, 101, 194"]}, "cell": "int_triangle/3", "stage": "C"}
+{"id": "po-int_triangle-t3-eval-001723", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 371 and c - a = 66. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 371 and c - a = 66. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "90, 125, 156", "ref": "90, 125, 156", "family": "po_int_triangle", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.004186, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 371, "g": 66, "kind": "obtuse"}, "seed": 205690953303004120, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 156, "valid_answers": ["90, 125, 156", "91, 123, 157", "92, 121, 158", "93, 119, 159", "94, 117, 160", "95, 115, 161", "96, 113, 162", "97, 111, 163", "98, 109, 164", "99, 107, 165", "100, 105, 166", "101, 103, 167"]}, "cell": "int_triangle/3", "stage": "C"}
+{"id": "po-int_triangle-t3-eval-000206", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 326 and c - a = 57. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 326 and c - a = 57. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "80, 109, 137", "ref": "80, 109, 137", "family": "po_int_triangle", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.004517, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 326, "g": 57, "kind": "obtuse"}, "seed": 205690953303000427, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 137, "valid_answers": ["80, 109, 137", "81, 107, 138", "82, 105, 139", "83, 103, 140", "84, 101, 141", "85, 99, 142", "86, 97, 143", "87, 95, 144", "88, 93, 145", "89, 91, 146"]}, "cell": "int_triangle/3", "stage": "C"}
+{"id": "po-int_triangle-t3-eval-000848", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 312 and c - a = 52. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 312 and c - a = 52. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "79, 102, 131", "ref": "79, 102, 131", "family": "po_int_triangle", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.003945, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 312, "g": 52, "kind": "obtuse"}, "seed": 205690953303001880, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 131, "valid_answers": ["79, 102, 131", "80, 100, 132", "81, 98, 133", "82, 96, 134", "83, 94, 135", "84, 92, 136", "85, 90, 137", "86, 88, 138"]}, "cell": "int_triangle/3", "stage": "C"}
+{"id": "po-int_triangle-t3-eval-001397", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 339 and c - a = 66. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 339 and c - a = 66. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "77, 119, 143", "ref": "77, 119, 143", "family": "po_int_triangle", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 0.006266, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 339, "g": 66, "kind": "obtuse"}, "seed": 205690953303003213, "version": "po-v0", "tier": 3, "n_raw": 15, "small": 143, "valid_answers": ["77, 119, 143", "78, 117, 144", "79, 115, 145", "80, 113, 146", "81, 111, 147", "82, 109, 148", "83, 107, 149", "84, 105, 150", "85, 103, 151", "86, 101, 152", "87, 99, 153", "88, 97, 154", "89, 95, 155", "90, 93, 156", "91, 91, 157"]}, "cell": "int_triangle/3", "stage": "C"}
+{"id": "po-int_triangle-t3-eval-000881", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 350 and c - a = 73. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 350 and c - a = 73. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "75, 127, 148", "ref": "75, 127, 148", "family": "po_int_triangle", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 0.007053, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 350, "g": 73, "kind": "obtuse"}, "seed": 205690953303001956, "version": "po-v0", "tier": 3, "n_raw": 18, "small": 148, "valid_answers": ["75, 127, 148", "76, 125, 149", "77, 123, 150", "78, 121, 151", "79, 119, 152", "80, 117, 153", "81, 115, 154", "82, 113, 155", "83, 111, 156", "84, 109, 157", "85, 107, 158", "86, 105, 159", "87, 103, 160", "88, 101, 161", "89, 99, 162", "90, 97, 163", "91, 95, 164", "92, 93, 165"]}, "cell": "int_triangle/3", "stage": "C"}
+{"id": "po-int_triangle-t3-eval-001948", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 308 and c - a = 48. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 308 and c - a = 48. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "81, 98, 129", "ref": "81, 98, 129", "family": "po_int_triangle", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.003036, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 308, "g": 48, "kind": "obtuse"}, "seed": 205690953303004770, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 129, "valid_answers": ["81, 98, 129", "82, 96, 130", "83, 94, 131", "84, 92, 132", "85, 90, 133", "86, 88, 134"]}, "cell": "int_triangle/3", "stage": "C"}
+{"id": "po-int_triangle-t3-eval-001562", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 275 and c - a = 39. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 275 and c - a = 39. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "76, 84, 115", "ref": "76, 84, 115", "family": "po_int_triangle", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001905, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 275, "g": 39, "kind": "obtuse"}, "seed": 205690953303003657, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 115, "valid_answers": ["76, 84, 115", "77, 82, 116", "78, 80, 117"]}, "cell": "int_triangle/3", "stage": "C"}
+{"id": "po-int_triangle-t3-eval-002363", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 387 and c - a = 81. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 387 and c - a = 81. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "83, 140, 164", "ref": "83, 140, 164", "family": "po_int_triangle", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.00641, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 387, "g": 81, "kind": "obtuse"}, "seed": 205690953303006114, "version": "po-v0", "tier": 3, "n_raw": 20, "small": 164, "valid_answers": ["83, 140, 164", "84, 138, 165", "85, 136, 166", "86, 134, 167", "87, 132, 168", "88, 130, 169", "89, 128, 170", "90, 126, 171", "91, 124, 172", "92, 122, 173", "93, 120, 174", "94, 118, 175", "95, 116, 176", "96, 114, 177", "97, 112, 178", "98, 110, 179", "99, 108, 180", "100, 106, 181", "101, 104, 182", "102, 102, 183"]}, "cell": "int_triangle/3", "stage": "C"}
+{"id": "po-int_triangle-t3-eval-000390", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 350 and c - a = 81. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 350 and c - a = 81. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "69, 131, 150", "ref": "69, 131, 150", "family": "po_int_triangle", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.008229, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 350, "g": 81, "kind": "obtuse"}, "seed": 205690953303000813, "version": "po-v0", "tier": 3, "n_raw": 21, "small": 150, "valid_answers": ["69, 131, 150", "70, 129, 151", "71, 127, 152", "72, 125, 153", "73, 123, 154", "74, 121, 155", "75, 119, 156", "76, 117, 157", "77, 115, 158", "78, 113, 159", "79, 111, 160", "80, 109, 161", "81, 107, 162", "82, 105, 163", "83, 103, 164", "84, 101, 165", "85, 99, 166", "86, 97, 167", "87, 95, 168", "88, 93, 169", "89, 91, 170"]}, "cell": "int_triangle/3", "stage": "C"}
+{"id": "po-int_triangle-t3-eval-000522", "task": "cx_po_int_triangle", "source": "cx_po_int_triangle", "split": "eval", "problem": "Find integers a <= b <= c that are the side lengths of a triangle with perimeter 363 and c - a = 56. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c that are the side lengths of a triangle with perimeter 363 and c - a = 56. The triangle must be obtuse. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "95, 117, 151", "ref": "95, 117, 151", "family": "po_int_triangle", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.002914, "solvable": true, "holdout": false, "meta": {"family": "po_int_triangle", "rung": "witness", "params": {"P": 363, "g": 56, "kind": "obtuse"}, "seed": 205690953303001100, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 151, "valid_answers": ["95, 117, 151", "96, 115, 152", "97, 113, 153", "98, 111, 154", "99, 109, 155", "100, 107, 156", "101, 105, 157", "102, 103, 158"]}, "cell": "int_triangle/3", "stage": "C"}
+{"id": "po-lattice_tri_area-t0-eval-001452", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (-2, -3), B = (3, -1) and C = (x, y) has area \\frac{35}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (-2, -3), B = (3, -1) and C = (x, y) has area \\frac{35}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 4", "ref": "-2, 4", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-2, -3], "B": [3, -1], "R": 6, "S2": 35}, "seed": 9453480300021223, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-2, 4", "3, 6"]}, "cell": "lattice_tri_area/0", "stage": "A"}
+{"id": "po-lattice_tri_area-t0-eval-000144", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (-3, 1), B = (1, -2) and C = (x, y) has area 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (-3, 1), B = (1, -2) and C = (x, y) has area 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -5", "ref": "-5, -5", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-3, 1], "B": [1, -2], "R": 6, "S2": 30}, "seed": 9453480300002020, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-5, -5", "3, 4"]}, "cell": "lattice_tri_area/0", "stage": "A"}
+{"id": "po-lattice_tri_area-t0-eval-000245", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (0, 3), B = (-3, -3) and C = (x, y) has area 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (0, 3), B = (-3, -3) and C = (x, y) has area 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, -5", "ref": "4, -5", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017751, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [0, 3], "B": [-3, -3], "R": 6, "S2": 48}, "seed": 9453480300003574, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["4, -5", "5, -3", "6, -1"]}, "cell": "lattice_tri_area/0", "stage": "A"}
+{"id": "po-lattice_tri_area-t0-eval-000665", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (1, 1), B = (0, -1) and C = (x, y) has area 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (1, 1), B = (0, -1) and C = (x, y) has area 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 3", "ref": "-6, 3", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017751, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [1, 1], "B": [0, -1], "R": 6, "S2": 16}, "seed": 9453480300009301, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-6, 3", "-5, 5", "6, -5"]}, "cell": "lattice_tri_area/0", "stage": "A"}
+{"id": "po-lattice_tri_area-t0-eval-000407", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (0, -2), B = (1, 2) and C = (x, y) has area \\frac{25}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (0, -2), B = (1, 2) and C = (x, y) has area \\frac{25}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -1", "ref": "-6, -1", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017751, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [0, -2], "B": [1, 2], "R": 6, "S2": 25}, "seed": 9453480300005767, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-6, -1", "-5, 3", "6, -3"]}, "cell": "lattice_tri_area/0", "stage": "A"}
+{"id": "po-lattice_tri_area-t0-eval-000831", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (2, 3), B = (-1, 1) and C = (x, y) has area \\frac{29}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (2, 3), B = (-1, 1) and C = (x, y) has area \\frac{29}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -6", "ref": "3, -6", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [2, 3], "B": [-1, 1], "R": 6, "S2": 29}, "seed": 9453480300012016, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 6, "valid_answers": ["3, -6", "6, -4"]}, "cell": "lattice_tri_area/0", "stage": "A"}
+{"id": "po-lattice_tri_area-t0-eval-000043", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (3, -2), B = (1, 1) and C = (x, y) has area \\frac{25}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (3, -2), B = (1, 1) and C = (x, y) has area \\frac{25}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -1", "ref": "-6, -1", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017751, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [3, -2], "B": [1, 1], "R": 6, "S2": 25}, "seed": 9453480300000507, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-6, -1", "-4, -4", "6, 6"]}, "cell": "lattice_tri_area/0", "stage": "A"}
+{"id": "po-lattice_tri_area-t0-eval-001250", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (3, 2), B = (-3, 3) and C = (x, y) has area \\frac{29}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (3, 2), B = (-3, 3) and C = (x, y) has area \\frac{29}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -2", "ref": "-2, -2", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [3, 2], "B": [-3, 3], "R": 6, "S2": 29}, "seed": 9453480300018109, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["-2, -2", "4, -3"]}, "cell": "lattice_tri_area/0", "stage": "A"}
+{"id": "po-lattice_tri_area-t0-eval-000882", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (-1, -3), B = (0, 2) and C = (x, y) has area 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (-1, -3), B = (0, 2) and C = (x, y) has area 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 2", "ref": "-6, 2", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017751, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-1, -3], "B": [0, 2], "R": 6, "S2": 30}, "seed": 9453480300012740, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-6, 2", "5, -3", "6, 2"]}, "cell": "lattice_tri_area/0", "stage": "A"}
+{"id": "po-lattice_tri_area-t0-eval-000525", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (-2, 2), B = (3, 0) and C = (x, y) has area 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (-2, 2), B = (3, 0) and C = (x, y) has area 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -4", "ref": "-2, -4", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017751, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-2, 2], "B": [3, 0], "R": 6, "S2": 30}, "seed": 9453480300007220, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-2, -4", "3, -6", "3, 6"]}, "cell": "lattice_tri_area/0", "stage": "A"}
+{"id": "po-lattice_tri_area-t0-eval-000461", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (1, -2), B = (-3, -3) and C = (x, y) has area \\frac{25}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (1, -2), B = (-3, -3) and C = (x, y) has area \\frac{25}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 3", "ref": "-4, 3", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017751, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [1, -2], "B": [-3, -3], "R": 6, "S2": 25}, "seed": 9453480300006470, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-4, 3", "0, 4", "4, 5"]}, "cell": "lattice_tri_area/0", "stage": "A"}
+{"id": "po-lattice_tri_area-t0-eval-001798", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (-2, -1), B = (3, 1) and C = (x, y) has area 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that the triangle with vertices A = (-2, -1), B = (3, 1) and C = (x, y) has area 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 5", "ref": "-5, 5", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-2, -1], "B": [3, 1], "R": 6, "S2": 36}, "seed": 9453480300026495, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-5, 5", "6, -5"]}, "cell": "lattice_tri_area/0", "stage": "A"}
+{"id": "po-lattice_tri_area-t1-eval-002199", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-6, 3), B = (4, -1) and C = (x, y) has area 33. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-6, 3), B = (4, -1) and C = (x, y) has area 33. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -4", "ref": "-5, -4", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.01662, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-6, 3], "B": [4, -1], "R": 9, "S2": 66}, "seed": 9453480301004760, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 5, "valid_answers": ["-5, -4", "-2, 8", "0, -6", "3, 6", "5, -8", "8, 4"]}, "cell": "lattice_tri_area/1", "stage": "C"}
+{"id": "po-lattice_tri_area-t1-eval-001777", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (1, 1), B = (-1, 5) and C = (x, y) has area 22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (1, 1), B = (-1, 5) and C = (x, y) has area 22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -1", "ref": "-9, -1", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.019391, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [1, 1], "B": [-1, 5], "R": 9, "S2": 44}, "seed": 9453480301003852, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 7, "valid_answers": ["-9, -1", "-8, -3", "-7, -5", "-6, -7", "-5, -9", "8, 9", "9, 7"]}, "cell": "lattice_tri_area/1", "stage": "C"}
+{"id": "po-lattice_tri_area-t1-eval-002386", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (1, -2), B = (2, 4) and C = (x, y) has area 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (1, -2), B = (2, 4) and C = (x, y) has area 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -6", "ref": "-1, -6", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.01662, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [1, -2], "B": [2, 4], "R": 9, "S2": 8}, "seed": 9453480301005172, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 0, "valid_answers": ["-1, -6", "0, 0", "1, 6", "2, -4", "3, 2", "4, 8"]}, "cell": "lattice_tri_area/1", "stage": "C"}
+{"id": "po-lattice_tri_area-t1-eval-001834", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-2, 1), B = (5, -2) and C = (x, y) has area \\frac{11}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-2, 1), B = (5, -2) and C = (x, y) has area \\frac{11}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 2", "ref": "-8, 2", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.01385, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-2, 1], "B": [5, -2], "R": 9, "S2": 11}, "seed": 9453480301003982, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 1, "valid_answers": ["-8, 2", "-3, 3", "-1, -1", "4, 0", "6, -4"]}, "cell": "lattice_tri_area/1", "stage": "C"}
+{"id": "po-lattice_tri_area-t1-eval-002488", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-6, -5), B = (3, 5) and C = (x, y) has area 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-6, -5), B = (3, 5) and C = (x, y) has area 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -7", "ref": "-8, -7", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.01108, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-6, -5], "B": [3, 5], "R": 9, "S2": 2}, "seed": 9453480301005375, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 3, "valid_answers": ["-8, -7", "-4, -3", "1, 3", "5, 7"]}, "cell": "lattice_tri_area/1", "stage": "C"}
+{"id": "po-lattice_tri_area-t1-eval-002364", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-5, 4), B = (6, 1) and C = (x, y) has area 58. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-5, 4), B = (6, 1) and C = (x, y) has area 58. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -6", "ref": "-7, -6", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00554, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-5, 4], "B": [6, 1], "R": 9, "S2": 116}, "seed": 9453480301005137, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 7, "valid_answers": ["-7, -6", "4, -9"]}, "cell": "lattice_tri_area/1", "stage": "C"}
+{"id": "po-lattice_tri_area-t1-eval-000145", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-1, -2), B = (-6, -4) and C = (x, y) has area 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-1, -2), B = (-6, -4) and C = (x, y) has area 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 6", "ref": "-5, 6", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.00831, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-1, -2], "B": [-6, -4], "R": 9, "S2": 48}, "seed": 9453480301000316, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 6, "valid_answers": ["-5, 6", "0, 8", "8, -8"]}, "cell": "lattice_tri_area/1", "stage": "C"}
+{"id": "po-lattice_tri_area-t1-eval-000411", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-4, -4), B = (-2, 1) and C = (x, y) has area \\frac{31}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-4, -4), B = (-2, 1) and C = (x, y) has area \\frac{31}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -1", "ref": "-9, -1", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.019391, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-4, -4], "B": [-2, 1], "R": 9, "S2": 31}, "seed": 9453480301000889, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 3, "valid_answers": ["-9, -1", "-7, 4", "-5, 9", "1, -7", "3, -2", "5, 3", "7, 8"]}, "cell": "lattice_tri_area/1", "stage": "C"}
+{"id": "po-lattice_tri_area-t1-eval-002425", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-5, 5), B = (0, 4) and C = (x, y) has area \\frac{65}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-5, 5), B = (0, 4) and C = (x, y) has area \\frac{65}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -8", "ref": "-5, -8", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00554, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-5, 5], "B": [0, 4], "R": 9, "S2": 65}, "seed": 9453480301005241, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 8, "valid_answers": ["-5, -8", "0, -9"]}, "cell": "lattice_tri_area/1", "stage": "C"}
+{"id": "po-lattice_tri_area-t1-eval-000366", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-6, 0), B = (4, -3) and C = (x, y) has area 35. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-6, 0), B = (4, -3) and C = (x, y) has area 35. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -7", "ref": "-6, -7", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.00831, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-6, 0], "B": [4, -3], "R": 9, "S2": 70}, "seed": 9453480301000793, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 4, "valid_answers": ["-6, -7", "-6, 7", "4, 4"]}, "cell": "lattice_tri_area/1", "stage": "C"}
+{"id": "po-lattice_tri_area-t1-eval-002043", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-6, 4), B = (-2, -1) and C = (x, y) has area 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (-6, 4), B = (-2, -1) and C = (x, y) has area 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 1", "ref": "-8, 1", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.019391, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-6, 4], "B": [-2, -1], "R": 9, "S2": 22}, "seed": 9453480301004437, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 2, "valid_answers": ["-8, 1", "-4, -4", "-4, 7", "0, -9", "0, 2", "4, -3", "8, -8"]}, "cell": "lattice_tri_area/1", "stage": "C"}
+{"id": "po-lattice_tri_area-t1-eval-000915", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (2, 0), B = (-3, 2) and C = (x, y) has area \\frac{11}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -9 <= x, y <= 9 such that the triangle with vertices A = (2, 0), B = (-3, 2) and C = (x, y) has area \\frac{11}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 1", "ref": "-6, 1", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.019391, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [2, 0], "B": [-3, 2], "R": 9, "S2": 11}, "seed": 9453480301002031, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 1, "valid_answers": ["-6, 1", "-5, 5", "-1, -1", "0, 3", "4, -3", "5, 1", "9, -5"]}, "cell": "lattice_tri_area/1", "stage": "C"}
+{"id": "po-lattice_tri_area-t2-eval-000799", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (-2, 0), B = (-6, -5) and C = (x, y) has area \\frac{11}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (-2, 0), B = (-6, -5) and C = (x, y) has area \\frac{11}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -6", "ref": "-9, -6", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.0144, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-2, 0], "B": [-6, -5], "R": 12, "S2": 11}, "seed": 9453480302001158, "version": "po-v0", "tier": 2, "n_raw": 9, "small": 1, "valid_answers": ["-9, -6", "-7, -9", "-5, -1", "-3, -4", "-1, 4", "1, 1", "3, 9", "5, 6", "9, 11"]}, "cell": "lattice_tri_area/2", "stage": "C"}
+{"id": "po-lattice_tri_area-t2-eval-001375", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (-4, 3), B = (8, 7) and C = (x, y) has area 62. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (-4, 3), B = (8, 7) and C = (x, y) has area 62. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, -10", "ref": "-12, -10", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.0176, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-4, 3], "B": [8, 7], "R": 12, "S2": 124}, "seed": 9453480302001969, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 5, "valid_answers": ["-12, -10", "-11, 11", "-9, -9", "-8, 12", "-6, -8", "-3, -7", "0, -6", "3, -5", "6, -4", "9, -3", "12, -2"]}, "cell": "lattice_tri_area/2", "stage": "C"}
+{"id": "po-lattice_tri_area-t2-eval-002296", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (-9, 7), B = (-10, 0) and C = (x, y) has area 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (-9, 7), B = (-10, 0) and C = (x, y) has area 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, 12", "ref": "-12, 12", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.008, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-9, 7], "B": [-10, 0], "R": 12, "S2": 26}, "seed": 9453480302003280, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 6, "valid_answers": ["-12, 12", "-8, -12", "-7, -5", "-6, 2", "-5, 9"]}, "cell": "lattice_tri_area/2", "stage": "C"}
+{"id": "po-lattice_tri_area-t2-eval-001495", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (5, -1), B = (8, 1) and C = (x, y) has area 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (5, -1), B = (8, 1) and C = (x, y) has area 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 5", "ref": "-10, 5", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0064, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [5, -1], "B": [8, 1], "R": 12, "S2": 48}, "seed": 9453480302002131, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 7, "valid_answers": ["-10, 5", "-7, 7", "-4, 9", "-1, 11"]}, "cell": "lattice_tri_area/2", "stage": "C"}
+{"id": "po-lattice_tri_area-t2-eval-001065", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (-2, -2), B = (9, 1) and C = (x, y) has area 33. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (-2, -2), B = (9, 1) and C = (x, y) has area 33. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -8", "ref": "-2, -8", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0064, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-2, -2], "B": [9, 1], "R": 12, "S2": 66}, "seed": 9453480302001528, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 4, "valid_answers": ["-2, -8", "-2, 4", "9, -5", "9, 7"]}, "cell": "lattice_tri_area/2", "stage": "C"}
+{"id": "po-lattice_tri_area-t2-eval-002099", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (-10, 7), B = (-1, 8) and C = (x, y) has area 39. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (-10, 7), B = (-1, 8) and C = (x, y) has area 39. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -1", "ref": "-4, -1", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0032, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-10, 7], "B": [-1, 8], "R": 12, "S2": 78}, "seed": 9453480302003003, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 4, "valid_answers": ["-4, -1", "5, 0"]}, "cell": "lattice_tri_area/2", "stage": "C"}
+{"id": "po-lattice_tri_area-t2-eval-002486", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (3, 2), B = (-4, 10) and C = (x, y) has area 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (3, 2), B = (-4, 10) and C = (x, y) has area 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 10", "ref": "-10, 10", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.008, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [3, 2], "B": [-4, 10], "R": 12, "S2": 48}, "seed": 9453480302003553, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 3, "valid_answers": ["-10, 10", "-3, 2", "2, 10", "4, -6", "9, 2"]}, "cell": "lattice_tri_area/2", "stage": "C"}
+{"id": "po-lattice_tri_area-t2-eval-001223", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (-6, 10), B = (4, 4) and C = (x, y) has area 29. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (-6, 10), B = (4, 4) and C = (x, y) has area 29. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 6", "ref": "-9, 6", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.0128, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-6, 10], "B": [4, 4], "R": 12, "S2": 58}, "seed": 9453480302001745, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 1, "valid_answers": ["-9, 6", "-4, 3", "1, 0", "2, 11", "6, -3", "7, 8", "11, -6", "12, 5"]}, "cell": "lattice_tri_area/2", "stage": "C"}
+{"id": "po-lattice_tri_area-t2-eval-002148", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (3, 3), B = (-6, 9) and C = (x, y) has area \\frac{165}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (3, 3), B = (-6, 9) and C = (x, y) has area \\frac{165}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-11, -6", "ref": "-11, -6", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0064, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [3, 3], "B": [-6, 9], "R": 12, "S2": 165}, "seed": 9453480302003067, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 8, "valid_answers": ["-11, -6", "-8, -8", "-5, -10", "-2, -12"]}, "cell": "lattice_tri_area/2", "stage": "C"}
+{"id": "po-lattice_tri_area-t2-eval-002016", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (0, -9), B = (9, -4) and C = (x, y) has area \\frac{175}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (0, -9), B = (9, -4) and C = (x, y) has area \\frac{175}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 6", "ref": "-8, 6", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0032, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [0, -9], "B": [9, -4], "R": 12, "S2": 175}, "seed": 9453480302002885, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 8, "valid_answers": ["-8, 6", "1, 11"]}, "cell": "lattice_tri_area/2", "stage": "C"}
+{"id": "po-lattice_tri_area-t2-eval-000993", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (-7, 5), B = (-10, 4) and C = (x, y) has area 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (-7, 5), B = (-10, 4) and C = (x, y) has area 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, 12", "ref": "-12, 12", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.0144, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-7, 5], "B": [-10, 4], "R": 12, "S2": 26}, "seed": 9453480302001421, "version": "po-v0", "tier": 2, "n_raw": 9, "small": 1, "valid_answers": ["-12, 12", "-11, -5", "-8, -4", "-5, -3", "-2, -2", "1, -1", "4, 0", "7, 1", "10, 2"]}, "cell": "lattice_tri_area/2", "stage": "C"}
+{"id": "po-lattice_tri_area-t2-eval-001515", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (3, 0), B = (10, -6) and C = (x, y) has area 45. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -12 <= x, y <= 12 such that the triangle with vertices A = (3, 0), B = (10, -6) and C = (x, y) has area 45. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, 0", "ref": "-12, 0", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.008, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [3, 0], "B": [10, -6], "R": 12, "S2": 90}, "seed": 9453480302002162, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 6, "valid_answers": ["-12, 0", "-5, -6", "2, -12", "4, 12", "11, 6"]}, "cell": "lattice_tri_area/2", "stage": "C"}
+{"id": "po-lattice_tri_area-t3-eval-001504", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (1, 0), B = (-13, 2) and C = (x, y) has area 114. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (1, 0), B = (-13, 2) and C = (x, y) has area 114. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-18, 19", "ref": "-18, 19", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.007139, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [1, 0], "B": [-13, 2], "R": 20, "S2": 228}, "seed": 9453480303001761, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 15, "valid_answers": ["-18, 19", "-15, -14", "-11, 18", "-8, -15", "-4, 17", "-1, -16", "3, 16", "6, -17", "10, 15", "13, -18", "17, 14", "20, -19"]}, "cell": "lattice_tri_area/3", "stage": "C"}
+{"id": "po-lattice_tri_area-t3-eval-000542", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (7, -11), B = (2, -14) and C = (x, y) has area \\frac{5}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (7, -11), B = (2, -14) and C = (x, y) has area \\frac{5}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -19", "ref": "-8, -19", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.006544, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [7, -11], "B": [2, -14], "R": 20, "S2": 5}, "seed": 9453480303000632, "version": "po-v0", "tier": 3, "n_raw": 11, "small": 10, "valid_answers": ["-8, -19", "-3, -18", "-3, -16", "2, -15", "2, -13", "7, -12", "7, -10", "12, -9", "12, -7", "17, -6", "17, -4"]}, "cell": "lattice_tri_area/3", "stage": "C"}
+{"id": "po-lattice_tri_area-t3-eval-002162", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (11, -5), B = (7, -2) and C = (x, y) has area \\frac{83}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (11, -5), B = (7, -2) and C = (x, y) has area \\frac{83}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-18, -4", "ref": "-18, -4", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.005949, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [11, -5], "B": [7, -2], "R": 20, "S2": 83}, "seed": 9453480303002543, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 10, "valid_answers": ["-18, -4", "-14, -7", "-10, -10", "-6, -13", "-2, -16", "2, -19", "8, 18", "12, 15", "16, 12", "20, 9"]}, "cell": "lattice_tri_area/3", "stage": "C"}
+{"id": "po-lattice_tri_area-t3-eval-002221", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (5, 13), B = (0, -5) and C = (x, y) has area \\frac{305}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (5, 13), B = (0, -5) and C = (x, y) has area \\frac{305}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-20, -16", "ref": "-20, -16", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002974, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [5, 13], "B": [0, -5], "R": 20, "S2": 305}, "seed": 9453480303002606, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 15, "valid_answers": ["-20, -16", "-15, 2", "-10, 20", "15, -12", "20, 6"]}, "cell": "lattice_tri_area/3", "stage": "C"}
+{"id": "po-lattice_tri_area-t3-eval-002023", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (-2, -6), B = (7, 7) and C = (x, y) has area \\frac{261}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (-2, -6), B = (7, 7) and C = (x, y) has area \\frac{261}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-20, -3", "ref": "-20, -3", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001785, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-2, -6], "B": [7, 7], "R": 20, "S2": 261}, "seed": 9453480303002379, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 11, "valid_answers": ["-20, -3", "-11, 10", "16, -9"]}, "cell": "lattice_tri_area/3", "stage": "C"}
+{"id": "po-lattice_tri_area-t3-eval-000037", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (-4, 15), B = (-9, 5) and C = (x, y) has area \\frac{105}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (-4, 15), B = (-9, 5) and C = (x, y) has area \\frac{105}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-20, 4", "ref": "-20, 4", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 30, "bucket": null, "guess_rate": 0.017847, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-4, 15], "B": [-9, 5], "R": 20, "S2": 105}, "seed": 9453480303000042, "version": "po-v0", "tier": 3, "n_raw": 30, "small": 1, "valid_answers": ["-20, 4", "-19, 6", "-18, 8", "-17, 10", "-16, 12", "-15, 14", "-14, 16", "-13, 18", "-12, 20", "-11, -20", "-10, -18", "-9, -16", "-8, -14", "-7, -12", "-6, -10", "-5, -8", "-4, -6", "-3, -4", "-2, -2", "-1, 0", "0, 2", "1, 4", "2, 6", "3, 8", "4, 10", "5, 12", "6, 14", "7, 16", "8, 18", "9, 20"]}, "cell": "lattice_tri_area/3", "stage": "C"}
+{"id": "po-lattice_tri_area-t3-eval-001691", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (-6, -3), B = (-15, -10) and C = (x, y) has area 40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (-6, -3), B = (-15, -10) and C = (x, y) has area 40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-20, -5", "ref": "-20, -5", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.004759, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-6, -3], "B": [-15, -10], "R": 20, "S2": 80}, "seed": 9453480303001991, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 8, "valid_answers": ["-20, -5", "-11, 2", "-10, -15", "-2, 9", "-1, -8", "7, 16", "8, -1", "17, 6"]}, "cell": "lattice_tri_area/3", "stage": "C"}
+{"id": "po-lattice_tri_area-t3-eval-002257", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (10, 12), B = (9, -10) and C = (x, y) has area 208. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (10, 12), B = (9, -10) and C = (x, y) has area 208. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -12", "ref": "-10, -12", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00119, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [10, 12], "B": [9, -10], "R": 20, "S2": 416}, "seed": 9453480303002652, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 10, "valid_answers": ["-10, -12", "-9, 10"]}, "cell": "lattice_tri_area/3", "stage": "C"}
+{"id": "po-lattice_tri_area-t3-eval-000006", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (-4, 2), B = (-2, 11) and C = (x, y) has area \\frac{169}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (-4, 2), B = (-2, 11) and C = (x, y) has area \\frac{169}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-19, 19", "ref": "-19, 19", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002974, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-4, 2], "B": [-2, 11], "R": 20, "S2": 169}, "seed": 9453480303000009, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 13, "valid_answers": ["-19, 19", "11, -15", "13, -6", "15, 3", "17, 12"]}, "cell": "lattice_tri_area/3", "stage": "C"}
+{"id": "po-lattice_tri_area-t3-eval-001890", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (-14, 14), B = (11, -8) and C = (x, y) has area 92. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (-14, 14), B = (11, -8) and C = (x, y) has area 92. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-11, 4", "ref": "-11, 4", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001785, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-14, 14], "B": [11, -8], "R": 20, "S2": 184}, "seed": 9453480303002223, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 8, "valid_answers": ["-11, 4", "8, 2", "14, -18"]}, "cell": "lattice_tri_area/3", "stage": "C"}
+{"id": "po-lattice_tri_area-t3-eval-000409", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (-14, -5), B = (-9, 11) and C = (x, y) has area \\frac{425}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (-14, -5), B = (-9, 11) and C = (x, y) has area \\frac{425}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, -10", "ref": "11, -10", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00119, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-14, -5], "B": [-9, 11], "R": 20, "S2": 425}, "seed": 9453480303000474, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 11, "valid_answers": ["11, -10", "16, 6"]}, "cell": "lattice_tri_area/3", "stage": "C"}
+{"id": "po-lattice_tri_area-t3-eval-000727", "task": "cx_po_lattice_tri_area", "source": "cx_po_lattice_tri_area", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (-4, 15), B = (13, -14) and C = (x, y) has area \\frac{7}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the triangle with vertices A = (-4, 15), B = (13, -14) and C = (x, y) has area \\frac{7}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 18", "ref": "-6, 18", "family": "po_lattice_tri_area", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00238, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-4, 15], "B": [13, -14], "R": 20, "S2": 7}, "seed": 9453480303000859, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 11, "valid_answers": ["-6, 18", "-2, 12", "11, -11", "15, -17"]}, "cell": "lattice_tri_area/3", "stage": "C"}
+{"id": "po-lin2_box-t0-eval-001359", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 6x - 9y = -24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 20 such that 6x - 9y = -24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 4", "ref": "2, 4", "family": "po_lin2_box", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.015873, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 6, "b": -9, "B": 20, "n": -24}, "seed": 20666920800001941, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 4, "valid_answers": ["2, 4", "5, 6", "8, 8", "11, 10", "14, 12", "17, 14", "20, 16"]}, "cell": "lin2_box/0", "stage": "A"}
+{"id": "po-lin2_box-t0-eval-001503", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 5x + 2y = 76. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 20 such that 5x + 2y = 76. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8, 18", "ref": "8, 18", "family": "po_lin2_box", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00907, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 5, "b": 2, "B": 20, "n": 76}, "seed": 20666920800002154, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 12, "valid_answers": ["8, 18", "10, 13", "12, 8", "14, 3"]}, "cell": "lin2_box/0", "stage": "A"}
+{"id": "po-lin2_box-t0-eval-000705", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 2x + 9y = 123. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 20 such that 2x + 9y = 123. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 13", "ref": "3, 13", "family": "po_lin2_box", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004535, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 2, "b": 9, "B": 20, "n": 123}, "seed": 20666920800000978, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 12, "valid_answers": ["3, 13", "12, 11"]}, "cell": "lin2_box/0", "stage": "A"}
+{"id": "po-lin2_box-t0-eval-000958", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 5x - 4y = -23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 20 such that 5x - 4y = -23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 7", "ref": "1, 7", "family": "po_lin2_box", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006803, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 5, "b": -4, "B": 20, "n": -23}, "seed": 20666920800001351, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 7, "valid_answers": ["1, 7", "5, 12", "9, 17"]}, "cell": "lin2_box/0", "stage": "A"}
+{"id": "po-lin2_box-t0-eval-000172", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 6x - 8y = -14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 20 such that 6x - 8y = -14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 4", "ref": "3, 4", "family": "po_lin2_box", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.011338, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 6, "b": -8, "B": 20, "n": -14}, "seed": 20666920800000225, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 4, "valid_answers": ["3, 4", "7, 7", "11, 10", "15, 13", "19, 16"]}, "cell": "lin2_box/0", "stage": "A"}
+{"id": "po-lin2_box-t0-eval-001328", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 2x - 7y = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 20 such that 2x - 7y = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 4", "ref": "5, 4", "family": "po_lin2_box", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006803, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 2, "b": -7, "B": 20, "n": -18}, "seed": 20666920800001899, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["5, 4", "12, 6", "19, 8"]}, "cell": "lin2_box/0", "stage": "A"}
+{"id": "po-lin2_box-t0-eval-002212", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 2x - 7y = 19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 20 such that 2x - 7y = 19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "13, 1", "ref": "13, 1", "family": "po_lin2_box", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004535, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 2, "b": -7, "B": 20, "n": 19}, "seed": 20666920800003288, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 13, "valid_answers": ["13, 1", "20, 3"]}, "cell": "lin2_box/0", "stage": "A"}
+{"id": "po-lin2_box-t0-eval-000719", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 6x + 9y = 234. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 20 such that 6x + 9y = 234. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9, 20", "ref": "9, 20", "family": "po_lin2_box", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00907, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 6, "b": 9, "B": 20, "n": 234}, "seed": 20666920800000998, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 16, "valid_answers": ["9, 20", "12, 18", "15, 16", "18, 14"]}, "cell": "lin2_box/0", "stage": "A"}
+{"id": "po-lin2_box-t0-eval-002406", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 5x + 2y = 102. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 20 such that 5x + 2y = 102. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "14, 16", "ref": "14, 16", "family": "po_lin2_box", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00907, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 5, "b": 2, "B": 20, "n": 102}, "seed": 20666920800003649, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 16, "valid_answers": ["14, 16", "16, 11", "18, 6", "20, 1"]}, "cell": "lin2_box/0", "stage": "A"}
+{"id": "po-lin2_box-t0-eval-002114", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 2x - 3y = -15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 20 such that 2x - 3y = -15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 5", "ref": "0, 5", "family": "po_lin2_box", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.015873, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 2, "b": -3, "B": 20, "n": -15}, "seed": 20666920800003130, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 5, "valid_answers": ["0, 5", "3, 7", "6, 9", "9, 11", "12, 13", "15, 15", "18, 17"]}, "cell": "lin2_box/0", "stage": "A"}
+{"id": "po-lin2_box-t0-eval-000870", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 9x - 2y = 119. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 20 such that 9x - 2y = 119. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "15, 8", "ref": "15, 8", "family": "po_lin2_box", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004535, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 9, "b": -2, "B": 20, "n": 119}, "seed": 20666920800001217, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 15, "valid_answers": ["15, 8", "17, 17"]}, "cell": "lin2_box/0", "stage": "A"}
+{"id": "po-lin2_box-t0-eval-001987", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 8x + 6y = 90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 20 such that 8x + 6y = 90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 15", "ref": "0, 15", "family": "po_lin2_box", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00907, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 8, "b": 6, "B": 20, "n": 90}, "seed": 20666920800002919, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 7, "valid_answers": ["0, 15", "3, 11", "6, 7", "9, 3"]}, "cell": "lin2_box/0", "stage": "A"}
+{"id": "po-lin2_box-t1-eval-000969", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 50 such that 19x + 8y = 659. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 50 such that 19x + 8y = 659. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "17, 42", "ref": "17, 42", "family": "po_lin2_box", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001153, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 19, "b": 8, "B": 50, "n": 659}, "seed": 20666920801001041, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 25, "valid_answers": ["17, 42", "25, 23", "33, 4"]}, "cell": "lin2_box/1", "stage": "B"}
+{"id": "po-lin2_box-t1-eval-000932", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 50 such that 15x - 2y = 34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 50 such that 15x - 2y = 34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 13", "ref": "4, 13", "family": "po_lin2_box", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001153, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 15, "b": -2, "B": 50, "n": 34}, "seed": 20666920801001002, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 13, "valid_answers": ["4, 13", "6, 28", "8, 43"]}, "cell": "lin2_box/1", "stage": "B"}
+{"id": "po-lin2_box-t1-eval-002090", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 50 such that 8x + 12y = 580. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 50 such that 8x + 12y = 580. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 47", "ref": "2, 47", "family": "po_lin2_box", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 17, "bucket": "9-20", "guess_rate": 0.006536, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 8, "b": 12, "B": 50, "n": 580}, "seed": 20666920801002219, "version": "po-v0", "tier": 1, "n_raw": 17, "small": 29, "valid_answers": ["2, 47", "5, 45", "8, 43", "11, 41", "14, 39", "17, 37", "20, 35", "23, 33", "26, 31", "29, 29", "32, 27", "35, 25", "38, 23", "41, 21", "44, 19", "47, 17", "50, 15"]}, "cell": "lin2_box/1", "stage": "B"}
+{"id": "po-lin2_box-t1-eval-000129", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 50 such that 6x + 19y = 636. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 50 such that 6x + 19y = 636. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 30", "ref": "11, 30", "family": "po_lin2_box", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001153, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 6, "b": 19, "B": 50, "n": 636}, "seed": 20666920801000142, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 30, "valid_answers": ["11, 30", "30, 24", "49, 18"]}, "cell": "lin2_box/1", "stage": "B"}
+{"id": "po-lin2_box-t1-eval-000353", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 50 such that 13x + 9y = 664. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 50 such that 13x + 9y = 664. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "22, 42", "ref": "22, 42", "family": "po_lin2_box", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001538, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 13, "b": 9, "B": 50, "n": 664}, "seed": 20666920801000379, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 31, "valid_answers": ["22, 42", "31, 29", "40, 16", "49, 3"]}, "cell": "lin2_box/1", "stage": "B"}
+{"id": "po-lin2_box-t1-eval-001666", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 50 such that 12x - 10y = 136. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 50 such that 12x - 10y = 136. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "13, 2", "ref": "13, 2", "family": "po_lin2_box", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.003076, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 12, "b": -10, "B": 50, "n": 136}, "seed": 20666920801001766, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 13, "valid_answers": ["13, 2", "18, 8", "23, 14", "28, 20", "33, 26", "38, 32", "43, 38", "48, 44"]}, "cell": "lin2_box/1", "stage": "B"}
+{"id": "po-lin2_box-t1-eval-002440", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 50 such that 14x + 10y = 548. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 50 such that 14x + 10y = 548. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 45", "ref": "7, 45", "family": "po_lin2_box", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.002691, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 14, "b": 10, "B": 50, "n": 548}, "seed": 20666920801002595, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 24, "valid_answers": ["7, 45", "12, 38", "17, 31", "22, 24", "27, 17", "32, 10", "37, 3"]}, "cell": "lin2_box/1", "stage": "B"}
+{"id": "po-lin2_box-t1-eval-000280", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 50 such that 4x - 17y = -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 50 such that 4x - 17y = -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 2", "ref": "3, 2", "family": "po_lin2_box", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001153, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 4, "b": -17, "B": 50, "n": -22}, "seed": 20666920801000303, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 3, "valid_answers": ["3, 2", "20, 6", "37, 10"]}, "cell": "lin2_box/1", "stage": "B"}
+{"id": "po-lin2_box-t1-eval-000514", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 50 such that 8x + 9y = 115. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 50 such that 8x + 9y = 115. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 11", "ref": "2, 11", "family": "po_lin2_box", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000769, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 8, "b": 9, "B": 50, "n": 115}, "seed": 20666920801000551, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 11, "valid_answers": ["2, 11", "11, 3"]}, "cell": "lin2_box/1", "stage": "B"}
+{"id": "po-lin2_box-t1-eval-000646", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 50 such that 4x - 17y = -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 50 such that 4x - 17y = -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 1", "ref": "2, 1", "family": "po_lin2_box", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001153, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 4, "b": -17, "B": 50, "n": -9}, "seed": 20666920801000688, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 2, "valid_answers": ["2, 1", "19, 5", "36, 9"]}, "cell": "lin2_box/1", "stage": "B"}
+{"id": "po-lin2_box-t1-eval-001865", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 50 such that 8x - 19y = 120. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 50 such that 8x - 19y = 120. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "15, 0", "ref": "15, 0", "family": "po_lin2_box", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000769, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 8, "b": -19, "B": 50, "n": 120}, "seed": 20666920801001973, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 15, "valid_answers": ["15, 0", "34, 8"]}, "cell": "lin2_box/1", "stage": "B"}
+{"id": "po-lin2_box-t1-eval-002147", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 50 such that 9x - 18y = -657. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 50 such that 9x - 18y = -657. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 37", "ref": "1, 37", "family": "po_lin2_box", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.005383, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 9, "b": -18, "B": 50, "n": -657}, "seed": 20666920801002277, "version": "po-v0", "tier": 1, "n_raw": 14, "small": 37, "valid_answers": ["1, 37", "3, 38", "5, 39", "7, 40", "9, 41", "11, 42", "13, 43", "15, 44", "17, 45", "19, 46", "21, 47", "23, 48", "25, 49", "27, 50"]}, "cell": "lin2_box/1", "stage": "B"}
+{"id": "po-lin2_box-t2-eval-000834", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 100 such that 24x + 2y = 2104. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 100 such that 24x + 2y = 2104. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "80, 92", "ref": "80, 92", "family": "po_lin2_box", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000784, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 24, "b": 2, "B": 100, "n": 2104}, "seed": 20666920802000885, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 81, "valid_answers": ["80, 92", "81, 80", "82, 68", "83, 56", "84, 44", "85, 32", "86, 20", "87, 8"]}, "cell": "lin2_box/2", "stage": "B"}
+{"id": "po-lin2_box-t2-eval-000180", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 100 such that 4x + 9y = 894. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 100 such that 4x + 9y = 894. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 98", "ref": "3, 98", "family": "po_lin2_box", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.001078, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 4, "b": 9, "B": 100, "n": 894}, "seed": 20666920802000188, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 70, "valid_answers": ["3, 98", "12, 94", "21, 90", "30, 86", "39, 82", "48, 78", "57, 74", "66, 70", "75, 66", "84, 62", "93, 58"]}, "cell": "lin2_box/2", "stage": "B"}
+{"id": "po-lin2_box-t2-eval-000113", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 100 such that 9x - 16y = 483. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 100 such that 9x - 16y = 483. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "59, 3", "ref": "59, 3", "family": "po_lin2_box", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000294, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 9, "b": -16, "B": 100, "n": 483}, "seed": 20666920802000119, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 59, "valid_answers": ["59, 3", "75, 12", "91, 21"]}, "cell": "lin2_box/2", "stage": "B"}
+{"id": "po-lin2_box-t2-eval-001269", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 100 such that 31x + 32y = 4139. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 100 such that 31x + 32y = 4139. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "53, 78", "ref": "53, 78", "family": "po_lin2_box", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000196, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 31, "b": 32, "B": 100, "n": 4139}, "seed": 20666920802001346, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 78, "valid_answers": ["53, 78", "85, 47"]}, "cell": "lin2_box/2", "stage": "B"}
+{"id": "po-lin2_box-t2-eval-001098", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 100 such that 3x - 14y = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 100 such that 3x - 14y = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 0", "ref": "3, 0", "family": "po_lin2_box", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000686, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 3, "b": -14, "B": 100, "n": 9}, "seed": 20666920802001166, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 3, "valid_answers": ["3, 0", "17, 3", "31, 6", "45, 9", "59, 12", "73, 15", "87, 18"]}, "cell": "lin2_box/2", "stage": "B"}
+{"id": "po-lin2_box-t2-eval-000373", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 100 such that 31x - 24y = -93. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 100 such that 31x - 24y = -93. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "21, 31", "ref": "21, 31", "family": "po_lin2_box", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000294, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 31, "b": -24, "B": 100, "n": -93}, "seed": 20666920802000394, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 31, "valid_answers": ["21, 31", "45, 62", "69, 93"]}, "cell": "lin2_box/2", "stage": "B"}
+{"id": "po-lin2_box-t2-eval-001300", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 100 such that 33x + 4y = 1441. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 100 such that 33x + 4y = 1441. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "33, 88", "ref": "33, 88", "family": "po_lin2_box", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000294, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 33, "b": 4, "B": 100, "n": 1441}, "seed": 20666920802001379, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 41, "valid_answers": ["33, 88", "37, 55", "41, 22"]}, "cell": "lin2_box/2", "stage": "B"}
+{"id": "po-lin2_box-t2-eval-002420", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 100 such that 6x - 4y = 200. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 100 such that 6x - 4y = 200. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "34, 1", "ref": "34, 1", "family": "po_lin2_box", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 34, "bucket": null, "guess_rate": 0.003333, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 6, "b": -4, "B": 100, "n": 200}, "seed": 20666920802002580, "version": "po-v0", "tier": 2, "n_raw": 34, "small": 34, "valid_answers": ["34, 1", "36, 4", "38, 7", "40, 10", "42, 13", "44, 16", "46, 19", "48, 22", "50, 25", "52, 28", "54, 31", "56, 34", "58, 37", "60, 40", "62, 43", "64, 46", "66, 49", "68, 52", "70, 55", "72, 58", "74, 61", "76, 64", "78, 67", "80, 70", "82, 73", "84, 76", "86, 79", "88, 82", "90, 85", "92, 88", "94, 91", "96, 94", "98, 97", "100, 100"]}, "cell": "lin2_box/2", "stage": "B"}
+{"id": "po-lin2_box-t2-eval-002333", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 100 such that 32x - 35y = 659. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 100 such that 32x - 35y = 659. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "37, 15", "ref": "37, 15", "family": "po_lin2_box", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000196, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 32, "b": -35, "B": 100, "n": 659}, "seed": 20666920802002483, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 37, "valid_answers": ["37, 15", "72, 47"]}, "cell": "lin2_box/2", "stage": "B"}
+{"id": "po-lin2_box-t2-eval-001604", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 100 such that 34x + 18y = 2174. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 100 such that 34x + 18y = 2174. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 100", "ref": "11, 100", "family": "po_lin2_box", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000588, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 34, "b": 18, "B": 100, "n": 2174}, "seed": 20666920802001700, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 47, "valid_answers": ["11, 100", "20, 83", "29, 66", "38, 49", "47, 32", "56, 15"]}, "cell": "lin2_box/2", "stage": "B"}
+{"id": "po-lin2_box-t2-eval-001940", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 100 such that 24x - 13y = -401. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 100 such that 24x - 13y = -401. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "12, 53", "ref": "12, 53", "family": "po_lin2_box", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000196, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 24, "b": -13, "B": 100, "n": -401}, "seed": 20666920802002061, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 53, "valid_answers": ["12, 53", "25, 77"]}, "cell": "lin2_box/2", "stage": "B"}
+{"id": "po-lin2_box-t2-eval-000266", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 100 such that 18x + 3y = 945. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 100 such that 18x + 3y = 945. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "36, 99", "ref": "36, 99", "family": "po_lin2_box", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 17, "bucket": "9-20", "guess_rate": 0.001667, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 18, "b": 3, "B": 100, "n": 945}, "seed": 20666920802000278, "version": "po-v0", "tier": 2, "n_raw": 17, "small": 45, "valid_answers": ["36, 99", "37, 93", "38, 87", "39, 81", "40, 75", "41, 69", "42, 63", "43, 57", "44, 51", "45, 45", "46, 39", "47, 33", "48, 27", "49, 21", "50, 15", "51, 9", "52, 3"]}, "cell": "lin2_box/2", "stage": "B"}
+{"id": "po-lin2_box-t3-eval-001759", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 200 such that 24x - 3y = 57. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 200 such that 24x - 3y = 57. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 5", "ref": "3, 5", "family": "po_lin2_box", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 25, "bucket": null, "guess_rate": 0.000619, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 24, "b": -3, "B": 200, "n": 57}, "seed": 20666920803001923, "version": "po-v0", "tier": 3, "n_raw": 25, "small": 5, "valid_answers": ["3, 5", "4, 13", "5, 21", "6, 29", "7, 37", "8, 45", "9, 53", "10, 61", "11, 69", "12, 77", "13, 85", "14, 93", "15, 101", "16, 109", "17, 117", "18, 125", "19, 133", "20, 141", "21, 149", "22, 157", "23, 165", "24, 173", "25, 181", "26, 189", "27, 197"]}, "cell": "lin2_box/3", "stage": "B"}
+{"id": "po-lin2_box-t3-eval-001856", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 200 such that 28x - 49y = -1295. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 200 such that 28x - 49y = -1295. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 27", "ref": "1, 27", "family": "po_lin2_box", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 29, "bucket": null, "guess_rate": 0.000718, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 28, "b": -49, "B": 200, "n": -1295}, "seed": 20666920803002029, "version": "po-v0", "tier": 3, "n_raw": 29, "small": 27, "valid_answers": ["1, 27", "8, 31", "15, 35", "22, 39", "29, 43", "36, 47", "43, 51", "50, 55", "57, 59", "64, 63", "71, 67", "78, 71", "85, 75", "92, 79", "99, 83", "106, 87", "113, 91", "120, 95", "127, 99", "134, 103", "141, 107", "148, 111", "155, 115", "162, 119", "169, 123", "176, 127", "183, 131", "190, 135", "197, 139"]}, "cell": "lin2_box/3", "stage": "B"}
+{"id": "po-lin2_box-t3-eval-000360", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 200 such that 44x + 26y = 2482. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 200 such that 44x + 26y = 2482. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 87", "ref": "5, 87", "family": "po_lin2_box", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 9.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 44, "b": 26, "B": 200, "n": 2482}, "seed": 20666920803000394, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 43, "valid_answers": ["5, 87", "18, 65", "31, 43", "44, 21"]}, "cell": "lin2_box/3", "stage": "B"}
+{"id": "po-lin2_box-t3-eval-002133", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 200 such that 13x - 60y = -9819. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 200 such that 13x - 60y = -9819. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "57, 176", "ref": "57, 176", "family": "po_lin2_box", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 13, "b": -60, "B": 200, "n": -9819}, "seed": 20666920803002325, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 176, "valid_answers": ["57, 176", "117, 189"]}, "cell": "lin2_box/3", "stage": "B"}
+{"id": "po-lin2_box-t3-eval-000921", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 200 such that 14x - 27y = -3357. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 200 such that 14x - 27y = -3357. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9, 129", "ref": "9, 129", "family": "po_lin2_box", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000149, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 14, "b": -27, "B": 200, "n": -3357}, "seed": 20666920803001019, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 129, "valid_answers": ["9, 129", "36, 143", "63, 157", "90, 171", "117, 185", "144, 199"]}, "cell": "lin2_box/3", "stage": "B"}
+{"id": "po-lin2_box-t3-eval-001029", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 200 such that 22x + 31y = 6978. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 200 such that 22x + 31y = 6978. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "41, 196", "ref": "41, 196", "family": "po_lin2_box", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000149, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 22, "b": 31, "B": 200, "n": 6978}, "seed": 20666920803001133, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 134, "valid_answers": ["41, 196", "72, 174", "103, 152", "134, 130", "165, 108", "196, 86"]}, "cell": "lin2_box/3", "stage": "B"}
+{"id": "po-lin2_box-t3-eval-000093", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 200 such that 72x + 30y = 18234. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 200 such that 72x + 30y = 18234. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "172, 195", "ref": "172, 195", "family": "po_lin2_box", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000149, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 72, "b": 30, "B": 200, "n": 18234}, "seed": 20666920803000106, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 182, "valid_answers": ["172, 195", "177, 183", "182, 171", "187, 159", "192, 147", "197, 135"]}, "cell": "lin2_box/3", "stage": "B"}
+{"id": "po-lin2_box-t3-eval-001553", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 200 such that 41x - 62y = 1687. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 200 such that 41x - 62y = 1687. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "85, 29", "ref": "85, 29", "family": "po_lin2_box", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 41, "b": -62, "B": 200, "n": 1687}, "seed": 20666920803001700, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 85, "valid_answers": ["85, 29", "147, 70"]}, "cell": "lin2_box/3", "stage": "B"}
+{"id": "po-lin2_box-t3-eval-001162", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 200 such that 45x + 39y = 7314. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 200 such that 45x + 39y = 7314. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 176", "ref": "10, 176", "family": "po_lin2_box", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.000297, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 45, "b": 39, "B": 200, "n": 7314}, "seed": 20666920803001277, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 88, "valid_answers": ["10, 176", "23, 161", "36, 146", "49, 131", "62, 116", "75, 101", "88, 86", "101, 71", "114, 56", "127, 41", "140, 26", "153, 11"]}, "cell": "lin2_box/3", "stage": "B"}
+{"id": "po-lin2_box-t3-eval-000981", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 200 such that 24x + 25y = 2052. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 200 such that 24x + 25y = 2052. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "23, 60", "ref": "23, 60", "family": "po_lin2_box", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 7.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 24, "b": 25, "B": 200, "n": 2052}, "seed": 20666920803001083, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 48, "valid_answers": ["23, 60", "48, 36", "73, 12"]}, "cell": "lin2_box/3", "stage": "B"}
+{"id": "po-lin2_box-t3-eval-001138", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 200 such that 25x - 42y = -3435. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 200 such that 25x - 42y = -3435. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "39, 105", "ref": "39, 105", "family": "po_lin2_box", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 9.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 25, "b": -42, "B": 200, "n": -3435}, "seed": 20666920803001252, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 105, "valid_answers": ["39, 105", "81, 130", "123, 155", "165, 180"]}, "cell": "lin2_box/3", "stage": "B"}
+{"id": "po-lin2_box-t3-eval-000110", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 200 such that 73x - 23y = 11961. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with 0 <= x, y <= 200 such that 73x - 23y = 11961. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "167, 10", "ref": "167, 10", "family": "po_lin2_box", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 73, "b": -23, "B": 200, "n": 11961}, "seed": 20666920803000125, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 167, "valid_answers": ["167, 10", "190, 83"]}, "cell": "lin2_box/3", "stage": "B"}
+{"id": "po-lin_commute-t0-eval-000166", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 6x + 6 and g(f(x)) = 6x - 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 6x + 6 and g(f(x)) = 6x - 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 3, -2, -1", "ref": "-3, 3, -2, -1", "family": "po_lin_commute", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 5, "A": 6, "P": 6, "Q": -7}, "seed": 423627283900000467, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["-3, 3, -2, -1", "2, -4, 3, 5"]}, "cell": "lin_commute/0", "stage": "B"}
+{"id": "po-lin_commute-t0-eval-000656", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 8x + 7 and g(f(x)) = 8x - 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 8x + 7 and g(f(x)) = 8x - 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 3, -2, -1", "ref": "-4, 3, -2, -1", "family": "po_lin_commute", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 5, "A": 8, "P": 7, "Q": -7}, "seed": 423627283900003597, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["-4, 3, -2, -1", "-2, 1, -4, -3", "2, -3, 4, 5", "4, -5, 2, 3"]}, "cell": "lin_commute/0", "stage": "B"}
+{"id": "po-lin_commute-t0-eval-000232", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -4x - 2 and g(f(x)) = -4x - 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -4x - 2 and g(f(x)) = -4x - 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -1, 4, 1", "ref": "-1, -1, 4, 1", "family": "po_lin_commute", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 5, "A": -4, "P": -2, "Q": -3}, "seed": 423627283900000719, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-1, -1, 4, 1", "4, 2, -1, -1"]}, "cell": "lin_commute/0", "stage": "B"}
+{"id": "po-lin_commute-t0-eval-000014", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -5x - 2 and g(f(x)) = -5x + 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -5x - 2 and g(f(x)) = -5x + 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 3, 1, 1", "ref": "-5, 3, 1, 1", "family": "po_lin_commute", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 5, "A": -5, "P": -2, "Q": 4}, "seed": 423627283900000026, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-5, 3, 1, 1", "1, -1, -5, -1"]}, "cell": "lin_commute/0", "stage": "B"}
+{"id": "po-lin_commute-t0-eval-000042", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -x + 6 and g(f(x)) = -x. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -x + 6 and g(f(x)) = -x. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 3, 1, -3", "ref": "-1, 3, 1, -3", "family": "po_lin_commute", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 5, "A": -1, "P": 6, "Q": 0}, "seed": 423627283900000076, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["-1, 3, 1, -3", "1, 3, -1, 3"]}, "cell": "lin_commute/0", "stage": "B"}
+{"id": "po-lin_commute-t0-eval-000827", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -3x + 7 and g(f(x)) = -3x - 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -3x + 7 and g(f(x)) = -3x - 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -2, 1, -3", "ref": "-3, -2, 1, -3", "family": "po_lin_commute", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 5, "A": -3, "P": 7, "Q": -5}, "seed": 423627283900006810, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["-3, -2, 1, -3", "1, 3, -3, 4"]}, "cell": "lin_commute/0", "stage": "B"}
+{"id": "po-lin_commute-t0-eval-000849", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 3x + 2 and g(f(x)) = 3x + 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 3x + 2 and g(f(x)) = 3x + 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -3, -3, -5", "ref": "-1, -3, -3, -5", "family": "po_lin_commute", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000205, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 5, "A": 3, "P": 2, "Q": 4}, "seed": 423627283900007511, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["-1, -3, -3, -5", "1, 1, 3, 1", "3, 5, 1, -1"]}, "cell": "lin_commute/0", "stage": "B"}
+{"id": "po-lin_commute-t0-eval-000199", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -3x + 3 and g(f(x)) = -3x - 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -3x + 3 and g(f(x)) = -3x - 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, 3, -3", "ref": "-1, 0, 3, -3", "family": "po_lin_commute", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 5, "A": -3, "P": 3, "Q": -3}, "seed": 423627283900000604, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["-1, 0, 3, -3", "3, 3, -1, 0"]}, "cell": "lin_commute/0", "stage": "B"}
+{"id": "po-lin_commute-t0-eval-000869", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 8x and g(f(x)) = 8x + 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 8x and g(f(x)) = 8x + 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -4, -2, -1", "ref": "-4, -4, -2, -1", "family": "po_lin_commute", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 5, "A": 8, "P": 0, "Q": 7}, "seed": 423627283900008226, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["-4, -4, -2, -1", "-2, -2, -4, -1", "2, 2, 4, -1", "4, 4, 2, -1"]}, "cell": "lin_commute/0", "stage": "B"}
+{"id": "po-lin_commute-t0-eval-000451", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -3x + 4 and g(f(x)) = -3x. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -3x + 4 and g(f(x)) = -3x. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 1, 1, -1", "ref": "-3, 1, 1, -1", "family": "po_lin_commute", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 5, "A": -3, "P": 4, "Q": 0}, "seed": 423627283900001810, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["-3, 1, 1, -1", "-1, 1, 3, -3", "1, 1, -3, 3", "3, 1, -1, 1"]}, "cell": "lin_commute/0", "stage": "B"}
+{"id": "po-lin_commute-t0-eval-000168", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 3x and g(f(x)) = 3x + 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 3x and g(f(x)) = 3x + 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -3, -1, -1", "ref": "-3, -3, -1, -1", "family": "po_lin_commute", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 5, "A": 3, "P": 0, "Q": 2}, "seed": 423627283900000470, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["-3, -3, -1, -1", "-1, -1, -3, -1", "1, 1, 3, -1", "3, 3, 1, -1"]}, "cell": "lin_commute/0", "stage": "B"}
+{"id": "po-lin_commute-t0-eval-000757", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 5x + 2 and g(f(x)) = 5x - 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 5x + 2 and g(f(x)) = 5x - 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 1, -5, -1", "ref": "-1, 1, -5, -1", "family": "po_lin_commute", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 5, "A": 5, "P": 2, "Q": -6}, "seed": 423627283900004985, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-1, 1, -5, -1", "1, -2, 5, 4"]}, "cell": "lin_commute/0", "stage": "B"}
+{"id": "po-lin_commute-t1-eval-001686", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -5x - 10 and g(f(x)) = -5x + 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -5x - 10 and g(f(x)) = -5x + 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 0, 1, 2", "ref": "-5, 0, 1, 2", "family": "po_lin_commute", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 8, "A": -5, "P": -10, "Q": 2}, "seed": 423627283901014538, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 5, "valid_answers": ["-5, 0, 1, 2", "1, -2, -5, -8"]}, "cell": "lin_commute/1", "stage": "B"}
+{"id": "po-lin_commute-t1-eval-000893", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -12x - 13 and g(f(x)) = -12x. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -12x - 13 and g(f(x)) = -12x. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -1, 2, 2", "ref": "-6, -1, 2, 2", "family": "po_lin_commute", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 9.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 8, "A": -12, "P": -13, "Q": 0}, "seed": 423627283901004271, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 4, "valid_answers": ["-6, -1, 2, 2", "-4, -1, 3, 3", "-3, -1, 4, 4", "-2, -1, 6, 6", "2, -1, -6, -6", "3, -1, -4, -4", "4, -1, -3, -3", "6, -1, -2, -2"]}, "cell": "lin_commute/1", "stage": "B"}
+{"id": "po-lin_commute-t1-eval-000551", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -x + 1 and g(f(x)) = -x + 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -x + 1 and g(f(x)) = -x + 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 2, 1, 1", "ref": "-1, 2, 1, 1", "family": "po_lin_commute", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 8, "A": -1, "P": 1, "Q": 3}, "seed": 423627283901002394, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 2, "valid_answers": ["-1, 2, 1, 1", "1, -1, -1, 2"]}, "cell": "lin_commute/1", "stage": "B"}
+{"id": "po-lin_commute-t1-eval-001008", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -7x - 3 and g(f(x)) = -7x + 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -7x - 3 and g(f(x)) = -7x + 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, 7, 3", "ref": "-1, 0, 7, 3", "family": "po_lin_commute", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 8, "A": -7, "P": -3, "Q": 3}, "seed": 423627283901005138, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 7, "valid_answers": ["-1, 0, 7, 3", "7, -3, -1, 0"]}, "cell": "lin_commute/1", "stage": "B"}
+{"id": "po-lin_commute-t1-eval-002003", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -8x + 4 and g(f(x)) = -8x + 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -8x + 4 and g(f(x)) = -8x + 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 1, 8, -3", "ref": "-1, 1, 8, -3", "family": "po_lin_commute", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 8, "A": -8, "P": 4, "Q": 5}, "seed": 423627283901029039, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 8, "valid_answers": ["-1, 1, 8, -3", "8, -4, -1, 1"]}, "cell": "lin_commute/1", "stage": "B"}
+{"id": "po-lin_commute-t1-eval-001984", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 3x - 7 and g(f(x)) = 3x + 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 3x - 7 and g(f(x)) = 3x + 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 2, -1, 3", "ref": "-3, 2, -1, 3", "family": "po_lin_commute", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 8, "A": 3, "P": -7, "Q": 1}, "seed": 423627283901027339, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 3, "valid_answers": ["-3, 2, -1, 3", "3, 5, 1, -4"]}, "cell": "lin_commute/1", "stage": "B"}
+{"id": "po-lin_commute-t1-eval-001986", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -7x - 1 and g(f(x)) = -7x + 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -7x - 1 and g(f(x)) = -7x + 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 1, 7, 2", "ref": "-1, 1, 7, 2", "family": "po_lin_commute", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 8, "A": -7, "P": -1, "Q": 9}, "seed": 423627283901027490, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 7, "valid_answers": ["-1, 1, 7, 2", "7, -8, -1, 1"]}, "cell": "lin_commute/1", "stage": "B"}
+{"id": "po-lin_commute-t1-eval-001261", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 25x + 12 and g(f(x)) = 25x - 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 25x + 12 and g(f(x)) = 25x - 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 2, -5, -2", "ref": "-5, 2, -5, -2", "family": "po_lin_commute", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 8, "A": 25, "P": 12, "Q": -12}, "seed": 423627283901007473, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 5, "valid_answers": ["-5, 2, -5, -2", "5, -3, 5, 3"]}, "cell": "lin_commute/1", "stage": "B"}
+{"id": "po-lin_commute-t1-eval-000330", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -2x + 1 and g(f(x)) = -2x + 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -2x + 1 and g(f(x)) = -2x + 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 3, 2, 2", "ref": "-1, 3, 2, 2", "family": "po_lin_commute", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 8, "A": -2, "P": 1, "Q": 8}, "seed": 423627283901001314, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 3, "valid_answers": ["-1, 3, 2, 2", "2, -5, -1, 3"]}, "cell": "lin_commute/1", "stage": "B"}
+{"id": "po-lin_commute-t1-eval-000189", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 9x and g(f(x)) = 9x. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 9x and g(f(x)) = 9x. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 0, -3, 0", "ref": "-3, 0, -3, 0", "family": "po_lin_commute", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 8, "A": 9, "P": 0, "Q": 0}, "seed": 423627283901000729, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 3, "valid_answers": ["-3, 0, -3, 0", "3, 0, 3, 0"]}, "cell": "lin_commute/1", "stage": "B"}
+{"id": "po-lin_commute-t1-eval-001565", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 40x + 3 and g(f(x)) = 40x + 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 40x + 3 and g(f(x)) = 40x + 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -5, -5, -1", "ref": "-8, -5, -5, -1", "family": "po_lin_commute", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 8, "A": 40, "P": 3, "Q": 24}, "seed": 423627283901012098, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 8, "valid_answers": ["-8, -5, -5, -1", "5, 3, 8, 0"]}, "cell": "lin_commute/1", "stage": "B"}
+{"id": "po-lin_commute-t1-eval-001804", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -24x - 5 and g(f(x)) = -24x + 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -24x - 5 and g(f(x)) = -24x + 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 3, 6, 2", "ref": "-4, 3, 6, 2", "family": "po_lin_commute", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 8, "A": -24, "P": -5, "Q": 20}, "seed": 423627283901018439, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 6, "valid_answers": ["-4, 3, 6, 2", "6, -5, -4, 0"]}, "cell": "lin_commute/1", "stage": "B"}
+{"id": "po-lin_commute-t2-eval-001208", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -8x - 4 and g(f(x)) = -8x + 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -8x - 4 and g(f(x)) = -8x + 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 12, 1, 2", "ref": "-8, 12, 1, 2", "family": "po_lin_commute", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 12, "A": -8, "P": -4, "Q": 14}, "seed": 423627283902007394, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 8, "valid_answers": ["-8, 12, 1, 2", "1, -2, -8, -2"]}, "cell": "lin_commute/2", "stage": "B"}
+{"id": "po-lin_commute-t2-eval-001630", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 9x + 3 and g(f(x)) = 9x + 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 9x + 3 and g(f(x)) = 9x + 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 0, 9, 3", "ref": "1, 0, 9, 3", "family": "po_lin_commute", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 12, "A": 9, "P": 3, "Q": 3}, "seed": 423627283902010293, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 9, "valid_answers": ["1, 0, 9, 3", "9, 3, 1, 0"]}, "cell": "lin_commute/2", "stage": "B"}
+{"id": "po-lin_commute-t2-eval-001671", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 9x - 3 and g(f(x)) = 9x - 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 9x - 3 and g(f(x)) = 9x - 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 6, -1, 1", "ref": "-9, 6, -1, 1", "family": "po_lin_commute", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 12, "A": 9, "P": -3, "Q": -5}, "seed": 423627283902010609, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 9, "valid_answers": ["-9, 6, -1, 1", "-1, 1, -9, 4"]}, "cell": "lin_commute/2", "stage": "B"}
+{"id": "po-lin_commute-t2-eval-002488", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -10x - 3 and g(f(x)) = -10x + 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -10x - 3 and g(f(x)) = -10x + 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 7, 1, 1", "ref": "-10, 7, 1, 1", "family": "po_lin_commute", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 12, "A": -10, "P": -3, "Q": 8}, "seed": 423627283902018408, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 10, "valid_answers": ["-10, 7, 1, 1", "1, -1, -10, -2"]}, "cell": "lin_commute/2", "stage": "B"}
+{"id": "po-lin_commute-t2-eval-000908", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 7x - 3 and g(f(x)) = 7x - 27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 7x - 3 and g(f(x)) = 7x - 27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 5, -7, 8", "ref": "-1, 5, -7, 8", "family": "po_lin_commute", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 12, "A": 7, "P": -3, "Q": -27}, "seed": 423627283902005209, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 7, "valid_answers": ["-1, 5, -7, 8", "1, -4, 7, 1"]}, "cell": "lin_commute/2", "stage": "B"}
+{"id": "po-lin_commute-t2-eval-001731", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -15x + 2 and g(f(x)) = -15x - 22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -15x + 2 and g(f(x)) = -15x - 22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -4, 5, -2", "ref": "-3, -4, 5, -2", "family": "po_lin_commute", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 12, "A": -15, "P": 2, "Q": -22}, "seed": 423627283902011155, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 5, "valid_answers": ["-3, -4, 5, -2", "5, 7, -3, -1"]}, "cell": "lin_commute/2", "stage": "B"}
+{"id": "po-lin_commute-t2-eval-000334", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -63x - 28 and g(f(x)) = -63x - 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -63x - 28 and g(f(x)) = -63x - 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -1, 7, 3", "ref": "-9, -1, 7, 3", "family": "po_lin_commute", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 12, "A": -63, "P": -28, "Q": -4}, "seed": 423627283902001779, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 9, "valid_answers": ["-9, -1, 7, 3", "7, 0, -9, -4"]}, "cell": "lin_commute/2", "stage": "B"}
+{"id": "po-lin_commute-t2-eval-001798", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 3x + 3 and g(f(x)) = 3x - 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 3x + 3 and g(f(x)) = 3x - 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 6, -1, 1", "ref": "-3, 6, -1, 1", "family": "po_lin_commute", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 12, "A": 3, "P": 3, "Q": -5}, "seed": 423627283902011683, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 3, "valid_answers": ["-3, 6, -1, 1", "-1, 1, -3, -2", "1, -4, 3, 7", "3, -9, 1, 4"]}, "cell": "lin_commute/2", "stage": "B"}
+{"id": "po-lin_commute-t2-eval-001898", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 6x - 2 and g(f(x)) = 6x + 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 6x - 2 and g(f(x)) = 6x + 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -5, -2, -1", "ref": "-3, -5, -2, -1", "family": "po_lin_commute", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 12, "A": 6, "P": -2, "Q": 9}, "seed": 423627283902012562, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 4, "valid_answers": ["-3, -5, -2, -1", "2, 4, 3, -3"]}, "cell": "lin_commute/2", "stage": "B"}
+{"id": "po-lin_commute-t2-eval-001655", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 21x + 6 and g(f(x)) = 21x - 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 21x + 6 and g(f(x)) = 21x - 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 6, -3, 0", "ref": "-7, 6, -3, 0", "family": "po_lin_commute", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 12, "A": 21, "P": 6, "Q": -18}, "seed": 423627283902010485, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 7, "valid_answers": ["-7, 6, -3, 0", "3, -3, 7, 3"]}, "cell": "lin_commute/2", "stage": "B"}
+{"id": "po-lin_commute-t2-eval-000801", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -4x - 5 and g(f(x)) = -4x - 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -4x - 5 and g(f(x)) = -4x - 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -5, 1, 0", "ref": "-4, -5, 1, 0", "family": "po_lin_commute", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 12, "A": -4, "P": -5, "Q": -5}, "seed": 423627283902004538, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 3, "valid_answers": ["-4, -5, 1, 0", "-2, -3, 2, 1", "-1, -2, 4, 3", "1, 0, -4, -5", "2, 1, -2, -3", "4, 3, -1, -2"]}, "cell": "lin_commute/2", "stage": "B"}
+{"id": "po-lin_commute-t2-eval-002112", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 21x + 14 and g(f(x)) = 21x - 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 21x + 14 and g(f(x)) = 21x - 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 0, -3, -2", "ref": "-7, 0, -3, -2", "family": "po_lin_commute", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 12, "A": 21, "P": 14, "Q": -2}, "seed": 423627283902014569, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 7, "valid_answers": ["-7, 0, -3, -2", "3, -1, 7, 5"]}, "cell": "lin_commute/2", "stage": "B"}
+{"id": "po-lin_commute-t3-eval-002231", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -12x + 3 and g(f(x)) = -12x - 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -12x + 3 and g(f(x)) = -12x - 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, -9, 1, -1", "ref": "-12, -9, 1, -1", "family": "po_lin_commute", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 20, "A": -12, "P": 3, "Q": -10}, "seed": 423627283903022248, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 12, "valid_answers": ["-12, -9, 1, -1", "1, 1, -12, 2"]}, "cell": "lin_commute/3", "stage": "B"}
+{"id": "po-lin_commute-t3-eval-000304", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 252x and g(f(x)) = 252x. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 252x and g(f(x)) = 252x. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-18, 0, -14, 0", "ref": "-18, 0, -14, 0", "family": "po_lin_commute", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 20, "A": 252, "P": 0, "Q": 0}, "seed": 423627283903002873, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 18, "valid_answers": ["-18, 0, -14, 0", "-14, 0, -18, 0", "14, 0, 18, 0", "18, 0, 14, 0"]}, "cell": "lin_commute/3", "stage": "B"}
+{"id": "po-lin_commute-t3-eval-002433", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 8x - 12 and g(f(x)) = 8x + 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 8x - 12 and g(f(x)) = 8x + 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 2, 8, -14", "ref": "1, 2, 8, -14", "family": "po_lin_commute", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 20, "A": 8, "P": -12, "Q": 2}, "seed": 423627283903024362, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 8, "valid_answers": ["1, 2, 8, -14", "8, 4, 1, -2"]}, "cell": "lin_commute/3", "stage": "B"}
+{"id": "po-lin_commute-t3-eval-001037", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 36x - 5 and g(f(x)) = 36x + 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 36x - 5 and g(f(x)) = 36x + 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-18, -5, -2, 0", "ref": "-18, -5, -2, 0", "family": "po_lin_commute", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 20, "A": 36, "P": -5, "Q": 10}, "seed": 423627283903009949, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 9, "valid_answers": ["-18, -5, -2, 0", "-4, -1, -9, 1", "3, 1, 12, -2"]}, "cell": "lin_commute/3", "stage": "B"}
+{"id": "po-lin_commute-t3-eval-002086", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 21x + 6 and g(f(x)) = 21x + 38. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 21x + 6 and g(f(x)) = 21x + 38. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -6, -7, -4", "ref": "-3, -6, -7, -4", "family": "po_lin_commute", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 20, "A": 21, "P": 6, "Q": 38}, "seed": 423627283903020504, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 7, "valid_answers": ["-3, -6, -7, -4", "7, 13, 3, -1"]}, "cell": "lin_commute/3", "stage": "B"}
+{"id": "po-lin_commute-t3-eval-000756", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 121x - 60 and g(f(x)) = 121x - 60. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 121x - 60 and g(f(x)) = 121x - 60. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-11, 6, -11, 6", "ref": "-11, 6, -11, 6", "family": "po_lin_commute", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 20, "A": 121, "P": -60, "Q": -60}, "seed": 423627283903007434, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 11, "valid_answers": ["-11, 6, -11, 6", "11, -5, 11, -5"]}, "cell": "lin_commute/3", "stage": "B"}
+{"id": "po-lin_commute-t3-eval-002451", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -6x + 11 and g(f(x)) = -6x + 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -6x + 11 and g(f(x)) = -6x + 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 5, 1, -1", "ref": "-6, 5, 1, -1", "family": "po_lin_commute", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 20, "A": -6, "P": 11, "Q": 4}, "seed": 423627283903024565, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 6, "valid_answers": ["-6, 5, 1, -1", "1, 1, -6, 10"]}, "cell": "lin_commute/3", "stage": "B"}
+{"id": "po-lin_commute-t3-eval-001921", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 2x and g(f(x)) = 2x + 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 2x and g(f(x)) = 2x + 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -4, -1, -2", "ref": "-2, -4, -1, -2", "family": "po_lin_commute", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 20, "A": 2, "P": 0, "Q": 2}, "seed": 423627283903018750, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 2, "valid_answers": ["-2, -4, -1, -2", "-1, -2, -2, -2", "1, 2, 2, -2", "2, 4, 1, -2"]}, "cell": "lin_commute/3", "stage": "B"}
+{"id": "po-lin_commute-t3-eval-002336", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 4x + 15 and g(f(x)) = 4x - 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 4x + 15 and g(f(x)) = 4x - 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 15, -1, 0", "ref": "-4, 15, -1, 0", "family": "po_lin_commute", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 20, "A": 4, "P": 15, "Q": -15}, "seed": 423627283903023281, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 5, "valid_answers": ["-4, 15, -1, 0", "-2, 5, -2, -5", "-1, 0, -4, -15", "2, -15, 2, 15"]}, "cell": "lin_commute/3", "stage": "B"}
+{"id": "po-lin_commute-t3-eval-001774", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 13x + 10 and g(f(x)) = 13x - 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 13x + 10 and g(f(x)) = 13x - 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -1, 13, 11", "ref": "1, -1, 13, 11", "family": "po_lin_commute", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 20, "A": 13, "P": 10, "Q": -2}, "seed": 423627283903017350, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 13, "valid_answers": ["1, -1, 13, 11", "13, -3, 1, 1"]}, "cell": "lin_commute/3", "stage": "B"}
+{"id": "po-lin_commute-t3-eval-002112", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 16x - 10 and g(f(x)) = 16x + 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = 16x - 10 and g(f(x)) = 16x + 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -2, -2, 1", "ref": "-8, -2, -2, 1", "family": "po_lin_commute", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 2e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 20, "A": 16, "P": -10, "Q": 5}, "seed": 423627283903020764, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 4, "valid_answers": ["-8, -2, -2, 1", "-2, 0, -8, 5", "1, 1, 16, -11", "4, 2, 4, -3", "16, 6, 1, -1"]}, "cell": "lin_commute/3", "stage": "B"}
+{"id": "po-lin_commute-t3-eval-000293", "task": "cx_po_lin_commute", "source": "cx_po_lin_commute", "split": "eval", "problem": "Find integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -19x - 6 and g(f(x)) = -19x - 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -20 <= a, b, c, d <= 20 such that f(x) = ax + b and g(x) = cx + d satisfy f(g(x)) = -19x - 6 and g(f(x)) = -19x - 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-19, -6, 1, 0", "ref": "-19, -6, 1, 0", "family": "po_lin_commute", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_lin_commute", "rung": "witness", "params": {"B": 20, "A": -19, "P": -6, "Q": -6}, "seed": 423627283903002779, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 19, "valid_answers": ["-19, -6, 1, 0", "1, 0, -19, -6"]}, "cell": "lin_commute/3", "stage": "B"}
+{"id": "po-lin_congruence-t0-eval-001347", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 470 such that 111x \\equiv 180 \\pmod{471}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 470 such that 111x \\equiv 180 \\pmod{471}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "78", "ref": "78", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006369, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 111, "b": 180, "m": 471, "M": 471}, "seed": 318040182800002172, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 78, "valid_answers": ["78", "235", "392"]}, "cell": "lin_congruence/0", "stage": "A"}
+{"id": "po-lin_congruence-t0-eval-002222", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 233 such that 80x \\equiv 32 \\pmod{234}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 233 such that 80x \\equiv 32 \\pmod{234}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "94", "ref": "94", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.008547, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 80, "b": 32, "m": 234, "M": 234}, "seed": 318040182800003583, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 94, "valid_answers": ["94", "211"]}, "cell": "lin_congruence/0", "stage": "A"}
+{"id": "po-lin_congruence-t0-eval-002036", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 623 such that 212x \\equiv 532 \\pmod{624}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 623 such that 212x \\equiv 532 \\pmod{624}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "29", "ref": "29", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00641, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 212, "b": 532, "m": 624, "M": 624}, "seed": 318040182800003291, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 29, "valid_answers": ["29", "185", "341", "497"]}, "cell": "lin_congruence/0", "stage": "A"}
+{"id": "po-lin_congruence-t0-eval-000485", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 411 such that 136x \\equiv 52 \\pmod{412}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 411 such that 136x \\equiv 52 \\pmod{412}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "64", "ref": "64", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.009709, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 136, "b": 52, "m": 412, "M": 412}, "seed": 318040182800000777, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 64, "valid_answers": ["64", "167", "270", "373"]}, "cell": "lin_congruence/0", "stage": "A"}
+{"id": "po-lin_congruence-t0-eval-000096", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 571 such that 564x \\equiv 416 \\pmod{572}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 571 such that 564x \\equiv 416 \\pmod{572}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "91", "ref": "91", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.006993, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 564, "b": 416, "m": 572, "M": 572}, "seed": 318040182800000156, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 91, "valid_answers": ["91", "234", "377", "520"]}, "cell": "lin_congruence/0", "stage": "A"}
+{"id": "po-lin_congruence-t0-eval-002028", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 431 such that 213x \\equiv 408 \\pmod{432}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 431 such that 213x \\equiv 408 \\pmod{432}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8", "ref": "8", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006944, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 213, "b": 408, "m": 432, "M": 432}, "seed": 318040182800003278, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 8, "valid_answers": ["8", "152", "296"]}, "cell": "lin_congruence/0", "stage": "A"}
+{"id": "po-lin_congruence-t0-eval-001004", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 293 such that 254x \\equiv 52 \\pmod{294}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 293 such that 254x \\equiv 52 \\pmod{294}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "131", "ref": "131", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.006803, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 254, "b": 52, "m": 294, "M": 294}, "seed": 318040182800001626, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 131, "valid_answers": ["131", "278"]}, "cell": "lin_congruence/0", "stage": "A"}
+{"id": "po-lin_congruence-t0-eval-001178", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 387 such that 46x \\equiv 56 \\pmod{388}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 387 such that 46x \\equiv 56 \\pmod{388}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "94", "ref": "94", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.005155, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 46, "b": 56, "m": 388, "M": 388}, "seed": 318040182800001892, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 94, "valid_answers": ["94", "288"]}, "cell": "lin_congruence/0", "stage": "A"}
+{"id": "po-lin_congruence-t0-eval-000952", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 699 such that 92x \\equiv 72 \\pmod{700}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 699 such that 92x \\equiv 72 \\pmod{700}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "16", "ref": "16", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005714, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 92, "b": 72, "m": 700, "M": 700}, "seed": 318040182800001551, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 16, "valid_answers": ["16", "191", "366", "541"]}, "cell": "lin_congruence/0", "stage": "A"}
+{"id": "po-lin_congruence-t0-eval-001945", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 135 such that 10x \\equiv 0 \\pmod{136}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 135 such that 10x \\equiv 0 \\pmod{136}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0", "ref": "0", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.014706, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 10, "b": 0, "m": 136, "M": 136}, "seed": 318040182800003139, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 0, "valid_answers": ["0", "68"]}, "cell": "lin_congruence/0", "stage": "A"}
+{"id": "po-lin_congruence-t0-eval-000487", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 596 such that 516x \\equiv 69 \\pmod{597}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 596 such that 516x \\equiv 69 \\pmod{597}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "36", "ref": "36", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.005025, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 516, "b": 69, "m": 597, "M": 597}, "seed": 318040182800000780, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 36, "valid_answers": ["36", "235", "434"]}, "cell": "lin_congruence/0", "stage": "A"}
+{"id": "po-lin_congruence-t0-eval-001202", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 145 such that 78x \\equiv 12 \\pmod{146}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 145 such that 78x \\equiv 12 \\pmod{146}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "17", "ref": "17", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.013699, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 78, "b": 12, "m": 146, "M": 146}, "seed": 318040182800001932, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 17, "valid_answers": ["17", "90"]}, "cell": "lin_congruence/0", "stage": "A"}
+{"id": "po-lin_congruence-t1-eval-000804", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 267 such that 200x \\equiv 244 \\pmod{268}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 267 such that 200x \\equiv 244 \\pmod{268}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "24", "ref": "24", "family": "po_lin_congruence", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.014925, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 200, "b": 244, "m": 268, "M": 268}, "seed": 318040182801001377, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 24, "valid_answers": ["24", "91", "158", "225"]}, "cell": "lin_congruence/1", "stage": "C"}
+{"id": "po-lin_congruence-t1-eval-001915", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 434 such that 141x \\equiv 24 \\pmod{435}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 434 such that 141x \\equiv 24 \\pmod{435}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "139", "ref": "139", "family": "po_lin_congruence", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006897, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 141, "b": 24, "m": 435, "M": 435}, "seed": 318040182801003246, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 139, "valid_answers": ["139", "284", "429"]}, "cell": "lin_congruence/1", "stage": "C"}
+{"id": "po-lin_congruence-t1-eval-001522", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 334 such that 45x \\equiv 210 \\pmod{335}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 334 such that 45x \\equiv 210 \\pmod{335}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "27", "ref": "27", "family": "po_lin_congruence", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.014925, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 45, "b": 210, "m": 335, "M": 335}, "seed": 318040182801002591, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 27, "valid_answers": ["27", "94", "161", "228", "295"]}, "cell": "lin_congruence/1", "stage": "C"}
+{"id": "po-lin_congruence-t1-eval-000146", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 678 such that 539x \\equiv 413 \\pmod{679}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 678 such that 539x \\equiv 413 \\pmod{679}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "31", "ref": "31", "family": "po_lin_congruence", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.010309, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 539, "b": 413, "m": 679, "M": 679}, "seed": 318040182801000246, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 31, "valid_answers": ["31", "128", "225", "322", "419", "516", "613"]}, "cell": "lin_congruence/1", "stage": "C"}
+{"id": "po-lin_congruence-t1-eval-002373", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 824 such that 395x \\equiv 15 \\pmod{825}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 824 such that 395x \\equiv 15 \\pmod{825}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "117", "ref": "117", "family": "po_lin_congruence", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.006061, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 395, "b": 15, "m": 825, "M": 825}, "seed": 318040182801003997, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 117, "valid_answers": ["117", "282", "447", "612", "777"]}, "cell": "lin_congruence/1", "stage": "C"}
+{"id": "po-lin_congruence-t1-eval-000074", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 1392 such that 294x \\equiv 217 \\pmod{1393}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 1392 such that 294x \\equiv 217 \\pmod{1393}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "195", "ref": "195", "family": "po_lin_congruence", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.005025, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 294, "b": 217, "m": 1393, "M": 1393}, "seed": 318040182801000119, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 195, "valid_answers": ["195", "394", "593", "792", "991", "1190", "1389"]}, "cell": "lin_congruence/1", "stage": "C"}
+{"id": "po-lin_congruence-t1-eval-002238", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 919 such that 655x \\equiv 905 \\pmod{920}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 919 such that 655x \\equiv 905 \\pmod{920}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7", "ref": "7", "family": "po_lin_congruence", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.005435, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 655, "b": 905, "m": 920, "M": 920}, "seed": 318040182801003777, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 7, "valid_answers": ["7", "191", "375", "559", "743"]}, "cell": "lin_congruence/1", "stage": "C"}
+{"id": "po-lin_congruence-t1-eval-002340", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 814 such that 650x \\equiv 50 \\pmod{815}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 814 such that 650x \\equiv 50 \\pmod{815}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "138", "ref": "138", "family": "po_lin_congruence", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.006135, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 650, "b": 50, "m": 815, "M": 815}, "seed": 318040182801003944, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 138, "valid_answers": ["138", "301", "464", "627", "790"]}, "cell": "lin_congruence/1", "stage": "C"}
+{"id": "po-lin_congruence-t1-eval-000070", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 1025 such that 480x \\equiv 360 \\pmod{1026}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 1025 such that 480x \\equiv 360 \\pmod{1026}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "129", "ref": "129", "family": "po_lin_congruence", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.005848, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 480, "b": 360, "m": 1026, "M": 1026}, "seed": 318040182801000109, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 129, "valid_answers": ["129", "300", "471", "642", "813", "984"]}, "cell": "lin_congruence/1", "stage": "C"}
+{"id": "po-lin_congruence-t1-eval-001146", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 643 such that 400x \\equiv 308 \\pmod{644}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 643 such that 400x \\equiv 308 \\pmod{644}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "70", "ref": "70", "family": "po_lin_congruence", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.006211, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 400, "b": 308, "m": 644, "M": 644}, "seed": 318040182801001937, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 70, "valid_answers": ["70", "231", "392", "553"]}, "cell": "lin_congruence/1", "stage": "C"}
+{"id": "po-lin_congruence-t1-eval-000567", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 1159 such that 616x \\equiv 832 \\pmod{1160}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 1159 such that 616x \\equiv 832 \\pmod{1160}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7", "ref": "7", "family": "po_lin_congruence", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.006897, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 616, "b": 832, "m": 1160, "M": 1160}, "seed": 318040182801000983, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 7, "valid_answers": ["7", "152", "297", "442", "587", "732", "877", "1022"]}, "cell": "lin_congruence/1", "stage": "C"}
+{"id": "po-lin_congruence-t1-eval-000044", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 443 such that 20x \\equiv 12 \\pmod{444}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 443 such that 20x \\equiv 12 \\pmod{444}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "45", "ref": "45", "family": "po_lin_congruence", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.009009, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 20, "b": 12, "m": 444, "M": 444}, "seed": 318040182801000073, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 45, "valid_answers": ["45", "156", "267", "378"]}, "cell": "lin_congruence/1", "stage": "C"}
+{"id": "po-lin_congruence-t2-eval-002485", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 1455 such that 559x \\equiv 260 \\pmod{1456}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 1455 such that 559x \\equiv 260 \\pmod{1456}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "76", "ref": "76", "family": "po_lin_congruence", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.008929, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 559, "b": 260, "m": 1456, "M": 1456}, "seed": 318040182802004204, "version": "po-v0", "tier": 2, "n_raw": 13, "small": 76, "valid_answers": ["76", "188", "300", "412", "524", "636", "748", "860", "972", "1084", "1196", "1308", "1420"]}, "cell": "lin_congruence/2", "stage": "C"}
+{"id": "po-lin_congruence-t2-eval-001415", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 3151 such that 2640x \\equiv 3008 \\pmod{3152}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 3151 such that 2640x \\equiv 3008 \\pmod{3152}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "68", "ref": "68", "family": "po_lin_congruence", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.005076, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 2640, "b": 3008, "m": 3152, "M": 3152}, "seed": 318040182802002409, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 68, "valid_answers": ["68", "265", "462", "659", "856", "1053", "1250", "1447", "1644", "1841", "2038", "2235", "2432", "2629", "2826", "3023"]}, "cell": "lin_congruence/2", "stage": "C"}
+{"id": "po-lin_congruence-t2-eval-001800", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 2918 such that 399x \\equiv 7 \\pmod{973}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 2918 such that 399x \\equiv 7 \\pmod{973}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "100", "ref": "100", "family": "po_lin_congruence", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.007194, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 399, "b": 7, "m": 973, "M": 2919}, "seed": 318040182802003015, "version": "po-v0", "tier": 2, "n_raw": 21, "small": 100, "valid_answers": ["100", "239", "378", "517", "656", "795", "934", "1073", "1212", "1351", "1490", "1629", "1768", "1907", "2046", "2185", "2324", "2463", "2602", "2741", "2880"]}, "cell": "lin_congruence/2", "stage": "C"}
+{"id": "po-lin_congruence-t2-eval-001545", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 8303 such that 132x \\equiv 1176 \\pmod{2076}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 8303 such that 132x \\equiv 1176 \\pmod{2076}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "119", "ref": "119", "family": "po_lin_congruence", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 48, "bucket": null, "guess_rate": 0.00578, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 132, "b": 1176, "m": 2076, "M": 8304}, "seed": 318040182802002609, "version": "po-v0", "tier": 2, "n_raw": 48, "small": 119, "valid_answers": ["119", "292", "465", "638", "811", "984", "1157", "1330", "1503", "1676", "1849", "2022", "2195", "2368", "2541", "2714", "2887", "3060", "3233", "3406", "3579", "3752", "3925", "4098", "4271", "4444", "4617", "4790", "4963", "5136", "5309", "5482", "5655", "5828", "6001", "6174", "6347", "6520", "6693", "6866", "7039", "7212", "7385", "7558", "7731", "7904", "8077", "8250"]}, "cell": "lin_congruence/2", "stage": "C"}
+{"id": "po-lin_congruence-t2-eval-001546", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 2133 such that 253x \\equiv 55 \\pmod{2134}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 2133 such that 253x \\equiv 55 \\pmod{2134}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "93", "ref": "93", "family": "po_lin_congruence", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.005155, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 253, "b": 55, "m": 2134, "M": 2134}, "seed": 318040182802002611, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 93, "valid_answers": ["93", "287", "481", "675", "869", "1063", "1257", "1451", "1645", "1839", "2033"]}, "cell": "lin_congruence/2", "stage": "C"}
+{"id": "po-lin_congruence-t2-eval-002315", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 496 such that 140x \\equiv 252 \\pmod{497}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 496 such that 140x \\equiv 252 \\pmod{497}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "16", "ref": "16", "family": "po_lin_congruence", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.014085, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 140, "b": 252, "m": 497, "M": 497}, "seed": 318040182802003913, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 16, "valid_answers": ["16", "87", "158", "229", "300", "371", "442"]}, "cell": "lin_congruence/2", "stage": "C"}
+{"id": "po-lin_congruence-t2-eval-000616", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 4916 such that 176x \\equiv 1353 \\pmod{1639}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 4916 such that 176x \\equiv 1353 \\pmod{1639}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "17", "ref": "17", "family": "po_lin_congruence", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 33, "bucket": null, "guess_rate": 0.006711, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 176, "b": 1353, "m": 1639, "M": 4917}, "seed": 318040182802001085, "version": "po-v0", "tier": 2, "n_raw": 33, "small": 17, "valid_answers": ["17", "166", "315", "464", "613", "762", "911", "1060", "1209", "1358", "1507", "1656", "1805", "1954", "2103", "2252", "2401", "2550", "2699", "2848", "2997", "3146", "3295", "3444", "3593", "3742", "3891", "4040", "4189", "4338", "4487", "4636", "4785"]}, "cell": "lin_congruence/2", "stage": "C"}
+{"id": "po-lin_congruence-t2-eval-000656", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 10259 such that 1875x \\equiv 345 \\pmod{2565}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 10259 such that 1875x \\equiv 345 \\pmod{2565}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "85", "ref": "85", "family": "po_lin_congruence", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 60, "bucket": null, "guess_rate": 0.005848, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 1875, "b": 345, "m": 2565, "M": 10260}, "seed": 318040182802001148, "version": "po-v0", "tier": 2, "n_raw": 60, "small": 85, "valid_answers": ["85", "256", "427", "598", "769", "940", "1111", "1282", "1453", "1624", "1795", "1966", "2137", "2308", "2479", "2650", "2821", "2992", "3163", "3334", "3505", "3676", "3847", "4018", "4189", "4360", "4531", "4702", "4873", "5044", "5215", "5386", "5557", "5728", "5899", "6070", "6241", "6412", "6583", "6754", "6925", "7096", "7267", "7438", "7609", "7780", "7951", "8122", "8293", "8464", "8635", "8806", "8977", "9148", "9319", "9490", "9661", "9832", "10003", "10174"]}, "cell": "lin_congruence/2", "stage": "C"}
+{"id": "po-lin_congruence-t2-eval-000276", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 4543 such that 2096x \\equiv 704 \\pmod{2272}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 4543 such that 2096x \\equiv 704 \\pmod{2272}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "138", "ref": "138", "family": "po_lin_congruence", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 32, "bucket": null, "guess_rate": 0.007042, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 2096, "b": 704, "m": 2272, "M": 4544}, "seed": 318040182802000466, "version": "po-v0", "tier": 2, "n_raw": 32, "small": 138, "valid_answers": ["138", "280", "422", "564", "706", "848", "990", "1132", "1274", "1416", "1558", "1700", "1842", "1984", "2126", "2268", "2410", "2552", "2694", "2836", "2978", "3120", "3262", "3404", "3546", "3688", "3830", "3972", "4114", "4256", "4398", "4540"]}, "cell": "lin_congruence/2", "stage": "C"}
+{"id": "po-lin_congruence-t2-eval-002069", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 993 such that 63x \\equiv 245 \\pmod{497}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 993 such that 63x \\equiv 245 \\pmod{497}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "67", "ref": "67", "family": "po_lin_congruence", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.014085, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 63, "b": 245, "m": 497, "M": 994}, "seed": 318040182802003489, "version": "po-v0", "tier": 2, "n_raw": 14, "small": 67, "valid_answers": ["67", "138", "209", "280", "351", "422", "493", "564", "635", "706", "777", "848", "919", "990"]}, "cell": "lin_congruence/2", "stage": "C"}
+{"id": "po-lin_congruence-t2-eval-002162", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 4931 such that 873x \\equiv 0 \\pmod{1233}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 4931 such that 873x \\equiv 0 \\pmod{1233}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0", "ref": "0", "family": "po_lin_congruence", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 36, "bucket": null, "guess_rate": 0.007299, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 873, "b": 0, "m": 1233, "M": 4932}, "seed": 318040182802003640, "version": "po-v0", "tier": 2, "n_raw": 36, "small": 0, "valid_answers": ["0", "137", "274", "411", "548", "685", "822", "959", "1096", "1233", "1370", "1507", "1644", "1781", "1918", "2055", "2192", "2329", "2466", "2603", "2740", "2877", "3014", "3151", "3288", "3425", "3562", "3699", "3836", "3973", "4110", "4247", "4384", "4521", "4658", "4795"]}, "cell": "lin_congruence/2", "stage": "C"}
+{"id": "po-lin_congruence-t2-eval-002343", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 2327 such that 426x \\equiv 1050 \\pmod{1164}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 2327 such that 426x \\equiv 1050 \\pmod{1164}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "191", "ref": "191", "family": "po_lin_congruence", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.005155, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 426, "b": 1050, "m": 1164, "M": 2328}, "seed": 318040182802003962, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 191, "valid_answers": ["191", "385", "579", "773", "967", "1161", "1355", "1549", "1743", "1937", "2131", "2325"]}, "cell": "lin_congruence/2", "stage": "C"}
+{"id": "po-lin_congruence-t3-eval-002421", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 4825 such that 3496x \\equiv 722 \\pmod{4826}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 4825 such that 3496x \\equiv 722 \\pmod{4826}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "43", "ref": "43", "family": "po_lin_congruence", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 38, "bucket": null, "guess_rate": 0.007874, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 3496, "b": 722, "m": 4826, "M": 4826}, "seed": 318040182803008906, "version": "po-v0", "tier": 3, "n_raw": 38, "small": 43, "valid_answers": ["43", "170", "297", "424", "551", "678", "805", "932", "1059", "1186", "1313", "1440", "1567", "1694", "1821", "1948", "2075", "2202", "2329", "2456", "2583", "2710", "2837", "2964", "3091", "3218", "3345", "3472", "3599", "3726", "3853", "3980", "4107", "4234", "4361", "4488", "4615", "4742"]}, "cell": "lin_congruence/3", "stage": "C"}
+{"id": "po-lin_congruence-t3-eval-002045", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 2911 such that 533x \\equiv 1209 \\pmod{1456}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 2911 such that 533x \\equiv 1209 \\pmod{1456}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5", "ref": "5", "family": "po_lin_congruence", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 26, "bucket": null, "guess_rate": 0.008929, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 533, "b": 1209, "m": 1456, "M": 2912}, "seed": 318040182803007480, "version": "po-v0", "tier": 3, "n_raw": 26, "small": 5, "valid_answers": ["5", "117", "229", "341", "453", "565", "677", "789", "901", "1013", "1125", "1237", "1349", "1461", "1573", "1685", "1797", "1909", "2021", "2133", "2245", "2357", "2469", "2581", "2693", "2805"]}, "cell": "lin_congruence/3", "stage": "C"}
+{"id": "po-lin_congruence-t3-eval-002449", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 6697 such that 4522x \\equiv 510 \\pmod{6698}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 6697 such that 4522x \\equiv 510 \\pmod{6698}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9", "ref": "9", "family": "po_lin_congruence", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 34, "bucket": null, "guess_rate": 0.005076, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 4522, "b": 510, "m": 6698, "M": 6698}, "seed": 318040182803009003, "version": "po-v0", "tier": 3, "n_raw": 34, "small": 9, "valid_answers": ["9", "206", "403", "600", "797", "994", "1191", "1388", "1585", "1782", "1979", "2176", "2373", "2570", "2767", "2964", "3161", "3358", "3555", "3752", "3949", "4146", "4343", "4540", "4737", "4934", "5131", "5328", "5525", "5722", "5919", "6116", "6313", "6510"]}, "cell": "lin_congruence/3", "stage": "C"}
+{"id": "po-lin_congruence-t3-eval-001489", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 6668 such that 5226x \\equiv 3627 \\pmod{6669}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 6668 such that 5226x \\equiv 3627 \\pmod{6669}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "150", "ref": "150", "family": "po_lin_congruence", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 39, "bucket": null, "guess_rate": 0.005848, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 5226, "b": 3627, "m": 6669, "M": 6669}, "seed": 318040182803005494, "version": "po-v0", "tier": 3, "n_raw": 39, "small": 150, "valid_answers": ["150", "321", "492", "663", "834", "1005", "1176", "1347", "1518", "1689", "1860", "2031", "2202", "2373", "2544", "2715", "2886", "3057", "3228", "3399", "3570", "3741", "3912", "4083", "4254", "4425", "4596", "4767", "4938", "5109", "5280", "5451", "5622", "5793", "5964", "6135", "6306", "6477", "6648"]}, "cell": "lin_congruence/3", "stage": "C"}
+{"id": "po-lin_congruence-t3-eval-000569", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 6143 such that 1080x \\equiv 1680 \\pmod{3072}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 6143 such that 1080x \\equiv 1680 \\pmod{3072}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "30", "ref": "30", "family": "po_lin_congruence", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 48, "bucket": null, "guess_rate": 0.007812, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 1080, "b": 1680, "m": 3072, "M": 6144}, "seed": 318040182803002159, "version": "po-v0", "tier": 3, "n_raw": 48, "small": 30, "valid_answers": ["30", "158", "286", "414", "542", "670", "798", "926", "1054", "1182", "1310", "1438", "1566", "1694", "1822", "1950", "2078", "2206", "2334", "2462", "2590", "2718", "2846", "2974", "3102", "3230", "3358", "3486", "3614", "3742", "3870", "3998", "4126", "4254", "4382", "4510", "4638", "4766", "4894", "5022", "5150", "5278", "5406", "5534", "5662", "5790", "5918", "6046"]}, "cell": "lin_congruence/3", "stage": "C"}
+{"id": "po-lin_congruence-t3-eval-000748", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 7963 such that 3212x \\equiv 3190 \\pmod{3982}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 7963 such that 3212x \\equiv 3190 \\pmod{3982}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "151", "ref": "151", "family": "po_lin_congruence", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 44, "bucket": null, "guess_rate": 0.005525, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 3212, "b": 3190, "m": 3982, "M": 7964}, "seed": 318040182803002834, "version": "po-v0", "tier": 3, "n_raw": 44, "small": 151, "valid_answers": ["151", "332", "513", "694", "875", "1056", "1237", "1418", "1599", "1780", "1961", "2142", "2323", "2504", "2685", "2866", "3047", "3228", "3409", "3590", "3771", "3952", "4133", "4314", "4495", "4676", "4857", "5038", "5219", "5400", "5581", "5762", "5943", "6124", "6305", "6486", "6667", "6848", "7029", "7210", "7391", "7572", "7753", "7934"]}, "cell": "lin_congruence/3", "stage": "C"}
+{"id": "po-lin_congruence-t3-eval-002078", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 4103 such that 3768x \\equiv 3504 \\pmod{4104}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 4103 such that 3768x \\equiv 3504 \\pmod{4104}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "14", "ref": "14", "family": "po_lin_congruence", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 24, "bucket": null, "guess_rate": 0.005848, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 3768, "b": 3504, "m": 4104, "M": 4104}, "seed": 318040182803007578, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 14, "valid_answers": ["14", "185", "356", "527", "698", "869", "1040", "1211", "1382", "1553", "1724", "1895", "2066", "2237", "2408", "2579", "2750", "2921", "3092", "3263", "3434", "3605", "3776", "3947"]}, "cell": "lin_congruence/3", "stage": "C"}
+{"id": "po-lin_congruence-t3-eval-002052", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 7499 such that 1260x \\equiv 1800 \\pmod{1875}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 7499 such that 1260x \\equiv 1800 \\pmod{1875}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "55", "ref": "55", "family": "po_lin_congruence", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 60, "bucket": null, "guess_rate": 0.008, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 1260, "b": 1800, "m": 1875, "M": 7500}, "seed": 318040182803007500, "version": "po-v0", "tier": 3, "n_raw": 60, "small": 55, "valid_answers": ["55", "180", "305", "430", "555", "680", "805", "930", "1055", "1180", "1305", "1430", "1555", "1680", "1805", "1930", "2055", "2180", "2305", "2430", "2555", "2680", "2805", "2930", "3055", "3180", "3305", "3430", "3555", "3680", "3805", "3930", "4055", "4180", "4305", "4430", "4555", "4680", "4805", "4930", "5055", "5180", "5305", "5430", "5555", "5680", "5805", "5930", "6055", "6180", "6305", "6430", "6555", "6680", "6805", "6930", "7055", "7180", "7305", "7430"]}, "cell": "lin_congruence/3", "stage": "C"}
+{"id": "po-lin_congruence-t3-eval-000458", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 3795 such that 2158x \\equiv 1846 \\pmod{3796}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 3795 such that 2158x \\equiv 1846 \\pmod{3796}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "29", "ref": "29", "family": "po_lin_congruence", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 26, "bucket": null, "guess_rate": 0.006849, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 2158, "b": 1846, "m": 3796, "M": 3796}, "seed": 318040182803001700, "version": "po-v0", "tier": 3, "n_raw": 26, "small": 29, "valid_answers": ["29", "175", "321", "467", "613", "759", "905", "1051", "1197", "1343", "1489", "1635", "1781", "1927", "2073", "2219", "2365", "2511", "2657", "2803", "2949", "3095", "3241", "3387", "3533", "3679"]}, "cell": "lin_congruence/3", "stage": "C"}
+{"id": "po-lin_congruence-t3-eval-000531", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 5579 such that 1836x \\equiv 1104 \\pmod{1860}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 5579 such that 1836x \\equiv 1104 \\pmod{1860}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "109", "ref": "109", "family": "po_lin_congruence", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 36, "bucket": null, "guess_rate": 0.006452, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 1836, "b": 1104, "m": 1860, "M": 5580}, "seed": 318040182803002024, "version": "po-v0", "tier": 3, "n_raw": 36, "small": 109, "valid_answers": ["109", "264", "419", "574", "729", "884", "1039", "1194", "1349", "1504", "1659", "1814", "1969", "2124", "2279", "2434", "2589", "2744", "2899", "3054", "3209", "3364", "3519", "3674", "3829", "3984", "4139", "4294", "4449", "4604", "4759", "4914", "5069", "5224", "5379", "5534"]}, "cell": "lin_congruence/3", "stage": "C"}
+{"id": "po-lin_congruence-t3-eval-002027", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 6863 such that 3250x \\equiv 1794 \\pmod{3432}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 6863 such that 3250x \\equiv 1794 \\pmod{3432}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9", "ref": "9", "family": "po_lin_congruence", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 52, "bucket": null, "guess_rate": 0.007576, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 3250, "b": 1794, "m": 3432, "M": 6864}, "seed": 318040182803007420, "version": "po-v0", "tier": 3, "n_raw": 52, "small": 9, "valid_answers": ["9", "141", "273", "405", "537", "669", "801", "933", "1065", "1197", "1329", "1461", "1593", "1725", "1857", "1989", "2121", "2253", "2385", "2517", "2649", "2781", "2913", "3045", "3177", "3309", "3441", "3573", "3705", "3837", "3969", "4101", "4233", "4365", "4497", "4629", "4761", "4893", "5025", "5157", "5289", "5421", "5553", "5685", "5817", "5949", "6081", "6213", "6345", "6477", "6609", "6741"]}, "cell": "lin_congruence/3", "stage": "C"}
+{"id": "po-lin_congruence-t3-eval-000356", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 6500 such that 1155x \\equiv 3828 \\pmod{6501}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 6500 such that 1155x \\equiv 3828 \\pmod{6501}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "99", "ref": "99", "family": "po_lin_congruence", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 33, "bucket": null, "guess_rate": 0.005076, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 1155, "b": 3828, "m": 6501, "M": 6501}, "seed": 318040182803001365, "version": "po-v0", "tier": 3, "n_raw": 33, "small": 99, "valid_answers": ["99", "296", "493", "690", "887", "1084", "1281", "1478", "1675", "1872", "2069", "2266", "2463", "2660", "2857", "3054", "3251", "3448", "3645", "3842", "4039", "4236", "4433", "4630", "4827", "5024", "5221", "5418", "5615", "5812", "6009", "6206", "6403"]}, "cell": "lin_congruence/3", "stage": "C"}
+{"id": "po-magic3-t0-eval-000254", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & -5 \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between -7 and 4, keeping the given entries, so that all rows, columns and both diagonals have sum -3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & -5 \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between -7 and 4, keeping the given entries, so that all rows, columns and both diagonals have sum -3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-5, 3, -1, 3, -1, -5, -1, -5, 3", "ref": "-5, 3, -1, 3, -1, -5, -1, -5, 3", "family": "po_magic3", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": -3, "lo": -7, "hi": 4, "fix": [[5, -5]], "distinct": false}, "seed": 339239530900000426, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 5, "valid_answers": ["-5, 3, -1, 3, -1, -5, -1, -5, 3", "-4, 1, 0, 3, -1, -5, -2, -3, 2", "-3, -1, 1, 3, -1, -5, -3, -1, 1", "-2, -3, 2, 3, -1, -5, -4, 1, 0", "-1, -5, 3, 3, -1, -5, -5, 3, -1"]}, "cell": "magic3/0", "stage": "C"}
+{"id": "po-magic3-t0-eval-001044", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & 6 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between 4 and 14, keeping the given entries, so that all rows, columns and both diagonals have sum 24. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & 6 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between 4 and 14, keeping the given entries, so that all rows, columns and both diagonals have sum 24. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "7, 6, 11, 12, 8, 4, 5, 10, 9", "ref": "7, 6, 11, 12, 8, 4, 5, 10, 9", "family": "po_magic3", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 24, "lo": 4, "hi": 14, "fix": [[1, 6]], "distinct": false}, "seed": 339239530900001936, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 10, "valid_answers": ["7, 6, 11, 12, 8, 4, 5, 10, 9", "8, 6, 10, 10, 8, 6, 6, 10, 8", "9, 6, 9, 8, 8, 8, 7, 10, 7", "10, 6, 8, 6, 8, 10, 8, 10, 6", "11, 6, 7, 4, 8, 12, 9, 10, 5"]}, "cell": "magic3/0", "stage": "C"}
+{"id": "po-magic3-t0-eval-001924", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} 7 & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between 2 and 11, keeping the given entries, so that all rows, columns and both diagonals have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} 7 & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between 2 and 11, keeping the given entries, so that all rows, columns and both diagonals have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "7, 2, 9, 8, 6, 4, 3, 10, 5", "ref": "7, 2, 9, 8, 6, 4, 3, 10, 5", "family": "po_magic3", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 18, "lo": 2, "hi": 11, "fix": [[0, 7]], "distinct": false}, "seed": 339239530900003694, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 7, "valid_answers": ["7, 2, 9, 8, 6, 4, 3, 10, 5", "7, 3, 8, 7, 6, 5, 4, 9, 5", "7, 4, 7, 6, 6, 6, 5, 8, 5", "7, 5, 6, 5, 6, 7, 6, 7, 5", "7, 6, 5, 4, 6, 8, 7, 6, 5", "7, 7, 4, 3, 6, 9, 8, 5, 5", "7, 8, 3, 2, 6, 10, 9, 4, 5"]}, "cell": "magic3/0", "stage": "C"}
+{"id": "po-magic3-t0-eval-000390", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ 10 & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between 3 and 15, keeping the given entries, so that all rows, columns and both diagonals have sum 24. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ 10 & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between 3 and 15, keeping the given entries, so that all rows, columns and both diagonals have sum 24. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "5, 12, 7, 10, 8, 6, 9, 4, 11", "ref": "5, 12, 7, 10, 8, 6, 9, 4, 11", "family": "po_magic3", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 24, "lo": 3, "hi": 15, "fix": [[3, 10]], "distinct": false}, "seed": 339239530900000681, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 10, "valid_answers": ["5, 12, 7, 10, 8, 6, 9, 4, 11", "6, 10, 8, 10, 8, 6, 8, 6, 10", "7, 8, 9, 10, 8, 6, 7, 8, 9", "8, 6, 10, 10, 8, 6, 6, 10, 8", "9, 4, 11, 10, 8, 6, 5, 12, 7"]}, "cell": "magic3/0", "stage": "C"}
+{"id": "po-magic3-t0-eval-001823", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} -1 & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between -4 and 5, keeping the given entries, so that all rows, columns and both diagonals have sum -3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} -1 & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between -4 and 5, keeping the given entries, so that all rows, columns and both diagonals have sum -3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-1, -4, 2, 2, -1, -4, -4, 2, -1", "ref": "-1, -4, 2, 2, -1, -4, -4, 2, -1", "family": "po_magic3", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": -3, "lo": -4, "hi": 5, "fix": [[0, -1]], "distinct": false}, "seed": 339239530900003479, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 1, "valid_answers": ["-1, -4, 2, 2, -1, -4, -4, 2, -1", "-1, -3, 1, 1, -1, -3, -3, 1, -1", "-1, -2, 0, 0, -1, -2, -2, 0, -1", "-1, -1, -1, -1, -1, -1, -1, -1, -1", "-1, 0, -2, -2, -1, 0, 0, -2, -1", "-1, 1, -3, -3, -1, 1, 1, -3, -1", "-1, 2, -4, -4, -1, 2, 2, -4, -1"]}, "cell": "magic3/0", "stage": "C"}
+{"id": "po-magic3-t0-eval-002373", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & 9 & \\square \\end{pmatrix} with integers between 1 and 11, keeping the given entries, so that all rows, columns and both diagonals have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & 9 & \\square \\end{pmatrix} with integers between 1 and 11, keeping the given entries, so that all rows, columns and both diagonals have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "5, 3, 10, 11, 6, 1, 2, 9, 7", "ref": "5, 3, 10, 11, 6, 1, 2, 9, 7", "family": "po_magic3", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 18, "lo": 1, "hi": 11, "fix": [[7, 9]], "distinct": false}, "seed": 339239530900004668, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 9, "valid_answers": ["5, 3, 10, 11, 6, 1, 2, 9, 7", "6, 3, 9, 9, 6, 3, 3, 9, 6", "7, 3, 8, 7, 6, 5, 4, 9, 5", "8, 3, 7, 5, 6, 7, 5, 9, 4", "9, 3, 6, 3, 6, 9, 6, 9, 3", "10, 3, 5, 1, 6, 11, 7, 9, 2"]}, "cell": "magic3/0", "stage": "C"}
+{"id": "po-magic3-t0-eval-000061", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & 8 \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between 4 and 13, keeping the given entries, so that all rows, columns and both diagonals have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & 8 \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between 4 and 13, keeping the given entries, so that all rows, columns and both diagonals have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "6, 8, 4, 4, 6, 8, 8, 4, 6", "ref": "6, 8, 4, 4, 6, 8, 8, 4, 6", "family": "po_magic3", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 18, "lo": 4, "hi": 13, "fix": [[5, 8]], "distinct": false}, "seed": 339239530900000113, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 8, "valid_answers": ["6, 8, 4, 4, 6, 8, 8, 4, 6", "7, 6, 5, 4, 6, 8, 7, 6, 5", "8, 4, 6, 4, 6, 8, 6, 8, 4"]}, "cell": "magic3/0", "stage": "C"}
+{"id": "po-magic3-t0-eval-001958", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & 1 \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between -7 and 3, keeping the given entries, so that all rows, columns and both diagonals have sum -6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & 1 \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between -7 and 3, keeping the given entries, so that all rows, columns and both diagonals have sum -6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-4, -3, 1, 3, -2, -7, -5, -1, 0", "ref": "-4, -3, 1, 3, -2, -7, -5, -1, 0", "family": "po_magic3", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": -6, "lo": -7, "hi": 3, "fix": [[2, 1]], "distinct": false}, "seed": 339239530900003750, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 5, "valid_answers": ["-4, -3, 1, 3, -2, -7, -5, -1, 0", "-3, -4, 1, 2, -2, -6, -5, 0, -1", "-2, -5, 1, 1, -2, -5, -5, 1, -2", "-1, -6, 1, 0, -2, -4, -5, 2, -3", "0, -7, 1, -1, -2, -3, -5, 3, -4"]}, "cell": "magic3/0", "stage": "C"}
+{"id": "po-magic3-t0-eval-001122", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & 1 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between -3 and 9, keeping the given entries, so that all rows, columns and both diagonals have sum 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & 1 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between -3 and 9, keeping the given entries, so that all rows, columns and both diagonals have sum 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "0, 1, 5, 7, 2, -3, -1, 3, 4", "ref": "0, 1, 5, 7, 2, -3, -1, 3, 4", "family": "po_magic3", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 6, "lo": -3, "hi": 9, "fix": [[1, 1]], "distinct": false}, "seed": 339239530900002091, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 3, "valid_answers": ["0, 1, 5, 7, 2, -3, -1, 3, 4", "1, 1, 4, 5, 2, -1, 0, 3, 3", "2, 1, 3, 3, 2, 1, 1, 3, 2", "3, 1, 2, 1, 2, 3, 2, 3, 1", "4, 1, 1, -1, 2, 5, 3, 3, 0", "5, 1, 0, -3, 2, 7, 4, 3, -1"]}, "cell": "magic3/0", "stage": "C"}
+{"id": "po-magic3-t0-eval-001077", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ 7 & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between 2 and 14, keeping the given entries, so that all rows, columns and both diagonals have sum 15. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ 7 & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between 2 and 14, keeping the given entries, so that all rows, columns and both diagonals have sum 15. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "3, 7, 5, 7, 5, 3, 5, 3, 7", "ref": "3, 7, 5, 7, 5, 3, 5, 3, 7", "family": "po_magic3", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 15, "lo": 2, "hi": 14, "fix": [[3, 7]], "distinct": false}, "seed": 339239530900002000, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 7, "valid_answers": ["3, 7, 5, 7, 5, 3, 5, 3, 7", "4, 5, 6, 7, 5, 3, 4, 5, 6", "5, 3, 7, 7, 5, 3, 3, 7, 5"]}, "cell": "magic3/0", "stage": "C"}
+{"id": "po-magic3-t0-eval-001683", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ 5 & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between 1 and 10, keeping the given entries, so that all rows, columns and both diagonals have sum 12. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ 5 & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with integers between 1 and 10, keeping the given entries, so that all rows, columns and both diagonals have sum 12. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "2, 7, 3, 5, 4, 3, 5, 1, 6", "ref": "2, 7, 3, 5, 4, 3, 5, 1, 6", "family": "po_magic3", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 12, "lo": 1, "hi": 10, "fix": [[3, 5]], "distinct": false}, "seed": 339239530900003193, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 5, "valid_answers": ["2, 7, 3, 5, 4, 3, 5, 1, 6", "3, 5, 4, 5, 4, 3, 4, 3, 5", "4, 3, 5, 5, 4, 3, 3, 5, 4", "5, 1, 6, 5, 4, 3, 2, 7, 3"]}, "cell": "magic3/0", "stage": "C"}
+{"id": "po-magic3-t0-eval-001045", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & -1 & \\square \\end{pmatrix} with integers between -7 and 4, keeping the given entries, so that all rows, columns and both diagonals have sum -3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & -1 & \\square \\end{pmatrix} with integers between -7 and 4, keeping the given entries, so that all rows, columns and both diagonals have sum -3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-3, -1, 1, 3, -1, -5, -3, -1, 1", "ref": "-3, -1, 1, 3, -1, -5, -3, -1, 1", "family": "po_magic3", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": -3, "lo": -7, "hi": 4, "fix": [[7, -1]], "distinct": false}, "seed": 339239530900001937, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 1, "valid_answers": ["-3, -1, 1, 3, -1, -5, -3, -1, 1", "-2, -1, 0, 1, -1, -3, -2, -1, 0", "-1, -1, -1, -1, -1, -1, -1, -1, -1", "0, -1, -2, -3, -1, 1, 0, -1, -2", "1, -1, -3, -5, -1, 3, 1, -1, -3"]}, "cell": "magic3/0", "stage": "C"}
+{"id": "po-magic3-t1-eval-000099", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & -4 \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -5 and 9, keeping the given entries, so that all rows, columns and both diagonals have sum 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & -4 \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -5 and 9, keeping the given entries, so that all rows, columns and both diagonals have sum 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-3, 4, 2, 6, 1, -4, 0, -2, 5", "ref": "-3, 4, 2, 6, 1, -4, 0, -2, 5", "family": "po_magic3", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 3, "lo": -5, "hi": 9, "fix": [[5, -4]], "distinct": true}, "seed": 339239530901000172, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 6, "valid_answers": ["-3, 4, 2, 6, 1, -4, 0, -2, 5", "-2, 2, 3, 6, 1, -4, -1, 0, 4", "-1, 0, 4, 6, 1, -4, -2, 2, 3", "0, -2, 5, 6, 1, -4, -3, 4, 2"]}, "cell": "magic3/1", "stage": "C"}
+{"id": "po-magic3-t1-eval-001268", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & 3 \\end{pmatrix} with distinct integers between -2 and 13, keeping the given entries, so that all rows, columns and both diagonals have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & 3 \\end{pmatrix} with distinct integers between -2 and 13, keeping the given entries, so that all rows, columns and both diagonals have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "9, -1, 10, 7, 6, 5, 2, 13, 3", "ref": "9, -1, 10, 7, 6, 5, 2, 13, 3", "family": "po_magic3", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 18, "lo": -2, "hi": 13, "fix": [[8, 3]], "distinct": true}, "seed": 339239530901002395, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 10, "valid_answers": ["9, -1, 10, 7, 6, 5, 2, 13, 3", "9, 1, 8, 5, 6, 7, 4, 11, 3", "9, 2, 7, 4, 6, 8, 5, 10, 3", "9, 4, 5, 2, 6, 10, 7, 8, 3", "9, 5, 4, 1, 6, 11, 8, 7, 3", "9, 7, 2, -1, 6, 13, 10, 5, 3"]}, "cell": "magic3/1", "stage": "C"}
+{"id": "po-magic3-t1-eval-002397", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & -3 & \\square \\end{pmatrix} with distinct integers between -5 and 10, keeping the given entries, so that all rows, columns and both diagonals have sum 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & -3 & \\square \\end{pmatrix} with distinct integers between -5 and 10, keeping the given entries, so that all rows, columns and both diagonals have sum 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-4, 7, 3, 9, 2, -5, 1, -3, 8", "ref": "-4, 7, 3, 9, 2, -5, 1, -3, 8", "family": "po_magic3", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 6, "lo": -5, "hi": 10, "fix": [[7, -3]], "distinct": true}, "seed": 339239530901004697, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 7, "valid_answers": ["-4, 7, 3, 9, 2, -5, 1, -3, 8", "-2, 7, 1, 5, 2, -1, 3, -3, 6", "-1, 7, 0, 3, 2, 1, 4, -3, 5", "0, 7, -1, 1, 2, 3, 5, -3, 4", "1, 7, -2, -1, 2, 5, 6, -3, 3", "3, 7, -4, -5, 2, 9, 8, -3, 1"]}, "cell": "magic3/1", "stage": "C"}
+{"id": "po-magic3-t1-eval-002220", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} -2 & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -6 and 6, keeping the given entries, so that all rows, columns and both diagonals have sum 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} -2 & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -6 and 6, keeping the given entries, so that all rows, columns and both diagonals have sum 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-2, 2, 3, 6, 1, -4, -1, 0, 4", "ref": "-2, 2, 3, 6, 1, -4, -1, 0, 4", "family": "po_magic3", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 3, "lo": -6, "hi": 6, "fix": [[0, -2]], "distinct": true}, "seed": 339239530901004337, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 5, "valid_answers": ["-2, 2, 3, 6, 1, -4, -1, 0, 4", "-2, 3, 2, 5, 1, -3, 0, -1, 4", "-2, 5, 0, 3, 1, -1, 2, -3, 4", "-2, 6, -1, 2, 1, 0, 3, -4, 4"]}, "cell": "magic3/1", "stage": "C"}
+{"id": "po-magic3-t1-eval-002499", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & 4 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between 2 and 14, keeping the given entries, so that all rows, columns and both diagonals have sum 21. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & 4 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between 2 and 14, keeping the given entries, so that all rows, columns and both diagonals have sum 21. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "6, 4, 11, 12, 7, 2, 3, 10, 8", "ref": "6, 4, 11, 12, 7, 2, 3, 10, 8", "family": "po_magic3", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 21, "lo": 2, "hi": 14, "fix": [[1, 4]], "distinct": true}, "seed": 339239530901004951, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 12, "valid_answers": ["6, 4, 11, 12, 7, 2, 3, 10, 8", "11, 4, 6, 2, 7, 12, 8, 10, 3"]}, "cell": "magic3/1", "stage": "C"}
+{"id": "po-magic3-t1-eval-000838", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & 6 \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between 2 and 16, keeping the given entries, so that all rows, columns and both diagonals have sum 24. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & 6 \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between 2 and 16, keeping the given entries, so that all rows, columns and both diagonals have sum 24. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "5, 13, 6, 9, 8, 7, 10, 3, 11", "ref": "5, 13, 6, 9, 8, 7, 10, 3, 11", "family": "po_magic3", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 24, "lo": 2, "hi": 16, "fix": [[2, 6]], "distinct": true}, "seed": 339239530901001535, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 13, "valid_answers": ["5, 13, 6, 9, 8, 7, 10, 3, 11", "11, 7, 6, 3, 8, 13, 10, 9, 5"]}, "cell": "magic3/1", "stage": "C"}
+{"id": "po-magic3-t1-eval-001863", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & 8 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -1 and 12, keeping the given entries, so that all rows, columns and both diagonals have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & 8 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -1 and 12, keeping the given entries, so that all rows, columns and both diagonals have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "3, 8, 7, 10, 6, 2, 5, 4, 9", "ref": "3, 8, 7, 10, 6, 2, 5, 4, 9", "family": "po_magic3", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 18, "lo": -1, "hi": 12, "fix": [[1, 8]], "distinct": true}, "seed": 339239530901003590, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 10, "valid_answers": ["3, 8, 7, 10, 6, 2, 5, 4, 9", "7, 8, 3, 2, 6, 10, 9, 4, 5"]}, "cell": "magic3/1", "stage": "C"}
+{"id": "po-magic3-t1-eval-000671", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & -6 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -8 and 7, keeping the given entries, so that all rows, columns and both diagonals have sum -3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & -6 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -8 and 7, keeping the given entries, so that all rows, columns and both diagonals have sum -3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-2, -6, 5, 6, -1, -8, -7, 4, 0", "ref": "-2, -6, 5, 6, -1, -8, -7, 4, 0", "family": "po_magic3", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": -3, "lo": -8, "hi": 7, "fix": [[1, -6]], "distinct": true}, "seed": 339239530901001219, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 6, "valid_answers": ["-2, -6, 5, 6, -1, -8, -7, 4, 0", "0, -6, 3, 2, -1, -4, -5, 4, -2", "1, -6, 2, 0, -1, -2, -4, 4, -3", "2, -6, 1, -2, -1, 0, -3, 4, -4", "3, -6, 0, -4, -1, 2, -2, 4, -5", "5, -6, -2, -8, -1, 6, 0, 4, -7"]}, "cell": "magic3/1", "stage": "C"}
+{"id": "po-magic3-t1-eval-000226", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & 4 & \\square \\end{pmatrix} with distinct integers between -6 and 9, keeping the given entries, so that all rows, columns and both diagonals have sum 0. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & 4 & \\square \\end{pmatrix} with distinct integers between -6 and 9, keeping the given entries, so that all rows, columns and both diagonals have sum 0. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-1, -4, 5, 6, 0, -6, -5, 4, 1", "ref": "-1, -4, 5, 6, 0, -6, -5, 4, 1", "family": "po_magic3", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 0, "lo": -6, "hi": 9, "fix": [[7, 4]], "distinct": true}, "seed": 339239530901000398, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 4, "valid_answers": ["-1, -4, 5, 6, 0, -6, -5, 4, 1", "1, -4, 3, 2, 0, -2, -3, 4, -1", "3, -4, 1, -2, 0, 2, -1, 4, -3", "5, -4, -1, -6, 0, 6, 1, 4, -5"]}, "cell": "magic3/1", "stage": "C"}
+{"id": "po-magic3-t1-eval-000902", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} 0 & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -6 and 7, keeping the given entries, so that all rows, columns and both diagonals have sum -3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} 0 & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -6 and 7, keeping the given entries, so that all rows, columns and both diagonals have sum -3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "0, -6, 3, 2, -1, -4, -5, 4, -2", "ref": "0, -6, 3, 2, -1, -4, -5, 4, -2", "family": "po_magic3", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": -3, "lo": -6, "hi": 7, "fix": [[0, 0]], "distinct": true}, "seed": 339239530901001651, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 5, "valid_answers": ["0, -6, 3, 2, -1, -4, -5, 4, -2", "0, -5, 2, 1, -1, -3, -4, 3, -2", "0, 1, -4, -5, -1, 3, 2, -3, -2", "0, 2, -5, -6, -1, 4, 3, -4, -2"]}, "cell": "magic3/1", "stage": "C"}
+{"id": "po-magic3-t1-eval-000191", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} 5 & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -1 and 13, keeping the given entries, so that all rows, columns and both diagonals have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} 5 & \\square & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -1 and 13, keeping the given entries, so that all rows, columns and both diagonals have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "5, 1, 12, 13, 6, -1, 0, 11, 7", "ref": "5, 1, 12, 13, 6, -1, 0, 11, 7", "family": "po_magic3", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 18, "lo": -1, "hi": 13, "fix": [[0, 5]], "distinct": true}, "seed": 339239530901000338, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 10, "valid_answers": ["5, 1, 12, 13, 6, -1, 0, 11, 7", "5, 2, 11, 12, 6, 0, 1, 10, 7", "5, 3, 10, 11, 6, 1, 2, 9, 7", "5, 4, 9, 10, 6, 2, 3, 8, 7", "5, 10, 3, 4, 6, 8, 9, 2, 7", "5, 11, 2, 3, 6, 9, 10, 1, 7", "5, 12, 1, 2, 6, 10, 11, 0, 7", "5, 13, 0, 1, 6, 11, 12, -1, 7"]}, "cell": "magic3/1", "stage": "C"}
+{"id": "po-magic3-t1-eval-000347", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & 3 \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -7 and 7, keeping the given entries, so that all rows, columns and both diagonals have sum 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & 3 \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -7 and 7, keeping the given entries, so that all rows, columns and both diagonals have sum 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-2, 2, 3, 6, 1, -4, -1, 0, 4", "ref": "-2, 2, 3, 6, 1, -4, -1, 0, 4", "family": "po_magic3", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "magic", "S": 3, "lo": -7, "hi": 7, "fix": [[2, 3]], "distinct": true}, "seed": 339239530901000625, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 6, "valid_answers": ["-2, 2, 3, 6, 1, -4, -1, 0, 4", "4, -4, 3, 0, 1, 2, -1, 6, -2"]}, "cell": "magic3/1", "stage": "C"}
+{"id": "po-magic3-t2-eval-000862", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} 13 & \\square & \\square \\\\ \\square & \\square & 9 \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between 2 and 13, keeping the given entries, so that all rows and columns have sum 24. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} 13 & \\square & \\square \\\\ \\square & \\square & 9 \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between 2 and 13, keeping the given entries, so that all rows and columns have sum 24. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "13, 3, 8, 5, 10, 9, 6, 11, 7", "ref": "13, 3, 8, 5, 10, 9, 6, 11, 7", "family": "po_magic3", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 24, "lo": 2, "hi": 13, "fix": [[0, 13], [5, 9]], "distinct": true}, "seed": 339239530902003812, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 13, "valid_answers": ["13, 3, 8, 5, 10, 9, 6, 11, 7", "13, 6, 5, 8, 7, 9, 3, 11, 10", "13, 7, 4, 3, 12, 9, 8, 5, 11", "13, 8, 3, 4, 11, 9, 7, 5, 12"]}, "cell": "magic3/2", "stage": "C"}
+{"id": "po-magic3-t2-eval-001778", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & 5 \\\\ \\square & \\square & \\square \\\\ \\square & 7 & \\square \\end{pmatrix} with distinct integers between -1 and 10, keeping the given entries, so that all rows and columns have sum 9. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & 5 \\\\ \\square & \\square & \\square \\\\ \\square & 7 & \\square \\end{pmatrix} with distinct integers between -1 and 10, keeping the given entries, so that all rows and columns have sum 9. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "1, 3, 5, 6, -1, 4, 2, 7, 0", "ref": "1, 3, 5, 6, -1, 4, 2, 7, 0", "family": "po_magic3", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 9, "lo": -1, "hi": 10, "fix": [[2, 5], [7, 7]], "distinct": true}, "seed": 339239530902007924, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 7, "valid_answers": ["1, 3, 5, 6, -1, 4, 2, 7, 0", "4, 0, 5, 6, 2, 1, -1, 7, 3"]}, "cell": "magic3/2", "stage": "C"}
+{"id": "po-magic3-t2-eval-000065", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & \\square \\\\ -1 & -3 & \\square \\end{pmatrix} with distinct integers between -7 and 3, keeping the given entries, so that all rows and columns have sum -3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & \\square \\\\ -1 & -3 & \\square \\end{pmatrix} with distinct integers between -7 and 3, keeping the given entries, so that all rows and columns have sum -3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-5, 2, 0, 3, -2, -4, -1, -3, 1", "ref": "-5, 2, 0, 3, -2, -4, -1, -3, 1", "family": "po_magic3", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": -3, "lo": -7, "hi": 3, "fix": [[6, -1], [7, -3]], "distinct": true}, "seed": 339239530902000255, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 5, "valid_answers": ["-5, 2, 0, 3, -2, -4, -1, -3, 1", "3, -2, -4, -5, 2, 0, -1, -3, 1"]}, "cell": "magic3/2", "stage": "C"}
+{"id": "po-magic3-t2-eval-000813", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & 4 \\\\ \\square & \\square & \\square \\\\ \\square & 0 & \\square \\end{pmatrix} with distinct integers between -2 and 9, keeping the given entries, so that all rows and columns have sum 9. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & 4 \\\\ \\square & \\square & \\square \\\\ \\square & 0 & \\square \\end{pmatrix} with distinct integers between -2 and 9, keeping the given entries, so that all rows and columns have sum 9. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-2, 7, 4, 8, 2, -1, 3, 0, 6", "ref": "-2, 7, 4, 8, 2, -1, 3, 0, 6", "family": "po_magic3", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 9, "lo": -2, "hi": 9, "fix": [[2, 4], [7, 0]], "distinct": true}, "seed": 339239530902003603, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 8, "valid_answers": ["-2, 7, 4, 8, 2, -1, 3, 0, 6", "-1, 6, 4, 8, 3, -2, 2, 0, 7"]}, "cell": "magic3/2", "stage": "C"}
+{"id": "po-magic3-t2-eval-000819", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & 8 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & 6 & \\square \\end{pmatrix} with distinct integers between 2 and 13, keeping the given entries, so that all rows and columns have sum 21. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & 8 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & 6 & \\square \\end{pmatrix} with distinct integers between 2 and 13, keeping the given entries, so that all rows and columns have sum 21. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "3, 8, 10, 5, 7, 9, 13, 6, 2", "ref": "3, 8, 10, 5, 7, 9, 13, 6, 2", "family": "po_magic3", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 21, "lo": 2, "hi": 13, "fix": [[1, 8], [7, 6]], "distinct": true}, "seed": 339239530902003625, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 12, "valid_answers": ["3, 8, 10, 5, 7, 9, 13, 6, 2", "4, 8, 9, 12, 7, 2, 5, 6, 10", "9, 8, 4, 2, 7, 12, 10, 6, 5", "10, 8, 3, 9, 7, 5, 2, 6, 13"]}, "cell": "magic3/2", "stage": "C"}
+{"id": "po-magic3-t2-eval-000424", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & 4 & \\square \\\\ \\square & \\square & \\square \\\\ 3 & \\square & \\square \\end{pmatrix} with distinct integers between 0 and 9, keeping the given entries, so that all rows and columns have sum 12. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & 4 & \\square \\\\ \\square & \\square & \\square \\\\ 3 & \\square & \\square \\end{pmatrix} with distinct integers between 0 and 9, keeping the given entries, so that all rows and columns have sum 12. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "2, 4, 6, 7, 0, 5, 3, 8, 1", "ref": "2, 4, 6, 7, 0, 5, 3, 8, 1", "family": "po_magic3", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 12, "lo": 0, "hi": 9, "fix": [[1, 4], [6, 3]], "distinct": true}, "seed": 339239530902001809, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 8, "valid_answers": ["2, 4, 6, 7, 0, 5, 3, 8, 1", "8, 4, 0, 1, 6, 5, 3, 2, 7"]}, "cell": "magic3/2", "stage": "C"}
+{"id": "po-magic3-t2-eval-000020", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} 9 & 0 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -1 and 10, keeping the given entries, so that all rows and columns have sum 15. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} 9 & 0 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -1 and 10, keeping the given entries, so that all rows and columns have sum 15. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "9, 0, 6, 2, 5, 8, 4, 10, 1", "ref": "9, 0, 6, 2, 5, 8, 4, 10, 1", "family": "po_magic3", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 15, "lo": -1, "hi": 10, "fix": [[0, 9], [1, 0]], "distinct": true}, "seed": 339239530902000072, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 10, "valid_answers": ["9, 0, 6, 2, 5, 8, 4, 10, 1", "9, 0, 6, 4, 10, 1, 2, 5, 8"]}, "cell": "magic3/2", "stage": "C"}
+{"id": "po-magic3-t2-eval-002147", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} -1 & \\square & \\square \\\\ \\square & \\square & -2 \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -3 and 7, keeping the given entries, so that all rows and columns have sum 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} -1 & \\square & \\square \\\\ \\square & \\square & -2 \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -3 and 7, keeping the given entries, so that all rows and columns have sum 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-1, 1, 6, 3, 5, -2, 4, 0, 2", "ref": "-1, 1, 6, 3, 5, -2, 4, 0, 2", "family": "po_magic3", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 6, "lo": -3, "hi": 7, "fix": [[0, -1], [5, -2]], "distinct": true}, "seed": 339239530902009602, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 6, "valid_answers": ["-1, 1, 6, 3, 5, -2, 4, 0, 2", "-1, 2, 5, 1, 7, -2, 6, -3, 3", "-1, 4, 3, 6, 2, -2, 1, 0, 5", "-1, 6, 1, 5, 3, -2, 2, -3, 7"]}, "cell": "magic3/2", "stage": "C"}
+{"id": "po-magic3-t2-eval-000555", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ -4 & -1 & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -6 and 4, keeping the given entries, so that all rows and columns have sum -3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ -4 & -1 & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -6 and 4, keeping the given entries, so that all rows and columns have sum -3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-2, 4, -5, -4, -1, 2, 3, -6, 0", "ref": "-2, 4, -5, -4, -1, 2, 3, -6, 0", "family": "po_magic3", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": -3, "lo": -6, "hi": 4, "fix": [[3, -4], [4, -1]], "distinct": true}, "seed": 339239530902002387, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 6, "valid_answers": ["-2, 4, -5, -4, -1, 2, 3, -6, 0", "3, -6, 0, -4, -1, 2, -2, 4, -5"]}, "cell": "magic3/2", "stage": "C"}
+{"id": "po-magic3-t2-eval-002130", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & 7 & \\square \\\\ 8 & \\square & \\square \\end{pmatrix} with distinct integers between 3 and 14, keeping the given entries, so that all rows and columns have sum 21. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & 7 & \\square \\\\ 8 & \\square & \\square \\end{pmatrix} with distinct integers between 3 and 14, keeping the given entries, so that all rows and columns have sum 21. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "4, 11, 6, 9, 7, 5, 8, 3, 10", "ref": "4, 11, 6, 9, 7, 5, 8, 3, 10", "family": "po_magic3", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 21, "lo": 3, "hi": 14, "fix": [[4, 7], [6, 8]], "distinct": true}, "seed": 339239530902009514, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 11, "valid_answers": ["4, 11, 6, 9, 7, 5, 8, 3, 10", "10, 5, 6, 3, 7, 11, 8, 9, 4"]}, "cell": "magic3/2", "stage": "C"}
+{"id": "po-magic3-t2-eval-000883", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & -3 \\\\ 0 & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -4 and 4, keeping the given entries, so that all rows and columns have sum 0. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & -3 \\\\ 0 & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -4 and 4, keeping the given entries, so that all rows and columns have sum 0. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "2, 1, -3, 0, -4, 4, -2, 3, -1", "ref": "2, 1, -3, 0, -4, 4, -2, 3, -1", "family": "po_magic3", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 0, "lo": -4, "hi": 4, "fix": [[2, -3], [3, 0]], "distinct": true}, "seed": 339239530902003894, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 4, "valid_answers": ["2, 1, -3, 0, -4, 4, -2, 3, -1", "4, -1, -3, 0, -2, 2, -4, 3, 1"]}, "cell": "magic3/2", "stage": "C"}
+{"id": "po-magic3-t2-eval-000142", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & \\square \\\\ 0 & \\square & -3 \\end{pmatrix} with distinct integers between -5 and 5, keeping the given entries, so that all rows and columns have sum 0. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & \\square & \\square \\\\ 0 & \\square & -3 \\end{pmatrix} with distinct integers between -5 and 5, keeping the given entries, so that all rows and columns have sum 0. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-5, 1, 4, 5, -4, -1, 0, 3, -3", "ref": "-5, 1, 4, 5, -4, -1, 0, 3, -3", "family": "po_magic3", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 0, "lo": -5, "hi": 5, "fix": [[6, 0], [8, -3]], "distinct": true}, "seed": 339239530902000546, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 5, "valid_answers": ["-5, 1, 4, 5, -4, -1, 0, 3, -3", "5, -4, -1, -5, 1, 4, 0, 3, -3"]}, "cell": "magic3/2", "stage": "C"}
+{"id": "po-magic3-t3-eval-001569", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} 3 & \\square & \\square \\\\ \\square & \\square & \\square \\\\ 5 & \\square & \\square \\end{pmatrix} with distinct integers between 0 and 13, keeping the given entries, so that all rows and columns have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} 3 & \\square & \\square \\\\ \\square & \\square & \\square \\\\ 5 & \\square & \\square \\end{pmatrix} with distinct integers between 0 and 13, keeping the given entries, so that all rows and columns have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "3, 2, 13, 10, 7, 1, 5, 9, 4", "ref": "3, 2, 13, 10, 7, 1, 5, 9, 4", "family": "po_magic3", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 18, "lo": 0, "hi": 13, "fix": [[0, 3], [6, 5]], "distinct": true}, "seed": 339239530903003327, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 10, "valid_answers": ["3, 2, 13, 10, 7, 1, 5, 9, 4", "3, 4, 11, 10, 1, 7, 5, 13, 0", "3, 4, 11, 10, 2, 6, 5, 12, 1", "3, 4, 11, 10, 8, 0, 5, 6, 7", "3, 6, 9, 10, 0, 8, 5, 12, 1", "3, 6, 9, 10, 1, 7, 5, 11, 2", "3, 7, 8, 10, 2, 6, 5, 9, 4", "3, 8, 7, 10, 6, 2, 5, 4, 9", "3, 9, 6, 10, 7, 1, 5, 2, 11", "3, 9, 6, 10, 8, 0, 5, 1, 12", "3, 11, 4, 10, 0, 8, 5, 7, 6", "3, 11, 4, 10, 6, 2, 5, 1, 12", "3, 11, 4, 10, 7, 1, 5, 0, 13", "3, 13, 2, 10, 1, 7, 5, 4, 9"]}, "cell": "magic3/3", "stage": "C"}
+{"id": "po-magic3-t3-eval-002351", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ 3 & \\square & \\square \\\\ \\square & 2 & \\square \\end{pmatrix} with distinct integers between -1 and 12, keeping the given entries, so that all rows and columns have sum 9. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ 3 & \\square & \\square \\\\ \\square & 2 & \\square \\end{pmatrix} with distinct integers between -1 and 12, keeping the given entries, so that all rows and columns have sum 9. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-1, 6, 4, 3, 1, 5, 7, 2, 0", "ref": "-1, 6, 4, 3, 1, 5, 7, 2, 0", "family": "po_magic3", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 9, "lo": -1, "hi": 12, "fix": [[3, 3], [7, 2]], "distinct": true}, "seed": 339239530903005048, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 7, "valid_answers": ["-1, 6, 4, 3, 1, 5, 7, 2, 0", "5, 0, 4, 3, 7, -1, 1, 2, 6"]}, "cell": "magic3/3", "stage": "C"}
+{"id": "po-magic3-t3-eval-000359", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & 7 & \\square \\\\ \\square & 5 & \\square \\end{pmatrix} with distinct integers between 3 and 16, keeping the given entries, so that all rows and columns have sum 21. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & 7 & \\square \\\\ \\square & 5 & \\square \\end{pmatrix} with distinct integers between 3 and 16, keeping the given entries, so that all rows and columns have sum 21. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "4, 9, 8, 11, 7, 3, 6, 5, 10", "ref": "4, 9, 8, 11, 7, 3, 6, 5, 10", "family": "po_magic3", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 21, "lo": 3, "hi": 16, "fix": [[4, 7], [7, 5]], "distinct": true}, "seed": 339239530903000813, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 11, "valid_answers": ["4, 9, 8, 11, 7, 3, 6, 5, 10", "8, 9, 4, 3, 7, 11, 10, 5, 6"]}, "cell": "magic3/3", "stage": "C"}
+{"id": "po-magic3-t3-eval-000511", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & 4 & \\square \\\\ \\square & 8 & \\square \\end{pmatrix} with distinct integers between 0 and 11, keeping the given entries, so that all rows and columns have sum 12. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & 4 & \\square \\\\ \\square & 8 & \\square \\end{pmatrix} with distinct integers between 0 and 11, keeping the given entries, so that all rows and columns have sum 12. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "5, 0, 7, 6, 4, 2, 1, 8, 3", "ref": "5, 0, 7, 6, 4, 2, 1, 8, 3", "family": "po_magic3", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 12, "lo": 0, "hi": 11, "fix": [[4, 4], [7, 8]], "distinct": true}, "seed": 339239530903001086, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 8, "valid_answers": ["5, 0, 7, 6, 4, 2, 1, 8, 3", "7, 0, 5, 2, 4, 6, 3, 8, 1"]}, "cell": "magic3/3", "stage": "C"}
+{"id": "po-magic3-t3-eval-000004", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} 7 & \\square & \\square \\\\ -3 & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -3 and 9, keeping the given entries, so that all rows and columns have sum 9. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} 7 & \\square & \\square \\\\ -3 & \\square & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -3 and 9, keeping the given entries, so that all rows and columns have sum 9. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "7, -1, 3, -3, 4, 8, 5, 6, -2", "ref": "7, -1, 3, -3, 4, 8, 5, 6, -2", "family": "po_magic3", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 9, "lo": -3, "hi": 9, "fix": [[0, 7], [3, -3]], "distinct": true}, "seed": 339239530903000012, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 8, "valid_answers": ["7, -1, 3, -3, 4, 8, 5, 6, -2", "7, 0, 2, -3, 3, 9, 5, 6, -2", "7, 0, 2, -3, 8, 4, 5, 1, 3", "7, 2, 0, -3, 4, 8, 5, 3, 1", "7, 2, 0, -3, 9, 3, 5, -2, 6", "7, 3, -1, -3, 8, 4, 5, -2, 6"]}, "cell": "magic3/3", "stage": "C"}
+{"id": "po-magic3-t3-eval-000506", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & 5 \\\\ \\square & \\square & \\square \\\\ \\square & 2 & \\square \\end{pmatrix} with distinct integers between 2 and 14, keeping the given entries, so that all rows and columns have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & 5 \\\\ \\square & \\square & \\square \\\\ \\square & 2 & \\square \\end{pmatrix} with distinct integers between 2 and 14, keeping the given entries, so that all rows and columns have sum 18. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "3, 10, 5, 8, 6, 4, 7, 2, 9", "ref": "3, 10, 5, 8, 6, 4, 7, 2, 9", "family": "po_magic3", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 18, "lo": 2, "hi": 14, "fix": [[2, 5], [7, 2]], "distinct": true}, "seed": 339239530903001074, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 10, "valid_answers": ["3, 10, 5, 8, 6, 4, 7, 2, 9", "4, 9, 5, 8, 7, 3, 6, 2, 10"]}, "cell": "magic3/3", "stage": "C"}
+{"id": "po-magic3-t3-eval-001737", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ 5 & \\square & \\square \\\\ \\square & 7 & \\square \\end{pmatrix} with distinct integers between -1 and 11, keeping the given entries, so that all rows and columns have sum 9. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ 5 & \\square & \\square \\\\ \\square & 7 & \\square \\end{pmatrix} with distinct integers between -1 and 11, keeping the given entries, so that all rows and columns have sum 9. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "1, 2, 6, 5, 0, 4, 3, 7, -1", "ref": "1, 2, 6, 5, 0, 4, 3, 7, -1", "family": "po_magic3", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 9, "lo": -1, "hi": 11, "fix": [[3, 5], [7, 7]], "distinct": true}, "seed": 339239530903003748, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 7, "valid_answers": ["1, 2, 6, 5, 0, 4, 3, 7, -1", "4, -1, 6, 5, 3, 1, 0, 7, 2"]}, "cell": "magic3/3", "stage": "C"}
+{"id": "po-magic3-t3-eval-001768", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & 4 \\\\ \\square & \\square & \\square \\\\ \\square & 1 & \\square \\end{pmatrix} with distinct integers between -3 and 11, keeping the given entries, so that all rows and columns have sum 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & 4 \\\\ \\square & \\square & \\square \\\\ \\square & 1 & \\square \\end{pmatrix} with distinct integers between -3 and 11, keeping the given entries, so that all rows and columns have sum 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-3, 5, 4, 7, 0, -1, 2, 1, 3", "ref": "-3, 5, 4, 7, 0, -1, 2, 1, 3", "family": "po_magic3", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 6, "lo": -3, "hi": 11, "fix": [[2, 4], [7, 1]], "distinct": true}, "seed": 339239530903003817, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 6, "valid_answers": ["-3, 5, 4, 7, 0, -1, 2, 1, 3", "-1, 3, 4, 7, 2, -3, 0, 1, 5", "0, 2, 4, -2, 3, 5, 8, 1, -3", "2, 0, 4, -2, 5, 3, 6, 1, -1", "3, -1, 4, -2, 6, 2, 5, 1, 0", "5, -3, 4, -2, 8, 0, 3, 1, 2"]}, "cell": "magic3/3", "stage": "C"}
+{"id": "po-magic3-t3-eval-001933", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & 2 \\\\ \\square & \\square & \\square \\\\ \\square & \\square & 6 \\end{pmatrix} with distinct integers between 0 and 11, keeping the given entries, so that all rows and columns have sum 12. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & 2 \\\\ \\square & \\square & \\square \\\\ \\square & \\square & 6 \\end{pmatrix} with distinct integers between 0 and 11, keeping the given entries, so that all rows and columns have sum 12. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "3, 7, 2, 8, 0, 4, 1, 5, 6", "ref": "3, 7, 2, 8, 0, 4, 1, 5, 6", "family": "po_magic3", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 12, "lo": 0, "hi": 11, "fix": [[2, 2], [8, 6]], "distinct": true}, "seed": 339239530903004191, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 8, "valid_answers": ["3, 7, 2, 8, 0, 4, 1, 5, 6", "7, 3, 2, 0, 8, 4, 5, 1, 6"]}, "cell": "magic3/3", "stage": "C"}
+{"id": "po-magic3-t3-eval-002468", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & 0 & -4 \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -6 and 6, keeping the given entries, so that all rows and columns have sum 0. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & \\square & \\square \\\\ \\square & 0 & -4 \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -6 and 6, keeping the given entries, so that all rows and columns have sum 0. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "-6, 5, 1, 4, 0, -4, 2, -5, 3", "ref": "-6, 5, 1, 4, 0, -4, 2, -5, 3", "family": "po_magic3", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 0, "lo": -6, "hi": 6, "fix": [[4, 0], [5, -4]], "distinct": true}, "seed": 339239530903005305, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 4, "valid_answers": ["-6, 5, 1, 4, 0, -4, 2, -5, 3", "-5, 6, -1, 4, 0, -4, 1, -6, 5", "-3, 2, 1, 4, 0, -4, -1, -2, 3", "-3, 5, -2, 4, 0, -4, -1, -5, 6", "-1, -5, 6, 4, 0, -4, -3, 5, -2", "-1, -2, 3, 4, 0, -4, -3, 2, 1", "1, -6, 5, 4, 0, -4, -5, 6, -1", "2, -5, 3, 4, 0, -4, -6, 5, 1"]}, "cell": "magic3/3", "stage": "C"}
+{"id": "po-magic3-t3-eval-002373", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} \\square & 2 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & -2 & \\square \\end{pmatrix} with distinct integers between -2 and 10, keeping the given entries, so that all rows and columns have sum 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} \\square & 2 & \\square \\\\ \\square & \\square & \\square \\\\ \\square & -2 & \\square \\end{pmatrix} with distinct integers between -2 and 10, keeping the given entries, so that all rows and columns have sum 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "0, 2, 4, 1, 6, -1, 5, -2, 3", "ref": "0, 2, 4, 1, 6, -1, 5, -2, 3", "family": "po_magic3", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 6, "lo": -2, "hi": 10, "fix": [[1, 2], [7, -2]], "distinct": true}, "seed": 339239530903005098, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 6, "valid_answers": ["0, 2, 4, 1, 6, -1, 5, -2, 3", "4, 2, 0, -1, 6, 1, 3, -2, 5"]}, "cell": "magic3/3", "stage": "C"}
+{"id": "po-magic3-t3-eval-000329", "task": "cx_po_magic3", "source": "cx_po_magic3", "split": "eval", "problem": "Fill the 3x3 grid \\begin{pmatrix} 0 & \\square & \\square \\\\ \\square & 4 & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -6 and 6, keeping the given entries, so that all rows and columns have sum 0. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.", "prompt": "Problem:\nFill the 3x3 grid \\begin{pmatrix} 0 & \\square & \\square \\\\ \\square & 4 & \\square \\\\ \\square & \\square & \\square \\end{pmatrix} with distinct integers between -6 and 6, keeping the given entries, so that all rows and columns have sum 0. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as the nine entries row by row, separated by commas.\n\n<think>\n", "answer": "0, -6, 6, -3, 4, -1, 3, 2, -5", "ref": "0, -6, 6, -3, 4, -1, 3, 2, -5", "family": "po_magic3", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_magic3", "rung": "witness", "params": {"kind": "semi", "S": 0, "lo": -6, "hi": 6, "fix": [[0, 0], [4, 4]], "distinct": true}, "seed": 339239530903000742, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 5, "valid_answers": ["0, -6, 6, -3, 4, -1, 3, 2, -5", "0, -5, 5, -3, 4, -1, 3, 1, -4", "0, -3, 3, -6, 4, 2, 6, -1, -5", "0, -3, 3, -5, 4, 1, 5, -1, -4", "0, -1, 1, 2, 4, -6, -2, -3, 5", "0, 2, -2, -1, 4, -3, 1, -6, 5"]}, "cell": "magic3/3", "stage": "C"}
+{"id": "po-mat2_commute-t0-eval-001991", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -7 and commutes with \\begin{pmatrix} 3 & -1 \\\\ -1 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -7 and commutes with \\begin{pmatrix} 3 & -1 \\\\ -1 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 2, 2, 1", "ref": "-3, 2, 2, 1", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00061, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 4, "M": [3, -1, -1, 1], "D": -7}, "seed": 319369941800002165, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["-3, 2, 2, 1", "-1, 2, 2, 3", "1, -2, -2, -3", "3, -2, -2, -1"]}, "cell": "mat2_commute/0", "stage": "A"}
+{"id": "po-mat2_commute-t0-eval-000018", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 4 and commutes with \\begin{pmatrix} 2 & -3 \\\\ -3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 4 and commutes with \\begin{pmatrix} 2 & -3 \\\\ -3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 0, 0, -2", "ref": "-2, 0, 0, -2", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000305, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 4, "M": [2, -3, -3, 0], "D": 4}, "seed": 319369941800000018, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["-2, 0, 0, -2", "2, 0, 0, 2"]}, "cell": "mat2_commute/0", "stage": "A"}
+{"id": "po-mat2_commute-t0-eval-000830", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 17 and commutes with \\begin{pmatrix} 2 & -3 \\\\ 3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 17 and commutes with \\begin{pmatrix} 2 & -3 \\\\ 3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 3, -3, -2", "ref": "-4, 3, -3, -2", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00061, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 4, "M": [2, -3, 3, 0], "D": 17}, "seed": 319369941800000864, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["-4, 3, -3, -2", "-2, -3, 3, -4", "2, 3, -3, 4", "4, -3, 3, 2"]}, "cell": "mat2_commute/0", "stage": "A"}
+{"id": "po-mat2_commute-t0-eval-001023", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -16 and commutes with \\begin{pmatrix} 0 & 2 \\\\ 1 & -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -16 and commutes with \\begin{pmatrix} 0 & 2 \\\\ 1 & -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -4, -2, 2", "ref": "-4, -4, -2, 2", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00061, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 4, "M": [0, 2, 1, -3], "D": -16}, "seed": 319369941800001076, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["-4, -4, -2, 2", "-2, -4, -2, 4", "2, 4, 2, -4", "4, 4, 2, -2"]}, "cell": "mat2_commute/0", "stage": "A"}
+{"id": "po-mat2_commute-t0-eval-001860", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 9 and commutes with \\begin{pmatrix} -1 & -3 \\\\ 2 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 9 and commutes with \\begin{pmatrix} -1 & -3 \\\\ 2 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -3, 2, -1", "ref": "-3, -3, 2, -1", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000914, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 4, "M": [-1, -3, 2, 1], "D": 9}, "seed": 319369941800002006, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 3, "valid_answers": ["-3, -3, 2, -1", "-3, 0, 0, -3", "-1, 3, -2, -3", "1, -3, 2, 3", "3, 0, 0, 3", "3, 3, -2, 1"]}, "cell": "mat2_commute/0", "stage": "A"}
+{"id": "po-mat2_commute-t0-eval-000065", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and commutes with \\begin{pmatrix} -2 & 0 \\\\ 0 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and commutes with \\begin{pmatrix} -2 & 0 \\\\ 0 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 0, 0, 2", "ref": "-3, 0, 0, 2", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00061, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 4, "M": [-2, 0, 0, 1], "D": -6}, "seed": 319369941800000066, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["-3, 0, 0, 2", "-2, 0, 0, 3", "2, 0, 0, -3", "3, 0, 0, -2"]}, "cell": "mat2_commute/0", "stage": "A"}
+{"id": "po-mat2_commute-t0-eval-002498", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -19 and commutes with \\begin{pmatrix} 2 & -1 \\\\ -1 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -19 and commutes with \\begin{pmatrix} 2 & -1 \\\\ -1 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -4, -4, 1", "ref": "-3, -4, -4, 1", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00061, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 4, "M": [2, -1, -1, 3], "D": -19}, "seed": 319369941800002758, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["-3, -4, -4, 1", "-1, -4, -4, 3", "1, 4, 4, -3", "3, 4, 4, -1"]}, "cell": "mat2_commute/0", "stage": "A"}
+{"id": "po-mat2_commute-t0-eval-001615", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 9 and commutes with \\begin{pmatrix} 1 & -3 \\\\ 1 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 9 and commutes with \\begin{pmatrix} 1 & -3 \\\\ 1 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 0, 0, -3", "ref": "-3, 0, 0, -3", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000305, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 4, "M": [1, -3, 1, -2], "D": 9}, "seed": 319369941800001727, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["-3, 0, 0, -3", "3, 0, 0, 3"]}, "cell": "mat2_commute/0", "stage": "A"}
+{"id": "po-mat2_commute-t0-eval-000675", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 14 and commutes with \\begin{pmatrix} 2 & -2 \\\\ 3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 14 and commutes with \\begin{pmatrix} 2 & -2 \\\\ 3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 2, -3, -2", "ref": "-4, 2, -3, -2", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00061, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 4, "M": [2, -2, 3, 0], "D": 14}, "seed": 319369941800000694, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["-4, 2, -3, -2", "-2, -2, 3, -4", "2, 2, -3, 4", "4, -2, 3, 2"]}, "cell": "mat2_commute/0", "stage": "A"}
+{"id": "po-mat2_commute-t0-eval-002319", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -2 and commutes with \\begin{pmatrix} 1 & -1 \\\\ 1 & -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -2 and commutes with \\begin{pmatrix} 1 & -1 \\\\ 1 & -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 1, -1, 1", "ref": "-3, 1, -1, 1", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00061, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 4, "M": [1, -1, 1, -3], "D": -2}, "seed": 319369941800002553, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["-3, 1, -1, 1", "-1, 1, -1, 3", "1, -1, 1, -3", "3, -1, 1, -1"]}, "cell": "mat2_commute/0", "stage": "A"}
+{"id": "po-mat2_commute-t0-eval-000388", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and commutes with \\begin{pmatrix} -3 & 1 \\\\ -2 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and commutes with \\begin{pmatrix} -3 & 1 \\\\ -2 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 1, -2, 2", "ref": "-4, 1, -2, 2", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00061, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 4, "M": [-3, 1, -2, 3], "D": -6}, "seed": 319369941800000398, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["-4, 1, -2, 2", "-2, 1, -2, 4", "2, -1, 2, -4", "4, -1, 2, -2"]}, "cell": "mat2_commute/0", "stage": "A"}
+{"id": "po-mat2_commute-t0-eval-000533", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 16 and commutes with \\begin{pmatrix} 3 & 3 \\\\ -2 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 16 and commutes with \\begin{pmatrix} 3 & 3 \\\\ -2 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 0, 0, -4", "ref": "-4, 0, 0, -4", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000305, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 4, "M": [3, 3, -2, 3], "D": 16}, "seed": 319369941800000547, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-4, 0, 0, -4", "4, 0, 0, 4"]}, "cell": "mat2_commute/0", "stage": "A"}
+{"id": "po-mat2_commute-t1-eval-001297", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -9 and commutes with \\begin{pmatrix} -3 & 1 \\\\ 2 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -9 and commutes with \\begin{pmatrix} -3 & 1 \\\\ 2 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 2, 4, 1", "ref": "-1, 2, 4, 1", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 6, "M": [-3, 1, 2, -2], "D": -9}, "seed": 319369941801001336, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["-1, 2, 4, 1", "1, -2, -4, -1"]}, "cell": "mat2_commute/1", "stage": "B"}
+{"id": "po-mat2_commute-t1-eval-000338", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -12 and commutes with \\begin{pmatrix} -3 & 2 \\\\ 2 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -12 and commutes with \\begin{pmatrix} -3 & 2 \\\\ 2 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 2, 2, 2", "ref": "-4, 2, 2, 2", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00014, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 6, "M": [-3, 2, 2, 3], "D": -12}, "seed": 319369941801000347, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 4, "valid_answers": ["-4, 2, 2, 2", "-2, 2, 2, 4", "2, -2, -2, -4", "4, -2, -2, -2"]}, "cell": "mat2_commute/1", "stage": "B"}
+{"id": "po-mat2_commute-t1-eval-001363", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 25 and commutes with \\begin{pmatrix} 3 & 0 \\\\ -3 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 25 and commutes with \\begin{pmatrix} 3 & 0 \\\\ -3 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 0, -6, -5", "ref": "-5, 0, -6, -5", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 26, "bucket": null, "guess_rate": 0.00091, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 6, "M": [3, 0, -3, 3], "D": 25}, "seed": 319369941801001406, "version": "po-v0", "tier": 1, "n_raw": 26, "small": 5, "valid_answers": ["-5, 0, -6, -5", "-5, 0, -5, -5", "-5, 0, -4, -5", "-5, 0, -3, -5", "-5, 0, -2, -5", "-5, 0, -1, -5", "-5, 0, 0, -5", "-5, 0, 1, -5", "-5, 0, 2, -5", "-5, 0, 3, -5", "-5, 0, 4, -5", "-5, 0, 5, -5", "-5, 0, 6, -5", "5, 0, -6, 5", "5, 0, -5, 5", "5, 0, -4, 5", "5, 0, -3, 5", "5, 0, -2, 5", "5, 0, -1, 5", "5, 0, 0, 5", "5, 0, 1, 5", "5, 0, 2, 5", "5, 0, 3, 5", "5, 0, 4, 5", "5, 0, 5, 5", "5, 0, 6, 5"]}, "cell": "mat2_commute/1", "stage": "B"}
+{"id": "po-mat2_commute-t1-eval-000951", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 4 and commutes with \\begin{pmatrix} 3 & -3 \\\\ -1 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 4 and commutes with \\begin{pmatrix} 3 & -3 \\\\ -1 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 0, 0, -2", "ref": "-2, 0, 0, -2", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 6, "M": [3, -3, -1, 1], "D": 4}, "seed": 319369941801000979, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 2, "valid_answers": ["-2, 0, 0, -2", "2, 0, 0, 2"]}, "cell": "mat2_commute/1", "stage": "B"}
+{"id": "po-mat2_commute-t1-eval-001968", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 25 and commutes with \\begin{pmatrix} -2 & 0 \\\\ 1 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 25 and commutes with \\begin{pmatrix} -2 & 0 \\\\ 1 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 0, 0, -5", "ref": "-5, 0, 0, -5", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 6, "M": [-2, 0, 1, 3], "D": 25}, "seed": 319369941801002059, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 5, "valid_answers": ["-5, 0, 0, -5", "5, 0, 0, 5"]}, "cell": "mat2_commute/1", "stage": "B"}
+{"id": "po-mat2_commute-t1-eval-000682", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -7 and commutes with \\begin{pmatrix} 2 & 1 \\\\ 3 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -7 and commutes with \\begin{pmatrix} 2 & 1 \\\\ 3 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -1, -3, 2", "ref": "-2, -1, -3, 2", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 6, "M": [2, 1, 3, -2], "D": -7}, "seed": 319369941801000700, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 3, "valid_answers": ["-2, -1, -3, 2", "2, 1, 3, -2"]}, "cell": "mat2_commute/1", "stage": "B"}
+{"id": "po-mat2_commute-t1-eval-000383", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 24 and commutes with \\begin{pmatrix} -1 & 0 \\\\ -1 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 24 and commutes with \\begin{pmatrix} -1 & 0 \\\\ -1 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 0, 2, -4", "ref": "-6, 0, 2, -4", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00014, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 6, "M": [-1, 0, -1, -2], "D": 24}, "seed": 319369941801000394, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 6, "valid_answers": ["-6, 0, 2, -4", "-4, 0, -2, -6", "4, 0, 2, 6", "6, 0, -2, 4"]}, "cell": "mat2_commute/1", "stage": "B"}
+{"id": "po-mat2_commute-t1-eval-000078", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -2 and commutes with \\begin{pmatrix} 1 & 0 \\\\ -1 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -2 and commutes with \\begin{pmatrix} 1 & 0 \\\\ -1 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 0, 1, 1", "ref": "-2, 0, 1, 1", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00014, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 6, "M": [1, 0, -1, -2], "D": -2}, "seed": 319369941801000082, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 2, "valid_answers": ["-2, 0, 1, 1", "-1, 0, 1, 2", "1, 0, -1, -2", "2, 0, -1, -1"]}, "cell": "mat2_commute/1", "stage": "B"}
+{"id": "po-mat2_commute-t1-eval-000573", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 25 and commutes with \\begin{pmatrix} 1 & -2 \\\\ 3 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 25 and commutes with \\begin{pmatrix} 1 & -2 \\\\ 3 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 0, 0, -5", "ref": "-5, 0, 0, -5", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.00021, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 6, "M": [1, -2, 3, 1], "D": 25}, "seed": 319369941801000590, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 5, "valid_answers": ["-5, 0, 0, -5", "-1, -4, 6, -1", "-1, 4, -6, -1", "1, -4, 6, 1", "1, 4, -6, 1", "5, 0, 0, 5"]}, "cell": "mat2_commute/1", "stage": "B"}
+{"id": "po-mat2_commute-t1-eval-001010", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 3 and commutes with \\begin{pmatrix} 1 & 1 \\\\ -3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 3 and commutes with \\begin{pmatrix} 1 & 1 \\\\ -3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -1, 3, 0", "ref": "-1, -1, 3, 0", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00014, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 6, "M": [1, 1, -3, 0], "D": 3}, "seed": 319369941801001038, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 3, "valid_answers": ["-1, -1, 3, 0", "0, -1, 3, 1", "0, 1, -3, -1", "1, 1, -3, 0"]}, "cell": "mat2_commute/1", "stage": "B"}
+{"id": "po-mat2_commute-t1-eval-000925", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 1 and commutes with \\begin{pmatrix} 1 & -3 \\\\ 2 & -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 1 and commutes with \\begin{pmatrix} 1 & -3 \\\\ 2 & -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, 0, -1", "ref": "-1, 0, 0, -1", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 6, "M": [1, -3, 2, -3], "D": 1}, "seed": 319369941801000952, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 1, "valid_answers": ["-1, 0, 0, -1", "1, 0, 0, 1"]}, "cell": "mat2_commute/1", "stage": "B"}
+{"id": "po-mat2_commute-t1-eval-001356", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 5 and commutes with \\begin{pmatrix} -1 & 1 \\\\ -3 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 5 and commutes with \\begin{pmatrix} -1 & 1 \\\\ -3 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -1, 3, -1", "ref": "-2, -1, 3, -1", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00014, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 6, "M": [-1, 1, -3, -2], "D": 5}, "seed": 319369941801001399, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 3, "valid_answers": ["-2, -1, 3, -1", "-1, 1, -3, -2", "1, -1, 3, 2", "2, 1, -3, 1"]}, "cell": "mat2_commute/1", "stage": "B"}
+{"id": "po-mat2_commute-t2-eval-002432", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -32 and commutes with \\begin{pmatrix} -3 & 0 \\\\ -1 & -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -32 and commutes with \\begin{pmatrix} -3 & 0 \\\\ -1 & -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 0, -6, 4", "ref": "-8, 0, -6, 4", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 8, "M": [-3, 0, -1, -1], "D": -32}, "seed": 319369941802002511, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 8, "valid_answers": ["-8, 0, -6, 4", "-4, 0, -6, 8", "4, 0, 6, -8", "8, 0, 6, -4"]}, "cell": "mat2_commute/2", "stage": "B"}
+{"id": "po-mat2_commute-t2-eval-000787", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -25 and commutes with \\begin{pmatrix} -3 & 1 \\\\ 1 & 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -25 and commutes with \\begin{pmatrix} -3 & 1 \\\\ 1 & 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 2, 2, 3", "ref": "-7, 2, 2, 3", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 8, "M": [-3, 1, 1, 2], "D": -25}, "seed": 319369941802000802, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 7, "valid_answers": ["-7, 2, 2, 3", "-3, 2, 2, 7", "3, -2, -2, -7", "7, -2, -2, -3"]}, "cell": "mat2_commute/2", "stage": "B"}
+{"id": "po-mat2_commute-t2-eval-000792", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 27 and commutes with \\begin{pmatrix} -2 & -1 \\\\ 3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 27 and commutes with \\begin{pmatrix} -2 & -1 \\\\ 3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -1, 3, -4", "ref": "-6, -1, 3, -4", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 8, "M": [-2, -1, 3, 0], "D": 27}, "seed": 319369941802000807, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 6, "valid_answers": ["-6, -1, 3, -4", "-4, 1, -3, -6", "4, -1, 3, 6", "6, 1, -3, 4"]}, "cell": "mat2_commute/2", "stage": "B"}
+{"id": "po-mat2_commute-t2-eval-002399", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 0 and commutes with \\begin{pmatrix} -2 & 1 \\\\ -2 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 0 and commutes with \\begin{pmatrix} -2 & 1 \\\\ -2 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 4, -8, 4", "ref": "-8, 4, -8, 4", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 17, "bucket": "9-20", "guess_rate": 0.000204, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 8, "M": [-2, 1, -2, 1], "D": 0}, "seed": 319369941802002477, "version": "po-v0", "tier": 2, "n_raw": 17, "small": 0, "valid_answers": ["-8, 4, -8, 4", "-6, 3, -6, 3", "-4, 2, -4, 2", "-4, 4, -8, 8", "-3, 3, -6, 6", "-2, 1, -2, 1", "-2, 2, -4, 4", "-1, 1, -2, 2", "0, 0, 0, 0", "1, -1, 2, -2", "2, -2, 4, -4", "2, -1, 2, -1", "3, -3, 6, -6", "4, -4, 8, -8", "4, -2, 4, -2", "6, -3, 6, -3", "8, -4, 8, -4"]}, "cell": "mat2_commute/2", "stage": "B"}
+{"id": "po-mat2_commute-t2-eval-002190", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -4 and commutes with \\begin{pmatrix} 3 & -3 \\\\ 0 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -4 and commutes with \\begin{pmatrix} 3 & -3 \\\\ 0 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 3, 0, 1", "ref": "-4, 3, 0, 1", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 8, "M": [3, -3, 0, -2], "D": -4}, "seed": 319369941802002254, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 4, "valid_answers": ["-4, 3, 0, 1", "-1, 3, 0, 4", "1, -3, 0, -4", "4, -3, 0, -1"]}, "cell": "mat2_commute/2", "stage": "B"}
+{"id": "po-mat2_commute-t2-eval-002430", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 1 and commutes with \\begin{pmatrix} -1 & 1 \\\\ -2 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 1 and commutes with \\begin{pmatrix} -1 & 1 \\\\ -2 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 2, -4, 1", "ref": "-7, 2, -4, 1", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 7.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 8, "M": [-1, 1, -2, 3], "D": 1}, "seed": 319369941802002509, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 1, "valid_answers": ["-7, 2, -4, 1", "-1, 0, 0, -1", "-1, 2, -4, 7", "1, -2, 4, -7", "1, 0, 0, 1", "7, -2, 4, -1"]}, "cell": "mat2_commute/2", "stage": "B"}
+{"id": "po-mat2_commute-t2-eval-001467", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 16 and commutes with \\begin{pmatrix} 3 & -2 \\\\ 3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 16 and commutes with \\begin{pmatrix} 3 & -2 \\\\ 3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 2, -3, -2", "ref": "-5, 2, -3, -2", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.00012, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 8, "M": [3, -2, 3, 0], "D": 16}, "seed": 319369941802001500, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 4, "valid_answers": ["-5, 2, -3, -2", "-4, 0, 0, -4", "-4, 4, -6, 2", "-2, -2, 3, -5", "-2, 4, -6, 4", "2, -4, 6, -4", "2, 2, -3, 5", "4, -4, 6, -2", "4, 0, 0, 4", "5, -2, 3, 2"]}, "cell": "mat2_commute/2", "stage": "B"}
+{"id": "po-mat2_commute-t2-eval-001649", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 36 and commutes with \\begin{pmatrix} 1 & -1 \\\\ 1 & -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 36 and commutes with \\begin{pmatrix} 1 & -1 \\\\ 1 & -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 2, -2, -4", "ref": "-8, 2, -2, -4", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.00012, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 8, "M": [1, -1, 1, -1], "D": 36}, "seed": 319369941802001686, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 6, "valid_answers": ["-8, 2, -2, -4", "-7, 1, -1, -5", "-6, 0, 0, -6", "-5, -1, 1, -7", "-4, -2, 2, -8", "4, 2, -2, 8", "5, 1, -1, 7", "6, 0, 0, 6", "7, -1, 1, 5", "8, -2, 2, 4"]}, "cell": "mat2_commute/2", "stage": "B"}
+{"id": "po-mat2_commute-t2-eval-000854", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and commutes with \\begin{pmatrix} -1 & -3 \\\\ -2 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and commutes with \\begin{pmatrix} -1 & -3 \\\\ -2 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 3, 2, 0", "ref": "-1, 3, 2, 0", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 8, "M": [-1, -3, -2, -2], "D": -6}, "seed": 319369941802000870, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 3, "valid_answers": ["-1, 3, 2, 0", "0, -3, -2, -1", "0, 3, 2, 1", "1, -3, -2, 0"]}, "cell": "mat2_commute/2", "stage": "B"}
+{"id": "po-mat2_commute-t2-eval-000496", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 49 and commutes with \\begin{pmatrix} 1 & -2 \\\\ -3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 49 and commutes with \\begin{pmatrix} 1 & -2 \\\\ -3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 0, 0, -7", "ref": "-7, 0, 0, -7", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 8, "M": [1, -2, -3, 0], "D": 49}, "seed": 319369941802000504, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 7, "valid_answers": ["-7, 0, 0, -7", "7, 0, 0, 7"]}, "cell": "mat2_commute/2", "stage": "B"}
+{"id": "po-mat2_commute-t2-eval-001560", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 3 and commutes with \\begin{pmatrix} -2 & 2 \\\\ -1 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 3 and commutes with \\begin{pmatrix} -2 & 2 \\\\ -1 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 4, -2, 1", "ref": "-5, 4, -2, 1", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 8, "M": [-2, 2, -1, 1], "D": 3}, "seed": 319369941802001595, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 5, "valid_answers": ["-5, 4, -2, 1", "-1, 4, -2, 5", "1, -4, 2, -5", "5, -4, 2, -1"]}, "cell": "mat2_commute/2", "stage": "B"}
+{"id": "po-mat2_commute-t2-eval-000222", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 12 and commutes with \\begin{pmatrix} 2 & -3 \\\\ 1 & 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 12 and commutes with \\begin{pmatrix} 2 & -3 \\\\ 1 & 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -3, 1, -3", "ref": "-3, -3, 1, -3", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 7.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 8, "M": [2, -3, 1, 2], "D": 12}, "seed": 319369941802000228, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 3, "valid_answers": ["-3, -3, 1, -3", "-3, 3, -1, -3", "0, -6, 2, 0", "0, 6, -2, 0", "3, -3, 1, 3", "3, 3, -1, 3"]}, "cell": "mat2_commute/2", "stage": "B"}
+{"id": "po-mat2_commute-t3-eval-000285", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 15 and commutes with \\begin{pmatrix} 0 & -3 \\\\ -1 & -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 15 and commutes with \\begin{pmatrix} 0 & -3 \\\\ -1 & -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 6, 2, -3", "ref": "-9, 6, 2, -3", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 4.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 10, "M": [0, -3, -1, -3], "D": 15}, "seed": 319369941803000287, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 6, "valid_answers": ["-9, 6, 2, -3", "-6, 3, 1, -3", "-3, -6, -2, -9", "-3, -3, -1, -6", "3, 3, 1, 6", "3, 6, 2, 9", "6, -3, -1, 3", "9, -6, -2, 3"]}, "cell": "mat2_commute/3", "stage": "B"}
+{"id": "po-mat2_commute-t3-eval-001321", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 68 and commutes with \\begin{pmatrix} -3 & -2 \\\\ -1 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 68 and commutes with \\begin{pmatrix} -3 & -2 \\\\ -1 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -2, -1, -7", "ref": "-10, -2, -1, -7", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 2.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 10, "M": [-3, -2, -1, 0], "D": 68}, "seed": 319369941803001345, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 10, "valid_answers": ["-10, -2, -1, -7", "-7, 2, 1, -10", "7, -2, -1, 10", "10, 2, 1, 7"]}, "cell": "mat2_commute/3", "stage": "B"}
+{"id": "po-mat2_commute-t3-eval-001415", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 73 and commutes with \\begin{pmatrix} -1 & -3 \\\\ 3 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 73 and commutes with \\begin{pmatrix} -1 & -3 \\\\ 3 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -9, 9, 2", "ref": "-4, -9, 9, 2", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 2.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 10, "M": [-1, -3, 3, 1], "D": 73}, "seed": 319369941803001442, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 9, "valid_answers": ["-4, -9, 9, 2", "-2, -9, 9, 4", "2, 9, -9, -4", "4, 9, -9, -2"]}, "cell": "mat2_commute/3", "stage": "B"}
+{"id": "po-mat2_commute-t3-eval-000772", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -8 and commutes with \\begin{pmatrix} -1 & -1 \\\\ -2 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -8 and commutes with \\begin{pmatrix} -1 & -1 \\\\ -2 & 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -2, -4, 0", "ref": "-8, -2, -4, 0", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 2.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 10, "M": [-1, -1, -2, 3], "D": -8}, "seed": 319369941803000786, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 8, "valid_answers": ["-8, -2, -4, 0", "0, -2, -4, 8", "0, 2, 4, -8", "8, 2, 4, 0"]}, "cell": "mat2_commute/3", "stage": "B"}
+{"id": "po-mat2_commute-t3-eval-000425", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 31 and commutes with \\begin{pmatrix} -3 & -3 \\\\ 1 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 31 and commutes with \\begin{pmatrix} -3 & -3 \\\\ 1 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -9, 3, -1", "ref": "-4, -9, 3, -1", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 2.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 10, "M": [-3, -3, 1, -2], "D": 31}, "seed": 319369941803000430, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 9, "valid_answers": ["-4, -9, 3, -1", "-1, 9, -3, -4", "1, -9, 3, 4", "4, 9, -3, 1"]}, "cell": "mat2_commute/3", "stage": "B"}
+{"id": "po-mat2_commute-t3-eval-000968", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 52 and commutes with \\begin{pmatrix} 1 & -1 \\\\ 3 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 52 and commutes with \\begin{pmatrix} 1 & -1 \\\\ 3 & -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 2, -6, -4", "ref": "-10, 2, -6, -4", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 2.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 10, "M": [1, -1, 3, -2], "D": 52}, "seed": 319369941803000983, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 10, "valid_answers": ["-10, 2, -6, -4", "-4, -2, 6, -10", "4, 2, -6, 10", "10, -2, 6, 4"]}, "cell": "mat2_commute/3", "stage": "B"}
+{"id": "po-mat2_commute-t3-eval-002355", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 41 and commutes with \\begin{pmatrix} 3 & -2 \\\\ 1 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 41 and commutes with \\begin{pmatrix} 3 & -2 \\\\ 1 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 8, -4, -1", "ref": "-9, 8, -4, -1", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 4.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 10, "M": [3, -2, 1, 1], "D": 41}, "seed": 319369941803002409, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 9, "valid_answers": ["-9, 8, -4, -1", "-9, 10, -5, 1", "-1, -8, 4, -9", "-1, 10, -5, 9", "1, -10, 5, -9", "1, 8, -4, 9", "9, -10, 5, -1", "9, -8, 4, 1"]}, "cell": "mat2_commute/3", "stage": "B"}
+{"id": "po-mat2_commute-t3-eval-000389", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 9 and commutes with \\begin{pmatrix} 3 & -2 \\\\ -3 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 9 and commutes with \\begin{pmatrix} 3 & -2 \\\\ -3 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 2, 3, -3", "ref": "-5, 2, 3, -3", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 3.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 10, "M": [3, -2, -3, 1], "D": 9}, "seed": 319369941803000393, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 3, "valid_answers": ["-5, 2, 3, -3", "-3, -2, -3, -5", "-3, 0, 0, -3", "3, 0, 0, 3", "3, 2, 3, 5", "5, -2, -3, 3"]}, "cell": "mat2_commute/3", "stage": "B"}
+{"id": "po-mat2_commute-t3-eval-002495", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 16 and commutes with \\begin{pmatrix} 2 & 2 \\\\ -1 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 16 and commutes with \\begin{pmatrix} 2 & 2 \\\\ -1 & 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -4, 2, -2", "ref": "-4, -4, 2, -2", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 5.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 10, "M": [2, 2, -1, 1], "D": 16}, "seed": 319369941803002553, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 4, "valid_answers": ["-4, -4, 2, -2", "-4, 0, 0, -4", "-2, -6, 3, 1", "-2, 4, -2, -4", "-1, -6, 3, 2", "1, 6, -3, -2", "2, -4, 2, 4", "2, 6, -3, -1", "4, 0, 0, 4", "4, 4, -2, 2"]}, "cell": "mat2_commute/3", "stage": "B"}
+{"id": "po-mat2_commute-t3-eval-000352", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 74 and commutes with \\begin{pmatrix} 0 & -3 \\\\ -2 & 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 74 and commutes with \\begin{pmatrix} 0 & -3 \\\\ -2 & 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -3, -2, -8", "ref": "-10, -3, -2, -8", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 2.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 10, "M": [0, -3, -2, 2], "D": 74}, "seed": 319369941803000356, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 10, "valid_answers": ["-10, -3, -2, -8", "-8, 3, 2, -10", "8, -3, -2, 10", "10, 3, 2, 8"]}, "cell": "mat2_commute/3", "stage": "B"}
+{"id": "po-mat2_commute-t3-eval-002114", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 97 and commutes with \\begin{pmatrix} -1 & -1 \\\\ 3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 97 and commutes with \\begin{pmatrix} -1 & -1 \\\\ 3 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -3, 9, -7", "ref": "-10, -3, 9, -7", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 2.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 10, "M": [-1, -1, 3, 0], "D": 97}, "seed": 319369941803002156, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 10, "valid_answers": ["-10, -3, 9, -7", "-7, 3, -9, -10", "7, -3, 9, 10", "10, 3, -9, 7"]}, "cell": "mat2_commute/3", "stage": "B"}
+{"id": "po-mat2_commute-t3-eval-000579", "task": "cx_po_mat2_commute", "source": "cx_po_mat2_commute", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 15 and commutes with \\begin{pmatrix} 3 & 3 \\\\ 1 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 15 and commutes with \\begin{pmatrix} 3 & 3 \\\\ 1 & 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -6, -2, -3", "ref": "-9, -6, -2, -3", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 4.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 10, "M": [3, 3, 1, 0], "D": 15}, "seed": 319369941803000588, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 6, "valid_answers": ["-9, -6, -2, -3", "-6, -3, -1, -3", "-3, 3, 1, -6", "-3, 6, 2, -9", "3, -6, -2, 9", "3, -3, -1, 6", "6, 3, 1, 3", "9, 6, 2, 3"]}, "cell": "mat2_commute/3", "stage": "B"}
+{"id": "po-mat2_eigvec-t0-eval-000869", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 0 and A \\begin{pmatrix} 0 \\end{pmatrix} = -2 \\begin{pmatrix} 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 0 and A \\begin{pmatrix} 0 \\end{pmatrix} = -2 \\begin{pmatrix} 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, -4, -2", "ref": "0, 0, -4, -2", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.001372, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 4, "v": [0, -3], "lam": -2, "D": 0}, "seed": 295698968000002159, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 2, "valid_answers": ["0, 0, -4, -2", "0, 0, -3, -2", "0, 0, -2, -2", "0, 0, -1, -2", "0, 0, 0, -2", "0, 0, 1, -2", "0, 0, 2, -2", "0, 0, 3, -2", "0, 0, 4, -2"]}, "cell": "mat2_eigvec/0", "stage": "B"}
+{"id": "po-mat2_eigvec-t0-eval-001310", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 2 and A \\begin{pmatrix} -2 \\end{pmatrix} = 2 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 2 and A \\begin{pmatrix} -2 \\end{pmatrix} = 2 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -4, 0, 1", "ref": "2, -4, 0, 1", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.001372, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 4, "v": [-2, 0], "lam": 2, "D": 2}, "seed": 295698968000004578, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 2, "valid_answers": ["2, -4, 0, 1", "2, -3, 0, 1", "2, -2, 0, 1", "2, -1, 0, 1", "2, 0, 0, 1", "2, 1, 0, 1", "2, 2, 0, 1", "2, 3, 0, 1", "2, 4, 0, 1"]}, "cell": "mat2_eigvec/0", "stage": "B"}
+{"id": "po-mat2_eigvec-t0-eval-000673", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -2 and A \\begin{pmatrix} 0 \\end{pmatrix} = 2 \\begin{pmatrix} 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -2 and A \\begin{pmatrix} 0 \\end{pmatrix} = 2 \\begin{pmatrix} 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, -4, 2", "ref": "-1, 0, -4, 2", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.001372, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 4, "v": [0, -1], "lam": 2, "D": -2}, "seed": 295698968000001484, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 2, "valid_answers": ["-1, 0, -4, 2", "-1, 0, -3, 2", "-1, 0, -2, 2", "-1, 0, -1, 2", "-1, 0, 0, 2", "-1, 0, 1, 2", "-1, 0, 2, 2", "-1, 0, 3, 2", "-1, 0, 4, 2"]}, "cell": "mat2_eigvec/0", "stage": "B"}
+{"id": "po-mat2_eigvec-t0-eval-000648", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 2 and A \\begin{pmatrix} -2 \\end{pmatrix} = -1 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 2 and A \\begin{pmatrix} -2 \\end{pmatrix} = -1 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -2, 0, -1", "ref": "-2, -2, 0, -1", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000305, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 4, "v": [-2, 1], "lam": -1, "D": 2}, "seed": 295698968000001394, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["-2, -2, 0, -1", "0, 2, -1, -3"]}, "cell": "mat2_eigvec/0", "stage": "B"}
+{"id": "po-mat2_eigvec-t0-eval-000526", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 1 and A \\begin{pmatrix} 1 \\end{pmatrix} = 1 \\begin{pmatrix} 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 1 and A \\begin{pmatrix} 1 \\end{pmatrix} = 1 \\begin{pmatrix} 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 1, -4, 3", "ref": "-1, 1, -4, 3", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000457, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 4, "v": [1, 2], "lam": 1, "D": 1}, "seed": 295698968000001074, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 1, "valid_answers": ["-1, 1, -4, 3", "1, 0, 0, 1", "3, -1, 4, -1"]}, "cell": "mat2_eigvec/0", "stage": "B"}
+{"id": "po-mat2_eigvec-t0-eval-000803", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -8 and A \\begin{pmatrix} -3 \\end{pmatrix} = -2 \\begin{pmatrix} -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -8 and A \\begin{pmatrix} -3 \\end{pmatrix} = -2 \\begin{pmatrix} -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -4, 0, 4", "ref": "-2, -4, 0, 4", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.001372, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 4, "v": [-3, 0], "lam": -2, "D": -8}, "seed": 295698968000001927, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 4, "valid_answers": ["-2, -4, 0, 4", "-2, -3, 0, 4", "-2, -2, 0, 4", "-2, -1, 0, 4", "-2, 0, 0, 4", "-2, 1, 0, 4", "-2, 2, 0, 4", "-2, 3, 0, 4", "-2, 4, 0, 4"]}, "cell": "mat2_eigvec/0", "stage": "B"}
+{"id": "po-mat2_eigvec-t0-eval-001017", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 2 and A \\begin{pmatrix} -2 \\end{pmatrix} = 1 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 2 and A \\begin{pmatrix} -2 \\end{pmatrix} = 1 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -2, 1, 3", "ref": "0, -2, 1, 3", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000305, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 4, "v": [-2, 1], "lam": 1, "D": 2}, "seed": 295698968000002808, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["0, -2, 1, 3", "2, 2, 0, 1"]}, "cell": "mat2_eigvec/0", "stage": "B"}
+{"id": "po-mat2_eigvec-t0-eval-001108", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 6 and A \\begin{pmatrix} 2 \\end{pmatrix} = 2 \\begin{pmatrix} 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 6 and A \\begin{pmatrix} 2 \\end{pmatrix} = 2 \\begin{pmatrix} 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -2, 4", "ref": "1, 1, -2, 4", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00061, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 4, "v": [2, 2], "lam": 2, "D": 6}, "seed": 295698968000003317, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["1, 1, -2, 4", "2, 0, -1, 3", "3, -1, 0, 2", "4, -2, 1, 1"]}, "cell": "mat2_eigvec/0", "stage": "B"}
+{"id": "po-mat2_eigvec-t0-eval-001666", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 2 and A \\begin{pmatrix} -2 \\end{pmatrix} = 1 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 2 and A \\begin{pmatrix} -2 \\end{pmatrix} = 1 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 2, -3, 4", "ref": "-1, 2, -3, 4", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000914, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 4, "v": [-2, -2], "lam": 1, "D": 2}, "seed": 295698968000009194, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 2, "valid_answers": ["-1, 2, -3, 4", "0, 1, -2, 3", "1, 0, -1, 2", "2, -1, 0, 1", "3, -2, 1, 0", "4, -3, 2, -1"]}, "cell": "mat2_eigvec/0", "stage": "B"}
+{"id": "po-mat2_eigvec-t0-eval-000139", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -1 and A \\begin{pmatrix} 2 \\end{pmatrix} = -1 \\begin{pmatrix} 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -1 and A \\begin{pmatrix} 2 \\end{pmatrix} = -1 \\begin{pmatrix} 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -4, 2, 3", "ref": "-3, -4, 2, 3", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000457, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 4, "v": [2, -1], "lam": -1, "D": -1}, "seed": 295698968000000247, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 1, "valid_answers": ["-3, -4, 2, 3", "-1, 0, 1, 1", "1, 4, 0, -1"]}, "cell": "mat2_eigvec/0", "stage": "B"}
+{"id": "po-mat2_eigvec-t0-eval-000377", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 6 and A \\begin{pmatrix} 2 \\end{pmatrix} = 3 \\begin{pmatrix} 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 6 and A \\begin{pmatrix} 2 \\end{pmatrix} = 3 \\begin{pmatrix} 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -4, 0, 2", "ref": "3, -4, 0, 2", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.001372, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 4, "v": [2, 0], "lam": 3, "D": 6}, "seed": 295698968000000730, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 3, "valid_answers": ["3, -4, 0, 2", "3, -3, 0, 2", "3, -2, 0, 2", "3, -1, 0, 2", "3, 0, 0, 2", "3, 1, 0, 2", "3, 2, 0, 2", "3, 3, 0, 2", "3, 4, 0, 2"]}, "cell": "mat2_eigvec/0", "stage": "B"}
+{"id": "po-mat2_eigvec-t0-eval-001508", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -8 and A \\begin{pmatrix} -2 \\end{pmatrix} = 2 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -8 and A \\begin{pmatrix} -2 \\end{pmatrix} = 2 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 4, 2, 0", "ref": "-2, 4, 2, 0", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000457, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 4, "v": [-2, -2], "lam": 2, "D": -8}, "seed": 295698968000006399, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["-2, 4, 2, 0", "-1, 3, 3, -1", "0, 2, 4, -2"]}, "cell": "mat2_eigvec/0", "stage": "B"}
+{"id": "po-mat2_eigvec-t1-eval-000061", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 4 and A \\begin{pmatrix} -1 \\end{pmatrix} = 2 \\begin{pmatrix} -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 4 and A \\begin{pmatrix} -1 \\end{pmatrix} = 2 \\begin{pmatrix} -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -1, 4, 4", "ref": "0, -1, 4, 4", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 6, "v": [-1, 2], "lam": 2, "D": 4}, "seed": 295698968001000079, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 2, "valid_answers": ["0, -1, 4, 4", "2, 0, 0, 2", "4, 1, -4, 0"]}, "cell": "mat2_eigvec/1", "stage": "B"}
+{"id": "po-mat2_eigvec-t1-eval-001102", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and A \\begin{pmatrix} -3 \\end{pmatrix} = 3 \\begin{pmatrix} -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and A \\begin{pmatrix} -3 \\end{pmatrix} = 3 \\begin{pmatrix} -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 6, 1, 0", "ref": "1, 6, 1, 0", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 6, "v": [-3, -1], "lam": 3, "D": -6}, "seed": 295698968001001848, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["1, 6, 1, 0", "4, -3, 2, -3"]}, "cell": "mat2_eigvec/1", "stage": "B"}
+{"id": "po-mat2_eigvec-t1-eval-000233", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 2 and A \\begin{pmatrix} 0 \\end{pmatrix} = -1 \\begin{pmatrix} 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 2 and A \\begin{pmatrix} 0 \\end{pmatrix} = -1 \\begin{pmatrix} 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 0, -6, -1", "ref": "-2, 0, -6, -1", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 6, "v": [0, 3], "lam": -1, "D": 2}, "seed": 295698968001000331, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 2, "valid_answers": ["-2, 0, -6, -1", "-2, 0, -5, -1", "-2, 0, -4, -1", "-2, 0, -3, -1", "-2, 0, -2, -1", "-2, 0, -1, -1", "-2, 0, 0, -1", "-2, 0, 1, -1", "-2, 0, 2, -1", "-2, 0, 3, -1", "-2, 0, 4, -1", "-2, 0, 5, -1", "-2, 0, 6, -1"]}, "cell": "mat2_eigvec/1", "stage": "B"}
+{"id": "po-mat2_eigvec-t1-eval-001986", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 6 and A \\begin{pmatrix} 1 \\end{pmatrix} = -1 \\begin{pmatrix} 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 6 and A \\begin{pmatrix} 1 \\end{pmatrix} = -1 \\begin{pmatrix} 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -5, 0, -1", "ref": "-6, -5, 0, -1", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.00021, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 6, "v": [1, -1], "lam": -1, "D": 6}, "seed": 295698968001004380, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 4, "valid_answers": ["-6, -5, 0, -1", "-5, -4, -1, -2", "-4, -3, -2, -3", "-3, -2, -3, -4", "-2, -1, -4, -5", "-1, 0, -5, -6"]}, "cell": "mat2_eigvec/1", "stage": "B"}
+{"id": "po-mat2_eigvec-t1-eval-000228", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -3 and A \\begin{pmatrix} -1 \\end{pmatrix} = 1 \\begin{pmatrix} -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -3 and A \\begin{pmatrix} -1 \\end{pmatrix} = 1 \\begin{pmatrix} -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 3, -4, 3", "ref": "-5, 3, -4, 3", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 6, "v": [-1, -2], "lam": 1, "D": -3}, "seed": 295698968001000323, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 3, "valid_answers": ["-5, 3, -4, 3", "-3, 2, 0, 1", "-1, 1, 4, -1"]}, "cell": "mat2_eigvec/1", "stage": "B"}
+{"id": "po-mat2_eigvec-t1-eval-000359", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 12 and A \\begin{pmatrix} -1 \\end{pmatrix} = -2 \\begin{pmatrix} -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 12 and A \\begin{pmatrix} -1 \\end{pmatrix} = -2 \\begin{pmatrix} -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -6, 0, -6", "ref": "-2, -6, 0, -6", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 6, "v": [-1, 0], "lam": -2, "D": 12}, "seed": 295698968001000523, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 6, "valid_answers": ["-2, -6, 0, -6", "-2, -5, 0, -6", "-2, -4, 0, -6", "-2, -3, 0, -6", "-2, -2, 0, -6", "-2, -1, 0, -6", "-2, 0, 0, -6", "-2, 1, 0, -6", "-2, 2, 0, -6", "-2, 3, 0, -6", "-2, 4, 0, -6", "-2, 5, 0, -6", "-2, 6, 0, -6"]}, "cell": "mat2_eigvec/1", "stage": "B"}
+{"id": "po-mat2_eigvec-t1-eval-001164", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -1 and A \\begin{pmatrix} 1 \\end{pmatrix} = -1 \\begin{pmatrix} 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -1 and A \\begin{pmatrix} 1 \\end{pmatrix} = -1 \\begin{pmatrix} 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 4, -6, 5", "ref": "-5, 4, -6, 5", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.000385, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 6, "v": [1, 1], "lam": -1, "D": -1}, "seed": 295698968001001992, "version": "po-v0", "tier": 1, "n_raw": 11, "small": 1, "valid_answers": ["-5, 4, -6, 5", "-4, 3, -5, 4", "-3, 2, -4, 3", "-2, 1, -3, 2", "-1, 0, -2, 1", "0, -1, -1, 0", "1, -2, 0, -1", "2, -3, 1, -2", "3, -4, 2, -3", "4, -5, 3, -4", "5, -6, 4, -5"]}, "cell": "mat2_eigvec/1", "stage": "B"}
+{"id": "po-mat2_eigvec-t1-eval-000754", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -24 and A \\begin{pmatrix} 3 \\end{pmatrix} = 3 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -24 and A \\begin{pmatrix} 3 \\end{pmatrix} = 3 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 6, 5, -2", "ref": "-3, 6, 5, -2", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 6, "v": [3, 3], "lam": 3, "D": -24}, "seed": 295698968001001184, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 6, "valid_answers": ["-3, 6, 5, -2", "-2, 5, 6, -3"]}, "cell": "mat2_eigvec/1", "stage": "B"}
+{"id": "po-mat2_eigvec-t1-eval-001538", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 9 and A \\begin{pmatrix} -1 \\end{pmatrix} = -3 \\begin{pmatrix} -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 9 and A \\begin{pmatrix} -1 \\end{pmatrix} = -3 \\begin{pmatrix} -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -1, 4, -1", "ref": "-5, -1, 4, -1", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 6, "v": [-1, 2], "lam": -3, "D": 9}, "seed": 295698968001002958, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 3, "valid_answers": ["-5, -1, 4, -1", "-3, 0, 0, -3", "-1, 1, -4, -5"]}, "cell": "mat2_eigvec/1", "stage": "B"}
+{"id": "po-mat2_eigvec-t1-eval-001155", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 3 and A \\begin{pmatrix} -1 \\end{pmatrix} = -1 \\begin{pmatrix} -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 3 and A \\begin{pmatrix} -1 \\end{pmatrix} = -1 \\begin{pmatrix} -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 5, -3, 2", "ref": "-6, 5, -3, 2", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000315, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 6, "v": [-1, -1], "lam": -1, "D": 3}, "seed": 295698968001001970, "version": "po-v0", "tier": 1, "n_raw": 9, "small": 2, "valid_answers": ["-6, 5, -3, 2", "-5, 4, -2, 1", "-4, 3, -1, 0", "-3, 2, 0, -1", "-2, 1, 1, -2", "-1, 0, 2, -3", "0, -1, 3, -4", "1, -2, 4, -5", "2, -3, 5, -6"]}, "cell": "mat2_eigvec/1", "stage": "B"}
+{"id": "po-mat2_eigvec-t1-eval-002070", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 0 and A \\begin{pmatrix} 0 \\end{pmatrix} = -3 \\begin{pmatrix} 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 0 and A \\begin{pmatrix} 0 \\end{pmatrix} = -3 \\begin{pmatrix} 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, -6, -3", "ref": "0, 0, -6, -3", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 6, "v": [0, -3], "lam": -3, "D": 0}, "seed": 295698968001004714, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 3, "valid_answers": ["0, 0, -6, -3", "0, 0, -5, -3", "0, 0, -4, -3", "0, 0, -3, -3", "0, 0, -2, -3", "0, 0, -1, -3", "0, 0, 0, -3", "0, 0, 1, -3", "0, 0, 2, -3", "0, 0, 3, -3", "0, 0, 4, -3", "0, 0, 5, -3", "0, 0, 6, -3"]}, "cell": "mat2_eigvec/1", "stage": "B"}
+{"id": "po-mat2_eigvec-t1-eval-000387", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and A \\begin{pmatrix} 1 \\end{pmatrix} = -3 \\begin{pmatrix} 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and A \\begin{pmatrix} 1 \\end{pmatrix} = -3 \\begin{pmatrix} 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -1, 6, 3", "ref": "-4, -1, 6, 3", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.00028, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 6, "v": [1, -1], "lam": -3, "D": -6}, "seed": 295698968001000566, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 3, "valid_answers": ["-4, -1, 6, 3", "-3, 0, 5, 2", "-2, 1, 4, 1", "-1, 2, 3, 0", "0, 3, 2, -1", "1, 4, 1, -2", "2, 5, 0, -3", "3, 6, -1, -4"]}, "cell": "mat2_eigvec/1", "stage": "B"}
+{"id": "po-mat2_eigvec-t2-eval-002045", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -9 and A \\begin{pmatrix} 3 \\end{pmatrix} = 3 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -9 and A \\begin{pmatrix} 3 \\end{pmatrix} = 3 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -8, 2, 5", "ref": "-5, -8, 2, 5", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.000132, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 8, "v": [3, -3], "lam": 3, "D": -9}, "seed": 295698968002003827, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 3, "valid_answers": ["-5, -8, 2, 5", "-4, -7, 1, 4", "-3, -6, 0, 3", "-2, -5, -1, 2", "-1, -4, -2, 1", "0, -3, -3, 0", "1, -2, -4, -1", "2, -1, -5, -2", "3, 0, -6, -3", "4, 1, -7, -4", "5, 2, -8, -5"]}, "cell": "mat2_eigvec/2", "stage": "B"}
+{"id": "po-mat2_eigvec-t2-eval-001457", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 1 and A \\begin{pmatrix} -2 \\end{pmatrix} = 1 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 1 and A \\begin{pmatrix} -2 \\end{pmatrix} = 1 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 8, -2, 5", "ref": "-3, 8, -2, 5", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 8, "v": [-2, -1], "lam": 1, "D": 1}, "seed": 295698968002002440, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 1, "valid_answers": ["-3, 8, -2, 5", "-1, 4, -1, 3", "1, 0, 0, 1", "3, -4, 1, -1", "5, -8, 2, -3"]}, "cell": "mat2_eigvec/2", "stage": "B"}
+{"id": "po-mat2_eigvec-t2-eval-001684", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 33 and A \\begin{pmatrix} 2 \\end{pmatrix} = 3 \\begin{pmatrix} 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 33 and A \\begin{pmatrix} 2 \\end{pmatrix} = 3 \\begin{pmatrix} 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 3, 5, 8", "ref": "6, 3, 5, 8", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 3.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 8, "v": [2, -2], "lam": 3, "D": 33}, "seed": 295698968002002974, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 7, "valid_answers": ["6, 3, 5, 8", "7, 4, 4, 7", "8, 5, 3, 6"]}, "cell": "mat2_eigvec/2", "stage": "B"}
+{"id": "po-mat2_eigvec-t2-eval-000460", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 4 and A \\begin{pmatrix} 0 \\end{pmatrix} = 1 \\begin{pmatrix} 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 4 and A \\begin{pmatrix} 0 \\end{pmatrix} = 1 \\begin{pmatrix} 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 0, -8, 1", "ref": "4, 0, -8, 1", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 17, "bucket": "9-20", "guess_rate": 0.000204, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 8, "v": [0, -2], "lam": 1, "D": 4}, "seed": 295698968002000633, "version": "po-v0", "tier": 2, "n_raw": 17, "small": 4, "valid_answers": ["4, 0, -8, 1", "4, 0, -7, 1", "4, 0, -6, 1", "4, 0, -5, 1", "4, 0, -4, 1", "4, 0, -3, 1", "4, 0, -2, 1", "4, 0, -1, 1", "4, 0, 0, 1", "4, 0, 1, 1", "4, 0, 2, 1", "4, 0, 3, 1", "4, 0, 4, 1", "4, 0, 5, 1", "4, 0, 6, 1", "4, 0, 7, 1", "4, 0, 8, 1"]}, "cell": "mat2_eigvec/2", "stage": "B"}
+{"id": "po-mat2_eigvec-t2-eval-001244", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and A \\begin{pmatrix} 3 \\end{pmatrix} = 1 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and A \\begin{pmatrix} 3 \\end{pmatrix} = 1 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 8, -1, 2", "ref": "-7, 8, -1, 2", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.00012, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 8, "v": [3, 3], "lam": 1, "D": -6}, "seed": 295698968002001993, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 4, "valid_answers": ["-7, 8, -1, 2", "-6, 7, 0, 1", "-5, 6, 1, 0", "-4, 5, 2, -1", "-3, 4, 3, -2", "-2, 3, 4, -3", "-1, 2, 5, -4", "0, 1, 6, -5", "1, 0, 7, -6", "2, -1, 8, -7"]}, "cell": "mat2_eigvec/2", "stage": "B"}
+{"id": "po-mat2_eigvec-t2-eval-002185", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 10 and A \\begin{pmatrix} -2 \\end{pmatrix} = -1 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 10 and A \\begin{pmatrix} -2 \\end{pmatrix} = -1 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 7, 2, -3", "ref": "-8, 7, 2, -3", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 7.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 8, "v": [-2, -2], "lam": -1, "D": 10}, "seed": 295698968002004188, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 6, "valid_answers": ["-8, 7, 2, -3", "-7, 6, 3, -4", "-6, 5, 4, -5", "-5, 4, 5, -6", "-4, 3, 6, -7", "-3, 2, 7, -8"]}, "cell": "mat2_eigvec/2", "stage": "B"}
+{"id": "po-mat2_eigvec-t2-eval-001736", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 0 and A \\begin{pmatrix} 3 \\end{pmatrix} = 2 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 0 and A \\begin{pmatrix} 3 \\end{pmatrix} = 2 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 8, -6, 8", "ref": "-6, 8, -6, 8", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 0.00018, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 8, "v": [3, 3], "lam": 2, "D": 0}, "seed": 295698968002003072, "version": "po-v0", "tier": 2, "n_raw": 15, "small": 1, "valid_answers": ["-6, 8, -6, 8", "-5, 7, -5, 7", "-4, 6, -4, 6", "-3, 5, -3, 5", "-2, 4, -2, 4", "-1, 3, -1, 3", "0, 2, 0, 2", "1, 1, 1, 1", "2, 0, 2, 0", "3, -1, 3, -1", "4, -2, 4, -2", "5, -3, 5, -3", "6, -4, 6, -4", "7, -5, 7, -5", "8, -6, 8, -6"]}, "cell": "mat2_eigvec/2", "stage": "B"}
+{"id": "po-mat2_eigvec-t2-eval-000293", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and A \\begin{pmatrix} -1 \\end{pmatrix} = -2 \\begin{pmatrix} -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and A \\begin{pmatrix} -1 \\end{pmatrix} = -2 \\begin{pmatrix} -1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -1, -6, 0", "ref": "1, -1, -6, 0", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 8, "v": [-1, -3], "lam": -2, "D": -6}, "seed": 295698968002000394, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 4, "valid_answers": ["1, -1, -6, 0", "4, -2, 3, -3"]}, "cell": "mat2_eigvec/2", "stage": "B"}
+{"id": "po-mat2_eigvec-t2-eval-001803", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -3 and A \\begin{pmatrix} 3 \\end{pmatrix} = -1 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -3 and A \\begin{pmatrix} 3 \\end{pmatrix} = -1 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 3, -4, 5", "ref": "-3, 3, -4, 5", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 8, "v": [3, 2], "lam": -1, "D": -3}, "seed": 295698968002003223, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 5, "valid_answers": ["-3, 3, -4, 5", "3, -6, 0, -1"]}, "cell": "mat2_eigvec/2", "stage": "B"}
+{"id": "po-mat2_eigvec-t2-eval-001987", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and A \\begin{pmatrix} -2 \\end{pmatrix} = -2 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -6 and A \\begin{pmatrix} -2 \\end{pmatrix} = -2 \\begin{pmatrix} -2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -3, 8, 6", "ref": "-5, -3, 8, 6", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.000144, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 8, "v": [-2, 2], "lam": -2, "D": -6}, "seed": 295698968002003670, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 3, "valid_answers": ["-5, -3, 8, 6", "-4, -2, 7, 5", "-3, -1, 6, 4", "-2, 0, 5, 3", "-1, 1, 4, 2", "0, 2, 3, 1", "1, 3, 2, 0", "2, 4, 1, -1", "3, 5, 0, -2", "4, 6, -1, -3", "5, 7, -2, -4", "6, 8, -3, -5"]}, "cell": "mat2_eigvec/2", "stage": "B"}
+{"id": "po-mat2_eigvec-t2-eval-000043", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -7 and A \\begin{pmatrix} 1 \\end{pmatrix} = 1 \\begin{pmatrix} 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -7 and A \\begin{pmatrix} 1 \\end{pmatrix} = 1 \\begin{pmatrix} 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -3, 3, 2", "ref": "-8, -3, 3, 2", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 8, "v": [1, -3], "lam": 1, "D": -7}, "seed": 295698968002000058, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 6, "valid_answers": ["-8, -3, 3, 2", "-5, -2, -6, -1"]}, "cell": "mat2_eigvec/2", "stage": "B"}
+{"id": "po-mat2_eigvec-t2-eval-002025", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -7 and A \\begin{pmatrix} 3 \\end{pmatrix} = -1 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -7 and A \\begin{pmatrix} 3 \\end{pmatrix} = -1 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -3, 3, 8", "ref": "-2, -3, 3, 8", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 8, "v": [3, -1], "lam": -1, "D": -7}, "seed": 295698968002003763, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 6, "valid_answers": ["-2, -3, 3, 8", "1, 6, 2, 5"]}, "cell": "mat2_eigvec/2", "stage": "B"}
+{"id": "po-mat2_eigvec-t3-eval-001950", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -30 and A \\begin{pmatrix} 2 \\end{pmatrix} = 3 \\begin{pmatrix} 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -30 and A \\begin{pmatrix} 2 \\end{pmatrix} = 3 \\begin{pmatrix} 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -10, 0, -10", "ref": "3, -10, 0, -10", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.000108, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 10, "v": [2, 0], "lam": 3, "D": -30}, "seed": 295698968003002827, "version": "po-v0", "tier": 3, "n_raw": 21, "small": 10, "valid_answers": ["3, -10, 0, -10", "3, -9, 0, -10", "3, -8, 0, -10", "3, -7, 0, -10", "3, -6, 0, -10", "3, -5, 0, -10", "3, -4, 0, -10", "3, -3, 0, -10", "3, -2, 0, -10", "3, -1, 0, -10", "3, 0, 0, -10", "3, 1, 0, -10", "3, 2, 0, -10", "3, 3, 0, -10", "3, 4, 0, -10", "3, 5, 0, -10", "3, 6, 0, -10", "3, 7, 0, -10", "3, 8, 0, -10", "3, 9, 0, -10", "3, 10, 0, -10"]}, "cell": "mat2_eigvec/3", "stage": "B"}
+{"id": "po-mat2_eigvec-t3-eval-002365", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -24 and A \\begin{pmatrix} 3 \\end{pmatrix} = -3 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -24 and A \\begin{pmatrix} 3 \\end{pmatrix} = -3 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 3, -4, 9", "ref": "-4, 3, -4, 9", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 10, "v": [3, 1], "lam": -3, "D": -24}, "seed": 295698968003003580, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 6, "valid_answers": ["-4, 3, -4, 9", "-1, -6, -3, 6"]}, "cell": "mat2_eigvec/3", "stage": "B"}
+{"id": "po-mat2_eigvec-t3-eval-000778", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -14 and A \\begin{pmatrix} -3 \\end{pmatrix} = 2 \\begin{pmatrix} -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -14 and A \\begin{pmatrix} -3 \\end{pmatrix} = 2 \\begin{pmatrix} -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 9, 2, -1", "ref": "-4, 9, 2, -1", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 10, "v": [-3, -2], "lam": 2, "D": -14}, "seed": 295698968003001014, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 7, "valid_answers": ["-4, 9, 2, -1", "2, 0, 6, -7"]}, "cell": "mat2_eigvec/3", "stage": "B"}
+{"id": "po-mat2_eigvec-t3-eval-001158", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -9 and A \\begin{pmatrix} -3 \\end{pmatrix} = 3 \\begin{pmatrix} -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -9 and A \\begin{pmatrix} -3 \\end{pmatrix} = 3 \\begin{pmatrix} -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 9, 1, 0", "ref": "0, 9, 1, 0", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 10, "v": [-3, -1], "lam": 3, "D": -9}, "seed": 295698968003001563, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 3, "valid_answers": ["0, 9, 1, 0", "3, 0, 2, -3", "6, -9, 3, -6"]}, "cell": "mat2_eigvec/3", "stage": "B"}
+{"id": "po-mat2_eigvec-t3-eval-000460", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -18 and A \\begin{pmatrix} -3 \\end{pmatrix} = -3 \\begin{pmatrix} -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -18 and A \\begin{pmatrix} -3 \\end{pmatrix} = -3 \\begin{pmatrix} -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 9, -4, 9", "ref": "-6, 9, -4, 9", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 10, "v": [-3, -1], "lam": -3, "D": -18}, "seed": 295698968003000565, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 6, "valid_answers": ["-6, 9, -4, 9", "-3, 0, -3, 6", "0, -9, -2, 3"]}, "cell": "mat2_eigvec/3", "stage": "B"}
+{"id": "po-mat2_eigvec-t3-eval-000928", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 4 and A \\begin{pmatrix} 0 \\end{pmatrix} = -1 \\begin{pmatrix} 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 4 and A \\begin{pmatrix} 0 \\end{pmatrix} = -1 \\begin{pmatrix} 0 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 0, -10, -1", "ref": "-4, 0, -10, -1", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.000108, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 10, "v": [0, -3], "lam": -1, "D": 4}, "seed": 295698968003001229, "version": "po-v0", "tier": 3, "n_raw": 21, "small": 4, "valid_answers": ["-4, 0, -10, -1", "-4, 0, -9, -1", "-4, 0, -8, -1", "-4, 0, -7, -1", "-4, 0, -6, -1", "-4, 0, -5, -1", "-4, 0, -4, -1", "-4, 0, -3, -1", "-4, 0, -2, -1", "-4, 0, -1, -1", "-4, 0, 0, -1", "-4, 0, 1, -1", "-4, 0, 2, -1", "-4, 0, 3, -1", "-4, 0, 4, -1", "-4, 0, 5, -1", "-4, 0, 6, -1", "-4, 0, 7, -1", "-4, 0, 8, -1", "-4, 0, 9, -1", "-4, 0, 10, -1"]}, "cell": "mat2_eigvec/3", "stage": "B"}
+{"id": "po-mat2_eigvec-t3-eval-000717", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -15 and A \\begin{pmatrix} -3 \\end{pmatrix} = -3 \\begin{pmatrix} -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -15 and A \\begin{pmatrix} -3 \\end{pmatrix} = -3 \\begin{pmatrix} -3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 3, -3, 6", "ref": "-4, 3, -3, 6", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 10, "v": [-3, -1], "lam": -3, "D": -15}, "seed": 295698968003000922, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 6, "valid_answers": ["-4, 3, -3, 6", "-1, -6, -2, 3"]}, "cell": "mat2_eigvec/3", "stage": "B"}
+{"id": "po-mat2_eigvec-t3-eval-001262", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 36 and A \\begin{pmatrix} 1 \\end{pmatrix} = -3 \\begin{pmatrix} 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 36 and A \\begin{pmatrix} 1 \\end{pmatrix} = -3 \\begin{pmatrix} 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 7, 2, -5", "ref": "-10, 7, 2, -5", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 3.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 10, "v": [1, 1], "lam": -3, "D": 36}, "seed": 295698968003001716, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 8, "valid_answers": ["-10, 7, 2, -5", "-9, 6, 3, -6", "-8, 5, 4, -7", "-7, 4, 5, -8", "-6, 3, 6, -9", "-5, 2, 7, -10"]}, "cell": "mat2_eigvec/3", "stage": "B"}
+{"id": "po-mat2_eigvec-t3-eval-000185", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 18 and A \\begin{pmatrix} 2 \\end{pmatrix} = 2 \\begin{pmatrix} 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 18 and A \\begin{pmatrix} 2 \\end{pmatrix} = 2 \\begin{pmatrix} 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 2, -4, 10", "ref": "1, 2, -4, 10", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 2.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 10, "v": [2, 1], "lam": 2, "D": 18}, "seed": 295698968003000218, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 6, "valid_answers": ["1, 2, -4, 10", "3, -2, -3, 8", "5, -6, -2, 6", "7, -10, -1, 4"]}, "cell": "mat2_eigvec/3", "stage": "B"}
+{"id": "po-mat2_eigvec-t3-eval-002499", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 9 and A \\begin{pmatrix} 2 \\end{pmatrix} = 1 \\begin{pmatrix} 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 9 and A \\begin{pmatrix} 2 \\end{pmatrix} = 1 \\begin{pmatrix} 2 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 0, -4, 9", "ref": "1, 0, -4, 9", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 10, "v": [2, 1], "lam": 1, "D": 9}, "seed": 295698968003003868, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 7, "valid_answers": ["1, 0, -4, 9", "3, -4, -3, 7", "5, -8, -2, 5"]}, "cell": "mat2_eigvec/3", "stage": "B"}
+{"id": "po-mat2_eigvec-t3-eval-000420", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -7 and A \\begin{pmatrix} 3 \\end{pmatrix} = 1 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -7 and A \\begin{pmatrix} 3 \\end{pmatrix} = 1 \\begin{pmatrix} 3 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -10, 2, 3", "ref": "-9, -10, 2, 3", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 6.7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 10, "v": [3, -3], "lam": 1, "D": -7}, "seed": 295698968003000518, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 4, "valid_answers": ["-9, -10, 2, 3", "-8, -9, 1, 2", "-7, -8, 0, 1", "-6, -7, -1, 0", "-5, -6, -2, -1", "-4, -5, -3, -2", "-3, -4, -4, -3", "-2, -3, -5, -4", "-1, -2, -6, -5", "0, -1, -7, -6", "1, 0, -8, -7", "2, 1, -9, -8", "3, 2, -10, -9"]}, "cell": "mat2_eigvec/3", "stage": "B"}
+{"id": "po-mat2_eigvec-t3-eval-002118", "task": "cx_po_mat2_eigvec", "source": "cx_po_mat2_eigvec", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 2 and A \\begin{pmatrix} 1 \\end{pmatrix} = 1 \\begin{pmatrix} 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 2 and A \\begin{pmatrix} 1 \\end{pmatrix} = 1 \\begin{pmatrix} 1 \\end{pmatrix}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -8, 9, 10", "ref": "-7, -8, 9, 10", "family": "po_mat2_eigvec", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 9.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_eigvec", "rung": "witness", "params": {"B": 10, "v": [1, -1], "lam": 1, "D": 2}, "seed": 295698968003003123, "version": "po-v0", "tier": 3, "n_raw": 18, "small": 2, "valid_answers": ["-7, -8, 9, 10", "-6, -7, 8, 9", "-5, -6, 7, 8", "-4, -5, 6, 7", "-3, -4, 5, 6", "-2, -3, 4, 5", "-1, -2, 3, 4", "0, -1, 2, 3", "1, 0, 1, 2", "2, 1, 0, 1", "3, 2, -1, 0", "4, 3, -2, -1", "5, 4, -3, -2", "6, 5, -4, -3", "7, 6, -5, -4", "8, 7, -6, -5", "9, 8, -7, -6", "10, 9, -8, -7"]}, "cell": "mat2_eigvec/3", "stage": "B"}
+{"id": "po-mat2_idempotent-t0-eval-000035", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 2a = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 2a = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -2, 1, 2", "ref": "-1, -2, 1, 2", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001666, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 3, "rel": [2, 0, 0, 0], "s": -2}, "seed": 354602262700000088, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 2, "valid_answers": ["-1, -2, 1, 2", "-1, -1, 2, 2", "-1, 1, -2, 2", "-1, 2, -1, 2"]}, "cell": "mat2_idempotent/0", "stage": "B"}
+{"id": "po-mat2_idempotent-t1-eval-000385", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and a + 2c = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and a + 2c = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -5, 4, 5", "ref": "-4, -5, 4, 5", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 5, "rel": [1, 0, 2, 0], "s": 4}, "seed": 354602262701001003, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 2, "valid_answers": ["-4, -5, 4, 5", "-2, -2, 3, 3", "0, 0, 2, 1", "2, -2, 1, -1"]}, "cell": "mat2_idempotent/1", "stage": "B"}
+{"id": "po-mat2_idempotent-t1-eval-000148", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and a + 3c = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and a + 3c = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -5, 4, 5", "ref": "-4, -5, 4, 5", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 5, "rel": [1, 0, 3, 0], "s": 8}, "seed": 354602262701000302, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 2, "valid_answers": ["-4, -5, 4, 5", "2, -1, 2, -1"]}, "cell": "mat2_idempotent/1", "stage": "B"}
+{"id": "po-mat2_idempotent-t1-eval-000823", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2a + 3b = 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2a + 3b = 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 2, -3, 3", "ref": "-2, 2, -3, 3", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 5, "rel": [-2, 3, 0, 0], "s": 10}, "seed": 354602262701005617, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 3, "valid_answers": ["-2, 2, -3, 3", "1, 4, 0, 0"]}, "cell": "mat2_idempotent/1", "stage": "B"}
+{"id": "po-mat2_idempotent-t1-eval-000875", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2b + c = -11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2b + c = -11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 4, -3, 4", "ref": "-3, 4, -3, 4", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 5, "rel": [0, -2, 1, 0], "s": -11}, "seed": 354602262701009727, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["-3, 4, -3, 4", "4, 4, -3, -3"]}, "cell": "mat2_idempotent/1", "stage": "B"}
+{"id": "po-mat2_idempotent-t1-eval-000199", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and b + 2c = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and b + 2c = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -3, 4, 4", "ref": "-3, -3, 4, 4", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 5, "rel": [0, 1, 2, 0], "s": 5}, "seed": 354602262701000429, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 4, "valid_answers": ["-3, -3, 4, 4", "0, 5, 0, 1", "1, 5, 0, 0", "4, -3, 4, -3"]}, "cell": "mat2_idempotent/1", "stage": "B"}
+{"id": "po-mat2_idempotent-t1-eval-000675", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and b + c = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and b + c = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, 3, 1", "ref": "0, 0, 3, 1", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 5, "rel": [0, 1, 1, 0], "s": 3}, "seed": 354602262701002854, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 3, "valid_answers": ["0, 0, 3, 1", "0, 3, 0, 1", "1, 0, 3, 0", "1, 3, 0, 0"]}, "cell": "mat2_idempotent/1", "stage": "B"}
+{"id": "po-mat2_idempotent-t1-eval-000603", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and a + 2c = -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and a + 2c = -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 4, -3, 4", "ref": "-3, 4, -3, 4", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 5, "rel": [1, 0, 2, 0], "s": -9}, "seed": 354602262701002158, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["-3, 4, -3, 4", "1, 0, -5, 0"]}, "cell": "mat2_idempotent/1", "stage": "B"}
+{"id": "po-mat2_idempotent-t1-eval-000256", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 3a - 2c = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 3a - 2c = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -1, 2, 2", "ref": "-1, -1, 2, 2", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 5, "rel": [3, 0, -2, 0], "s": -7}, "seed": 354602262701000563, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 2, "valid_answers": ["-1, -1, 2, 2", "1, 0, 5, 0"]}, "cell": "mat2_idempotent/1", "stage": "B"}
+{"id": "po-mat2_idempotent-t1-eval-000658", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 3a - 2b = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 3a - 2b = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -2, 0, 1", "ref": "0, -2, 0, 1", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000205, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 5, "rel": [3, -2, 0, 0], "s": 4}, "seed": 354602262701002707, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 2, "valid_answers": ["0, -2, 0, 1", "2, 1, -2, -1", "4, 4, -3, -3"]}, "cell": "mat2_idempotent/1", "stage": "B"}
+{"id": "po-mat2_idempotent-t1-eval-000669", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and b - 2c = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and b - 2c = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 2, -1, 2", "ref": "-1, 2, -1, 2", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.00041, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 5, "rel": [0, 1, -2, 0], "s": 4}, "seed": 354602262701002807, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 2, "valid_answers": ["-1, 2, -1, 2", "0, 0, -2, 1", "0, 4, 0, 1", "1, 0, -2, 0", "1, 4, 0, 0", "2, 2, -1, -1"]}, "cell": "mat2_idempotent/1", "stage": "B"}
+{"id": "po-mat2_idempotent-t1-eval-000232", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2b - c = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2b - c = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -3, 2, 3", "ref": "-2, -3, 2, 3", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.00041, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 5, "rel": [0, -2, -1, 0], "s": 4}, "seed": 354602262701000501, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 2, "valid_answers": ["-2, -3, 2, 3", "0, -2, 0, 1", "0, 0, -4, 1", "1, -2, 0, 0", "1, 0, -4, 0", "3, -3, 2, -2"]}, "cell": "mat2_idempotent/1", "stage": "B"}
+{"id": "po-mat2_idempotent-t1-eval-000341", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -b + 2c = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -b + 2c = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, 3, 1", "ref": "0, 0, 3, 1", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 5, "rel": [0, -1, 2, 0], "s": 6}, "seed": 354602262701000804, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 3, "valid_answers": ["0, 0, 3, 1", "1, 0, 3, 0"]}, "cell": "mat2_idempotent/1", "stage": "B"}
+{"id": "po-mat2_idempotent-t2-eval-001180", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 3b + 2c = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 3b + 2c = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -6, 0, 1", "ref": "0, -6, 0, 1", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 8, "rel": [0, 3, 2, 0], "s": -18}, "seed": 354602262702003809, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 6, "valid_answers": ["0, -6, 0, 1", "1, -6, 0, 0"]}, "cell": "mat2_idempotent/2", "stage": "B"}
+{"id": "po-mat2_idempotent-t2-eval-000902", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 2b - c = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 2b - c = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 6, 0, 1", "ref": "0, 6, 0, 1", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 8, "rel": [0, 2, -1, 0], "s": 12}, "seed": 354602262702002334, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 6, "valid_answers": ["0, 6, 0, 1", "1, 6, 0, 0"]}, "cell": "mat2_idempotent/2", "stage": "B"}
+{"id": "po-mat2_idempotent-t2-eval-001345", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and a + 2c = -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and a + 2c = -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 4, -3, 4", "ref": "-3, 4, -3, 4", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 3.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 8, "rel": [1, 0, 2, 0], "s": -9}, "seed": 354602262702005046, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 4, "valid_answers": ["-3, 4, -3, 4", "1, 0, -5, 0", "3, 1, -6, -2"]}, "cell": "mat2_idempotent/2", "stage": "B"}
+{"id": "po-mat2_idempotent-t2-eval-000398", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -a + 3c = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -a + 3c = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 2, -3, 3", "ref": "-2, 2, -3, 3", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 8, "rel": [-1, 0, 3, 0], "s": -7}, "seed": 354602262702000750, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 2, "valid_answers": ["-2, 2, -3, 3", "1, 0, -2, 0"]}, "cell": "mat2_idempotent/2", "stage": "B"}
+{"id": "po-mat2_idempotent-t2-eval-000960", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 3b - c = -21. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 3b - c = -21. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -5, 6, 6", "ref": "-5, -5, 6, 6", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 8, "rel": [0, 3, -1, 0], "s": -21}, "seed": 354602262702002577, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 6, "valid_answers": ["-5, -5, 6, 6", "0, -7, 0, 1", "1, -7, 0, 0", "6, -5, 6, -5"]}, "cell": "mat2_idempotent/2", "stage": "B"}
+{"id": "po-mat2_idempotent-t2-eval-001186", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 3a + c = 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 3a + c = 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 0, 8, 0", "ref": "1, 0, 8, 0", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 3.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 8, "rel": [3, 0, 1, 0], "s": 11}, "seed": 354602262702003860, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 3, "valid_answers": ["1, 0, 8, 0", "3, -3, 2, -2", "5, 5, -4, -4"]}, "cell": "mat2_idempotent/2", "stage": "B"}
+{"id": "po-mat2_idempotent-t2-eval-000253", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2a + b = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2a + b = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -6, 0, 1", "ref": "0, -6, 0, 1", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 7.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 8, "rel": [-2, 1, 0, 0], "s": -6}, "seed": 354602262702000449, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 2, "valid_answers": ["0, -6, 0, 1", "1, -4, 0, 0", "2, -2, 1, -1", "4, 2, -6, -3", "5, 4, -5, -4", "6, 6, -5, -5"]}, "cell": "mat2_idempotent/2", "stage": "B"}
+{"id": "po-mat2_idempotent-t2-eval-001258", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and b + c = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and b + c = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -3, 0, 1", "ref": "0, -3, 0, 1", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 8, "rel": [0, 1, 1, 0], "s": -3}, "seed": 354602262702004311, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 3, "valid_answers": ["0, -3, 0, 1", "0, 0, -3, 1", "1, -3, 0, 0", "1, 0, -3, 0"]}, "cell": "mat2_idempotent/2", "stage": "B"}
+{"id": "po-mat2_idempotent-t2-eval-000573", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2a + 3b = 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2a + 3b = 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 2, -3, 3", "ref": "-2, 2, -3, 3", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 3.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 8, "rel": [-2, 3, 0, 0], "s": 10}, "seed": 354602262702001216, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 3, "valid_answers": ["-2, 2, -3, 3", "1, 4, 0, 0", "4, 6, -2, -3"]}, "cell": "mat2_idempotent/2", "stage": "B"}
+{"id": "po-mat2_idempotent-t2-eval-000783", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2a - 2b = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2a - 2b = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -1, 6, 3", "ref": "-2, -1, 6, 3", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 8, "rel": [-2, -2, 0, 0], "s": 6}, "seed": 354602262702001823, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 2, "valid_answers": ["-2, -1, 6, 3", "-1, -2, 1, 2", "0, -3, 0, 1", "1, -4, 0, 0", "3, -6, 1, -2"]}, "cell": "mat2_idempotent/2", "stage": "B"}
+{"id": "po-mat2_idempotent-t2-eval-000001", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 2a + c = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 2a + c = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -4, 5, 5", "ref": "-4, -4, 5, 5", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 7.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 8, "rel": [2, 0, 1, 0], "s": -3}, "seed": 354602262702000002, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 2, "valid_answers": ["-4, -4, 5, 5", "-3, -4, 3, 4", "-2, -6, 1, 3", "-1, 2, -1, 2", "0, 0, -3, 1", "1, 0, -5, 0"]}, "cell": "mat2_idempotent/2", "stage": "B"}
+{"id": "po-mat2_idempotent-t2-eval-001170", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 3a + 2c = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 3a + 2c = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, 2, 1", "ref": "0, 0, 2, 1", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 3.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 8, "rel": [3, 0, 2, 0], "s": 4}, "seed": 354602262702003753, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 2, "valid_answers": ["0, 0, 2, 1", "2, 2, -1, -1", "4, 3, -4, -3"]}, "cell": "mat2_idempotent/2", "stage": "B"}
+{"id": "po-mat2_idempotent-t3-eval-001697", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 2a + 2b + 3c = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 2a + 2b + 3c = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 10, 0, 1", "ref": "0, 10, 0, 1", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 12, "rel": [2, 2, 3, 0], "s": 20}, "seed": 354602262703002522, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 6, "valid_answers": ["0, 10, 0, 1", "1, 0, 6, 0", "1, 9, 0, 0", "6, -5, 6, -5"]}, "cell": "mat2_idempotent/3", "stage": "B"}
+{"id": "po-mat2_idempotent-t3-eval-001292", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -a + 3b + 3c = 35. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -a + 3b + 3c = 35. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 0, 12, 0", "ref": "1, 0, 12, 0", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 12, "rel": [-1, 3, 3, 0], "s": 35}, "seed": 354602262703001828, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 12, "valid_answers": ["1, 0, 12, 0", "1, 12, 0, 0"]}, "cell": "mat2_idempotent/3", "stage": "B"}
+{"id": "po-mat2_idempotent-t3-eval-000020", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 3a - 2b - 2c = -25. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 3a - 2b - 2c = -25. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -10, 9, 10", "ref": "-9, -10, 9, 10", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 12, "rel": [3, -2, -2, 0], "s": -25}, "seed": 354602262703000021, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 10, "valid_answers": ["-9, -10, 9, 10", "-9, 9, -10, 10"]}, "cell": "mat2_idempotent/3", "stage": "B"}
+{"id": "po-mat2_idempotent-t3-eval-000624", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 2a + 2b + c = -17. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 2a + 2b + c = -17. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -6, 7, 7", "ref": "-6, -6, 7, 7", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 12, "rel": [2, 2, 1, 0], "s": -17}, "seed": 354602262703000807, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 6, "valid_answers": ["-6, -6, 7, 7", "-5, -6, 5, 6"]}, "cell": "mat2_idempotent/3", "stage": "B"}
+{"id": "po-mat2_idempotent-t3-eval-000813", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -a - b - c = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -a - b - c = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -6, 12, 9", "ref": "-8, -6, 12, 9", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 3.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 12, "rel": [-1, -1, -1, 0], "s": 2}, "seed": 354602262703001080, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 2, "valid_answers": ["-8, -6, 12, 9", "-8, 12, -6, 9", "-3, -3, 4, 4", "-3, 4, -3, 4", "-1, -2, 1, 2", "-1, 1, -2, 2", "0, -2, 0, 1", "0, 0, -2, 1", "1, -3, 0, 0", "1, 0, -3, 0", "3, -6, 1, -2", "3, 1, -6, -2"]}, "cell": "mat2_idempotent/3", "stage": "B"}
+{"id": "po-mat2_idempotent-t3-eval-000295", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 2a + 3b + c = 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and 2a + 3b + c = 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 6, -1, 3", "ref": "-2, 6, -1, 3", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 1.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 12, "rel": [2, 3, 1, 0], "s": 13}, "seed": 354602262703000371, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 3, "valid_answers": ["-2, 6, -1, 3", "1, 0, 11, 0", "3, 3, -2, -2", "4, 3, -4, -3", "6, -3, 10, -5"]}, "cell": "mat2_idempotent/3", "stage": "B"}
+{"id": "po-mat2_idempotent-t3-eval-002094", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -a - 2b - 2c = 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -a - 2b - 2c = 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -7, 0, 1", "ref": "0, -7, 0, 1", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 12, "rel": [-1, -2, -2, 0], "s": 14}, "seed": 354602262703003326, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 7, "valid_answers": ["0, -7, 0, 1", "0, 0, -7, 1"]}, "cell": "mat2_idempotent/3", "stage": "B"}
+{"id": "po-mat2_idempotent-t3-eval-001750", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and a - 2b - c = 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and a - 2b - c = 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, -11, 1", "ref": "0, 0, -11, 1", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 1.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 12, "rel": [1, -2, -1, 0], "s": 11}, "seed": 354602262703002619, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 5, "valid_answers": ["0, 0, -11, 1", "1, -5, 0, 0", "1, 0, -10, 0", "5, -5, 4, -4", "5, 2, -10, -4"]}, "cell": "mat2_idempotent/3", "stage": "B"}
+{"id": "po-mat2_idempotent-t3-eval-000829", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2a + b + c = -13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2a + b + c = -13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -11, 0, 0", "ref": "1, -11, 0, 0", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 12, "rel": [-2, 1, 1, 0], "s": -13}, "seed": 354602262703001101, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 6, "valid_answers": ["1, -11, 0, 0", "1, 0, -11, 0", "6, -6, 5, -5", "6, 5, -6, -5", "7, -6, 7, -6", "7, 7, -6, -6"]}, "cell": "mat2_idempotent/3", "stage": "B"}
+{"id": "po-mat2_idempotent-t3-eval-001526", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2a + 3b - c = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2a + 3b - c = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, -5, 1", "ref": "0, 0, -5, 1", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 12, "rel": [-2, 3, -1, 0], "s": 5}, "seed": 354602262703002221, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 3, "valid_answers": ["0, 0, -5, 1", "1, 0, -7, 0", "3, 3, -2, -2", "4, 3, -4, -3"]}, "cell": "mat2_idempotent/3", "stage": "B"}
+{"id": "po-mat2_idempotent-t3-eval-001681", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -a + 2b + 2c = 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -a + 2b + 2c = 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-11, -11, 12, 12", "ref": "-11, -11, 12, 12", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 12, "rel": [-1, 2, 2, 0], "s": 13}, "seed": 354602262703002494, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 7, "valid_answers": ["-11, -11, 12, 12", "-11, 12, -11, 12", "1, 0, 7, 0", "1, 7, 0, 0"]}, "cell": "mat2_idempotent/3", "stage": "B"}
+{"id": "po-mat2_idempotent-t3-eval-000409", "task": "cx_po_mat2_idempotent", "source": "cx_po_mat2_idempotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2a + b + 3c = -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = A, A \\ne 0 and A \\ne I and -2a + b + 3c = -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -9, 0, 1", "ref": "0, -9, 0, 1", "family": "po_mat2_idempotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_idempotent", "rung": "witness", "params": {"B": 12, "rel": [-2, 1, 3, 0], "s": -9}, "seed": 354602262703000510, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 2, "valid_answers": ["0, -9, 0, 1", "0, 0, -3, 1", "1, -7, 0, 0", "2, 1, -2, -1", "3, -6, 1, -2", "3, 3, -2, -2"]}, "cell": "mat2_idempotent/3", "stage": "B"}
+{"id": "po-mat2_involution-t0-eval-000046", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -b = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -b = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 2, 0, 1", "ref": "-1, 2, 0, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000833, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 3, "rel": [0, -1, 0, 0], "s": -2}, "seed": 243261926600000432, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["-1, 2, 0, 1", "1, 2, 0, -1"]}, "cell": "mat2_involution/0", "stage": "C"}
+{"id": "po-mat2_involution-t0-eval-000040", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 3c = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 3c = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, 2, 1", "ref": "-1, 0, 2, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000833, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 3, "rel": [0, 0, 3, 0], "s": 6}, "seed": 243261926600000324, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["-1, 0, 2, 1", "1, 0, 2, -1"]}, "cell": "mat2_involution/0", "stage": "C"}
+{"id": "po-mat2_involution-t0-eval-000025", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and c = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and c = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -1, 3, 2", "ref": "-2, -1, 3, 2", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001666, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 3, "rel": [0, 0, 1, 0], "s": 3}, "seed": 243261926600000140, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["-2, -1, 3, 2", "-1, 0, 3, 1", "1, 0, 3, -1", "2, -1, 3, -2"]}, "cell": "mat2_involution/0", "stage": "C"}
+{"id": "po-mat2_involution-t0-eval-000002", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -c = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -c = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 1, -3, 2", "ref": "-2, 1, -3, 2", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001666, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 3, "rel": [0, 0, -1, 0], "s": 3}, "seed": 243261926600000018, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["-2, 1, -3, 2", "-1, 0, -3, 1", "1, 0, -3, -1", "2, 1, -3, -2"]}, "cell": "mat2_involution/0", "stage": "C"}
+{"id": "po-mat2_involution-t0-eval-000047", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and b = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and b = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -2, 0, 1", "ref": "-1, -2, 0, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000833, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 3, "rel": [0, 1, 0, 0], "s": -2}, "seed": 243261926600000443, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["-1, -2, 0, 1", "1, -2, 0, -1"]}, "cell": "mat2_involution/0", "stage": "C"}
+{"id": "po-mat2_involution-t0-eval-000037", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and c = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and c = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, -2, 1", "ref": "-1, 0, -2, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000833, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 3, "rel": [0, 0, 1, 0], "s": -2}, "seed": 243261926600000219, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["-1, 0, -2, 1", "1, 0, -2, -1"]}, "cell": "mat2_involution/0", "stage": "C"}
+{"id": "po-mat2_involution-t0-eval-000010", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -3, 1, 2", "ref": "-2, -3, 1, 2", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001666, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 3, "rel": [-1, 0, 0, 0], "s": 2}, "seed": 243261926600000039, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["-2, -3, 1, 2", "-2, -1, 3, 2", "-2, 1, -3, 2", "-2, 3, -1, 2"]}, "cell": "mat2_involution/0", "stage": "C"}
+{"id": "po-mat2_involution-t0-eval-000017", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 3c = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 3c = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, -2, 1", "ref": "-1, 0, -2, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000833, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 3, "rel": [0, 0, 3, 0], "s": -6}, "seed": 243261926600000068, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["-1, 0, -2, 1", "1, 0, -2, -1"]}, "cell": "mat2_involution/0", "stage": "C"}
+{"id": "po-mat2_involution-t1-eval-000020", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 2c = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 2c = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 3, -1, 2", "ref": "-2, 3, -1, 2", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 5, "rel": [-1, 0, 2, 0], "s": 0}, "seed": 243261926601000043, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 3, "valid_answers": ["-2, 3, -1, 2", "2, -3, 1, -2"]}, "cell": "mat2_involution/1", "stage": "C"}
+{"id": "po-mat2_involution-t1-eval-000140", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2b - c = -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2b - c = -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -1, 3, 2", "ref": "-2, -1, 3, 2", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 5, "rel": [0, 2, -1, 0], "s": -5}, "seed": 243261926601000326, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 3, "valid_answers": ["-2, -1, 3, 2", "-1, 0, 5, 1", "1, 0, 5, -1", "2, -1, 3, -2"]}, "cell": "mat2_involution/1", "stage": "C"}
+{"id": "po-mat2_involution-t1-eval-000337", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2b + c = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2b + c = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, 4, 1", "ref": "-1, 0, 4, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 5, "rel": [0, 2, 1, 0], "s": 4}, "seed": 243261926601000951, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 2, "valid_answers": ["-1, 0, 4, 1", "-1, 2, 0, 1", "1, 0, 4, -1", "1, 2, 0, -1"]}, "cell": "mat2_involution/1", "stage": "C"}
+{"id": "po-mat2_involution-t1-eval-000044", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2b - 2c = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2b - 2c = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -3, 1, 2", "ref": "-2, -3, 1, 2", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000546, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 5, "rel": [0, 2, -2, 0], "s": -8}, "seed": 243261926601000106, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 3, "valid_answers": ["-2, -3, 1, 2", "-2, -1, 3, 2", "-1, -4, 0, 1", "-1, 0, 4, 1", "1, -4, 0, -1", "1, 0, 4, -1", "2, -3, 1, -2", "2, -1, 3, -2"]}, "cell": "mat2_involution/1", "stage": "C"}
+{"id": "po-mat2_involution-t1-eval-000000", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -b + 3c = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -b + 3c = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -3, 1, 2", "ref": "-2, -3, 1, 2", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 5, "rel": [0, -1, 3, 0], "s": 6}, "seed": 243261926601000003, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 2, "valid_answers": ["-2, -3, 1, 2", "-1, 0, 2, 1", "1, 0, 2, -1", "2, -3, 1, -2"]}, "cell": "mat2_involution/1", "stage": "C"}
+{"id": "po-mat2_involution-t1-eval-000317", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -2b - c = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -2b - c = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -4, 2, 3", "ref": "-3, -4, 2, 3", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 5, "rel": [0, -2, -1, 0], "s": 6}, "seed": 243261926601000879, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 3, "valid_answers": ["-3, -4, 2, 3", "-1, -3, 0, 1", "1, -3, 0, -1", "3, -4, 2, -3"]}, "cell": "mat2_involution/1", "stage": "C"}
+{"id": "po-mat2_involution-t1-eval-000510", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2b - 2c = 16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2b - 2c = 16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 3, -5, 4", "ref": "-4, 3, -5, 4", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 5, "rel": [0, 2, -2, 0], "s": 16}, "seed": 243261926601001799, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 5, "valid_answers": ["-4, 3, -5, 4", "-4, 5, -3, 4", "4, 3, -5, -4", "4, 5, -3, -4"]}, "cell": "mat2_involution/1", "stage": "C"}
+{"id": "po-mat2_involution-t1-eval-000682", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 2c = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 2c = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, 4, 1", "ref": "-1, 0, 4, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 5, "rel": [-1, 0, 2, 0], "s": 9}, "seed": 243261926601004030, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["-1, 0, 4, 1", "1, 0, 5, -1"]}, "cell": "mat2_involution/1", "stage": "C"}
+{"id": "po-mat2_involution-t1-eval-000594", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 2c = -11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 2c = -11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 0, -5, -1", "ref": "1, 0, -5, -1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 5, "rel": [-1, 0, 2, 0], "s": -11}, "seed": 243261926601002475, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["1, 0, -5, -1", "3, 2, -4, -3"]}, "cell": "mat2_involution/1", "stage": "C"}
+{"id": "po-mat2_involution-t1-eval-000121", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -2a + 2c = -10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -2a + 2c = -10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 0, -4, -1", "ref": "1, 0, -4, -1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000205, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 5, "rel": [-2, 0, 2, 0], "s": -10}, "seed": 243261926601000273, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 3, "valid_answers": ["1, 0, -4, -1", "2, 1, -3, -2", "3, 4, -2, -3"]}, "cell": "mat2_involution/1", "stage": "C"}
+{"id": "po-mat2_involution-t1-eval-000307", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -2a - 2c = -14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -2a - 2c = -14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -2, 4, -3", "ref": "3, -2, 4, -3", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 5, "rel": [-2, 0, -2, 0], "s": -14}, "seed": 243261926601000852, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["3, -2, 4, -3", "4, -5, 3, -4"]}, "cell": "mat2_involution/1", "stage": "C"}
+{"id": "po-mat2_involution-t1-eval-000438", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 2b = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 2b = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -1, 3, 2", "ref": "-2, -1, 3, 2", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 5, "rel": [-1, 2, 0, 0], "s": 0}, "seed": 243261926601001416, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 3, "valid_answers": ["-2, -1, 3, 2", "2, 1, -3, -2"]}, "cell": "mat2_involution/1", "stage": "C"}
+{"id": "po-mat2_involution-t2-eval-000757", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and b - c = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and b - c = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -7, 0, 1", "ref": "-1, -7, 0, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 8, "rel": [0, 1, -1, 0], "s": -7}, "seed": 243261926602001716, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 7, "valid_answers": ["-1, -7, 0, 1", "-1, 0, 7, 1", "1, -7, 0, -1", "1, 0, 7, -1"]}, "cell": "mat2_involution/2", "stage": "C"}
+{"id": "po-mat2_involution-t2-eval-000495", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 3b + c = -23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 3b + c = -23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -8, 1, 3", "ref": "-3, -8, 1, 3", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 8, "rel": [0, 3, 1, 0], "s": -23}, "seed": 243261926602000957, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 8, "valid_answers": ["-3, -8, 1, 3", "3, -8, 1, -3"]}, "cell": "mat2_involution/2", "stage": "C"}
+{"id": "po-mat2_involution-t2-eval-001514", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 3a + 3b = -27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 3a + 3b = -27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -4, 6, 5", "ref": "-5, -4, 6, 5", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 3.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 8, "rel": [3, 3, 0, 0], "s": -27}, "seed": 243261926602007742, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 5, "valid_answers": ["-5, -4, 6, 5", "-4, -5, 3, 4", "-1, -8, 0, 1"]}, "cell": "mat2_involution/2", "stage": "C"}
+{"id": "po-mat2_involution-t2-eval-000176", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and a + 3b = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and a + 3b = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -2, 0, 1", "ref": "-1, -2, 0, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 3.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 8, "rel": [1, 3, 0, 0], "s": -7}, "seed": 243261926602000290, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 2, "valid_answers": ["-1, -2, 0, 1", "2, -3, 1, -2", "5, -4, 6, -5"]}, "cell": "mat2_involution/2", "stage": "C"}
+{"id": "po-mat2_involution-t2-eval-000076", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 3b - 2c = -24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 3b - 2c = -24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -4, 6, 5", "ref": "-5, -4, 6, 5", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 8, "rel": [0, 3, -2, 0], "s": -24}, "seed": 243261926602000120, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 6, "valid_answers": ["-5, -4, 6, 5", "-1, -8, 0, 1", "1, -8, 0, -1", "5, -4, 6, -5"]}, "cell": "mat2_involution/2", "stage": "C"}
+{"id": "po-mat2_involution-t2-eval-000154", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2a - 2c = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2a - 2c = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -6, 4, 5", "ref": "-5, -6, 4, 5", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 3.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 8, "rel": [2, 0, -2, 0], "s": -18}, "seed": 243261926602000239, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 5, "valid_answers": ["-5, -6, 4, 5", "-4, -3, 5, 4", "-1, 0, 8, 1"]}, "cell": "mat2_involution/2", "stage": "C"}
+{"id": "po-mat2_involution-t2-eval-000699", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and b + c = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and b + c = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, 3, 1", "ref": "-1, 0, 3, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 8, "rel": [0, 1, 1, 0], "s": 3}, "seed": 243261926602001536, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 3, "valid_answers": ["-1, 0, 3, 1", "-1, 3, 0, 1", "1, 0, 3, -1", "1, 3, 0, -1"]}, "cell": "mat2_involution/2", "stage": "C"}
+{"id": "po-mat2_involution-t2-eval-000801", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -b + c = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -b + c = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, -7, 1", "ref": "-1, 0, -7, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 8, "rel": [0, -1, 1, 0], "s": -7}, "seed": 243261926602001868, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 7, "valid_answers": ["-1, 0, -7, 1", "-1, 7, 0, 1", "1, 0, -7, -1", "1, 7, 0, -1"]}, "cell": "mat2_involution/2", "stage": "C"}
+{"id": "po-mat2_involution-t2-eval-001360", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -2a + 3b = -23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -2a + 3b = -23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -7, 0, -1", "ref": "1, -7, 0, -1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 8, "rel": [-2, 3, 0, 0], "s": -23}, "seed": 243261926602005065, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 5, "valid_answers": ["1, -7, 0, -1", "4, -5, 3, -4"]}, "cell": "mat2_involution/2", "stage": "C"}
+{"id": "po-mat2_involution-t2-eval-000879", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 3a + b = -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 3a + b = -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 6, -4, 5", "ref": "-5, 6, -4, 5", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 8, "rel": [3, 1, 0, 0], "s": -9}, "seed": 243261926602002175, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 3, "valid_answers": ["-5, 6, -4, 5", "-4, 3, -5, 4", "-2, -3, 1, 2", "-1, -6, 0, 1"]}, "cell": "mat2_involution/2", "stage": "C"}
+{"id": "po-mat2_involution-t2-eval-000249", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -2a - c = 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -2a - c = 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 2, -4, 3", "ref": "-3, 2, -4, 3", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 8, "rel": [-2, 0, -1, 0], "s": 10}, "seed": 243261926602000434, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 4, "valid_answers": ["-3, 2, -4, 3", "-1, 0, -8, 1"]}, "cell": "mat2_involution/2", "stage": "C"}
+{"id": "po-mat2_involution-t2-eval-001347", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 2b = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 2b = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 3, -1, 2", "ref": "-2, 3, -1, 2", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 8, "rel": [-1, 2, 0, 0], "s": 8}, "seed": 243261926602004923, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 3, "valid_answers": ["-2, 3, -1, 2", "6, 7, -5, -6"]}, "cell": "mat2_involution/2", "stage": "C"}
+{"id": "po-mat2_involution-t3-eval-002152", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 3b + 3c = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 3b + 3c = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -3, 8, 5", "ref": "-5, -3, 8, 5", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 12, "rel": [-1, 3, 3, 0], "s": 20}, "seed": 243261926603003939, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 7, "valid_answers": ["-5, -3, 8, 5", "-5, 8, -3, 5", "1, 0, 7, -1", "1, 7, 0, -1"]}, "cell": "mat2_involution/3", "stage": "C"}
+{"id": "po-mat2_involution-t3-eval-000720", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2a - 2b + 2c = -60. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2a - 2b + 2c = -60. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 9, -11, 10", "ref": "-10, 9, -11, 10", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 12, "rel": [2, -2, 2, 0], "s": -60}, "seed": 243261926603001060, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 11, "valid_answers": ["-10, 9, -11, 10", "-10, 11, -9, 10"]}, "cell": "mat2_involution/3", "stage": "C"}
+{"id": "po-mat2_involution-t3-eval-002261", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and a - b + 3c = -11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and a - b + 3c = -11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 10, 0, 1", "ref": "-1, 10, 0, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 12, "rel": [1, -1, 3, 0], "s": -11}, "seed": 243261926603004221, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 4, "valid_answers": ["-1, 10, 0, 1", "1, 0, -4, -1", "1, 12, 0, -1", "3, 2, -4, -3"]}, "cell": "mat2_involution/3", "stage": "C"}
+{"id": "po-mat2_involution-t3-eval-002345", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 3b + 2c = -25. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 3b + 2c = -25. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -8, 0, -1", "ref": "1, -8, 0, -1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 12, "rel": [-1, 3, 2, 0], "s": -25}, "seed": 243261926603004495, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 8, "valid_answers": ["1, -8, 0, -1", "1, 0, -12, -1", "3, -8, 1, -3", "10, -11, 9, -10"]}, "cell": "mat2_involution/3", "stage": "C"}
+{"id": "po-mat2_involution-t3-eval-001923", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -2a + 2b + 2c = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -2a + 2b + 2c = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -10, 0, 1", "ref": "-1, -10, 0, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 12, "rel": [-2, 2, 2, 0], "s": -18}, "seed": 243261926603003410, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 8, "valid_answers": ["-1, -10, 0, 1", "-1, 0, -10, 1", "1, -8, 0, -1", "1, 0, -8, -1", "7, -8, 6, -7", "7, 6, -8, -7", "11, -10, 12, -11", "11, 12, -10, -11"]}, "cell": "mat2_involution/3", "stage": "C"}
+{"id": "po-mat2_involution-t3-eval-000476", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -2a - 2b + 2c = 46. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -2a - 2b + 2c = 46. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -12, 4, 7", "ref": "-7, -12, 4, 7", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 12, "rel": [-2, -2, 2, 0], "s": 46}, "seed": 243261926603000671, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 12, "valid_answers": ["-7, -12, 4, 7", "-7, -4, 12, 7"]}, "cell": "mat2_involution/3", "stage": "C"}
+{"id": "po-mat2_involution-t3-eval-000201", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a - b + c = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a - b + c = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -4, 0, 1", "ref": "-1, -4, 0, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 12, "rel": [-1, -1, 1, 0], "s": 5}, "seed": 243261926603000264, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 4, "valid_answers": ["-1, -4, 0, 1", "-1, 0, 4, 1", "1, -6, 0, -1", "1, 0, 6, -1", "5, -6, 4, -5", "5, -4, 6, -5"]}, "cell": "mat2_involution/3", "stage": "C"}
+{"id": "po-mat2_involution-t3-eval-001090", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 3a + b - c = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 3a + b - c = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, -9, 1", "ref": "-1, 0, -9, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 12, "rel": [3, 1, -1, 0], "s": 6}, "seed": 243261926603001657, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 3, "valid_answers": ["-1, 0, -9, 1", "-1, 9, 0, 1", "1, 0, -3, -1", "1, 3, 0, -1", "6, -7, 5, -6", "6, -5, 7, -6"]}, "cell": "mat2_involution/3", "stage": "C"}
+{"id": "po-mat2_involution-t3-eval-000809", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2a + 2b + 3c = -29. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2a + 2b + 3c = -29. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 7, -9, 8", "ref": "-8, 7, -9, 8", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 12, "rel": [2, 2, 3, 0], "s": -29}, "seed": 243261926603001204, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 9, "valid_answers": ["-8, 7, -9, 8", "-1, 0, -9, 1"]}, "cell": "mat2_involution/3", "stage": "C"}
+{"id": "po-mat2_involution-t3-eval-000794", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2a - b + c = -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and 2a - b + c = -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, -7, 1", "ref": "-1, 0, -7, 1", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 12, "rel": [2, -1, 1, 0], "s": -9}, "seed": 243261926603001178, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 7, "valid_answers": ["-1, 0, -7, 1", "-1, 7, 0, 1", "1, 0, -11, -1", "1, 11, 0, -1"]}, "cell": "mat2_involution/3", "stage": "C"}
+{"id": "po-mat2_involution-t3-eval-001337", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 2b - 2c = 35. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + 2b - 2c = 35. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 6, -8, 7", "ref": "-7, 6, -8, 7", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 12, "rel": [-1, 2, -2, 0], "s": 35}, "seed": 243261926603002091, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 8, "valid_answers": ["-7, 6, -8, 7", "-7, 8, -6, 7"]}, "cell": "mat2_involution/3", "stage": "C"}
+{"id": "po-mat2_involution-t3-eval-000430", "task": "cx_po_mat2_involution", "source": "cx_po_mat2_involution", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + b + c = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = I and A \\ne \\pm I and -a + b + c = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -8, 6, 7", "ref": "-7, -8, 6, 7", "family": "po_mat2_involution", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 2.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_involution", "rung": "witness", "params": {"B": 12, "rel": [-1, 1, 1, 0], "s": 5}, "seed": 243261926603000602, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 4, "valid_answers": ["-7, -8, 6, 7", "-7, 6, -8, 7", "-3, -2, 4, 3", "-3, 4, -2, 3", "-1, 0, 4, 1", "-1, 4, 0, 1", "1, 0, 6, -1", "1, 6, 0, -1", "5, -2, 12, -5", "5, 12, -2, -5"]}, "cell": "mat2_involution/3", "stage": "C"}
+{"id": "po-mat2_nilpotent-t1-eval-000476", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a + 2b = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a + 2b = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 4, 0, 0", "ref": "0, 4, 0, 0", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 5, "rel": [3, 2, 0, 0], "s": 8}, "seed": 92186124901002857, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["0, 4, 0, 0", "2, 1, -4, -2"]}, "cell": "mat2_nilpotent/1", "stage": "B"}
+{"id": "po-mat2_nilpotent-t1-eval-000153", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3b - c = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3b - c = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 1, -4, 2", "ref": "-2, 1, -4, 2", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 5, "rel": [0, 3, -1, 0], "s": 7}, "seed": 92186124901000432, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["-2, 1, -4, 2", "2, 1, -4, -2"]}, "cell": "mat2_nilpotent/1", "stage": "B"}
+{"id": "po-mat2_nilpotent-t1-eval-000481", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 2a + b = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 2a + b = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 2, -2, 2", "ref": "-2, 2, -2, 2", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 5, "rel": [2, 1, 0, 0], "s": -2}, "seed": 92186124901003021, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 2, "valid_answers": ["-2, 2, -2, 2", "0, -2, 0, 0"]}, "cell": "mat2_nilpotent/1", "stage": "B"}
+{"id": "po-mat2_nilpotent-t1-eval-000085", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 2a + c = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 2a + c = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -4, 4, 4", "ref": "-4, -4, 4, 4", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 5, "rel": [2, 0, 1, 0], "s": -4}, "seed": 92186124901000227, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["-4, -4, 4, 4", "0, 0, -4, 0"]}, "cell": "mat2_nilpotent/1", "stage": "B"}
+{"id": "po-mat2_nilpotent-t1-eval-000347", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and a + b = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and a + b = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -2, 2, 2", "ref": "-2, -2, 2, 2", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 5, "rel": [1, 1, 0, 0], "s": -4}, "seed": 92186124901001329, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 2, "valid_answers": ["-2, -2, 2, 2", "0, -4, 0, 0"]}, "cell": "mat2_nilpotent/1", "stage": "B"}
+{"id": "po-mat2_nilpotent-t1-eval-000333", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a + c = -10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a + c = -10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -5, 5, 5", "ref": "-5, -5, 5, 5", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 5, "rel": [3, 0, 1, 0], "s": -10}, "seed": 92186124901001239, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["-5, -5, 5, 5", "-2, 1, -4, 2"]}, "cell": "mat2_nilpotent/1", "stage": "B"}
+{"id": "po-mat2_nilpotent-t1-eval-000209", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a + 2b = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a + 2b = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -1, 4, 2", "ref": "-2, -1, 4, 2", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 5, "rel": [3, 2, 0, 0], "s": -8}, "seed": 92186124901000643, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["-2, -1, 4, 2", "0, -4, 0, 0"]}, "cell": "mat2_nilpotent/1", "stage": "B"}
+{"id": "po-mat2_nilpotent-t1-eval-000213", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2b + c = -12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2b + c = -12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 4, -4, 4", "ref": "-4, 4, -4, 4", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 5, "rel": [0, -2, 1, 0], "s": -12}, "seed": 92186124901000653, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["-4, 4, -4, 4", "4, 4, -4, -4"]}, "cell": "mat2_nilpotent/1", "stage": "B"}
+{"id": "po-mat2_nilpotent-t1-eval-000415", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2b - c = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2b - c = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -3, 3, 3", "ref": "-3, -3, 3, 3", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000205, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 5, "rel": [0, -2, -1, 0], "s": 3}, "seed": 92186124901001950, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 3, "valid_answers": ["-3, -3, 3, 3", "0, 0, -3, 0", "3, -3, 3, -3"]}, "cell": "mat2_nilpotent/1", "stage": "B"}
+{"id": "po-mat2_nilpotent-t1-eval-000150", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and a - 2c = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and a - 2c = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 4, -4, 4", "ref": "-4, 4, -4, 4", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000205, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 5, "rel": [1, 0, -2, 0], "s": 4}, "seed": 92186124901000426, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 2, "valid_answers": ["-4, 4, -4, 4", "0, 0, -2, 0", "2, 4, -1, -2"]}, "cell": "mat2_nilpotent/1", "stage": "B"}
+{"id": "po-mat2_nilpotent-t1-eval-000212", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -b + c = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -b + c = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -4, 4, 4", "ref": "-4, -4, 4, 4", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 5, "rel": [0, -1, 1, 0], "s": 8}, "seed": 92186124901000651, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["-4, -4, 4, 4", "4, -4, 4, -4"]}, "cell": "mat2_nilpotent/1", "stage": "B"}
+{"id": "po-mat2_nilpotent-t1-eval-000267", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and b - 2c = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and b - 2c = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 4, -4, 4", "ref": "-4, 4, -4, 4", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 5, "rel": [0, 1, -2, 0], "s": 12}, "seed": 92186124901000864, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["-4, 4, -4, 4", "4, 4, -4, -4"]}, "cell": "mat2_nilpotent/1", "stage": "B"}
+{"id": "po-mat2_nilpotent-t2-eval-000178", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and a + b = -12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and a + b = -12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -6, 6, 6", "ref": "-6, -6, 6, 6", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 8, "rel": [1, 1, 0, 0], "s": -12}, "seed": 92186124902000379, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 6, "valid_answers": ["-6, -6, 6, 6", "-4, -8, 2, 4"]}, "cell": "mat2_nilpotent/2", "stage": "B"}
+{"id": "po-mat2_nilpotent-t2-eval-000202", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 2b - c = -12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 2b - c = -12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -4, 4, 4", "ref": "-4, -4, 4, 4", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 8, "rel": [0, 2, -1, 0], "s": -12}, "seed": 92186124902000458, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 4, "valid_answers": ["-4, -4, 4, 4", "-4, -2, 8, 4", "0, -6, 0, 0", "4, -4, 4, -4", "4, -2, 8, -4"]}, "cell": "mat2_nilpotent/2", "stage": "B"}
+{"id": "po-mat2_nilpotent-t2-eval-000156", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a + 3c = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a + 3c = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -2, 8, 4", "ref": "-4, -2, 8, 4", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 3.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 8, "rel": [3, 0, 3, 0], "s": 12}, "seed": 92186124902000333, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 2, "valid_answers": ["-4, -2, 8, 4", "0, 0, 4, 0", "2, -2, 2, -2"]}, "cell": "mat2_nilpotent/2", "stage": "B"}
+{"id": "po-mat2_nilpotent-t2-eval-001011", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2b - 2c = -14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2b - 2c = -14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, 7, 0", "ref": "0, 0, 7, 0", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 8, "rel": [0, -2, -2, 0], "s": -14}, "seed": 92186124902008393, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 7, "valid_answers": ["0, 0, 7, 0", "0, 7, 0, 0"]}, "cell": "mat2_nilpotent/2", "stage": "B"}
+{"id": "po-mat2_nilpotent-t2-eval-000155", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2a + 3b = 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2a + 3b = 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 3, -3, 3", "ref": "-3, 3, -3, 3", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 8, "rel": [-2, 3, 0, 0], "s": 15}, "seed": 92186124902000332, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 3, "valid_answers": ["-3, 3, -3, 3", "0, 5, 0, 0"]}, "cell": "mat2_nilpotent/2", "stage": "B"}
+{"id": "po-mat2_nilpotent-t2-eval-000538", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2a + b = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2a + b = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -6, 6, 6", "ref": "-6, -6, 6, 6", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 8, "rel": [-2, 1, 0, 0], "s": 6}, "seed": 92186124902001579, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 2, "valid_answers": ["-6, -6, 6, 6", "-4, -2, 8, 4", "-2, 2, -2, 2", "0, 6, 0, 0"]}, "cell": "mat2_nilpotent/2", "stage": "B"}
+{"id": "po-mat2_nilpotent-t2-eval-000575", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2a + 2c = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2a + 2c = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -8, 2, 4", "ref": "-4, -8, 2, 4", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 8, "rel": [-2, 0, 2, 0], "s": 12}, "seed": 92186124902001805, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 3, "valid_answers": ["-4, -8, 2, 4", "-3, -3, 3, 3", "-2, -1, 4, 2", "0, 0, 6, 0"]}, "cell": "mat2_nilpotent/2", "stage": "B"}
+{"id": "po-mat2_nilpotent-t2-eval-000840", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -b + 2c = 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -b + 2c = 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -8, 8, 8", "ref": "-8, -8, 8, 8", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 8, "rel": [0, -1, 2, 0], "s": 24}, "seed": 92186124902003769, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 8, "valid_answers": ["-8, -8, 8, 8", "8, -8, 8, -8"]}, "cell": "mat2_nilpotent/2", "stage": "B"}
+{"id": "po-mat2_nilpotent-t2-eval-000662", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2b + c = 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2b + c = 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -5, 5, 5", "ref": "-5, -5, 5, 5", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 8, "rel": [0, -2, 1, 0], "s": 15}, "seed": 92186124902002369, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 5, "valid_answers": ["-5, -5, 5, 5", "5, -5, 5, -5"]}, "cell": "mat2_nilpotent/2", "stage": "B"}
+{"id": "po-mat2_nilpotent-t2-eval-000805", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -a + 3b = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -a + 3b = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 5, -5, 5", "ref": "-5, 5, -5, 5", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 8, "rel": [-1, 3, 0, 0], "s": 20}, "seed": 92186124902003424, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 5, "valid_answers": ["-5, 5, -5, 5", "4, 8, -2, -4"]}, "cell": "mat2_nilpotent/2", "stage": "B"}
+{"id": "po-mat2_nilpotent-t2-eval-000958", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a - b = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a - b = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -7, 0, 0", "ref": "0, -7, 0, 0", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 8, "rel": [3, -1, 0, 0], "s": 7}, "seed": 92186124902005664, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 4, "valid_answers": ["0, -7, 0, 0", "2, -1, 4, -2"]}, "cell": "mat2_nilpotent/2", "stage": "B"}
+{"id": "po-mat2_nilpotent-t2-eval-000420", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -b + 2c = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -b + 2c = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, -4, 0", "ref": "0, 0, -4, 0", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 8, "rel": [0, -1, 2, 0], "s": -8}, "seed": 92186124902001083, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 4, "valid_answers": ["0, 0, -4, 0", "0, 8, 0, 0"]}, "cell": "mat2_nilpotent/2", "stage": "B"}
+{"id": "po-mat2_nilpotent-t3-eval-000997", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2a + 3b - c = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2a + 3b - c = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 2, -2, 2", "ref": "-2, 2, -2, 2", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 12, "rel": [-2, 3, -1, 0], "s": 12}, "seed": 92186124903001603, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 2, "valid_answers": ["-2, 2, -2, 2", "0, 0, -12, 0", "0, 4, 0, 0", "6, 6, -6, -6"]}, "cell": "mat2_nilpotent/3", "stage": "B"}
+{"id": "po-mat2_nilpotent-t3-eval-001665", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2a - b - 2c = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2a - b - 2c = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, 9, 0", "ref": "0, 0, 9, 0", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 8e-06, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 12, "rel": [-2, -1, -2, 0], "s": -18}, "seed": 92186124903003114, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 6, "valid_answers": ["0, 0, 9, 0", "6, -6, 6, -6", "6, 12, -3, -6"]}, "cell": "mat2_nilpotent/3", "stage": "B"}
+{"id": "po-mat2_nilpotent-t3-eval-001275", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2a + b + 3c = 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -2a + b + 3c = 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -9, 4, 6", "ref": "-6, -9, 4, 6", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 12, "rel": [-2, 1, 3, 0], "s": 15}, "seed": 92186124903002201, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 4, "valid_answers": ["-6, -9, 4, 6", "-6, 12, -3, 6", "-2, -1, 4, 2", "0, 0, 5, 0"]}, "cell": "mat2_nilpotent/3", "stage": "B"}
+{"id": "po-mat2_nilpotent-t3-eval-002000", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a + b - 2c = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a + b - 2c = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -8, 0, 0", "ref": "0, -8, 0, 0", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 12, "rel": [3, 1, -2, 0], "s": -8}, "seed": 92186124903004066, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 4, "valid_answers": ["0, -8, 0, 0", "0, 0, 4, 0"]}, "cell": "mat2_nilpotent/3", "stage": "B"}
+{"id": "po-mat2_nilpotent-t3-eval-001155", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 2a - b + c = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 2a - b + c = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -8, 2, 4", "ref": "-4, -8, 2, 4", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 12, "rel": [2, -1, 1, 0], "s": 2}, "seed": 92186124903001954, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 2, "valid_answers": ["-4, -8, 2, 4", "-4, -2, 8, 4", "0, -2, 0, 0", "0, 0, 2, 0"]}, "cell": "mat2_nilpotent/3", "stage": "B"}
+{"id": "po-mat2_nilpotent-t3-eval-000671", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and a - b - c = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and a - b - c = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -6, 0, 0", "ref": "0, -6, 0, 0", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 12, "rel": [1, -1, -1, 0], "s": 6}, "seed": 92186124903000986, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 6, "valid_answers": ["0, -6, 0, 0", "0, 0, -6, 0", "6, -6, 6, -6", "6, 6, -6, -6"]}, "cell": "mat2_nilpotent/3", "stage": "B"}
+{"id": "po-mat2_nilpotent-t3-eval-001293", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a - 2b + 2c = 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a - 2b + 2c = 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -7, 0, 0", "ref": "0, -7, 0, 0", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 8e-06, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 12, "rel": [3, -2, 2, 0], "s": 14}, "seed": 92186124903002250, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 2, "valid_answers": ["0, -7, 0, 0", "0, 0, 7, 0", "2, -2, 2, -2"]}, "cell": "mat2_nilpotent/3", "stage": "B"}
+{"id": "po-mat2_nilpotent-t3-eval-001319", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a - b + c = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a - b + c = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -8, 0, 0", "ref": "0, -8, 0, 0", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 8e-06, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 12, "rel": [3, -1, 1, 0], "s": 8}, "seed": 92186124903002305, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 8, "valid_answers": ["0, -8, 0, 0", "0, 0, 8, 0", "8, 8, -8, -8"]}, "cell": "mat2_nilpotent/3", "stage": "B"}
+{"id": "po-mat2_nilpotent-t3-eval-001858", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -a + 3b + 3c = -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -a + 3b + 3c = -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, -8, 2, -4", "ref": "4, -8, 2, -4", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 12, "rel": [-1, 3, 3, 0], "s": -22}, "seed": 92186124903003675, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 8, "valid_answers": ["4, -8, 2, -4", "4, 2, -8, -4"]}, "cell": "mat2_nilpotent/3", "stage": "B"}
+{"id": "po-mat2_nilpotent-t3-eval-001345", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and a - 2b + 3c = -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and a - 2b + 3c = -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 10, 0, 0", "ref": "0, 10, 0, 0", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 12, "rel": [1, -2, 3, 0], "s": -20}, "seed": 92186124903002354, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 5, "valid_answers": ["0, 10, 0, 0", "5, 5, -5, -5"]}, "cell": "mat2_nilpotent/3", "stage": "B"}
+{"id": "po-mat2_nilpotent-t3-eval-000909", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a - b + c = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 3a - b + c = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -12, 3, 6", "ref": "-6, -12, 3, 6", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 1.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 12, "rel": [3, -1, 1, 0], "s": -3}, "seed": 92186124903001423, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 3, "valid_answers": ["-6, -12, 3, 6", "-6, -3, 12, 6", "-3, -3, 3, 3", "0, 0, -3, 0", "0, 3, 0, 0"]}, "cell": "mat2_nilpotent/3", "stage": "B"}
+{"id": "po-mat2_nilpotent-t3-eval-001333", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -a + 3b + 2c = -14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -a + 3b + 2c = -14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, -7, 0", "ref": "0, 0, -7, 0", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 8e-06, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 12, "rel": [-1, 3, 2, 0], "s": -14}, "seed": 92186124903002338, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 7, "valid_answers": ["0, 0, -7, 0", "4, 2, -8, -4", "7, -7, 7, -7"]}, "cell": "mat2_nilpotent/3", "stage": "B"}
+{"id": "po-mat2_trdet-t0-eval-000214", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -4 and determinant -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -4 and determinant -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -3, -2, -2", "ref": "-2, -3, -2, -2", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001666, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 3, "t": -4, "D": -2}, "seed": 128182186700002217, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["-2, -3, -2, -2", "-2, -2, -3, -2", "-2, 2, 3, -2", "-2, 3, 2, -2"]}, "cell": "mat2_trdet/0", "stage": "C"}
+{"id": "po-mat2_trdet-t0-eval-000011", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -4 and determinant -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -4 and determinant -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -3, -2, -1", "ref": "-3, -3, -2, -1", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.003332, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 3, "t": -4, "D": -3}, "seed": 128182186700000012, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 3, "valid_answers": ["-3, -3, -2, -1", "-3, -2, -3, -1", "-3, 2, 3, -1", "-3, 3, 2, -1", "-1, -3, -2, -3", "-1, -2, -3, -3", "-1, 2, 3, -3", "-1, 3, 2, -3"]}, "cell": "mat2_trdet/0", "stage": "C"}
+{"id": "po-mat2_trdet-t0-eval-000160", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -4 and determinant 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -4 and determinant 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -3, -1, -1", "ref": "-3, -3, -1, -1", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.004165, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 3, "t": -4, "D": 0}, "seed": 128182186700000431, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 2, "valid_answers": ["-3, -3, -1, -1", "-3, -1, -3, -1", "-3, 1, 3, -1", "-3, 3, 1, -1", "-2, -2, -2, -2", "-2, 2, 2, -2", "-1, -3, -1, -3", "-1, -1, -3, -3", "-1, 1, 3, -3", "-1, 3, 1, -3"]}, "cell": "mat2_trdet/0", "stage": "C"}
+{"id": "po-mat2_trdet-t0-eval-000105", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 3 and determinant -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 3 and determinant -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -3, -3, 2", "ref": "1, -3, -3, 2", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001666, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 3, "t": 3, "D": -7}, "seed": 128182186700000185, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["1, -3, -3, 2", "1, 3, 3, 2", "2, -3, -3, 1", "2, 3, 3, 1"]}, "cell": "mat2_trdet/0", "stage": "C"}
+{"id": "po-mat2_trdet-t0-eval-000039", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 6 and determinant 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 6 and determinant 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -3, 0, 3", "ref": "3, -3, 0, 3", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.005414, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 3, "t": 6, "D": 9}, "seed": 128182186700000049, "version": "po-v0", "tier": 0, "n_raw": 13, "small": 3, "valid_answers": ["3, -3, 0, 3", "3, -2, 0, 3", "3, -1, 0, 3", "3, 0, -3, 3", "3, 0, -2, 3", "3, 0, -1, 3", "3, 0, 0, 3", "3, 0, 1, 3", "3, 0, 2, 3", "3, 0, 3, 3", "3, 1, 0, 3", "3, 2, 0, 3", "3, 3, 0, 3"]}, "cell": "mat2_trdet/0", "stage": "C"}
+{"id": "po-mat2_trdet-t0-eval-000167", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 2 and determinant -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 2 and determinant -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -1, 1, 3", "ref": "-1, -1, 1, 3", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.006664, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 3, "t": 2, "D": -2}, "seed": 128182186700000505, "version": "po-v0", "tier": 0, "n_raw": 16, "small": 2, "valid_answers": ["-1, -1, 1, 3", "-1, 1, -1, 3", "0, -2, -1, 2", "0, -1, -2, 2", "0, 1, 2, 2", "0, 2, 1, 2", "1, -3, -1, 1", "1, -1, -3, 1", "1, 1, 3, 1", "1, 3, 1, 1", "2, -2, -1, 0", "2, -1, -2, 0", "2, 1, 2, 0", "2, 2, 1, 0", "3, -1, 1, -1", "3, 1, -1, -1"]}, "cell": "mat2_trdet/0", "stage": "C"}
+{"id": "po-mat2_trdet-t0-eval-000224", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -2 and determinant -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -2 and determinant -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -3, -3, -1", "ref": "-1, -3, -3, -1", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000833, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 3, "t": -2, "D": -8}, "seed": 128182186700003425, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["-1, -3, -3, -1", "-1, 3, 3, -1"]}, "cell": "mat2_trdet/0", "stage": "C"}
+{"id": "po-mat2_trdet-t0-eval-000195", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 2 and determinant -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 2 and determinant -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -3, -3, 1", "ref": "1, -3, -3, 1", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000833, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 3, "t": 2, "D": -8}, "seed": 128182186700000907, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["1, -3, -3, 1", "1, 3, 3, 1"]}, "cell": "mat2_trdet/0", "stage": "C"}
+{"id": "po-mat2_trdet-t0-eval-000052", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -3 and determinant 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -3 and determinant 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -3, 3, 0", "ref": "-3, -3, 3, 0", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001666, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 3, "t": -3, "D": 9}, "seed": 128182186700000075, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["-3, -3, 3, 0", "-3, 3, -3, 0", "0, -3, 3, -3", "0, 3, -3, -3"]}, "cell": "mat2_trdet/0", "stage": "C"}
+{"id": "po-mat2_trdet-t0-eval-000005", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 3 and determinant 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 3 and determinant 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -3, 0, 3", "ref": "0, -3, 0, 3", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 34, "bucket": null, "guess_rate": 0.014161, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 3, "t": 3, "D": 0}, "seed": 128182186700000005, "version": "po-v0", "tier": 0, "n_raw": 34, "small": 2, "valid_answers": ["0, -3, 0, 3", "0, -2, 0, 3", "0, -1, 0, 3", "0, 0, -3, 3", "0, 0, -2, 3", "0, 0, -1, 3", "0, 0, 0, 3", "0, 0, 1, 3", "0, 0, 2, 3", "0, 0, 3, 3", "0, 1, 0, 3", "0, 2, 0, 3", "0, 3, 0, 3", "1, -2, -1, 2", "1, -1, -2, 2", "1, 1, 2, 2", "1, 2, 1, 2", "2, -2, -1, 1", "2, -1, -2, 1", "2, 1, 2, 1", "2, 2, 1, 1", "3, -3, 0, 0", "3, -2, 0, 0", "3, -1, 0, 0", "3, 0, -3, 0", "3, 0, -2, 0", "3, 0, -1, 0", "3, 0, 0, 0", "3, 0, 1, 0", "3, 0, 2, 0", "3, 0, 3, 0", "3, 1, 0, 0", "3, 2, 0, 0", "3, 3, 0, 0"]}, "cell": "mat2_trdet/0", "stage": "C"}
+{"id": "po-mat2_trdet-t0-eval-000121", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -5 and determinant -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -5 and determinant -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -3, -3, -2", "ref": "-3, -3, -3, -2", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001666, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 3, "t": -5, "D": -3}, "seed": 128182186700000222, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["-3, -3, -3, -2", "-3, 3, 3, -2", "-2, -3, -3, -3", "-2, 3, 3, -3"]}, "cell": "mat2_trdet/0", "stage": "C"}
+{"id": "po-mat2_trdet-t0-eval-000163", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 6 and determinant 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 6 and determinant 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -2, 2, 3", "ref": "3, -2, 2, 3", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000833, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 3, "t": 6, "D": 13}, "seed": 128182186700000458, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["3, -2, 2, 3", "3, 2, -2, 3"]}, "cell": "mat2_trdet/0", "stage": "C"}
+{"id": "po-mat2_trdet-t1-eval-000606", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 3 and determinant -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 3 and determinant -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -1, 1, 4", "ref": "-1, -1, 1, 4", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.001366, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 5, "t": 3, "D": -3}, "seed": 128182186701001616, "version": "po-v0", "tier": 1, "n_raw": 20, "small": 3, "valid_answers": ["-1, -1, 1, 4", "-1, 1, -1, 4", "0, -3, -1, 3", "0, -1, -3, 3", "0, 1, 3, 3", "0, 3, 1, 3", "1, -5, -1, 2", "1, -1, -5, 2", "1, 1, 5, 2", "1, 5, 1, 2", "2, -5, -1, 1", "2, -1, -5, 1", "2, 1, 5, 1", "2, 5, 1, 1", "3, -3, -1, 0", "3, -1, -3, 0", "3, 1, 3, 0", "3, 3, 1, 0", "4, -1, 1, -1", "4, 1, -1, -1"]}, "cell": "mat2_trdet/1", "stage": "B"}
+{"id": "po-mat2_trdet-t1-eval-000010", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 2 and determinant -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 2 and determinant -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -5, -1, 5", "ref": "-3, -5, -1, 5", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 24, "bucket": null, "guess_rate": 0.001639, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 5, "t": 2, "D": -20}, "seed": 128182186701000011, "version": "po-v0", "tier": 1, "n_raw": 24, "small": 4, "valid_answers": ["-3, -5, -1, 5", "-3, -1, -5, 5", "-3, 1, 5, 5", "-3, 5, 1, 5", "-2, -4, -3, 4", "-2, -3, -4, 4", "-2, 3, 4, 4", "-2, 4, 3, 4", "0, -5, -4, 2", "0, -4, -5, 2", "0, 4, 5, 2", "0, 5, 4, 2", "2, -5, -4, 0", "2, -4, -5, 0", "2, 4, 5, 0", "2, 5, 4, 0", "4, -4, -3, -2", "4, -3, -4, -2", "4, 3, 4, -2", "4, 4, 3, -2", "5, -5, -1, -3", "5, -1, -5, -3", "5, 1, 5, -3", "5, 5, 1, -3"]}, "cell": "mat2_trdet/1", "stage": "B"}
+{"id": "po-mat2_trdet-t1-eval-000803", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 10 and determinant 16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 10 and determinant 16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, -3, -3, 5", "ref": "5, -3, -3, 5", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000137, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 5, "t": 10, "D": 16}, "seed": 128182186701004632, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 5, "valid_answers": ["5, -3, -3, 5", "5, 3, 3, 5"]}, "cell": "mat2_trdet/1", "stage": "B"}
+{"id": "po-mat2_trdet-t1-eval-000110", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 4 and determinant -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 4 and determinant -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -4, -1, 5", "ref": "-1, -4, -1, 5", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 24, "bucket": null, "guess_rate": 0.001639, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 5, "t": 4, "D": -9}, "seed": 128182186701000149, "version": "po-v0", "tier": 1, "n_raw": 24, "small": 4, "valid_answers": ["-1, -4, -1, 5", "-1, -2, -2, 5", "-1, -1, -4, 5", "-1, 1, 4, 5", "-1, 2, 2, 5", "-1, 4, 1, 5", "0, -3, -3, 4", "0, 3, 3, 4", "1, -4, -3, 3", "1, -3, -4, 3", "1, 3, 4, 3", "1, 4, 3, 3", "3, -4, -3, 1", "3, -3, -4, 1", "3, 3, 4, 1", "3, 4, 3, 1", "4, -3, -3, 0", "4, 3, 3, 0", "5, -4, -1, -1", "5, -2, -2, -1", "5, -1, -4, -1", "5, 1, 4, -1", "5, 2, 2, -1", "5, 4, 1, -1"]}, "cell": "mat2_trdet/1", "stage": "B"}
+{"id": "po-mat2_trdet-t1-eval-000124", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -7 and determinant 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -7 and determinant 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -4, 1, -2", "ref": "-5, -4, 1, -2", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.001366, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 5, "t": -7, "D": 14}, "seed": 128182186701000171, "version": "po-v0", "tier": 1, "n_raw": 20, "small": 4, "valid_answers": ["-5, -4, 1, -2", "-5, -2, 2, -2", "-5, -1, 4, -2", "-5, 1, -4, -2", "-5, 2, -2, -2", "-5, 4, -1, -2", "-4, -2, 1, -3", "-4, -1, 2, -3", "-4, 1, -2, -3", "-4, 2, -1, -3", "-3, -2, 1, -4", "-3, -1, 2, -4", "-3, 1, -2, -4", "-3, 2, -1, -4", "-2, -4, 1, -5", "-2, -2, 2, -5", "-2, -1, 4, -5", "-2, 1, -4, -5", "-2, 2, -2, -5", "-2, 4, -1, -5"]}, "cell": "mat2_trdet/1", "stage": "B"}
+{"id": "po-mat2_trdet-t1-eval-000609", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 9 and determinant 36. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 9 and determinant 36. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, -4, 4, 5", "ref": "4, -4, 4, 5", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 5, "t": 9, "D": 36}, "seed": 128182186701001627, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 5, "valid_answers": ["4, -4, 4, 5", "4, 4, -4, 5", "5, -4, 4, 4", "5, 4, -4, 4"]}, "cell": "mat2_trdet/1", "stage": "B"}
+{"id": "po-mat2_trdet-t1-eval-000001", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -4 and determinant -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -4 and determinant -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -2, -1, 1", "ref": "-5, -2, -1, 1", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.001093, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 5, "t": -4, "D": -7}, "seed": 128182186701000001, "version": "po-v0", "tier": 1, "n_raw": 16, "small": 5, "valid_answers": ["-5, -2, -1, 1", "-5, -1, -2, 1", "-5, 1, 2, 1", "-5, 2, 1, 1", "-3, -5, -2, -1", "-3, -2, -5, -1", "-3, 2, 5, -1", "-3, 5, 2, -1", "-1, -5, -2, -3", "-1, -2, -5, -3", "-1, 2, 5, -3", "-1, 5, 2, -3", "1, -2, -1, -5", "1, -1, -2, -5", "1, 1, 2, -5", "1, 2, 1, -5"]}, "cell": "mat2_trdet/1", "stage": "B"}
+{"id": "po-mat2_trdet-t1-eval-000563", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -3 and determinant 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -3 and determinant 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -5, 5, 2", "ref": "-5, -5, 5, 2", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.00082, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 5, "t": -3, "D": 15}, "seed": 128182186701001371, "version": "po-v0", "tier": 1, "n_raw": 12, "small": 5, "valid_answers": ["-5, -5, 5, 2", "-5, 5, -5, 2", "-3, -5, 3, 0", "-3, -3, 5, 0", "-3, 3, -5, 0", "-3, 5, -3, 0", "0, -5, 3, -3", "0, -3, 5, -3", "0, 3, -5, -3", "0, 5, -3, -3", "2, -5, 5, -5", "2, 5, -5, -5"]}, "cell": "mat2_trdet/1", "stage": "B"}
+{"id": "po-mat2_trdet-t1-eval-000074", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -1 and determinant -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -1 and determinant -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -2, -1, 4", "ref": "-5, -2, -1, 4", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 28, "bucket": null, "guess_rate": 0.001912, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 5, "t": -1, "D": -22}, "seed": 128182186701000096, "version": "po-v0", "tier": 1, "n_raw": 28, "small": 4, "valid_answers": ["-5, -2, -1, 4", "-5, -1, -2, 4", "-5, 1, 2, 4", "-5, 2, 1, 4", "-4, -5, -2, 3", "-4, -2, -5, 3", "-4, 2, 5, 3", "-4, 5, 2, 3", "-3, -4, -4, 2", "-3, 4, 4, 2", "-2, -5, -4, 1", "-2, -4, -5, 1", "-2, 4, 5, 1", "-2, 5, 4, 1", "1, -5, -4, -2", "1, -4, -5, -2", "1, 4, 5, -2", "1, 5, 4, -2", "2, -4, -4, -3", "2, 4, 4, -3", "3, -5, -2, -4", "3, -2, -5, -4", "3, 2, 5, -4", "3, 5, 2, -4", "4, -2, -1, -5", "4, -1, -2, -5", "4, 1, 2, -5", "4, 2, 1, -5"]}, "cell": "mat2_trdet/1", "stage": "B"}
+{"id": "po-mat2_trdet-t1-eval-000196", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 2 and determinant -15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 2 and determinant -15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -5, 0, 5", "ref": "-3, -5, 0, 5", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 60, "bucket": null, "guess_rate": 0.004098, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 5, "t": 2, "D": -15}, "seed": 128182186701000272, "version": "po-v0", "tier": 1, "n_raw": 60, "small": 4, "valid_answers": ["-3, -5, 0, 5", "-3, -4, 0, 5", "-3, -3, 0, 5", "-3, -2, 0, 5", "-3, -1, 0, 5", "-3, 0, -5, 5", "-3, 0, -4, 5", "-3, 0, -3, 5", "-3, 0, -2, 5", "-3, 0, -1, 5", "-3, 0, 0, 5", "-3, 0, 1, 5", "-3, 0, 2, 5", "-3, 0, 3, 5", "-3, 0, 4, 5", "-3, 0, 5, 5", "-3, 1, 0, 5", "-3, 2, 0, 5", "-3, 3, 0, 5", "-3, 4, 0, 5", "-3, 5, 0, 5", "-1, -4, -3, 3", "-1, -3, -4, 3", "-1, 3, 4, 3", "-1, 4, 3, 3", "0, -5, -3, 2", "0, -3, -5, 2", "0, 3, 5, 2", "0, 5, 3, 2", "1, -4, -4, 1", "1, 4, 4, 1", "2, -5, -3, 0", "2, -3, -5, 0", "2, 3, 5, 0", "2, 5, 3, 0", "3, -4, -3, -1", "3, -3, -4, -1", "3, 3, 4, -1", "3, 4, 3, -1", "5, -5, 0, -3", "5, -4, 0, -3", "5, -3, 0, -3", "5, -2, 0, -3", "5, -1, 0, -3", "5, 0, -5, -3", "5, 0, -4, -3", "5, 0, -3, -3", "5, 0, -2, -3", "5, 0, -1, -3", "5, 0, 0, -3", "5, 0, 1, -3", "5, 0, 2, -3", "5, 0, 3, -3", "5, 0, 4, -3", "5, 0, 5, -3", "5, 1, 0, -3", "5, 2, 0, -3", "5, 3, 0, -3", "5, 4, 0, -3", "5, 5, 0, -3"]}, "cell": "mat2_trdet/1", "stage": "B"}
+{"id": "po-mat2_trdet-t1-eval-000876", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 7 and determinant 21. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 7 and determinant 21. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -3, 3, 4", "ref": "3, -3, 3, 4", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 5, "t": 7, "D": 21}, "seed": 128182186701009589, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 4, "valid_answers": ["3, -3, 3, 4", "3, 3, -3, 4", "4, -3, 3, 3", "4, 3, -3, 3"]}, "cell": "mat2_trdet/1", "stage": "B"}
+{"id": "po-mat2_trdet-t1-eval-000000", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -4 and determinant 19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -4 and determinant 19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -4, 4, -1", "ref": "-3, -4, 4, -1", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000546, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 5, "t": -4, "D": 19}, "seed": 128182186701000000, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 4, "valid_answers": ["-3, -4, 4, -1", "-3, 4, -4, -1", "-2, -5, 3, -2", "-2, -3, 5, -2", "-2, 3, -5, -2", "-2, 5, -3, -2", "-1, -4, 4, -3", "-1, 4, -4, -3"]}, "cell": "mat2_trdet/1", "stage": "B"}
+{"id": "po-mat2_trdet-t2-eval-000717", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 2 and determinant -32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 2 and determinant -32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -3, 1, 7", "ref": "-5, -3, 1, 7", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 24, "bucket": null, "guess_rate": 0.000474, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 7, "t": 2, "D": -32}, "seed": 128182186702001217, "version": "po-v0", "tier": 2, "n_raw": 24, "small": 6, "valid_answers": ["-5, -3, 1, 7", "-5, -1, 3, 7", "-5, 1, -3, 7", "-5, 3, -1, 7", "-4, -4, -2, 6", "-4, -2, -4, 6", "-4, 2, 4, 6", "-4, 4, 2, 6", "-2, -6, -4, 4", "-2, -4, -6, 4", "-2, 4, 6, 4", "-2, 6, 4, 4", "4, -6, -4, -2", "4, -4, -6, -2", "4, 4, 6, -2", "4, 6, 4, -2", "6, -4, -2, -4", "6, -2, -4, -4", "6, 2, 4, -4", "6, 4, 2, -4", "7, -3, 1, -5", "7, -1, 3, -5", "7, 1, -3, -5", "7, 3, -1, -5"]}, "cell": "mat2_trdet/2", "stage": "B"}
+{"id": "po-mat2_trdet-t2-eval-001735", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 8 and determinant 25. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 8 and determinant 25. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -6, 3, 7", "ref": "1, -6, 3, 7", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 0.000356, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 7, "t": 8, "D": 25}, "seed": 128182186702006126, "version": "po-v0", "tier": 2, "n_raw": 18, "small": 4, "valid_answers": ["1, -6, 3, 7", "1, -3, 6, 7", "1, 3, -6, 7", "1, 6, -3, 7", "3, -5, 2, 5", "3, -2, 5, 5", "3, 2, -5, 5", "3, 5, -2, 5", "4, -3, 3, 4", "4, 3, -3, 4", "5, -5, 2, 3", "5, -2, 5, 3", "5, 2, -5, 3", "5, 5, -2, 3", "7, -6, 3, 1", "7, -3, 6, 1", "7, 3, -6, 1", "7, 6, -3, 1"]}, "cell": "mat2_trdet/2", "stage": "B"}
+{"id": "po-mat2_trdet-t2-eval-001919", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 6 and determinant -14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 6 and determinant -14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -7, -1, 7", "ref": "-1, -7, -1, 7", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.000316, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 7, "t": 6, "D": -14}, "seed": 128182186702008758, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 7, "valid_answers": ["-1, -7, -1, 7", "-1, -1, -7, 7", "-1, 1, 7, 7", "-1, 7, 1, 7", "0, -7, -2, 6", "0, -2, -7, 6", "0, 2, 7, 6", "0, 7, 2, 6", "6, -7, -2, 0", "6, -2, -7, 0", "6, 2, 7, 0", "6, 7, 2, 0", "7, -7, -1, -1", "7, -1, -7, -1", "7, 1, 7, -1", "7, 7, 1, -1"]}, "cell": "mat2_trdet/2", "stage": "B"}
+{"id": "po-mat2_trdet-t2-eval-000775", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 3 and determinant 51. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 3 and determinant 51. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -7, 7, 2", "ref": "1, -7, 7, 2", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 7.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 7, "t": 3, "D": 51}, "seed": 128182186702001349, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 7, "valid_answers": ["1, -7, 7, 2", "1, 7, -7, 2", "2, -7, 7, 1", "2, 7, -7, 1"]}, "cell": "mat2_trdet/2", "stage": "B"}
+{"id": "po-mat2_trdet-t2-eval-001986", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -1 and determinant -58. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -1 and determinant -58. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -4, -4, 6", "ref": "-7, -4, -4, 6", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.000237, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 7, "t": -1, "D": -58}, "seed": 128182186702010130, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 7, "valid_answers": ["-7, -4, -4, 6", "-7, 4, 4, 6", "-6, -7, -4, 5", "-6, -4, -7, 5", "-6, 4, 7, 5", "-6, 7, 4, 5", "5, -7, -4, -6", "5, -4, -7, -6", "5, 4, 7, -6", "5, 7, 4, -6", "6, -4, -4, -7", "6, 4, 4, -7"]}, "cell": "mat2_trdet/2", "stage": "B"}
+{"id": "po-mat2_trdet-t2-eval-000609", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 2 and determinant 32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 2 and determinant 32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -7, 5, 3", "ref": "-1, -7, 5, 3", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000158, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 7, "t": 2, "D": 32}, "seed": 128182186702000978, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 7, "valid_answers": ["-1, -7, 5, 3", "-1, -5, 7, 3", "-1, 5, -7, 3", "-1, 7, -5, 3", "3, -7, 5, -1", "3, -5, 7, -1", "3, 5, -7, -1", "3, 7, -5, -1"]}, "cell": "mat2_trdet/2", "stage": "B"}
+{"id": "po-mat2_trdet-t2-eval-000356", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 4 and determinant -33. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 4 and determinant -33. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -6, -2, 7", "ref": "-3, -6, -2, 7", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 36, "bucket": null, "guess_rate": 0.000711, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 7, "t": 4, "D": -33}, "seed": 128182186702000514, "version": "po-v0", "tier": 2, "n_raw": 36, "small": 6, "valid_answers": ["-3, -6, -2, 7", "-3, -4, -3, 7", "-3, -3, -4, 7", "-3, -2, -6, 7", "-3, 2, 6, 7", "-3, 3, 4, 7", "-3, 4, 3, 7", "-3, 6, 2, 7", "-2, -7, -3, 6", "-2, -3, -7, 6", "-2, 3, 7, 6", "-2, 7, 3, 6", "-1, -7, -4, 5", "-1, -4, -7, 5", "-1, 4, 7, 5", "-1, 7, 4, 5", "1, -6, -6, 3", "1, 6, 6, 3", "3, -6, -6, 1", "3, 6, 6, 1", "5, -7, -4, -1", "5, -4, -7, -1", "5, 4, 7, -1", "5, 7, 4, -1", "6, -7, -3, -2", "6, -3, -7, -2", "6, 3, 7, -2", "6, 7, 3, -2", "7, -6, -2, -3", "7, -4, -3, -3", "7, -3, -4, -3", "7, -2, -6, -3", "7, 2, 6, -3", "7, 3, 4, -3", "7, 4, 3, -3", "7, 6, 2, -3"]}, "cell": "mat2_trdet/2", "stage": "B"}
+{"id": "po-mat2_trdet-t2-eval-002036", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 4 and determinant 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 4 and determinant 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -7, 7, 5", "ref": "-1, -7, 7, 5", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 7.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 7, "t": 4, "D": 44}, "seed": 128182186702011084, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 7, "valid_answers": ["-1, -7, 7, 5", "-1, 7, -7, 5", "5, -7, 7, -1", "5, 7, -7, -1"]}, "cell": "mat2_trdet/2", "stage": "B"}
+{"id": "po-mat2_trdet-t2-eval-001591", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -1 and determinant -40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -1 and determinant -40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -2, 1, 6", "ref": "-7, -2, 1, 6", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 32, "bucket": null, "guess_rate": 0.000632, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 7, "t": -1, "D": -40}, "seed": 128182186702004870, "version": "po-v0", "tier": 2, "n_raw": 32, "small": 5, "valid_answers": ["-7, -2, 1, 6", "-7, -1, 2, 6", "-7, 1, -2, 6", "-7, 2, -1, 6", "-6, -5, -2, 5", "-6, -2, -5, 5", "-6, 2, 5, 5", "-6, 5, 2, 5", "-5, -5, -4, 4", "-5, -4, -5, 4", "-5, 4, 5, 4", "-5, 5, 4, 4", "-4, -7, -4, 3", "-4, -4, -7, 3", "-4, 4, 7, 3", "-4, 7, 4, 3", "3, -7, -4, -4", "3, -4, -7, -4", "3, 4, 7, -4", "3, 7, 4, -4", "4, -5, -4, -5", "4, -4, -5, -5", "4, 4, 5, -5", "4, 5, 4, -5", "5, -5, -2, -6", "5, -2, -5, -6", "5, 2, 5, -6", "5, 5, 2, -6", "6, -2, 1, -7", "6, -1, 2, -7", "6, 1, -2, -7", "6, 2, -1, -7"]}, "cell": "mat2_trdet/2", "stage": "B"}
+{"id": "po-mat2_trdet-t2-eval-000769", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -6 and determinant 29. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -6 and determinant 29. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -6, 6, 1", "ref": "-7, -6, 6, 1", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 24, "bucket": null, "guess_rate": 0.000474, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 7, "t": -6, "D": 29}, "seed": 128182186702001337, "version": "po-v0", "tier": 2, "n_raw": 24, "small": 5, "valid_answers": ["-7, -6, 6, 1", "-7, 6, -6, 1", "-5, -6, 4, -1", "-5, -4, 6, -1", "-5, 4, -6, -1", "-5, 6, -4, -1", "-4, -7, 3, -2", "-4, -3, 7, -2", "-4, 3, -7, -2", "-4, 7, -3, -2", "-3, -5, 4, -3", "-3, -4, 5, -3", "-3, 4, -5, -3", "-3, 5, -4, -3", "-2, -7, 3, -4", "-2, -3, 7, -4", "-2, 3, -7, -4", "-2, 7, -3, -4", "-1, -6, 4, -5", "-1, -4, 6, -5", "-1, 4, -6, -5", "-1, 6, -4, -5", "1, -6, 6, -7", "1, 6, -6, -7"]}, "cell": "mat2_trdet/2", "stage": "B"}
+{"id": "po-mat2_trdet-t2-eval-000509", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -5 and determinant 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -5 and determinant 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -7, 2, 1", "ref": "-6, -7, 2, 1", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 36, "bucket": null, "guess_rate": 0.000711, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 7, "t": -5, "D": 8}, "seed": 128182186702000794, "version": "po-v0", "tier": 2, "n_raw": 36, "small": 3, "valid_answers": ["-6, -7, 2, 1", "-6, -2, 7, 1", "-6, 2, -7, 1", "-6, 7, -2, 1", "-5, -4, 2, 0", "-5, -2, 4, 0", "-5, 2, -4, 0", "-5, 4, -2, 0", "-4, -4, 1, -1", "-4, -2, 2, -1", "-4, -1, 4, -1", "-4, 1, -4, -1", "-4, 2, -2, -1", "-4, 4, -1, -1", "-3, -2, 1, -2", "-3, -1, 2, -2", "-3, 1, -2, -2", "-3, 2, -1, -2", "-2, -2, 1, -3", "-2, -1, 2, -3", "-2, 1, -2, -3", "-2, 2, -1, -3", "-1, -4, 1, -4", "-1, -2, 2, -4", "-1, -1, 4, -4", "-1, 1, -4, -4", "-1, 2, -2, -4", "-1, 4, -1, -4", "0, -4, 2, -5", "0, -2, 4, -5", "0, 2, -4, -5", "0, 4, -2, -5", "1, -7, 2, -6", "1, -2, 7, -6", "1, 2, -7, -6", "1, 7, -2, -6"]}, "cell": "mat2_trdet/2", "stage": "B"}
+{"id": "po-mat2_trdet-t2-eval-000272", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 6 and determinant 19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 6 and determinant 19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -7, 2, 5", "ref": "1, -7, 2, 5", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.000237, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 7, "t": 6, "D": 19}, "seed": 128182186702000376, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 5, "valid_answers": ["1, -7, 2, 5", "1, -2, 7, 5", "1, 2, -7, 5", "1, 7, -2, 5", "3, -5, 2, 3", "3, -2, 5, 3", "3, 2, -5, 3", "3, 5, -2, 3", "5, -7, 2, 1", "5, -2, 7, 1", "5, 2, -7, 1", "5, 7, -2, 1"]}, "cell": "mat2_trdet/2", "stage": "B"}
+{"id": "po-mat2_trdet-t3-eval-001247", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 13 and determinant 38. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 13 and determinant 38. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -8, 1, 10", "ref": "3, -8, 1, 10", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 44, "bucket": null, "guess_rate": 0.000226, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 10, "t": 13, "D": 38}, "seed": 128182186703002198, "version": "po-v0", "tier": 3, "n_raw": 44, "small": 7, "valid_answers": ["3, -8, 1, 10", "3, -4, 2, 10", "3, -2, 4, 10", "3, -1, 8, 10", "3, 1, -8, 10", "3, 2, -4, 10", "3, 4, -2, 10", "3, 8, -1, 10", "4, -2, 1, 9", "4, -1, 2, 9", "4, 1, -2, 9", "4, 2, -1, 9", "5, -2, -1, 8", "5, -1, -2, 8", "5, 1, 2, 8", "5, 2, 1, 8", "6, -4, -1, 7", "6, -2, -2, 7", "6, -1, -4, 7", "6, 1, 4, 7", "6, 2, 2, 7", "6, 4, 1, 7", "7, -4, -1, 6", "7, -2, -2, 6", "7, -1, -4, 6", "7, 1, 4, 6", "7, 2, 2, 6", "7, 4, 1, 6", "8, -2, -1, 5", "8, -1, -2, 5", "8, 1, 2, 5", "8, 2, 1, 5", "9, -2, 1, 4", "9, -1, 2, 4", "9, 1, -2, 4", "9, 2, -1, 4", "10, -8, 1, 3", "10, -4, 2, 3", "10, -2, 4, 3", "10, -1, 8, 3", "10, 1, -8, 3", "10, 2, -4, 3", "10, 4, -2, 3", "10, 8, -1, 3"]}, "cell": "mat2_trdet/3", "stage": "C"}
+{"id": "po-mat2_trdet-t3-eval-001306", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 15 and determinant 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 15 and determinant 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, -7, -7, 10", "ref": "5, -7, -7, 10", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 2.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 10, "t": 15, "D": 1}, "seed": 128182186703002315, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 10, "valid_answers": ["5, -7, -7, 10", "5, 7, 7, 10", "10, -7, -7, 5", "10, 7, 7, 5"]}, "cell": "mat2_trdet/3", "stage": "C"}
+{"id": "po-mat2_trdet-t3-eval-001979", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 9 and determinant 108. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 9 and determinant 108. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -10, 10, 8", "ref": "1, -10, 10, 8", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 6.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 10, "t": 9, "D": 108}, "seed": 128182186703003941, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 10, "valid_answers": ["1, -10, 10, 8", "1, 10, -10, 8", "3, -10, 9, 6", "3, -9, 10, 6", "3, 9, -10, 6", "3, 10, -9, 6", "6, -10, 9, 3", "6, -9, 10, 3", "6, 9, -10, 3", "6, 10, -9, 3", "8, -10, 10, 1", "8, 10, -10, 1"]}, "cell": "mat2_trdet/3", "stage": "C"}
+{"id": "po-mat2_trdet-t3-eval-000940", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -12 and determinant -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -12 and determinant -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -5, -5, -2", "ref": "-10, -5, -5, -2", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 28, "bucket": null, "guess_rate": 0.000144, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 10, "t": -12, "D": -5}, "seed": 128182186703001589, "version": "po-v0", "tier": 3, "n_raw": 28, "small": 8, "valid_answers": ["-10, -5, -5, -2", "-10, 5, 5, -2", "-9, -8, -4, -3", "-9, -4, -8, -3", "-9, 4, 8, -3", "-9, 8, 4, -3", "-7, -10, -4, -5", "-7, -8, -5, -5", "-7, -5, -8, -5", "-7, -4, -10, -5", "-7, 4, 10, -5", "-7, 5, 8, -5", "-7, 8, 5, -5", "-7, 10, 4, -5", "-5, -10, -4, -7", "-5, -8, -5, -7", "-5, -5, -8, -7", "-5, -4, -10, -7", "-5, 4, 10, -7", "-5, 5, 8, -7", "-5, 8, 5, -7", "-5, 10, 4, -7", "-3, -8, -4, -9", "-3, -4, -8, -9", "-3, 4, 8, -9", "-3, 8, 4, -9", "-2, -5, -5, -10", "-2, 5, 5, -10"]}, "cell": "mat2_trdet/3", "stage": "C"}
+{"id": "po-mat2_trdet-t3-eval-001089", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -2 and determinant 72. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -2 and determinant 72. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -10, 8, 2", "ref": "-4, -10, 8, 2", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 8.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 10, "t": -2, "D": 72}, "seed": 128182186703001854, "version": "po-v0", "tier": 3, "n_raw": 16, "small": 9, "valid_answers": ["-4, -10, 8, 2", "-4, -8, 10, 2", "-4, 8, -10, 2", "-4, 10, -8, 2", "-2, -9, 8, 0", "-2, -8, 9, 0", "-2, 8, -9, 0", "-2, 9, -8, 0", "0, -9, 8, -2", "0, -8, 9, -2", "0, 8, -9, -2", "0, 9, -8, -2", "2, -10, 8, -4", "2, -8, 10, -4", "2, 8, -10, -4", "2, 10, -8, -4"]}, "cell": "mat2_trdet/3", "stage": "C"}
+{"id": "po-mat2_trdet-t3-eval-000483", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 13 and determinant 26. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 13 and determinant 26. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -4, -1, 10", "ref": "3, -4, -1, 10", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 48, "bucket": null, "guess_rate": 0.000247, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 10, "t": 13, "D": 26}, "seed": 128182186703000766, "version": "po-v0", "tier": 3, "n_raw": 48, "small": 7, "valid_answers": ["3, -4, -1, 10", "3, -2, -2, 10", "3, -1, -4, 10", "3, 1, 4, 10", "3, 2, 2, 10", "3, 4, 1, 10", "4, -10, -1, 9", "4, -5, -2, 9", "4, -2, -5, 9", "4, -1, -10, 9", "4, 1, 10, 9", "4, 2, 5, 9", "4, 5, 2, 9", "4, 10, 1, 9", "5, -7, -2, 8", "5, -2, -7, 8", "5, 2, 7, 8", "5, 7, 2, 8", "6, -8, -2, 7", "6, -4, -4, 7", "6, -2, -8, 7", "6, 2, 8, 7", "6, 4, 4, 7", "6, 8, 2, 7", "7, -8, -2, 6", "7, -4, -4, 6", "7, -2, -8, 6", "7, 2, 8, 6", "7, 4, 4, 6", "7, 8, 2, 6", "8, -7, -2, 5", "8, -2, -7, 5", "8, 2, 7, 5", "8, 7, 2, 5", "9, -10, -1, 4", "9, -5, -2, 4", "9, -2, -5, 4", "9, -1, -10, 4", "9, 1, 10, 4", "9, 2, 5, 4", "9, 5, 2, 4", "9, 10, 1, 4", "10, -4, -1, 3", "10, -2, -2, 3", "10, -1, -4, 3", "10, 1, 4, 3", "10, 2, 2, 3", "10, 4, 1, 3"]}, "cell": "mat2_trdet/3", "stage": "C"}
+{"id": "po-mat2_trdet-t3-eval-001197", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -3 and determinant 63. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -3 and determinant 63. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -9, 9, 3", "ref": "-6, -9, 9, 3", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 6.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 10, "t": -3, "D": 63}, "seed": 128182186703002094, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 9, "valid_answers": ["-6, -9, 9, 3", "-6, 9, -9, 3", "-3, -9, 7, 0", "-3, -7, 9, 0", "-3, 7, -9, 0", "-3, 9, -7, 0", "0, -9, 7, -3", "0, -7, 9, -3", "0, 7, -9, -3", "0, 9, -7, -3", "3, -9, 9, -6", "3, 9, -9, -6"]}, "cell": "mat2_trdet/3", "stage": "C"}
+{"id": "po-mat2_trdet-t3-eval-001818", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 14 and determinant 112. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 14 and determinant 112. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, -9, 8, 10", "ref": "4, -9, 8, 10", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 8.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 10, "t": 14, "D": 112}, "seed": 128182186703003520, "version": "po-v0", "tier": 3, "n_raw": 16, "small": 8, "valid_answers": ["4, -9, 8, 10", "4, -8, 9, 10", "4, 8, -9, 10", "4, 9, -8, 10", "6, -8, 8, 8", "6, 8, -8, 8", "7, -9, 7, 7", "7, -7, 9, 7", "7, 7, -9, 7", "7, 9, -7, 7", "8, -8, 8, 6", "8, 8, -8, 6", "10, -9, 8, 4", "10, -8, 9, 4", "10, 8, -9, 4", "10, 9, -8, 4"]}, "cell": "mat2_trdet/3", "stage": "C"}
+{"id": "po-mat2_trdet-t3-eval-002445", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -7 and determinant -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -7 and determinant -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -9, 3, 3", "ref": "-10, -9, 3, 3", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 52, "bucket": null, "guess_rate": 0.000267, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 10, "t": -7, "D": -3}, "seed": 128182186703005333, "version": "po-v0", "tier": 3, "n_raw": 52, "small": 5, "valid_answers": ["-10, -9, 3, 3", "-10, -3, 9, 3", "-10, 3, -9, 3", "-10, 9, -3, 3", "-9, -5, 3, 2", "-9, -3, 5, 2", "-9, 3, -5, 2", "-9, 5, -3, 2", "-8, -5, 1, 1", "-8, -1, 5, 1", "-8, 1, -5, 1", "-8, 5, -1, 1", "-7, -3, -1, 0", "-7, -1, -3, 0", "-7, 1, 3, 0", "-7, 3, 1, 0", "-6, -9, -1, -1", "-6, -3, -3, -1", "-6, -1, -9, -1", "-6, 1, 9, -1", "-6, 3, 3, -1", "-6, 9, 1, -1", "-4, -5, -3, -3", "-4, -3, -5, -3", "-4, 3, 5, -3", "-4, 5, 3, -3", "-3, -5, -3, -4", "-3, -3, -5, -4", "-3, 3, 5, -4", "-3, 5, 3, -4", "-1, -9, -1, -6", "-1, -3, -3, -6", "-1, -1, -9, -6", "-1, 1, 9, -6", "-1, 3, 3, -6", "-1, 9, 1, -6", "0, -3, -1, -7", "0, -1, -3, -7", "0, 1, 3, -7", "0, 3, 1, -7", "1, -5, 1, -8", "1, -1, 5, -8", "1, 1, -5, -8", "1, 5, -1, -8", "2, -5, 3, -9", "2, -3, 5, -9", "2, 3, -5, -9", "2, 5, -3, -9", "3, -9, 3, -10", "3, -3, 9, -10", "3, 3, -9, -10", "3, 9, -3, -10"]}, "cell": "mat2_trdet/3", "stage": "C"}
+{"id": "po-mat2_trdet-t3-eval-001914", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -8 and determinant 33. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -8 and determinant 33. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -7, 6, 1", "ref": "-9, -7, 6, 1", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 32, "bucket": null, "guess_rate": 0.000165, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 10, "t": -8, "D": 33}, "seed": 128182186703003773, "version": "po-v0", "tier": 3, "n_raw": 32, "small": 6, "valid_answers": ["-9, -7, 6, 1", "-9, -6, 7, 1", "-9, 6, -7, 1", "-9, 7, -6, 1", "-6, -7, 3, -2", "-6, -3, 7, -2", "-6, 3, -7, -2", "-6, 7, -3, -2", "-5, -9, 2, -3", "-5, -6, 3, -3", "-5, -3, 6, -3", "-5, -2, 9, -3", "-5, 2, -9, -3", "-5, 3, -6, -3", "-5, 6, -3, -3", "-5, 9, -2, -3", "-3, -9, 2, -5", "-3, -6, 3, -5", "-3, -3, 6, -5", "-3, -2, 9, -5", "-3, 2, -9, -5", "-3, 3, -6, -5", "-3, 6, -3, -5", "-3, 9, -2, -5", "-2, -7, 3, -6", "-2, -3, 7, -6", "-2, 3, -7, -6", "-2, 7, -3, -6", "1, -7, 6, -9", "1, -6, 7, -9", "1, 6, -7, -9", "1, 7, -6, -9"]}, "cell": "mat2_trdet/3", "stage": "C"}
+{"id": "po-mat2_trdet-t3-eval-000407", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 5 and determinant -41. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace 5 and determinant -41. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -9, 1, 10", "ref": "-5, -9, 1, 10", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 44, "bucket": null, "guess_rate": 0.000226, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 10, "t": 5, "D": -41}, "seed": 128182186703000630, "version": "po-v0", "tier": 3, "n_raw": 44, "small": 7, "valid_answers": ["-5, -9, 1, 10", "-5, -3, 3, 10", "-5, -1, 9, 10", "-5, 1, -9, 10", "-5, 3, -3, 10", "-5, 9, -1, 10", "-4, -5, -1, 9", "-4, -1, -5, 9", "-4, 1, 5, 9", "-4, 5, 1, 9", "-2, -9, -3, 7", "-2, -3, -9, 7", "-2, 3, 9, 7", "-2, 9, 3, 7", "-1, -7, -5, 6", "-1, -5, -7, 6", "-1, 5, 7, 6", "-1, 7, 5, 6", "1, -9, -5, 4", "1, -5, -9, 4", "1, 5, 9, 4", "1, 9, 5, 4", "4, -9, -5, 1", "4, -5, -9, 1", "4, 5, 9, 1", "4, 9, 5, 1", "6, -7, -5, -1", "6, -5, -7, -1", "6, 5, 7, -1", "6, 7, 5, -1", "7, -9, -3, -2", "7, -3, -9, -2", "7, 3, 9, -2", "7, 9, 3, -2", "9, -5, -1, -4", "9, -1, -5, -4", "9, 1, 5, -4", "9, 5, 1, -4", "10, -9, 1, -5", "10, -3, 3, -5", "10, -1, 9, -5", "10, 1, -9, -5", "10, 3, -3, -5", "10, 9, -1, -5"]}, "cell": "mat2_trdet/3", "stage": "C"}
+{"id": "po-mat2_trdet-t3-eval-001261", "task": "cx_po_mat2_trdet", "source": "cx_po_mat2_trdet", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -7 and determinant -25. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has trace -7 and determinant -25. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -5, 1, 3", "ref": "-10, -5, 1, 3", "family": "po_mat2_trdet", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 28, "bucket": null, "guess_rate": 0.000144, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_trdet", "rung": "witness", "params": {"B": 10, "t": -7, "D": -25}, "seed": 128182186703002224, "version": "po-v0", "tier": 3, "n_raw": 28, "small": 7, "valid_answers": ["-10, -5, 1, 3", "-10, -1, 5, 3", "-10, 1, -5, 3", "-10, 5, -1, 3", "-9, -7, -1, 2", "-9, -1, -7, 2", "-9, 1, 7, 2", "-9, 7, 1, 2", "-7, -5, -5, 0", "-7, 5, 5, 0", "-5, -7, -5, -2", "-5, -5, -7, -2", "-5, 5, 7, -2", "-5, 7, 5, -2", "-2, -7, -5, -5", "-2, -5, -7, -5", "-2, 5, 7, -5", "-2, 7, 5, -5", "0, -5, -5, -7", "0, 5, 5, -7", "2, -7, -1, -9", "2, -1, -7, -9", "2, 1, 7, -9", "2, 7, 1, -9", "3, -5, 1, -10", "3, -1, 5, -10", "3, 1, -5, -10", "3, 5, -1, -10"]}, "cell": "mat2_trdet/3", "stage": "C"}
+{"id": "po-mat3_det-t0-eval-001916", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} 1 & 2 & 2 \\\\ 0 & -2 & 1 \\\\ x & y & z \\end{pmatrix} is 28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} 1 & 2 & 2 \\\\ 0 & -2 & 1 \\\\ x & y & z \\end{pmatrix} is 28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -4, -3", "ref": "3, -4, -3", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.009602, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [1, 2, 2, 0, -2, 1, -2, -2, -1], "pos": [6, 7, 8], "D": 28}, "seed": 261600747300013148, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 4, "valid_answers": ["3, -4, -3", "3, -2, -4", "4, -4, 0", "4, -2, -1", "4, 0, -2", "4, 2, -3", "4, 4, -4"]}, "cell": "mat3_det/0", "stage": "A"}
+{"id": "po-mat3_det-t0-eval-000408", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -2 & -1 & -2 \\\\ x & y & z \\\\ -2 & -1 & 0 \\end{pmatrix} is 22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -2 & -1 & -2 \\\\ x & y & z \\\\ -2 & -1 & 0 \\end{pmatrix} is 22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -4, -4", "ref": "3, -4, -4", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.012346, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [-2, -1, -2, 2, 2, 1, -2, -1, 0], "pos": [3, 4, 5], "D": 22}, "seed": 261600747300002756, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 4, "valid_answers": ["3, -4, -4", "3, -4, -3", "3, -4, -2", "3, -4, -1", "3, -4, 0", "3, -4, 1", "3, -4, 2", "3, -4, 3", "3, -4, 4"]}, "cell": "mat3_det/0", "stage": "A"}
+{"id": "po-mat3_det-t0-eval-001358", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -2 & 0 & 1 \\\\ x & y & z \\\\ -2 & -1 & -2 \\end{pmatrix} is 23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -2 & 0 & 1 \\\\ x & y & z \\\\ -2 & -1 & -2 \\end{pmatrix} is 23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 2, -4", "ref": "-3, 2, -4", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.012346, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [-2, 0, 1, 1, 0, -2, -2, -1, -2], "pos": [3, 4, 5], "D": 23}, "seed": 261600747300009252, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 3, "valid_answers": ["-3, 2, -4", "-3, 3, -1", "-3, 4, 2", "-1, 3, -2", "-1, 4, 1", "1, 3, -3", "1, 4, 0", "3, 3, -4", "3, 4, -1"]}, "cell": "mat3_det/0", "stage": "A"}
+{"id": "po-mat3_det-t0-eval-000865", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -1 & 2 & 0 \\\\ x & y & z \\\\ 1 & -1 & 2 \\end{pmatrix} is -13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -1 & 2 & 0 \\\\ x & y & z \\\\ 1 & -1 & 2 \\end{pmatrix} is -13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 3, -3", "ref": "1, 3, -3", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.019204, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [-1, 2, 0, 1, 2, -2, 1, -1, 2], "pos": [3, 4, 5], "D": -13}, "seed": 261600747300005933, "version": "po-v0", "tier": 0, "n_raw": 14, "small": 2, "valid_answers": ["1, 3, -3", "1, 4, -1", "2, 1, -3", "2, 2, -1", "2, 3, 1", "2, 4, 3", "3, -1, -3", "3, 0, -1", "3, 1, 1", "3, 2, 3", "4, -3, -3", "4, -2, -1", "4, -1, 1", "4, 0, 3"]}, "cell": "mat3_det/0", "stage": "A"}
+{"id": "po-mat3_det-t0-eval-000345", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} 0 & -1 & 1 \\\\ -1 & 0 & -1 \\\\ x & y & z \\end{pmatrix} is -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} 0 & -1 & 1 \\\\ -1 & 0 & -1 \\\\ x & y & z \\end{pmatrix} is -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 1, 4", "ref": "-4, 1, 4", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.013717, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [0, -1, 1, -1, 0, -1, 1, -2, 1], "pos": [6, 7, 8], "D": -9}, "seed": 261600747300002386, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 3, "valid_answers": ["-4, 1, 4", "-4, 2, 3", "-4, 3, 2", "-4, 4, 1", "-3, 2, 4", "-3, 3, 3", "-3, 4, 2", "-2, 3, 4", "-2, 4, 3", "-1, 4, 4"]}, "cell": "mat3_det/0", "stage": "A"}
+{"id": "po-mat3_det-t0-eval-000589", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -1 & -2 & 1 \\\\ x & y & z \\\\ -1 & 0 & -2 \\end{pmatrix} is -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -1 & -2 & 1 \\\\ x & y & z \\\\ -1 & 0 & -2 \\end{pmatrix} is -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -4, -4", "ref": "0, -4, -4", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.013717, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [-1, -2, 1, -2, 1, 1, -1, 0, -2], "pos": [3, 4, 5], "D": -20}, "seed": 261600747300003903, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 3, "valid_answers": ["0, -4, -4", "1, -4, -2", "2, -4, 0", "2, -2, -3", "3, -4, 2", "3, -2, -1", "3, 0, -4", "4, -4, 4", "4, -2, 1", "4, 0, -2"]}, "cell": "mat3_det/0", "stage": "A"}
+{"id": "po-mat3_det-t0-eval-002462", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} 0 & -2 & 1 \\\\ x & y & z \\\\ -2 & 2 & 2 \\end{pmatrix} is 34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} 0 & -2 & 1 \\\\ x & y & z \\\\ -2 & 2 & 2 \\end{pmatrix} is 34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 3, 4", "ref": "2, 3, 4", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.010974, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [0, -2, 1, -1, 2, -1, -2, 2, 2], "pos": [3, 4, 5], "D": 34}, "seed": 261600747300016830, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 3, "valid_answers": ["2, 3, 4", "3, 0, 4", "3, 2, 3", "3, 4, 2", "4, -3, 4", "4, -1, 3", "4, 1, 2", "4, 3, 1"]}, "cell": "mat3_det/0", "stage": "A"}
+{"id": "po-mat3_det-t0-eval-000546", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -1 & 1 & -2 \\\\ -2 & 1 & -1 \\\\ x & y & z \\end{pmatrix} is 19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -1 & 1 & -2 \\\\ -2 & 1 & -1 \\\\ x & y & z \\end{pmatrix} is 19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 4, 4", "ref": "3, 4, 4", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002743, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [-1, 1, -2, -2, 1, -1, 1, 2, 2], "pos": [6, 7, 8], "D": 19}, "seed": 261600747300003639, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["3, 4, 4", "4, 4, 3"]}, "cell": "mat3_det/0", "stage": "A"}
+{"id": "po-mat3_det-t0-eval-001181", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -2 & -2 & 1 \\\\ x & y & z \\\\ 1 & -2 & 0 \\end{pmatrix} is -13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -2 & -2 & 1 \\\\ x & y & z \\\\ 1 & -2 & 0 \\end{pmatrix} is -13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -3, 4", "ref": "-4, -3, 4", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.016461, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [-2, -2, 1, -2, 1, 1, 1, -2, 0], "pos": [3, 4, 5], "D": -13}, "seed": 261600747300007902, "version": "po-v0", "tier": 0, "n_raw": 12, "small": 2, "valid_answers": ["-4, -3, 4", "-4, 3, 3", "-3, 1, 3", "-2, -1, 3", "-1, -3, 3", "-1, 3, 2", "0, 1, 2", "1, -1, 2", "2, -3, 2", "2, 3, 1", "3, 1, 1", "4, -1, 1"]}, "cell": "mat3_det/0", "stage": "A"}
+{"id": "po-mat3_det-t0-eval-000934", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} x & y & z \\\\ 1 & -2 & 0 \\\\ 0 & -1 & -2 \\end{pmatrix} is 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} x & y & z \\\\ 1 & -2 & 0 \\\\ 0 & -1 & -2 \\end{pmatrix} is 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 3, -3", "ref": "1, 3, -3", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.019204, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [-1, 1, 2, 1, -2, 0, 0, -1, -2], "pos": [0, 1, 2], "D": 13}, "seed": 261600747300006388, "version": "po-v0", "tier": 0, "n_raw": 14, "small": 2, "valid_answers": ["1, 3, -3", "1, 4, -1", "2, 1, -3", "2, 2, -1", "2, 3, 1", "2, 4, 3", "3, -1, -3", "3, 0, -1", "3, 1, 1", "3, 2, 3", "4, -3, -3", "4, -2, -1", "4, -1, 1", "4, 0, 3"]}, "cell": "mat3_det/0", "stage": "A"}
+{"id": "po-mat3_det-t0-eval-002155", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -2 & -2 & 1 \\\\ -1 & 1 & 1 \\\\ x & y & z \\end{pmatrix} is 22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -2 & -2 & 1 \\\\ -1 & 1 & 1 \\\\ x & y & z \\end{pmatrix} is 22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -2, -3", "ref": "-4, -2, -3", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.009602, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [-2, -2, 1, -1, 1, 1, 2, 2, -2], "pos": [6, 7, 8], "D": 22}, "seed": 261600747300014866, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 3, "valid_answers": ["-4, -2, -3", "-4, 2, -2", "-3, -3, -4", "-3, 1, -3", "-2, 0, -4", "-2, 4, -3", "-1, 3, -4"]}, "cell": "mat3_det/0", "stage": "A"}
+{"id": "po-mat3_det-t0-eval-001034", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -2 & -2 & -1 \\\\ -2 & 0 & -1 \\\\ x & y & z \\end{pmatrix} is -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -2 & -2 & -1 \\\\ -2 & 0 & -1 \\\\ x & y & z \\end{pmatrix} is -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -4, 4", "ref": "-3, -4, 4", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.012346, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [-2, -2, -1, -2, 0, -1, -1, 2, -1], "pos": [6, 7, 8], "D": -22}, "seed": 261600747300006993, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 4, "valid_answers": ["-3, -4, 4", "-3, -3, 4", "-3, -2, 4", "-3, -1, 4", "-3, 0, 4", "-3, 1, 4", "-3, 2, 4", "-3, 3, 4", "-3, 4, 4"]}, "cell": "mat3_det/0", "stage": "A"}
+{"id": "po-mat3_det-t1-eval-001034", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -2 & -2 & -3 \\\\ 2 & -2 & 3 \\\\ x & y & z \\end{pmatrix} is 68. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -2 & -2 & -3 \\\\ 2 & -2 & 3 \\\\ x & y & z \\end{pmatrix} is 68. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -6, 1", "ref": "-5, -6, 1", "family": "po_mat3_det", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 26, "bucket": null, "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [-2, -2, -3, 2, -2, 3, 3, -1, 1], "pos": [6, 7, 8], "D": 68}, "seed": 261600747301001710, "version": "po-v0", "tier": 1, "n_raw": 26, "small": 4, "valid_answers": ["-5, -6, 1", "-5, -5, 1", "-5, -4, 1", "-5, -3, 1", "-5, -2, 1", "-5, -1, 1", "-5, 0, 1", "-5, 1, 1", "-5, 2, 1", "-5, 3, 1", "-5, 4, 1", "-5, 5, 1", "-5, 6, 1", "-3, -6, 4", "-3, -5, 4", "-3, -4, 4", "-3, -3, 4", "-3, -2, 4", "-3, -1, 4", "-3, 0, 4", "-3, 1, 4", "-3, 2, 4", "-3, 3, 4", "-3, 4, 4", "-3, 5, 4", "-3, 6, 4"]}, "cell": "mat3_det/1", "stage": "C"}
+{"id": "po-mat3_det-t1-eval-002079", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -3 & -2 & 0 \\\\ x & y & z \\\\ 3 & 0 & -3 \\end{pmatrix} is 48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -3 & -2 & 0 \\\\ x & y & z \\\\ 3 & 0 & -3 \\end{pmatrix} is 48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -2, -5", "ref": "-6, -2, -5", "family": "po_mat3_det", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 38, "bucket": null, "guess_rate": 0.017296, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [-3, -2, 0, 1, 0, -2, 3, 0, -3], "pos": [3, 4, 5], "D": 48}, "seed": 261600747301003470, "version": "po-v0", "tier": 1, "n_raw": 38, "small": 3, "valid_answers": ["-6, -2, -5", "-6, 0, -2", "-6, 2, 1", "-6, 4, 4", "-5, -2, -6", "-5, 0, -3", "-5, 2, 0", "-5, 4, 3", "-5, 6, 6", "-4, 0, -4", "-4, 2, -1", "-4, 4, 2", "-4, 6, 5", "-3, 0, -5", "-3, 2, -2", "-3, 4, 1", "-3, 6, 4", "-2, 0, -6", "-2, 2, -3", "-2, 4, 0", "-2, 6, 3", "-1, 2, -4", "-1, 4, -1", "-1, 6, 2", "0, 2, -5", "0, 4, -2", "0, 6, 1", "1, 2, -6", "1, 4, -3", "1, 6, 0", "2, 4, -4", "2, 6, -1", "3, 4, -5", "3, 6, -2", "4, 4, -6", "4, 6, -3", "5, 6, -4", "6, 6, -5"]}, "cell": "mat3_det/1", "stage": "C"}
+{"id": "po-mat3_det-t1-eval-001272", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -1 & 3 & 0 \\\\ x & y & z \\\\ -1 & 2 & 3 \\end{pmatrix} is -13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -1 & 3 & 0 \\\\ x & y & z \\\\ -1 & 2 & 3 \\end{pmatrix} is -13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 6, 4", "ref": "-1, 6, 4", "family": "po_mat3_det", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 0.008193, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [-1, 3, 0, -3, 2, -2, -1, 2, 3], "pos": [3, 4, 5], "D": -13}, "seed": 261600747301002114, "version": "po-v0", "tier": 1, "n_raw": 18, "small": 1, "valid_answers": ["-1, 6, 4", "0, 3, 4", "0, 4, 1", "0, 5, -2", "0, 6, -5", "1, 0, 4", "1, 1, 1", "1, 2, -2", "1, 3, -5", "2, -3, 4", "2, -2, 1", "2, -1, -2", "2, 0, -5", "3, -6, 4", "3, -5, 1", "3, -4, -2", "3, -3, -5", "4, -6, -5"]}, "cell": "mat3_det/1", "stage": "C"}
+{"id": "po-mat3_det-t1-eval-000953", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} x & y & z \\\\ -1 & -2 & 1 \\\\ -2 & 0 & -1 \\end{pmatrix} is 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} x & y & z \\\\ -1 & -2 & 1 \\\\ -2 & 0 & -1 \\end{pmatrix} is 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -6, -2", "ref": "-6, -6, -2", "family": "po_mat3_det", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 36, "bucket": null, "guess_rate": 0.016386, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [2, 0, -3, -1, -2, 1, -2, 0, -1], "pos": [0, 1, 2], "D": 14}, "seed": 261600747301001570, "version": "po-v0", "tier": 1, "n_raw": 36, "small": 2, "valid_answers": ["-6, -6, -2", "-6, -2, -5", "-5, -4, -3", "-5, 0, -6", "-4, -6, -1", "-4, -2, -4", "-3, -4, -2", "-3, 0, -5", "-2, -6, 0", "-2, -2, -3", "-2, 2, -6", "-1, -4, -1", "-1, 0, -4", "0, -6, 1", "0, -2, -2", "0, 2, -5", "1, -4, 0", "1, 0, -3", "1, 4, -6", "2, -6, 2", "2, -2, -1", "2, 2, -4", "3, -4, 1", "3, 0, -2", "3, 4, -5", "4, -6, 3", "4, -2, 0", "4, 2, -3", "4, 6, -6", "5, -4, 2", "5, 0, -1", "5, 4, -4", "6, -6, 4", "6, -2, 1", "6, 2, -2", "6, 6, -5"]}, "cell": "mat3_det/1", "stage": "C"}
+{"id": "po-mat3_det-t1-eval-001874", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 3 & -1 & 2 \\\\ 2 & -1 & -3 \\\\ x & y & z \\end{pmatrix} is -47. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 3 & -1 & 2 \\\\ 2 & -1 & -3 \\\\ x & y & z \\end{pmatrix} is -47. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -1, 4", "ref": "-6, -1, 4", "family": "po_mat3_det", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.005917, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [3, -1, 2, 2, -1, -3, 2, 3, 3], "pos": [6, 7, 8], "D": -47}, "seed": 261600747301003114, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 3, "valid_answers": ["-6, -1, 4", "-5, -2, -4", "-4, -2, 1", "-3, -2, 6", "-2, -3, -2", "-1, -3, 3", "0, -4, -5", "1, -4, 0", "2, -4, 5", "3, -5, -3", "4, -5, 2", "5, -6, -6", "6, -6, -1"]}, "cell": "mat3_det/1", "stage": "C"}
+{"id": "po-mat3_det-t1-eval-000512", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} x & y & z \\\\ -2 & 2 & 3 \\\\ -2 & -2 & -1 \\end{pmatrix} is 64. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} x & y & z \\\\ -2 & 2 & 3 \\\\ -2 & -2 & -1 \\end{pmatrix} is 64. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -6, 5", "ref": "-6, -6, 5", "family": "po_mat3_det", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 35, "bucket": null, "guess_rate": 0.015931, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [-3, -1, -2, -2, 2, 3, -2, -2, -1], "pos": [0, 1, 2], "D": 64}, "seed": 261600747301000853, "version": "po-v0", "tier": 1, "n_raw": 35, "small": 4, "valid_answers": ["-6, -6, 5", "-6, -5, 6", "-4, -6, 4", "-4, -5, 5", "-4, -4, 6", "-2, -6, 3", "-2, -5, 4", "-2, -4, 5", "-2, -3, 6", "0, -6, 2", "0, -5, 3", "0, -4, 4", "0, -3, 5", "0, -2, 6", "2, -6, 1", "2, -5, 2", "2, -4, 3", "2, -3, 4", "2, -2, 5", "2, -1, 6", "4, -6, 0", "4, -5, 1", "4, -4, 2", "4, -3, 3", "4, -2, 4", "4, -1, 5", "4, 0, 6", "6, -6, -1", "6, -5, 0", "6, -4, 1", "6, -3, 2", "6, -2, 3", "6, -1, 4", "6, 0, 5", "6, 1, 6"]}, "cell": "mat3_det/1", "stage": "C"}
+{"id": "po-mat3_det-t1-eval-002055", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -2 & -2 & 2 \\\\ -1 & -2 & -2 \\\\ x & y & z \\end{pmatrix} is 40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -2 & -2 & 2 \\\\ -1 & -2 & -2 \\\\ x & y & z \\end{pmatrix} is 40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -6, 6", "ref": "-1, -6, 6", "family": "po_mat3_det", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 28, "bucket": null, "guess_rate": 0.012745, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [-2, -2, 2, -1, -2, -2, 2, 1, -3], "pos": [6, 7, 8], "D": 40}, "seed": 261600747301003433, "version": "po-v0", "tier": 1, "n_raw": 28, "small": 3, "valid_answers": ["-1, -6, 6", "0, -6, 2", "0, -5, 5", "1, -6, -2", "1, -5, 1", "1, -4, 4", "2, -6, -6", "2, -5, -3", "2, -4, 0", "2, -3, 3", "2, -2, 6", "3, -4, -4", "3, -3, -1", "3, -2, 2", "3, -1, 5", "4, -3, -5", "4, -2, -2", "4, -1, 1", "4, 0, 4", "5, -2, -6", "5, -1, -3", "5, 0, 0", "5, 1, 3", "5, 2, 6", "6, 0, -4", "6, 1, -1", "6, 2, 2", "6, 3, 5"]}, "cell": "mat3_det/1", "stage": "C"}
+{"id": "po-mat3_det-t1-eval-000457", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 3 & -1 & -2 \\\\ -1 & 2 & 1 \\\\ x & y & z \\end{pmatrix} is 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 3 & -1 & -2 \\\\ -1 & 2 & 1 \\\\ x & y & z \\end{pmatrix} is 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -6, 6", "ref": "-6, -6, 6", "family": "po_mat3_det", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 30, "bucket": null, "guess_rate": 0.013655, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [3, -1, -2, -1, 2, 1, 3, 1, 0], "pos": [6, 7, 8], "D": 18}, "seed": 261600747301000767, "version": "po-v0", "tier": 1, "n_raw": 30, "small": 2, "valid_answers": ["-6, -6, 6", "-5, -3, 6", "-4, -5, 5", "-4, 0, 6", "-3, -2, 5", "-3, 3, 6", "-2, -4, 4", "-2, 1, 5", "-2, 6, 6", "-1, -6, 3", "-1, -1, 4", "-1, 4, 5", "0, -3, 3", "0, 2, 4", "1, -5, 2", "1, 0, 3", "1, 5, 4", "2, -2, 2", "2, 3, 3", "3, -4, 1", "3, 1, 2", "3, 6, 3", "4, -6, 0", "4, -1, 1", "4, 4, 2", "5, -3, 0", "5, 2, 1", "6, -5, -1", "6, 0, 0", "6, 5, 1"]}, "cell": "mat3_det/1", "stage": "C"}
+{"id": "po-mat3_det-t1-eval-001075", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -2 & 3 & 0 \\\\ 1 & -1 & 2 \\\\ x & y & z \\end{pmatrix} is 19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -2 & 3 & 0 \\\\ 1 & -1 & 2 \\\\ x & y & z \\end{pmatrix} is 19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 5, -5", "ref": "-1, 5, -5", "family": "po_mat3_det", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 23, "bucket": null, "guess_rate": 0.010469, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [-2, 3, 0, 1, -1, 2, 0, -3, -2], "pos": [6, 7, 8], "D": 19}, "seed": 261600747301001776, "version": "po-v0", "tier": 1, "n_raw": 23, "small": 2, "valid_answers": ["-1, 5, -5", "-1, 6, -1", "0, 4, -3", "0, 5, 1", "0, 6, 5", "1, 2, -5", "1, 3, -1", "1, 4, 3", "2, 1, -3", "2, 2, 1", "2, 3, 5", "3, -1, -5", "3, 0, -1", "3, 1, 3", "4, -2, -3", "4, -1, 1", "4, 0, 5", "5, -4, -5", "5, -3, -1", "5, -2, 3", "6, -5, -3", "6, -4, 1", "6, -3, 5"]}, "cell": "mat3_det/1", "stage": "C"}
+{"id": "po-mat3_det-t1-eval-002391", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -2 & 1 & 3 \\\\ x & y & z \\\\ -2 & 3 & 2 \\end{pmatrix} is 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -2 & 1 & 3 \\\\ x & y & z \\\\ -2 & 3 & 2 \\end{pmatrix} is 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 6, 6", "ref": "-5, 6, 6", "family": "po_mat3_det", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 24, "bucket": null, "guess_rate": 0.010924, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [-2, 1, 3, -2, -2, 3, -2, 3, 2], "pos": [3, 4, 5], "D": 1}, "seed": 261600747301004015, "version": "po-v0", "tier": 1, "n_raw": 24, "small": 1, "valid_answers": ["-5, 6, 6", "-3, -1, 6", "-3, 1, 5", "-3, 3, 4", "-3, 5, 3", "-1, -6, 5", "-1, -4, 4", "-1, -2, 3", "-1, 0, 2", "-1, 2, 1", "-1, 4, 0", "-1, 6, -1", "1, -5, 1", "1, -3, 0", "1, -1, -1", "1, 1, -2", "1, 3, -3", "1, 5, -4", "3, -6, -2", "3, -4, -3", "3, -2, -4", "3, 0, -5", "3, 2, -6", "5, -5, -6"]}, "cell": "mat3_det/1", "stage": "C"}
+{"id": "po-mat3_det-t1-eval-000466", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -3 & 3 & 0 \\\\ -1 & -3 & -3 \\\\ x & y & z \\end{pmatrix} is -33. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -3 & 3 & 0 \\\\ -1 & -3 & -3 \\\\ x & y & z \\end{pmatrix} is -33. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 3, -5", "ref": "-6, 3, -5", "family": "po_mat3_det", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 34, "bucket": null, "guess_rate": 0.015476, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [-3, 3, 0, -1, -3, -3, 3, 1, -3], "pos": [6, 7, 8], "D": -33}, "seed": 261600747301000781, "version": "po-v0", "tier": 1, "n_raw": 34, "small": 2, "valid_answers": ["-6, 3, -5", "-5, 2, -5", "-5, 6, -2", "-4, 1, -5", "-4, 5, -2", "-3, 0, -5", "-3, 4, -2", "-2, -1, -5", "-2, 3, -2", "-1, -2, -5", "-1, 2, -2", "-1, 6, 1", "0, -3, -5", "0, 1, -2", "0, 5, 1", "1, -4, -5", "1, 0, -2", "1, 4, 1", "2, -5, -5", "2, -1, -2", "2, 3, 1", "3, -6, -5", "3, -2, -2", "3, 2, 1", "3, 6, 4", "4, -3, -2", "4, 1, 1", "4, 5, 4", "5, -4, -2", "5, 0, 1", "5, 4, 4", "6, -5, -2", "6, -1, 1", "6, 3, 4"]}, "cell": "mat3_det/1", "stage": "C"}
+{"id": "po-mat3_det-t1-eval-001501", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 2 & 2 & -2 \\\\ x & y & z \\\\ 0 & -3 & -3 \\end{pmatrix} is -96. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 2 & 2 & -2 \\\\ x & y & z \\\\ 0 & -3 & -3 \\end{pmatrix} is -96. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -2, -6", "ref": "-6, -2, -6", "family": "po_mat3_det", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 25, "bucket": null, "guess_rate": 0.011379, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [2, 2, -2, -3, 3, -1, 0, -3, -3], "pos": [3, 4, 5], "D": -96}, "seed": 261600747301002487, "version": "po-v0", "tier": 1, "n_raw": 25, "small": 4, "valid_answers": ["-6, -2, -6", "-6, -1, -5", "-6, 0, -4", "-6, 1, -3", "-6, 2, -2", "-6, 3, -1", "-6, 4, 0", "-6, 5, 1", "-6, 6, 2", "-5, 0, -6", "-5, 1, -5", "-5, 2, -4", "-5, 3, -3", "-5, 4, -2", "-5, 5, -1", "-5, 6, 0", "-4, 2, -6", "-4, 3, -5", "-4, 4, -4", "-4, 5, -3", "-4, 6, -2", "-3, 4, -6", "-3, 5, -5", "-3, 6, -4", "-2, 6, -6"]}, "cell": "mat3_det/1", "stage": "C"}
+{"id": "po-mat3_det-t2-eval-001847", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} 1 & -4 & -4 \\\\ x & y & z \\\\ 1 & 1 & 4 \\end{pmatrix} is 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} 1 & -4 & -4 \\\\ x & y & z \\\\ 1 & 1 & 4 \\end{pmatrix} is 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 8, -9", "ref": "-9, 8, -9", "family": "po_mat3_det", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 32, "bucket": null, "guess_rate": 0.004665, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 9, "M": [1, -4, -4, 3, 0, 3, 1, 1, 4], "pos": [3, 4, 5], "D": 1}, "seed": 261600747302003090, "version": "po-v0", "tier": 2, "n_raw": 32, "small": 1, "valid_answers": ["-9, 8, -9", "-8, 9, -5", "-7, 5, -9", "-6, 6, -5", "-5, 2, -9", "-5, 7, -1", "-4, 3, -5", "-4, 8, 3", "-3, -1, -9", "-3, 4, -1", "-3, 9, 7", "-2, 0, -5", "-2, 5, 3", "-1, -4, -9", "-1, 1, -1", "-1, 6, 7", "0, -3, -5", "0, 2, 3", "1, -7, -9", "1, -2, -1", "1, 3, 7", "2, -6, -5", "2, -1, 3", "3, -5, -1", "3, 0, 7", "4, -9, -5", "4, -4, 3", "5, -8, -1", "5, -3, 7", "6, -7, 3", "7, -6, 7", "9, -9, 7"]}, "cell": "mat3_det/2", "stage": "B"}
+{"id": "po-mat3_det-t2-eval-000122", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} x & y & z \\\\ 3 & -1 & 2 \\\\ -3 & -2 & 0 \\end{pmatrix} is 59. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} x & y & z \\\\ 3 & -1 & 2 \\\\ -3 & -2 & 0 \\end{pmatrix} is 59. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -7, -5", "ref": "-7, -7, -5", "family": "po_mat3_det", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 30, "bucket": null, "guess_rate": 0.004374, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 9, "M": [-2, 1, 1, 3, -1, 2, -3, -2, 0], "pos": [0, 1, 2], "D": 59}, "seed": 261600747302000201, "version": "po-v0", "tier": 2, "n_raw": 30, "small": 4, "valid_answers": ["-7, -7, -5", "-7, -4, -7", "-7, -1, -9", "-4, -8, -3", "-4, -5, -5", "-4, -2, -7", "-4, 1, -9", "-1, -9, -1", "-1, -6, -3", "-1, -3, -5", "-1, 0, -7", "-1, 3, -9", "2, -7, -1", "2, -4, -3", "2, -1, -5", "2, 2, -7", "2, 5, -9", "5, -8, 1", "5, -5, -1", "5, -2, -3", "5, 1, -5", "5, 4, -7", "5, 7, -9", "8, -9, 3", "8, -6, 1", "8, -3, -1", "8, 0, -3", "8, 3, -5", "8, 6, -7", "8, 9, -9"]}, "cell": "mat3_det/2", "stage": "B"}
+{"id": "po-mat3_det-t2-eval-002312", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} x & y & z \\\\ 3 & -1 & -3 \\\\ -1 & -3 & 2 \\end{pmatrix} is -14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} x & y & z \\\\ 3 & -1 & -3 \\\\ -1 & -3 & 2 \\end{pmatrix} is -14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 4, 9", "ref": "-8, 4, 9", "family": "po_mat3_det", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 31, "bucket": null, "guess_rate": 0.00452, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 9, "M": [2, 3, 4, 3, -1, -3, -1, -3, 2], "pos": [0, 1, 2], "D": -14}, "seed": 261600747302003849, "version": "po-v0", "tier": 2, "n_raw": 31, "small": 1, "valid_answers": ["-8, 4, 9", "-7, 7, 7", "-6, 0, 8", "-5, -7, 9", "-5, 3, 6", "-4, -4, 7", "-4, 6, 4", "-3, -1, 5", "-3, 9, 2", "-2, -8, 6", "-2, 2, 3", "-1, -5, 4", "-1, 5, 1", "0, -2, 2", "0, 8, -1", "1, -9, 3", "1, 1, 0", "2, -6, 1", "2, 4, -2", "3, -3, -1", "3, 7, -4", "4, 0, -3", "5, -7, -2", "5, 3, -5", "6, -4, -4", "6, 6, -7", "7, -1, -6", "7, 9, -9", "8, -8, -5", "8, 2, -8", "9, -5, -7"]}, "cell": "mat3_det/2", "stage": "B"}
+{"id": "po-mat3_det-t2-eval-000948", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} -2 & -4 & -2 \\\\ x & y & z \\\\ 4 & -3 & -2 \\end{pmatrix} is 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} -2 & -4 & -2 \\\\ x & y & z \\\\ 4 & -3 & -2 \\end{pmatrix} is 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -7, -5", "ref": "-9, -7, -5", "family": "po_mat3_det", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 33, "bucket": null, "guess_rate": 0.004811, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 9, "M": [-2, -4, -2, 3, 4, -4, 4, -3, -2], "pos": [3, 4, 5], "D": 44}, "seed": 261600747302001582, "version": "po-v0", "tier": 2, "n_raw": 33, "small": 2, "valid_answers": ["-9, -7, -5", "-9, 4, 1", "-8, -5, -4", "-8, 6, 2", "-7, -3, -3", "-7, 8, 3", "-6, -1, -2", "-5, 1, -1", "-4, -8, -6", "-4, 3, 0", "-3, -6, -5", "-3, 5, 1", "-2, -4, -4", "-2, 7, 2", "-1, -2, -3", "-1, 9, 3", "0, 0, -2", "1, -9, -7", "1, 2, -1", "2, -7, -6", "2, 4, 0", "3, -5, -5", "3, 6, 1", "4, -3, -4", "4, 8, 2", "5, -1, -3", "6, 1, -2", "7, -8, -7", "7, 3, -1", "8, -6, -6", "8, 5, 0", "9, -4, -5", "9, 7, 1"]}, "cell": "mat3_det/2", "stage": "B"}
+{"id": "po-mat3_det-t2-eval-000078", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} -1 & 1 & 3 \\\\ x & y & z \\\\ -4 & -4 & 1 \\end{pmatrix} is -24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} -1 & 1 & 3 \\\\ x & y & z \\\\ -4 & -4 & 1 \\end{pmatrix} is -24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -7, 8", "ref": "-9, -7, 8", "family": "po_mat3_det", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 25, "bucket": null, "guess_rate": 0.003645, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 9, "M": [-1, 1, 3, 0, -2, -4, -4, -4, 1], "pos": [3, 4, 5], "D": -24}, "seed": 261600747302000122, "version": "po-v0", "tier": 2, "n_raw": 25, "small": 1, "valid_answers": ["-9, -7, 8", "-8, -8, 5", "-7, -9, 2", "-5, -3, 7", "-4, -4, 4", "-3, -5, 1", "-2, -6, -2", "-2, 2, 9", "-1, -7, -5", "-1, 1, 6", "0, -8, -8", "0, 0, 3", "1, -1, 0", "2, -2, -3", "2, 6, 8", "3, -3, -6", "3, 5, 5", "4, -4, -9", "4, 4, 2", "5, 3, -1", "6, 2, -4", "7, 1, -7", "7, 9, 4", "8, 8, 1", "9, 7, -2"]}, "cell": "mat3_det/2", "stage": "B"}
+{"id": "po-mat3_det-t2-eval-000575", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} 2 & 1 & 4 \\\\ 2 & -1 & 3 \\\\ x & y & z \\end{pmatrix} is -69. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} 2 & 1 & 4 \\\\ 2 & -1 & 3 \\\\ x & y & z \\end{pmatrix} is -69. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -9, -3", "ref": "-9, -9, -3", "family": "po_mat3_det", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 27, "bucket": null, "guess_rate": 0.003936, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 9, "M": [2, 1, 4, 2, -1, 3, -2, 4, 0], "pos": [6, 7, 8], "D": -69}, "seed": 261600747302000967, "version": "po-v0", "tier": 2, "n_raw": 27, "small": 6, "valid_answers": ["-9, -9, -3", "-9, -7, -2", "-9, -5, -1", "-9, -3, 0", "-9, -1, 1", "-9, 1, 2", "-9, 3, 3", "-9, 5, 4", "-9, 7, 5", "-9, 9, 6", "-7, -8, 1", "-7, -6, 2", "-7, -4, 3", "-7, -2, 4", "-7, 0, 5", "-7, 2, 6", "-7, 4, 7", "-7, 6, 8", "-7, 8, 9", "-5, -9, 4", "-5, -7, 5", "-5, -5, 6", "-5, -3, 7", "-5, -1, 8", "-5, 1, 9", "-3, -8, 8", "-3, -6, 9"]}, "cell": "mat3_det/2", "stage": "B"}
+{"id": "po-mat3_det-t2-eval-001620", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} -1 & 2 & -2 \\\\ x & y & z \\\\ 0 & -1 & 2 \\end{pmatrix} is -28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} -1 & 2 & -2 \\\\ x & y & z \\\\ 0 & -1 & 2 \\end{pmatrix} is -28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 9, 8", "ref": "1, 9, 8", "family": "po_mat3_det", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 45, "bucket": null, "guess_rate": 0.006561, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 9, "M": [-1, 2, -2, 3, -4, -1, 0, -1, 2], "pos": [3, 4, 5], "D": -28}, "seed": 261600747302002710, "version": "po-v0", "tier": 2, "n_raw": 45, "small": 6, "valid_answers": ["1, 9, 8", "2, 8, 8", "2, 9, 6", "3, 7, 8", "3, 8, 6", "3, 9, 4", "4, 6, 8", "4, 7, 6", "4, 8, 4", "4, 9, 2", "5, 5, 8", "5, 6, 6", "5, 7, 4", "5, 8, 2", "5, 9, 0", "6, 4, 8", "6, 5, 6", "6, 6, 4", "6, 7, 2", "6, 8, 0", "6, 9, -2", "7, 3, 8", "7, 4, 6", "7, 5, 4", "7, 6, 2", "7, 7, 0", "7, 8, -2", "7, 9, -4", "8, 2, 8", "8, 3, 6", "8, 4, 4", "8, 5, 2", "8, 6, 0", "8, 7, -2", "8, 8, -4", "8, 9, -6", "9, 1, 8", "9, 2, 6", "9, 3, 4", "9, 4, 2", "9, 5, 0", "9, 6, -2", "9, 7, -4", "9, 8, -6", "9, 9, -8"]}, "cell": "mat3_det/2", "stage": "B"}
+{"id": "po-mat3_det-t2-eval-000209", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} -3 & 2 & -1 \\\\ x & y & z \\\\ 1 & -3 & -1 \\end{pmatrix} is 58. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} -3 & 2 & -1 \\\\ x & y & z \\\\ 1 & -3 & -1 \\end{pmatrix} is 58. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 8, -8", "ref": "-6, 8, -8", "family": "po_mat3_det", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 30, "bucket": null, "guess_rate": 0.004374, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 9, "M": [-3, 2, -1, -4, -1, 3, 1, -3, -1], "pos": [3, 4, 5], "D": 58}, "seed": 261600747302000368, "version": "po-v0", "tier": 2, "n_raw": 30, "small": 5, "valid_answers": ["-6, 8, -8", "-5, 5, -9", "-4, 9, -6", "-3, 6, -7", "-2, 3, -8", "-1, 0, -9", "-1, 7, -5", "0, 4, -6", "1, 1, -7", "1, 8, -3", "2, -2, -8", "2, 5, -4", "3, -5, -9", "3, 2, -5", "3, 9, -1", "4, -1, -6", "4, 6, -2", "5, -4, -7", "5, 3, -3", "6, -7, -8", "6, 0, -4", "6, 7, 0", "7, -3, -5", "7, 4, -1", "8, -6, -6", "8, 1, -2", "8, 8, 2", "9, -9, -7", "9, -2, -3", "9, 5, 1"]}, "cell": "mat3_det/2", "stage": "B"}
+{"id": "po-mat3_det-t2-eval-002234", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} x & y & z \\\\ -3 & -4 & 4 \\\\ -3 & 4 & -2 \\end{pmatrix} is 42. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} x & y & z \\\\ -3 & -4 & 4 \\\\ -3 & 4 & -2 \\end{pmatrix} is 42. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -9, 8", "ref": "-9, -9, 8", "family": "po_mat3_det", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 34, "bucket": null, "guess_rate": 0.004957, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 9, "M": [-3, 2, 2, -3, -4, 4, -3, 4, -2], "pos": [0, 1, 2], "D": 42}, "seed": 261600747302003703, "version": "po-v0", "tier": 2, "n_raw": 34, "small": 1, "valid_answers": ["-9, -9, 8", "-9, -5, 5", "-9, -1, 2", "-9, 3, -1", "-9, 7, -4", "-6, -9, 7", "-6, -5, 4", "-6, -1, 1", "-6, 3, -2", "-6, 7, -5", "-3, -9, 6", "-3, -5, 3", "-3, -1, 0", "-3, 3, -3", "-3, 7, -6", "0, -9, 5", "0, -5, 2", "0, -1, -1", "0, 3, -4", "0, 7, -7", "3, -9, 4", "3, -5, 1", "3, -1, -2", "3, 3, -5", "3, 7, -8", "6, -9, 3", "6, -5, 0", "6, -1, -3", "6, 3, -6", "6, 7, -9", "9, -9, 2", "9, -5, -1", "9, -1, -4", "9, 3, -7"]}, "cell": "mat3_det/2", "stage": "B"}
+{"id": "po-mat3_det-t2-eval-000455", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} x & y & z \\\\ 0 & -4 & -3 \\\\ 4 & 3 & 0 \\end{pmatrix} is -93. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} x & y & z \\\\ 0 & -4 & -3 \\\\ 4 & 3 & 0 \\end{pmatrix} is -93. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -7, -6", "ref": "-9, -7, -6", "family": "po_mat3_det", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.002916, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 9, "M": [-4, -4, -3, 0, -4, -3, 4, 3, 0], "pos": [0, 1, 2], "D": -93}, "seed": 261600747302000769, "version": "po-v0", "tier": 2, "n_raw": 20, "small": 3, "valid_answers": ["-9, -7, -6", "-9, -3, -3", "-9, 1, 0", "-9, 5, 3", "-9, 9, 6", "-5, -8, -9", "-5, -4, -6", "-5, 0, -3", "-5, 4, 0", "-5, 8, 3", "-1, -5, -9", "-1, -1, -6", "-1, 3, -3", "-1, 7, 0", "3, -2, -9", "3, 2, -6", "3, 6, -3", "7, 1, -9", "7, 5, -6", "7, 9, -3"]}, "cell": "mat3_det/2", "stage": "B"}
+{"id": "po-mat3_det-t2-eval-000895", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} 4 & 0 & 2 \\\\ x & y & z \\\\ 0 & -3 & -4 \\end{pmatrix} is -40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} 4 & 0 & 2 \\\\ x & y & z \\\\ 0 & -3 & -4 \\end{pmatrix} is -40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 1, -6", "ref": "-8, 1, -6", "family": "po_mat3_det", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 38, "bucket": null, "guess_rate": 0.00554, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 9, "M": [4, 0, 2, 1, -4, -4, 0, -3, -4], "pos": [3, 4, 5], "D": -40}, "seed": 261600747302001488, "version": "po-v0", "tier": 2, "n_raw": 38, "small": 2, "valid_answers": ["-8, 1, -6", "-8, 4, -2", "-8, 7, 2", "-6, -2, -9", "-6, 1, -5", "-6, 4, -1", "-6, 7, 3", "-4, -2, -8", "-4, 1, -4", "-4, 4, 0", "-4, 7, 4", "-2, -2, -7", "-2, 1, -3", "-2, 4, 1", "-2, 7, 5", "0, -2, -6", "0, 1, -2", "0, 4, 2", "0, 7, 6", "2, -5, -9", "2, -2, -5", "2, 1, -1", "2, 4, 3", "2, 7, 7", "4, -5, -8", "4, -2, -4", "4, 1, 0", "4, 4, 4", "4, 7, 8", "6, -5, -7", "6, -2, -3", "6, 1, 1", "6, 4, 5", "6, 7, 9", "8, -5, -6", "8, -2, -2", "8, 1, 2", "8, 4, 6"]}, "cell": "mat3_det/2", "stage": "B"}
+{"id": "po-mat3_det-t2-eval-000134", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} 1 & -3 & -1 \\\\ -3 & 2 & 2 \\\\ x & y & z \\end{pmatrix} is 25. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -9 <= x, y, z <= 9 such that the determinant of \\begin{pmatrix} 1 & -3 & -1 \\\\ -3 & 2 & 2 \\\\ x & y & z \\end{pmatrix} is 25. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -4, 1", "ref": "-9, -4, 1", "family": "po_mat3_det", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 50, "bucket": null, "guess_rate": 0.00729, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 9, "M": [1, -3, -1, -3, 2, 2, 4, -3, 3], "pos": [6, 7, 8], "D": 25}, "seed": 261600747302000223, "version": "po-v0", "tier": 2, "n_raw": 50, "small": 3, "valid_answers": ["-9, -4, 1", "-9, 3, 2", "-8, -7, 0", "-8, 0, 1", "-8, 7, 2", "-7, -3, 0", "-7, 4, 1", "-6, -6, -1", "-6, 1, 0", "-6, 8, 1", "-5, -9, -2", "-5, -2, -1", "-5, 5, 0", "-4, -5, -2", "-4, 2, -1", "-4, 9, 0", "-3, -8, -3", "-3, -1, -2", "-3, 6, -1", "-2, -4, -3", "-2, 3, -2", "-1, -7, -4", "-1, 0, -3", "-1, 7, -2", "0, -3, -4", "0, 4, -3", "1, -6, -5", "1, 1, -4", "1, 8, -3", "2, -9, -6", "2, -2, -5", "2, 5, -4", "3, -5, -6", "3, 2, -5", "3, 9, -4", "4, -8, -7", "4, -1, -6", "4, 6, -5", "5, -4, -7", "5, 3, -6", "6, -7, -8", "6, 0, -7", "6, 7, -6", "7, -3, -8", "7, 4, -7", "8, -6, -9", "8, 1, -8", "8, 8, -7", "9, -2, -9", "9, 5, -8"]}, "cell": "mat3_det/2", "stage": "B"}
+{"id": "po-mat3_det-t3-eval-002109", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} x & -2 & 1 \\\\ 3 & -2 & y \\\\ 1 & z & -2 \\end{pmatrix} is -30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} x & -2 & 1 \\\\ 3 & -2 & y \\\\ 1 & z & -2 \\end{pmatrix} is -30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -2, 0", "ref": "-6, -2, 0", "family": "po_mat3_det", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 0.010014, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [2, -2, 1, 3, -2, 0, 1, 3, -2], "pos": [0, 5, 7], "D": -30}, "seed": 261600747303002667, "version": "po-v0", "tier": 3, "n_raw": 22, "small": 2, "valid_answers": ["-6, -2, 0", "-5, -1, 1", "-5, 0, 0", "-4, -1, 6", "-4, 2, 0", "-3, -2, 4", "-3, 1, -1", "-3, 4, 0", "-2, -4, 4", "-2, -3, 6", "-2, 0, -4", "-2, 1, -2", "-2, 6, 0", "0, 1, -6", "0, 4, -4", "1, -6, -4", "1, -3, -5", "1, 6, 4", "2, -1, -6", "2, 4, 4", "3, -2, -4", "4, -5, -2"]}, "cell": "mat3_det/3", "stage": "C"}
+{"id": "po-mat3_det-t3-eval-001816", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 2 & x & -1 \\\\ 2 & 1 & y \\\\ z & -2 & 0 \\end{pmatrix} is -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 2 & x & -1 \\\\ 2 & 1 & y \\\\ z & -2 & 0 \\end{pmatrix} is -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -3, 0", "ref": "-6, -3, 0", "family": "po_mat3_det", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 29, "bucket": null, "guess_rate": 0.0132, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [2, -2, -1, 2, 1, -1, -2, -2, 0], "pos": [1, 5, 6], "D": -8}, "seed": 261600747303002292, "version": "po-v0", "tier": 3, "n_raw": 29, "small": 3, "valid_answers": ["-6, -3, 0", "-5, -3, 0", "-5, 1, 4", "-4, -3, 0", "-3, -3, 0", "-3, -1, -2", "-3, 2, 4", "-3, 3, 3", "-2, -3, 0", "-2, 4, 4", "-1, -3, 0", "-1, -1, -4", "0, -4, 4", "0, -3, 0", "0, -2, -4", "1, -5, -2", "1, -3, 0", "1, -2, 4", "1, 3, -6", "2, -3, 0", "2, 2, -4", "3, -3, 0", "3, -1, 4", "3, 1, -4", "3, 5, -2", "4, -3, 0", "5, -3, 0", "5, -1, 2", "6, -3, 0"]}, "cell": "mat3_det/3", "stage": "C"}
+{"id": "po-mat3_det-t3-eval-001899", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -3 & x & -2 \\\\ y & -1 & 0 \\\\ 1 & -1 & z \\end{pmatrix} is 40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -3 & x & -2 \\\\ y & -1 & 0 \\\\ 1 & -1 & z \\end{pmatrix} is 40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -6, -2", "ref": "-5, -6, -2", "family": "po_mat3_det", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.008648, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [-3, 2, -2, 3, -1, 0, 1, -1, 0], "pos": [1, 3, 8], "D": 40}, "seed": 261600747303002399, "version": "po-v0", "tier": 3, "n_raw": 19, "small": 3, "valid_answers": ["-5, -6, -2", "-5, -3, -4", "-5, 1, 5", "-5, 3, 2", "-3, 3, 3", "-2, -6, -6", "-2, 3, 4", "-2, 6, 2", "-1, 3, 6", "-1, 5, 4", "1, -6, 6", "2, -5, 4", "3, -3, 4", "3, 3, -6", "3, 6, -2", "4, -6, 2", "4, 3, -4", "5, 3, -3", "5, 4, -2"]}, "cell": "mat3_det/3", "stage": "C"}
+{"id": "po-mat3_det-t3-eval-000497", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 3 & -3 & x \\\\ -3 & y & 1 \\\\ z & 0 & 0 \\end{pmatrix} is 30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 3 & -3 & x \\\\ -3 & y & 1 \\\\ z & 0 & 0 \\end{pmatrix} is 30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -2, -2", "ref": "-6, -2, -2", "family": "po_mat3_det", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 26, "bucket": null, "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [3, -3, 0, -3, 0, 1, 0, 0, 0], "pos": [2, 4, 6], "D": 30}, "seed": 261600747303000640, "version": "po-v0", "tier": 3, "n_raw": 26, "small": 4, "valid_answers": ["-6, -2, -2", "-6, 3, 2", "-4, -3, -2", "-4, 2, 6", "-3, -4, -2", "-3, -1, -5", "-3, 3, 5", "-3, 6, 2", "-2, -6, -2", "-2, -1, -6", "-2, 4, 6", "-1, -3, -5", "-1, -2, -6", "1, 2, -6", "1, 3, -5", "2, -4, 6", "2, 1, -6", "2, 6, -2", "3, -6, 2", "3, -3, 5", "3, 1, -5", "3, 4, -2", "4, -2, 6", "4, 3, -2", "6, -3, 2", "6, 2, -2"]}, "cell": "mat3_det/3", "stage": "C"}
+{"id": "po-mat3_det-t3-eval-000672", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 0 & -2 & x \\\\ 1 & y & -1 \\\\ z & -2 & 3 \\end{pmatrix} is 36. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 0 & -2 & x \\\\ 1 & y & -1 \\\\ z & -2 & 3 \\end{pmatrix} is 36. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -6, -1", "ref": "-4, -6, -1", "family": "po_mat3_det", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.003641, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [0, -2, 1, 1, -1, -1, 1, -2, 3], "pos": [2, 4, 6], "D": 36}, "seed": 261600747303000858, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 3, "valid_answers": ["-4, -6, -1", "-4, 5, 1", "-3, -2, -6", "-3, 2, 3", "-1, 5, 4", "1, -6, 4", "5, 2, -5", "6, -2, 3"]}, "cell": "mat3_det/3", "stage": "C"}
+{"id": "po-mat3_det-t3-eval-001714", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -3 & 0 & x \\\\ 3 & y & -2 \\\\ z & 1 & 0 \\end{pmatrix} is -13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} -3 & 0 & x \\\\ 3 & y & -2 \\\\ z & 1 & 0 \\end{pmatrix} is -13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -4, 1", "ref": "-1, -4, 1", "family": "po_mat3_det", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.004552, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [-3, 0, 3, 3, 2, -2, 0, 1, 0], "pos": [2, 4, 6], "D": -13}, "seed": 261600747303002157, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 2, "valid_answers": ["-1, -4, 1", "-1, -2, 2", "-1, -1, 4", "-1, 1, -4", "-1, 2, -2", "-1, 4, -1", "1, -5, -2", "1, -2, -5", "1, 2, 5", "1, 5, 2"]}, "cell": "mat3_det/3", "stage": "C"}
+{"id": "po-mat3_det-t3-eval-000872", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 3 & 0 & x \\\\ y & -1 & -1 \\\\ 2 & z & -1 \\end{pmatrix} is -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 3 & 0 & x \\\\ y & -1 & -1 \\\\ 2 & z & -1 \\end{pmatrix} is -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -6, 0", "ref": "-5, -6, 0", "family": "po_mat3_det", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 32, "bucket": null, "guess_rate": 0.014565, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [3, 0, 2, 0, -1, -1, 2, -1, -1], "pos": [2, 3, 7], "D": -7}, "seed": 261600747303001101, "version": "po-v0", "tier": 3, "n_raw": 32, "small": 2, "valid_answers": ["-5, -6, 0", "-5, -5, 0", "-5, -4, 0", "-5, -3, 0", "-5, -2, 0", "-5, -1, 0", "-5, 0, 0", "-5, 1, 0", "-5, 2, 0", "-5, 3, 0", "-5, 4, 0", "-5, 5, 0", "-5, 6, 0", "-4, 1, 2", "-2, 0, -2", "-2, 1, -6", "-2, 2, 6", "-2, 3, 2", "-1, -5, -1", "-1, -1, -2", "-1, 1, -4", "-1, 5, 4", "1, -6, 4", "1, -5, 6", "1, -1, -6", "1, 0, -4", "1, 1, -3", "1, 3, -2", "2, -5, 2", "2, 2, -2", "4, -3, 2", "4, 0, -6"]}, "cell": "mat3_det/3", "stage": "C"}
+{"id": "po-mat3_det-t3-eval-001717", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} x & 0 & -3 \\\\ -1 & 0 & y \\\\ -2 & z & 0 \\end{pmatrix} is 198. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} x & 0 & -3 \\\\ -1 & 0 & y \\\\ -2 & z & 0 \\end{pmatrix} is 198. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -6, -6", "ref": "-6, -6, -6", "family": "po_mat3_det", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.002731, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [-2, 0, -3, -1, 0, -2, -2, -2, 0], "pos": [0, 5, 7], "D": 198}, "seed": 261600747303002163, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 6, "valid_answers": ["-6, -6, -6", "-6, 5, 6", "-5, 6, 6", "5, -6, 6", "6, -5, 6", "6, 6, -6"]}, "cell": "mat3_det/3", "stage": "C"}
+{"id": "po-mat3_det-t3-eval-000275", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 2 & -3 & x \\\\ -3 & y & 1 \\\\ z & -1 & -3 \\end{pmatrix} is 54. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} 2 & -3 & x \\\\ -3 & y & 1 \\\\ z & -1 & -3 \\end{pmatrix} is 54. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -1, -5", "ref": "-2, -1, -5", "family": "po_mat3_det", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.003641, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [2, -3, 3, -3, -2, 1, -1, -1, -3], "pos": [2, 4, 6], "D": 54}, "seed": 261600747303000365, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 4, "valid_answers": ["-2, -1, -5", "1, -5, -4", "1, -4, -2", "2, -4, -1", "2, 1, -5", "4, -5, -1", "5, -1, 2", "5, 1, -2"]}, "cell": "mat3_det/3", "stage": "C"}
+{"id": "po-mat3_det-t3-eval-001970", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} x & 1 & -2 \\\\ -3 & y & -3 \\\\ 2 & -1 & z \\end{pmatrix} is 30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} x & 1 & -2 \\\\ -3 & y & -3 \\\\ 2 & -1 & z \\end{pmatrix} is 30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 6, 0", "ref": "-6, 6, 0", "family": "po_mat3_det", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.005462, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [2, 1, -2, -3, -1, -3, 2, -1, -2], "pos": [0, 4, 8], "D": 30}, "seed": 261600747303002491, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 4, "valid_answers": ["-6, 6, 0", "-4, -6, 2", "-4, 3, -2", "-2, -6, 4", "-2, 4, -4", "-1, 6, -5", "0, 6, 6", "2, 3, 4", "3, -6, -5", "4, 4, 2", "5, -6, -3", "5, 6, 1"]}, "cell": "mat3_det/3", "stage": "C"}
+{"id": "po-mat3_det-t3-eval-002421", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} x & 3 & 0 \\\\ -2 & y & 1 \\\\ 1 & 3 & z \\end{pmatrix} is 27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} x & 3 & 0 \\\\ -2 & y & 1 \\\\ 1 & 3 & z \\end{pmatrix} is 27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 0, 1", "ref": "-6, 0, 1", "family": "po_mat3_det", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 36, "bucket": null, "guess_rate": 0.016386, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [3, 3, 0, -2, 3, 1, 1, 3, 3], "pos": [0, 4, 8], "D": 27}, "seed": 261600747303003044, "version": "po-v0", "tier": 3, "n_raw": 36, "small": 3, "valid_answers": ["-6, 0, 1", "-6, 2, -1", "-5, 3, -1", "-4, 0, 2", "-4, 1, 6", "-4, 2, -6", "-4, 3, -2", "-3, -3, 1", "-3, 1, 5", "-3, 3, -5", "-2, -6, 1", "-2, 0, 3", "-2, 6, -3", "-1, -1, 3", "0, -6, 4", "0, -5, 4", "0, -4, 4", "0, -3, 4", "0, -2, 4", "0, -1, 4", "0, 0, 4", "0, 1, 4", "0, 2, 4", "0, 3, 4", "0, 4, 4", "0, 5, 4", "0, 6, 4", "1, 3, 3", "2, -6, -5", "2, 0, 5", "2, 2, 3", "4, -6, -2", "4, -3, -6", "4, 0, 6", "4, 3, 2", "6, 6, 1"]}, "cell": "mat3_det/3", "stage": "C"}
+{"id": "po-mat3_det-t3-eval-001488", "task": "cx_po_mat3_det", "source": "cx_po_mat3_det", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} x & 2 & 0 \\\\ 1 & y & -1 \\\\ -1 & -1 & z \\end{pmatrix} is -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that the determinant of \\begin{pmatrix} x & 2 & 0 \\\\ 1 & y & -1 \\\\ -1 & -1 & z \\end{pmatrix} is -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -2, -1", "ref": "-5, -2, -1", "family": "po_mat3_det", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 28, "bucket": null, "guess_rate": 0.012745, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 6, "M": [3, 2, 0, 1, -3, -1, -1, -1, 1], "pos": [0, 4, 8], "D": -1}, "seed": 261600747303001875, "version": "po-v0", "tier": 3, "n_raw": 28, "small": 1, "valid_answers": ["-5, -2, -1", "-5, 0, 4", "-3, -1, -6", "-3, 0, 3", "-1, -6, -1", "-1, -4, -2", "-1, -3, -4", "-1, -1, 4", "-1, 0, 2", "-1, 2, 1", "1, 0, 1", "1, 1, 2", "1, 3, -2", "1, 4, -1", "3, -6, 0", "3, -5, 0", "3, -4, 0", "3, -3, 0", "3, -2, 0", "3, -1, 0", "3, 0, 0", "3, 1, 0", "3, 2, 0", "3, 3, 0", "3, 4, 0", "3, 5, 0", "3, 6, 0", "5, 0, -1"]}, "cell": "mat3_det/3", "stage": "C"}
+{"id": "po-mobius-t0-eval-000997", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(1) = -1 and f(3) = -\\frac{5}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(1) = -1 and f(3) = -\\frac{5}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 2, 0, -3", "ref": "1, 2, 0, -3", "family": "po_mobius", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0025, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 3, "pts": [[1, "-1"], [3, "-5/3"]]}, "seed": 68114079300004264, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 2, "valid_answers": ["1, 2, 0, -3", "2, -1, -1, 0", "3, 1, -1, -3"]}, "cell": "mobius/0", "stage": "C"}
+{"id": "po-mobius-t0-eval-000335", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = -3 and f(2) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = -3 and f(2) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -2, -3, -2", "ref": "1, -2, -3, -2", "family": "po_mobius", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.005, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 3, "pts": [[-1, "-3"], [2, "0"]]}, "seed": 68114079300000751, "version": "po-v0", "tier": 0, "n_raw": 12, "small": 2, "valid_answers": ["1, -2, -3, -2", "1, -2, -2, -1", "1, -2, -1, 0", "1, -2, 0, 1", "1, -2, 1, 2", "1, -2, 2, 3"]}, "cell": "mobius/0", "stage": "C"}
+{"id": "po-mobius-t0-eval-001024", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = -\\frac{5}{7} and f(2) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = -\\frac{5}{7} and f(2) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -2, -3, -2", "ref": "1, -2, -3, -2", "family": "po_mobius", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001667, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 3, "pts": [[-3, "-5/7"], [2, "0"]]}, "seed": 68114079300004676, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 2, "valid_answers": ["1, -2, -3, -2", "1, -2, -2, 1"]}, "cell": "mobius/0", "stage": "C"}
+{"id": "po-mobius-t0-eval-001161", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(1) = -1 and f(3) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(1) = -1 and f(3) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, 1, -2", "ref": "0, 1, 1, -2", "family": "po_mobius", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.003333, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 3, "pts": [[1, "-1"], [3, "1"]]}, "seed": 68114079300007923, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 2, "valid_answers": ["0, 1, 1, -2", "1, -2, 0, 1", "1, 0, 2, -3", "2, -3, 1, 0"]}, "cell": "mobius/0", "stage": "C"}
+{"id": "po-mobius-t0-eval-001169", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = -\\frac{1}{2} and f(1) = \\frac{1}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = -\\frac{1}{2} and f(1) = \\frac{1}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, 2, 0", "ref": "0, 1, 2, 0", "family": "po_mobius", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001667, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 3, "pts": [[-1, "-1/2"], [1, "1/2"]]}, "seed": 68114079300008129, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 2, "valid_answers": ["0, 1, 2, 0", "1, 0, 0, 2"]}, "cell": "mobius/0", "stage": "C"}
+{"id": "po-mobius-t0-eval-000493", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = -\\frac{5}{3} and f(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = -\\frac{5}{3} and f(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -2, -1, 0", "ref": "1, -2, -1, 0", "family": "po_mobius", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0025, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 3, "pts": [[-3, "-5/3"], [1, "1"]]}, "seed": 68114079300001220, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 2, "valid_answers": ["1, -2, -1, 0", "2, 1, 0, 3", "3, -1, -1, 3"]}, "cell": "mobius/0", "stage": "C"}
+{"id": "po-mobius-t0-eval-000552", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = 0 and f(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = 0 and f(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 0, -2, 3", "ref": "1, 0, -2, 3", "family": "po_mobius", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.008333, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 3, "pts": [[0, "0"], [1, "1"]]}, "seed": 68114079300001450, "version": "po-v0", "tier": 0, "n_raw": 24, "small": 1, "valid_answers": ["1, 0, -2, 3", "1, 0, -1, 2", "1, 0, 0, 1", "1, 0, 2, -1", "1, 0, 3, -2", "2, 0, -1, 3", "2, 0, 1, 1", "2, 0, 3, -1", "3, 0, 1, 2", "3, 0, 2, 1"]}, "cell": "mobius/0", "stage": "C"}
+{"id": "po-mobius-t0-eval-000598", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(1) = -1 and f(3) = -\\frac{7}{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(1) = -1 and f(3) = -\\frac{7}{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 1, -1, -2", "ref": "2, 1, -1, -2", "family": "po_mobius", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001667, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 3, "pts": [[1, "-1"], [3, "-7/5"]]}, "seed": 68114079300001631, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 2, "valid_answers": ["2, 1, -1, -2", "3, -2, -2, 1"]}, "cell": "mobius/0", "stage": "C"}
+{"id": "po-mobius-t0-eval-000062", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = 2 and f(2) = \\frac{2}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = 2 and f(2) = \\frac{2}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 2, 1, 1", "ref": "0, 2, 1, 1", "family": "po_mobius", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001667, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 3, "pts": [[0, "2"], [2, "2/3"]]}, "seed": 68114079300000113, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 2, "valid_answers": ["0, 2, 1, 1", "2, -2, 2, -1"]}, "cell": "mobius/0", "stage": "C"}
+{"id": "po-mobius-t0-eval-000818", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-2) = -\\frac{1}{2} and f(-1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-2) = -\\frac{1}{2} and f(-1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -1, 0", "ref": "1, 1, -1, 0", "family": "po_mobius", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.004167, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 3, "pts": [[-2, "-1/2"], [-1, "0"]]}, "seed": 68114079300002739, "version": "po-v0", "tier": 0, "n_raw": 14, "small": 1, "valid_answers": ["1, 1, -1, 0", "1, 1, 0, 2", "2, 2, -3, -2", "2, 2, -1, 2", "3, 3, -2, 2"]}, "cell": "mobius/0", "stage": "C"}
+{"id": "po-mobius-t0-eval-000910", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = -1 and f(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = -1 and f(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 2, 1, 1", "ref": "0, 2, 1, 1", "family": "po_mobius", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.003333, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 3, "pts": [[-3, "-1"], [1, "1"]]}, "seed": 68114079300003349, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 2, "valid_answers": ["0, 2, 1, 1", "1, -3, -2, 0", "1, 1, 0, 2", "2, 0, -1, 3"]}, "cell": "mobius/0", "stage": "C"}
+{"id": "po-mobius-t0-eval-000679", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = -\\frac{1}{2} and f(2) = -\\frac{5}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -3 <= a, b, c, d <= 3 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = -\\frac{1}{2} and f(2) = -\\frac{5}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 1, -1, -2", "ref": "2, 1, -1, -2", "family": "po_mobius", "domain": "function", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001667, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 3, "pts": [[0, "-1/2"], [2, "-5/4"]]}, "seed": 68114079300001992, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 2, "valid_answers": ["2, 1, -1, -2", "3, -1, -3, 2"]}, "cell": "mobius/0", "stage": "C"}
+{"id": "po-mobius-t1-eval-001680", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(2) = \\frac{1}{2} and f(3) = \\frac{3}{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(2) = \\frac{1}{2} and f(3) = \\frac{3}{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 0, 1, 2", "ref": "1, 0, 1, 2", "family": "po_mobius", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000546, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 5, "pts": [[2, "1/2"], [3, "3/5"]]}, "seed": 68114079301007685, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 2, "valid_answers": ["1, 0, 1, 2", "2, -3, 3, -4", "3, -3, 4, -2", "4, -3, 5, 0"]}, "cell": "mobius/1", "stage": "B"}
+{"id": "po-mobius-t1-eval-001633", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-2) = -1 and f(3) = -\\frac{1}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-2) = -1 and f(3) = -\\frac{1}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, -1, -3", "ref": "0, 1, -1, -3", "family": "po_mobius", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000683, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 5, "pts": [[-2, "-1"], [3, "-1/6"]]}, "seed": 68114079301006702, "version": "po-v0", "tier": 1, "n_raw": 12, "small": 2, "valid_answers": ["0, 1, -1, -3", "1, -2, -2, 0", "1, -1, -3, -3", "2, -5, -3, 3", "2, -3, -5, -3"]}, "cell": "mobius/1", "stage": "B"}
+{"id": "po-mobius-t1-eval-001348", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = \\frac{1}{4} and f(0) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = \\frac{1}{4} and f(0) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 2, -3, -1", "ref": "0, 2, -3, -1", "family": "po_mobius", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000546, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 5, "pts": [[-3, "1/4"], [0, "-2"]]}, "seed": 68114079301003861, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 2, "valid_answers": ["0, 2, -3, -1", "1, 2, 1, -1", "1, 4, -2, -2", "2, 2, 5, -1"]}, "cell": "mobius/1", "stage": "B"}
+{"id": "po-mobius-t1-eval-001026", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = -\\frac{5}{4} and f(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = -\\frac{5}{4} and f(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 5, 3, 5", "ref": "0, 5, 3, 5", "family": "po_mobius", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000956, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 5, "pts": [[-3, "-5/4"], [0, "1"]]}, "seed": 68114079301002291, "version": "po-v0", "tier": 1, "n_raw": 18, "small": 2, "valid_answers": ["0, 5, 3, 5", "1, -2, -2, -2", "2, 1, -1, 1", "3, -1, -3, -1", "3, 4, 0, 4", "4, -3, -5, -3", "5, 5, -1, 5"]}, "cell": "mobius/1", "stage": "B"}
+{"id": "po-mobius-t1-eval-001459", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-2) = 4 and f(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-2) = 4 and f(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 4, 1, 3", "ref": "0, 4, 1, 3", "family": "po_mobius", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.001093, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 5, "pts": [[-2, "4"], [1, "1"]]}, "seed": 68114079301004661, "version": "po-v0", "tier": 1, "n_raw": 20, "small": 2, "valid_answers": ["0, 4, 1, 3", "1, -2, 0, -1", "2, 0, 1, 1", "3, -2, 1, 0", "3, 2, 2, 3", "4, 4, 3, 5", "5, -2, 2, 1", "5, 2, 3, 4"]}, "cell": "mobius/1", "stage": "B"}
+{"id": "po-mobius-t1-eval-001068", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-2) = \\frac{1}{4} and f(-1) = \\frac{1}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-2) = \\frac{1}{4} and f(-1) = \\frac{1}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, -1, 2", "ref": "0, 1, -1, 2", "family": "po_mobius", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 5, "pts": [[-2, "1/4"], [-1, "1/3"]]}, "seed": 68114079301002444, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 2, "valid_answers": ["0, 1, -1, 2", "1, 0, 5, 2"]}, "cell": "mobius/1", "stage": "B"}
+{"id": "po-mobius-t1-eval-001175", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = \\frac{5}{4} and f(0) = \\frac{1}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = \\frac{5}{4} and f(0) = \\frac{1}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -2, 0, -4", "ref": "1, -2, 0, -4", "family": "po_mobius", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.00041, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 5, "pts": [[-3, "5/4"], [0, "1/2"]]}, "seed": 68114079301002900, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 2, "valid_answers": ["1, -2, 0, -4", "2, 1, 2, 2", "3, -1, 2, -2"]}, "cell": "mobius/1", "stage": "B"}
+{"id": "po-mobius-t1-eval-000566", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = 0 and f(1) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = 0 and f(1) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 0, -5, 4", "ref": "2, 0, -5, 4", "family": "po_mobius", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.001913, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 5, "pts": [[0, "0"], [1, "-2"]]}, "seed": 68114079301000983, "version": "po-v0", "tier": 1, "n_raw": 34, "small": 2, "valid_answers": ["2, 0, -5, 4", "2, 0, -4, 3", "2, 0, -3, 2", "2, 0, -2, 1", "2, 0, 0, -1", "2, 0, 1, -2", "2, 0, 2, -3", "2, 0, 3, -4", "2, 0, 4, -5", "4, 0, -5, 3", "4, 0, -3, 1", "4, 0, -1, -1", "4, 0, 1, -3", "4, 0, 3, -5"]}, "cell": "mobius/1", "stage": "B"}
+{"id": "po-mobius-t1-eval-001766", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = -\\frac{3}{2} and f(1) = -\\frac{1}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = -\\frac{3}{2} and f(1) = -\\frac{1}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 3, -2, -4", "ref": "0, 3, -2, -4", "family": "po_mobius", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.00082, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 5, "pts": [[-1, "-3/2"], [1, "-1/2"]]}, "seed": 68114079301009898, "version": "po-v0", "tier": 1, "n_raw": 16, "small": 2, "valid_answers": ["0, 3, -2, -4", "1, -2, 0, 2", "2, -1, -2, 0", "3, 0, -4, -2", "4, -5, -2, 4", "5, -4, -4, 2"]}, "cell": "mobius/1", "stage": "B"}
+{"id": "po-mobius-t1-eval-001136", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(2) = 0 and f(3) = -\\frac{1}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(2) = 0 and f(3) = -\\frac{1}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -2, -3, 3", "ref": "1, -2, -3, 3", "family": "po_mobius", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000683, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 5, "pts": [[2, "0"], [3, "-1/6"]]}, "seed": 68114079301002709, "version": "po-v0", "tier": 1, "n_raw": 12, "small": 2, "valid_answers": ["1, -2, -3, 3", "1, -2, -2, 0", "1, -2, -1, -3", "2, -4, -5, 3", "2, -4, -3, -3"]}, "cell": "mobius/1", "stage": "B"}
+{"id": "po-mobius-t1-eval-000487", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = \\frac{1}{2} and f(0) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = \\frac{1}{2} and f(0) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, -1, -1", "ref": "0, 1, -1, -1", "family": "po_mobius", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 0.002459, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 5, "pts": [[-3, "1/2"], [0, "-1"]]}, "seed": 68114079301000819, "version": "po-v0", "tier": 1, "n_raw": 54, "small": 1, "valid_answers": ["0, 1, -1, -1", "1, -3, 5, 3", "1, -2, 4, 2", "1, -1, 3, 1", "1, 1, 1, -1", "1, 2, 0, -2", "1, 4, -2, -4", "1, 5, -3, -5", "2, -1, 5, 1", "2, 1, 3, -1", "2, 3, 1, -3", "2, 5, -1, -5", "3, 1, 5, -1", "3, 2, 4, -2", "3, 4, 2, -4", "3, 5, 1, -5", "4, 3, 5, -3", "4, 5, 3, -5"]}, "cell": "mobius/1", "stage": "B"}
+{"id": "po-mobius-t1-eval-000809", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = \\frac{1}{3} and f(1) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -5 <= a, b, c, d <= 5 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = \\frac{1}{3} and f(1) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, -2, 1", "ref": "0, 1, -2, 1", "family": "po_mobius", "domain": "function", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.001639, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 5, "pts": [[-1, "1/3"], [1, "-1"]]}, "seed": 68114079301001629, "version": "po-v0", "tier": 1, "n_raw": 28, "small": 2, "valid_answers": ["0, 1, -2, 1", "1, -2, 5, -4", "1, 0, 1, -2", "1, 2, -3, 0", "1, 3, -5, 1", "2, -1, 4, -5", "2, 1, 0, -3", "2, 3, -4, -1", "3, 1, 1, -5", "3, 2, -1, -4", "3, 4, -5, -2", "4, 3, -2, -5"]}, "cell": "mobius/1", "stage": "B"}
+{"id": "po-mobius-t2-eval-000767", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = \\frac{11}{9} and f(-1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = \\frac{11}{9} and f(-1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -5, 1, -6", "ref": "2, -5, 1, -6", "family": "po_mobius", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000237, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 7, "pts": [[-3, "11/9"], [-1, "1"]]}, "seed": 68114079302001021, "version": "po-v0", "tier": 2, "n_raw": 14, "small": 3, "valid_answers": ["2, -5, 1, -6", "3, -2, 2, -3", "4, 1, 3, 0", "5, 4, 4, 3", "6, 7, 5, 6", "7, -1, 5, -3"]}, "cell": "mobius/2", "stage": "B"}
+{"id": "po-mobius-t2-eval-000143", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = -\\frac{2}{3} and f(3) = \\frac{5}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = -\\frac{2}{3} and f(3) = \\frac{5}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 2, 3, -3", "ref": "1, 2, 3, -3", "family": "po_mobius", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000119, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 7, "pts": [[0, "-2/3"], [3, "5/6"]]}, "seed": 68114079302000184, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 3, "valid_answers": ["1, 2, 3, -3", "3, -4, 0, 6", "4, -2, 3, 3"]}, "cell": "mobius/2", "stage": "B"}
+{"id": "po-mobius-t2-eval-000151", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = -\\frac{1}{3} and f(1) = -\\frac{1}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = -\\frac{1}{3} and f(1) = -\\frac{1}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, -3, -3", "ref": "0, 1, -3, -3", "family": "po_mobius", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 7, "pts": [[0, "-1/3"], [1, "-1/6"]]}, "seed": 68114079302000195, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 3, "valid_answers": ["0, 1, -3, -3", "1, -2, 0, 6"]}, "cell": "mobius/2", "stage": "B"}
+{"id": "po-mobius-t2-eval-000161", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = -1 and f(1) = \\frac{1}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = -1 and f(1) = \\frac{1}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, 2, 1", "ref": "0, 1, 2, 1", "family": "po_mobius", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 0.000869, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 7, "pts": [[-1, "-1"], [1, "1/3"]]}, "seed": 68114079302000208, "version": "po-v0", "tier": 2, "n_raw": 56, "small": 2, "valid_answers": ["0, 1, 2, 1", "1, -4, -7, -2", "1, -3, -5, -1", "1, -2, -3, 0", "1, 0, 1, 2", "1, 2, 5, 4", "1, 3, 7, 5", "2, -3, -4, 1", "2, -1, 0, 3", "2, 1, 4, 5", "3, -5, -7, 1", "3, -4, -5, 2", "3, -2, -1, 4", "3, -1, 1, 5", "3, 1, 5, 7", "4, -5, -6, 3", "4, -3, -2, 5", "4, -1, 2, 7", "5, -6, -7, 4", "5, -4, -3, 6", "5, -3, -1, 7", "6, -5, -4, 7"]}, "cell": "mobius/2", "stage": "B"}
+{"id": "po-mobius-t2-eval-000483", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = -1 and f(2) = \\frac{5}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = -1 and f(2) = \\frac{5}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 5, 3, -2", "ref": "0, 5, 3, -2", "family": "po_mobius", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.000395, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 7, "pts": [[-1, "-1"], [2, "5/4"]]}, "seed": 68114079302000638, "version": "po-v0", "tier": 2, "n_raw": 26, "small": 2, "valid_answers": ["0, 5, 3, -2", "1, -7, -4, 4", "1, 3, 2, 0", "2, 1, 1, 2", "3, -1, 0, 4", "3, 4, 3, 2", "4, -3, -1, 6", "4, 7, 5, 2", "5, 0, 1, 6", "7, 6, 5, 6"]}, "cell": "mobius/2", "stage": "B"}
+{"id": "po-mobius-t2-eval-001014", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = \\frac{4}{3} and f(3) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = \\frac{4}{3} and f(3) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -3, -4, -7", "ref": "1, -3, -4, -7", "family": "po_mobius", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 17, "bucket": "9-20", "guess_rate": 0.000672, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 7, "pts": [[-1, "4/3"], [3, "0"]]}, "seed": 68114079302001390, "version": "po-v0", "tier": 2, "n_raw": 42, "small": 3, "valid_answers": ["1, -3, -4, -7", "1, -3, -3, -6", "1, -3, -2, -5", "1, -3, -1, -4", "1, -3, 0, -3", "1, -3, 1, -2", "1, -3, 2, -1", "1, -3, 3, 0", "1, -3, 4, 1", "1, -3, 5, 2", "1, -3, 6, 3", "1, -3, 7, 4", "2, -6, -1, -7", "2, -6, 1, -5", "2, -6, 3, -3", "2, -6, 5, -1", "2, -6, 7, 1"]}, "cell": "mobius/2", "stage": "B"}
+{"id": "po-mobius-t2-eval-000015", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = 0 and f(2) = -\\frac{9}{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = 0 and f(2) = -\\frac{9}{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 3, -6, 7", "ref": "3, 3, -6, 7", "family": "po_mobius", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.000435, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 7, "pts": [[-1, "0"], [2, "-9/5"]]}, "seed": 68114079302000019, "version": "po-v0", "tier": 2, "n_raw": 28, "small": 3, "valid_answers": ["3, 3, -6, 7", "3, 3, -5, 5", "3, 3, -4, 3", "3, 3, -3, 1", "3, 3, -2, -1", "3, 3, -1, -3", "3, 3, 0, -5", "3, 3, 1, -7", "6, 6, -7, 4", "6, 6, -5, 0", "6, 6, -3, -4"]}, "cell": "mobius/2", "stage": "B"}
+{"id": "po-mobius-t2-eval-000835", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(1) = \\frac{1}{3} and f(2) = \\frac{1}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(1) = \\frac{1}{3} and f(2) = \\frac{1}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, 1, 2", "ref": "0, 1, 1, 2", "family": "po_mobius", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000158, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 7, "pts": [[1, "1/3"], [2, "1/4"]]}, "seed": 68114079302001123, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 2, "valid_answers": ["0, 1, 1, 2", "1, 0, 5, -2", "1, 1, 6, 0", "1, 2, 7, 2"]}, "cell": "mobius/2", "stage": "B"}
+{"id": "po-mobius-t2-eval-001756", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = \\frac{1}{3} and f(-2) = \\frac{1}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = \\frac{1}{3} and f(-2) = \\frac{1}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, -1, 0", "ref": "0, 1, -1, 0", "family": "po_mobius", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000356, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 7, "pts": [[-3, "1/3"], [-2, "1/2"]]}, "seed": 68114079302002619, "version": "po-v0", "tier": 2, "n_raw": 30, "small": 1, "valid_answers": ["0, 1, -1, 0", "1, -2, 7, 6", "1, -1, 6, 6", "1, 0, 5, 6", "1, 1, 4, 6", "1, 4, 1, 6", "1, 5, 0, 6", "1, 6, -1, 6", "1, 7, -2, 6"]}, "cell": "mobius/2", "stage": "B"}
+{"id": "po-mobius-t2-eval-000024", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = 1 and f(2) = -\\frac{1}{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = 1 and f(2) = -\\frac{1}{9}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, -5, 1", "ref": "0, 1, -5, 1", "family": "po_mobius", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000316, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 7, "pts": [[0, "1"], [2, "-1/9"]]}, "seed": 68114079302000029, "version": "po-v0", "tier": 2, "n_raw": 18, "small": 3, "valid_answers": ["0, 1, -5, 1", "1, -3, 6, -3", "1, -1, -4, -1", "2, -5, 7, -5", "2, -3, -3, -3", "3, -5, -2, -5", "3, -4, -7, -4", "4, -7, -1, -7"]}, "cell": "mobius/2", "stage": "B"}
+{"id": "po-mobius-t2-eval-000439", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(1) = 2 and f(3) = \\frac{4}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(1) = 2 and f(3) = \\frac{4}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -7, 0, -3", "ref": "1, -7, 0, -3", "family": "po_mobius", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000237, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 7, "pts": [[1, "2"], [3, "4/3"]]}, "seed": 68114079302000582, "version": "po-v0", "tier": 2, "n_raw": 24, "small": 1, "valid_answers": ["1, -7, 0, -3", "1, 1, 1, 0", "3, -5, 2, -3", "5, -3, 4, -3", "6, -2, 5, -3", "7, -1, 6, -3"]}, "cell": "mobius/2", "stage": "B"}
+{"id": "po-mobius-t2-eval-000205", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = -\\frac{3}{2} and f(1) = -\\frac{5}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = -\\frac{3}{2} and f(1) = -\\frac{5}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -6, 0, 4", "ref": "1, -6, 0, 4", "family": "po_mobius", "domain": "function", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000119, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 7, "pts": [[0, "-3/2"], [1, "-5/4"]]}, "seed": 68114079302000265, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 3, "valid_answers": ["1, -6, 0, 4", "2, 3, -2, -2", "7, 3, -6, -2"]}, "cell": "mobius/2", "stage": "B"}
+{"id": "po-mobius-t3-eval-002015", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = \\frac{9}{4} and f(0) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = \\frac{9}{4} and f(0) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -3, 1, -1", "ref": "2, -3, 1, -1", "family": "po_mobius", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 4.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 9, "pts": [[-3, "9/4"], [0, "3"]]}, "seed": 68114079303002820, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 3, "valid_answers": ["2, -3, 1, -1", "5, 6, 2, 2", "7, 3, 3, 1"]}, "cell": "mobius/3", "stage": "B"}
+{"id": "po-mobius-t3-eval-001129", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-2) = \\frac{1}{3} and f(2) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-2) = \\frac{1}{3} and f(2) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, -1, 1", "ref": "0, 1, -1, 1", "family": "po_mobius", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 26, "bucket": null, "guess_rate": 0.000399, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 9, "pts": [[-2, "1/3"], [2, "-1"]]}, "seed": 68114079303001446, "version": "po-v0", "tier": 3, "n_raw": 80, "small": 1, "valid_answers": ["0, 1, -1, 1", "1, -5, 6, -9", "1, -4, 5, -8", "1, -3, 4, -7", "1, -1, 2, -5", "1, 0, 1, -4", "1, 1, 0, -3", "1, 3, -2, -1", "1, 4, -3, 0", "1, 5, -4, 1", "1, 6, -5, 2", "1, 7, -6, 3", "1, 8, -7, 4", "1, 9, -8, 5", "2, -1, 3, -9", "2, 1, 1, -7", "2, 3, -1, -5", "2, 5, -3, -3", "2, 7, -5, -1", "2, 9, -7, 1", "3, 4, -1, -8", "3, 5, -2, -7", "3, 7, -4, -5", "3, 8, -5, -4", "4, 7, -3, -9", "4, 9, -5, -7"]}, "cell": "mobius/3", "stage": "B"}
+{"id": "po-mobius-t3-eval-001825", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-2) = 1 and f(1) = -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-2) = 1 and f(1) = -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 5, -2, 1", "ref": "0, 5, -2, 1", "family": "po_mobius", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.000322, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 9, "pts": [[-2, "1"], [1, "-5"]]}, "seed": 68114079303002514, "version": "po-v0", "tier": 3, "n_raw": 54, "small": 3, "valid_answers": ["0, 5, -2, 1", "1, -6, 3, -2", "1, 4, -1, 0", "1, 9, -3, 1", "2, -7, 4, -3", "2, 3, 0, -1", "3, -8, 5, -4", "3, 2, 1, -2", "3, 7, -1, -1", "4, -9, 6, -5", "4, 1, 2, -3", "5, 0, 3, -4", "5, 5, 1, -3", "6, -1, 4, -5", "7, -2, 5, -6", "7, 3, 3, -5", "7, 8, 1, -4", "8, -3, 6, -7", "8, 7, 2, -5", "9, -4, 7, -8", "9, 1, 5, -7"]}, "cell": "mobius/3", "stage": "B"}
+{"id": "po-mobius-t3-eval-000568", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = -\\frac{7}{6} and f(-2) = -\\frac{4}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = -\\frac{7}{6} and f(-2) = -\\frac{4}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -8, -3, 3", "ref": "2, -8, -3, 3", "family": "po_mobius", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 6.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 9, "pts": [[-3, "-7/6"], [-2, "-4/3"]]}, "seed": 68114079303000675, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 3, "valid_answers": ["2, -8, -3, 3", "3, 2, -3, -3", "5, -6, -6, 0", "8, -4, -9, -3"]}, "cell": "mobius/3", "stage": "B"}
+{"id": "po-mobius-t3-eval-001618", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = \\frac{1}{5} and f(3) = -\\frac{5}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = \\frac{1}{5} and f(3) = -\\frac{5}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 2, -2, 3", "ref": "1, 2, -2, 3", "family": "po_mobius", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 9.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 9, "pts": [[-1, "1/5"], [3, "-5/3"]]}, "seed": 68114079303002185, "version": "po-v0", "tier": 3, "n_raw": 16, "small": 3, "valid_answers": ["1, 2, -2, 3", "3, 1, 1, -9", "4, 3, -1, -6", "6, 7, -5, 0", "7, 9, -7, 3", "9, 8, -4, -9"]}, "cell": "mobius/3", "stage": "B"}
+{"id": "po-mobius-t3-eval-001034", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-2) = -\\frac{2}{3} and f(2) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-2) = -\\frac{2}{3} and f(2) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -2, -7, -8", "ref": "1, -2, -7, -8", "family": "po_mobius", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.000246, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 9, "pts": [[-2, "-2/3"], [2, "0"]]}, "seed": 68114079303001313, "version": "po-v0", "tier": 3, "n_raw": 46, "small": 2, "valid_answers": ["1, -2, -7, -8", "1, -2, -6, -6", "1, -2, -5, -4", "1, -2, -4, -2", "1, -2, -3, 0", "1, -2, -2, 2", "1, -2, -1, 4", "1, -2, 0, 6", "1, -2, 1, 8", "2, -4, -9, -6", "2, -4, -7, -2", "2, -4, -5, 2", "3, -6, -8, 2", "3, -6, -7, 4", "3, -6, -5, 8", "4, -8, -9, 6"]}, "cell": "mobius/3", "stage": "B"}
+{"id": "po-mobius-t3-eval-001473", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = -\\frac{12}{7} and f(0) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = -\\frac{12}{7} and f(0) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -3, -2, 1", "ref": "3, -3, -2, 1", "family": "po_mobius", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 4.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 9, "pts": [[-3, "-12/7"], [0, "-3"]]}, "seed": 68114079303001964, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 3, "valid_answers": ["3, -3, -2, 1", "6, 6, -3, -2", "9, 3, -5, -1"]}, "cell": "mobius/3", "stage": "B"}
+{"id": "po-mobius-t3-eval-002422", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = -\\frac{1}{3} and f(1) = -\\frac{1}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = -\\frac{1}{3} and f(1) = -\\frac{1}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, 1, -3", "ref": "0, 1, 1, -3", "family": "po_mobius", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.000307, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 9, "pts": [[0, "-1/3"], [1, "-1/2"]]}, "seed": 68114079303003500, "version": "po-v0", "tier": 3, "n_raw": 52, "small": 3, "valid_answers": ["0, 1, 1, -3", "1, -3, -5, 9", "1, -2, -4, 6", "1, 1, -1, -3", "1, 2, 0, -6", "1, 3, 1, -9", "2, -3, -7, 9", "2, -1, -5, 3", "2, 1, -3, -3", "2, 3, -1, -9", "3, -2, -8, 6", "3, -1, -7, 3", "3, 1, -5, -3", "3, 2, -4, -6", "4, -1, -9, 3", "4, 1, -7, -3", "4, 3, -5, -9", "5, 1, -9, -3", "5, 2, -8, -6", "5, 3, -7, -9"]}, "cell": "mobius/3", "stage": "B"}
+{"id": "po-mobius-t3-eval-001188", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = \\frac{1}{2} and f(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(0) = \\frac{1}{2} and f(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 4, -7, 8", "ref": "0, 4, -7, 8", "family": "po_mobius", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.000184, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 9, "pts": [[0, "1/2"], [1, "4"]]}, "seed": 68114079303001530, "version": "po-v0", "tier": 3, "n_raw": 30, "small": 3, "valid_answers": ["0, 4, -7, 8", "1, 3, -5, 6", "2, 2, -3, 4", "3, 1, -1, 2", "5, -1, 3, -2", "5, 3, -4, 6", "6, -2, 5, -4", "7, -3, 7, -6", "7, 1, 0, 2", "8, -4, 9, -8", "8, 4, -5, 8", "9, -1, 4, -2"]}, "cell": "mobius/3", "stage": "B"}
+{"id": "po-mobius-t3-eval-000482", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = \\frac{3}{5} and f(0) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-3) = \\frac{3}{5} and f(0) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 9, -4, 3", "ref": "0, 9, -4, 3", "family": "po_mobius", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000107, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 9, "pts": [[-3, "3/5"], [0, "3"]]}, "seed": 68114079303000573, "version": "po-v0", "tier": 3, "n_raw": 22, "small": 3, "valid_answers": ["0, 9, -4, 3", "1, -3, 3, -1", "1, 6, -1, 2", "2, 3, 2, 1", "4, -3, 8, -1", "5, 3, 7, 1", "7, 6, 9, 2"]}, "cell": "mobius/3", "stage": "B"}
+{"id": "po-mobius-t3-eval-000871", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = 1 and f(2) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(-1) = 1 and f(2) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 4, -1, 3", "ref": "0, 4, -1, 3", "family": "po_mobius", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 24, "bucket": null, "guess_rate": 0.000368, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 9, "pts": [[-1, "1"], [2, "4"]]}, "seed": 68114079303001093, "version": "po-v0", "tier": 3, "n_raw": 70, "small": 2, "valid_answers": ["0, 4, -1, 3", "1, -6, 2, -5", "1, 2, 0, 1", "1, 6, -1, 4", "2, -8, 3, -7", "2, 0, 1, -1", "2, 8, -1, 5", "3, -2, 2, -3", "3, 2, 1, 0", "4, -4, 3, -5", "5, -6, 4, -7", "5, -2, 3, -4", "5, 2, 2, -1", "5, 6, 1, 2", "6, -8, 5, -9", "6, 8, 1, 3", "7, -6, 5, -8", "7, -2, 4, -5", "7, 2, 3, -2", "7, 6, 2, 1", "8, -4, 5, -7", "8, 4, 3, -1", "9, -2, 5, -6", "9, 2, 4, -3"]}, "cell": "mobius/3", "stage": "B"}
+{"id": "po-mobius-t3-eval-000241", "task": "cx_po_mobius", "source": "cx_po_mobius", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(1) = -\\frac{1}{2} and f(3) = -\\frac{1}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 and ad - bc \\ne 0 such that f(x) = \\frac{ax + b}{cx + d} satisfies f(1) = -\\frac{1}{2} and f(3) = -\\frac{1}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 2, -1, -3", "ref": "0, 2, -1, -3", "family": "po_mobius", "domain": "function", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000138, "solvable": true, "holdout": false, "meta": {"family": "po_mobius", "rung": "witness", "params": {"B": 9, "pts": [[1, "-1/2"], [3, "-1/3"]]}, "seed": 68114079303000275, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 3, "valid_answers": ["0, 2, -1, -3", "1, -5, -1, 9", "1, 1, -4, 0", "1, 3, -5, -3", "1, 5, -6, -6", "1, 7, -7, -9", "2, -4, -5, 9", "2, 0, -7, 3", "2, 4, -9, -3"]}, "cell": "mobius/3", "stage": "B"}
+{"id": "po-moment_weights-t0-eval-000070", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 15 and sum of squares 41. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 15 and sum of squares 41. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 5, 1, 0, 2", "ref": "0, 5, 1, 0, 2", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.008081, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 8, "M1": 15, "M2": 41}, "seed": 75305729600000176, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["0, 5, 1, 0, 2", "1, 3, 1, 2, 1", "2, 0, 4, 1, 1", "2, 1, 1, 4, 0"]}, "cell": "moment_weights/0", "stage": "C"}
+{"id": "po-moment_weights-t0-eval-000130", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 11 and sum of squares 19. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 11 and sum of squares 19. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 6, 1, 1, 0", "ref": "0, 6, 1, 1, 0", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00404, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 8, "M1": 11, "M2": 19}, "seed": 75305729600001435, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["0, 6, 1, 1, 0", "1, 3, 4, 0, 0"]}, "cell": "moment_weights/0", "stage": "C"}
+{"id": "po-moment_weights-t0-eval-000004", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 23 and sum of squares 83. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 23 and sum of squares 83. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 3, 0, 0, 5", "ref": "0, 3, 0, 0, 5", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00404, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 8, "M1": 23, "M2": 83}, "seed": 75305729600000007, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["0, 3, 0, 0, 5", "1, 1, 0, 2, 4"]}, "cell": "moment_weights/0", "stage": "C"}
+{"id": "po-moment_weights-t0-eval-000103", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 20 and sum of squares 60. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 20 and sum of squares 60. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 2, 2, 2, 2", "ref": "0, 2, 2, 2, 2", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00404, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 8, "M1": 20, "M2": 60}, "seed": 75305729600000411, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["0, 2, 2, 2, 2", "1, 0, 2, 4, 1"]}, "cell": "moment_weights/0", "stage": "C"}
+{"id": "po-moment_weights-t0-eval-000049", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 17 and sum of squares 59. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 17 and sum of squares 59. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 2, 0, 1, 3", "ref": "2, 2, 0, 1, 3", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00404, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 8, "M1": 17, "M2": 59}, "seed": 75305729600000092, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["2, 2, 0, 1, 3", "3, 0, 0, 3, 2"]}, "cell": "moment_weights/0", "stage": "C"}
+{"id": "po-moment_weights-t0-eval-000110", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 11 and sum of squares 23. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 11 and sum of squares 23. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 7, 0, 0, 1", "ref": "0, 7, 0, 0, 1", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006061, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 8, "M1": 11, "M2": 23}, "seed": 75305729600000492, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["0, 7, 0, 0, 1", "1, 5, 0, 2, 0", "2, 2, 3, 1, 0"]}, "cell": "moment_weights/0", "stage": "C"}
+{"id": "po-moment_weights-t0-eval-000066", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 21 and sum of squares 63. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 21 and sum of squares 63. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, 3, 2, 2", "ref": "0, 1, 3, 2, 2", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006061, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 8, "M1": 21, "M2": 63}, "seed": 75305729600000167, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["0, 1, 3, 2, 2", "0, 2, 0, 5, 1", "1, 0, 0, 7, 0"]}, "cell": "moment_weights/0", "stage": "C"}
+{"id": "po-moment_weights-t0-eval-000056", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 20 and sum of squares 62. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 20 and sum of squares 62. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 2, 3, 0, 3", "ref": "0, 2, 3, 0, 3", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.008081, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 8, "M1": 20, "M2": 62}, "seed": 75305729600000115, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["0, 2, 3, 0, 3", "0, 3, 0, 3, 2", "1, 0, 3, 2, 2", "1, 1, 0, 5, 1"]}, "cell": "moment_weights/0", "stage": "C"}
+{"id": "po-moment_weights-t0-eval-000116", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 15 and sum of squares 33. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 15 and sum of squares 33. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 3, 3, 2, 0", "ref": "0, 3, 3, 2, 0", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00404, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 8, "M1": 15, "M2": 33}, "seed": 75305729600000589, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["0, 3, 3, 2, 0", "1, 0, 6, 1, 0"]}, "cell": "moment_weights/0", "stage": "C"}
+{"id": "po-moment_weights-t0-eval-000036", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 18 and sum of squares 58. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 18 and sum of squares 58. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 2, 2, 0, 3", "ref": "1, 2, 2, 0, 3", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00404, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 8, "M1": 18, "M2": 58}, "seed": 75305729600000070, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["1, 2, 2, 0, 3", "2, 0, 2, 2, 2"]}, "cell": "moment_weights/0", "stage": "C"}
+{"id": "po-moment_weights-t0-eval-000000", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 15 and sum of squares 43. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 15 and sum of squares 43. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 3, 2, 0, 2", "ref": "1, 3, 2, 0, 2", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00404, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 8, "M1": 15, "M2": 43}, "seed": 75305729600000001, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["1, 3, 2, 0, 2", "2, 1, 2, 2, 1"]}, "cell": "moment_weights/0", "stage": "C"}
+{"id": "po-moment_weights-t0-eval-000063", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 18 and sum of squares 48. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 8 numbers, each equal to 0, 1, 2, 3 or 4, has sum 18 and sum of squares 48. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 2, 3, 2, 1", "ref": "0, 2, 3, 2, 1", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006061, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 8, "M1": 18, "M2": 48}, "seed": 75305729600000149, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["0, 2, 3, 2, 1", "0, 3, 0, 5, 0", "1, 0, 3, 4, 0"]}, "cell": "moment_weights/0", "stage": "C"}
+{"id": "po-moment_weights-t1-eval-000024", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 19 and sum of squares 57. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 19 and sum of squares 57. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 5, 1, 0, 3", "ref": "3, 5, 1, 0, 3", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.002198, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 12, "M1": 19, "M2": 57}, "seed": 75305729601000027, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 4, "valid_answers": ["3, 5, 1, 0, 3", "4, 3, 1, 2, 2", "5, 0, 4, 1, 2", "5, 1, 1, 4, 1"]}, "cell": "moment_weights/1", "stage": "C"}
+{"id": "po-moment_weights-t1-eval-000108", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 22 and sum of squares 54. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 22 and sum of squares 54. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 6, 4, 0, 2", "ref": "0, 6, 4, 0, 2", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.003297, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 12, "M1": 22, "M2": 54}, "seed": 75305729601000187, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 4, "valid_answers": ["0, 6, 4, 0, 2", "0, 7, 1, 3, 1", "1, 4, 4, 2, 1", "1, 5, 1, 5, 0", "2, 1, 7, 1, 1", "2, 2, 4, 4, 0"]}, "cell": "moment_weights/1", "stage": "C"}
+{"id": "po-moment_weights-t1-eval-000231", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 24 and sum of squares 80. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 24 and sum of squares 80. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 3, 1, 1, 4", "ref": "3, 3, 1, 1, 4", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001648, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 12, "M1": 24, "M2": 80}, "seed": 75305729601000740, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 4, "valid_answers": ["3, 3, 1, 1, 4", "4, 0, 4, 0, 4", "4, 1, 1, 3, 3"]}, "cell": "moment_weights/1", "stage": "C"}
+{"id": "po-moment_weights-t1-eval-000044", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 27 and sum of squares 91. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 27 and sum of squares 91. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 3, 2, 0, 5", "ref": "2, 3, 2, 0, 5", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001099, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 12, "M1": 27, "M2": 91}, "seed": 75305729601000054, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["2, 3, 2, 0, 5", "3, 1, 2, 2, 4"]}, "cell": "moment_weights/1", "stage": "C"}
+{"id": "po-moment_weights-t1-eval-000116", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 15 and sum of squares 39. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 15 and sum of squares 39. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 7, 0, 0, 2", "ref": "3, 7, 0, 0, 2", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002747, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 12, "M1": 15, "M2": 39}, "seed": 75305729601000204, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 5, "valid_answers": ["3, 7, 0, 0, 2", "4, 5, 0, 2, 1", "5, 2, 3, 1, 1", "5, 3, 0, 4, 0", "6, 0, 3, 3, 0"]}, "cell": "moment_weights/1", "stage": "C"}
+{"id": "po-moment_weights-t1-eval-000336", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 11 and sum of squares 23. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 11 and sum of squares 23. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 7, 0, 0, 1", "ref": "4, 7, 0, 0, 1", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001648, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 12, "M1": 11, "M2": 23}, "seed": 75305729601004437, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 5, "valid_answers": ["4, 7, 0, 0, 1", "5, 5, 0, 2, 0", "6, 2, 3, 1, 0"]}, "cell": "moment_weights/1", "stage": "C"}
+{"id": "po-moment_weights-t1-eval-000216", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 17 and sum of squares 43. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 17 and sum of squares 43. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 7, 1, 0, 2", "ref": "2, 7, 1, 0, 2", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002747, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 12, "M1": 17, "M2": 43}, "seed": 75305729601000619, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 4, "valid_answers": ["2, 7, 1, 0, 2", "3, 5, 1, 2, 1", "4, 2, 4, 1, 1", "4, 3, 1, 4, 0", "5, 0, 4, 3, 0"]}, "cell": "moment_weights/1", "stage": "C"}
+{"id": "po-moment_weights-t1-eval-000114", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 21 and sum of squares 53. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 21 and sum of squares 53. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 8, 1, 1, 2", "ref": "0, 8, 1, 1, 2", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.003846, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 12, "M1": 21, "M2": 53}, "seed": 75305729601000201, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 4, "valid_answers": ["0, 8, 1, 1, 2", "1, 5, 4, 0, 2", "1, 6, 1, 3, 1", "2, 3, 4, 2, 1", "2, 4, 1, 5, 0", "3, 0, 7, 1, 1", "3, 1, 4, 4, 0"]}, "cell": "moment_weights/1", "stage": "C"}
+{"id": "po-moment_weights-t1-eval-000155", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 30 and sum of squares 86. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 30 and sum of squares 86. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, 7, 1, 3", "ref": "0, 1, 7, 1, 3", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002747, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 12, "M1": 30, "M2": 86}, "seed": 75305729601000316, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 4, "valid_answers": ["0, 1, 7, 1, 3", "0, 2, 4, 4, 2", "0, 3, 1, 7, 1", "1, 0, 4, 6, 1", "1, 1, 1, 9, 0"]}, "cell": "moment_weights/1", "stage": "C"}
+{"id": "po-moment_weights-t1-eval-000186", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 14 and sum of squares 42. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 14 and sum of squares 42. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 2, 2, 0, 2", "ref": "6, 2, 2, 0, 2", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001099, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 12, "M1": 14, "M2": 42}, "seed": 75305729601000451, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 6, "valid_answers": ["6, 2, 2, 0, 2", "7, 0, 2, 2, 1"]}, "cell": "moment_weights/1", "stage": "C"}
+{"id": "po-moment_weights-t1-eval-000192", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 32 and sum of squares 112. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 32 and sum of squares 112. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 3, 1, 1, 6", "ref": "1, 3, 1, 1, 6", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001648, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 12, "M1": 32, "M2": 112}, "seed": 75305729601000474, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 5, "valid_answers": ["1, 3, 1, 1, 6", "2, 0, 4, 0, 6", "2, 1, 1, 3, 5"]}, "cell": "moment_weights/1", "stage": "C"}
+{"id": "po-moment_weights-t1-eval-000335", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 18 and sum of squares 36. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 12 numbers, each equal to 0, 1, 2, 3 or 4, has sum 18 and sum of squares 36. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 8, 3, 0, 1", "ref": "0, 8, 3, 0, 1", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002747, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 12, "M1": 18, "M2": 36}, "seed": 75305729601004399, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 6, "valid_answers": ["0, 8, 3, 0, 1", "0, 9, 0, 3, 0", "1, 6, 3, 2, 0", "2, 3, 6, 1, 0", "3, 0, 9, 0, 0"]}, "cell": "moment_weights/1", "stage": "C"}
+{"id": "po-moment_weights-t2-eval-000544", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 24 and sum of squares 48. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 24 and sum of squares 48. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 16, 0, 0, 2", "ref": "2, 16, 0, 0, 2", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000847, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 20, "M1": 24, "M2": 48}, "seed": 75305729602003357, "version": "po-v0", "tier": 2, "n_raw": 9, "small": 6, "valid_answers": ["2, 16, 0, 0, 2", "3, 14, 0, 2, 1", "4, 11, 3, 1, 1", "4, 12, 0, 4, 0", "5, 8, 6, 0, 1", "5, 9, 3, 3, 0", "6, 6, 6, 2, 0", "7, 3, 9, 1, 0", "8, 0, 12, 0, 0"]}, "cell": "moment_weights/2", "stage": "C"}
+{"id": "po-moment_weights-t2-eval-000410", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 54 and sum of squares 166. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 54 and sum of squares 166. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, 11, 1, 7", "ref": "0, 1, 11, 1, 7", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000753, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 20, "M1": 54, "M2": 166}, "seed": 75305729602001275, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 7, "valid_answers": ["0, 1, 11, 1, 7", "0, 2, 8, 4, 6", "0, 3, 5, 7, 5", "0, 4, 2, 10, 4", "1, 0, 8, 6, 5", "1, 1, 5, 9, 4", "1, 2, 2, 12, 3", "2, 0, 2, 14, 2"]}, "cell": "moment_weights/2", "stage": "C"}
+{"id": "po-moment_weights-t2-eval-000653", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 19 and sum of squares 35. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 19 and sum of squares 35. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 11, 2, 0, 1", "ref": "6, 11, 2, 0, 1", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000376, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 20, "M1": 19, "M2": 35}, "seed": 75305729602006803, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 8, "valid_answers": ["6, 11, 2, 0, 1", "7, 9, 2, 2, 0", "8, 6, 5, 1, 0", "9, 3, 8, 0, 0"]}, "cell": "moment_weights/2", "stage": "C"}
+{"id": "po-moment_weights-t2-eval-000499", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 47 and sum of squares 165. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 47 and sum of squares 165. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 4, 2, 1, 9", "ref": "4, 4, 2, 1, 9", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000376, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 20, "M1": 47, "M2": 165}, "seed": 75305729602002409, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 7, "valid_answers": ["4, 4, 2, 1, 9", "5, 1, 5, 0, 9", "5, 2, 2, 3, 8", "6, 0, 2, 5, 7"]}, "cell": "moment_weights/2", "stage": "C"}
+{"id": "po-moment_weights-t2-eval-000391", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 45 and sum of squares 127. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 45 and sum of squares 127. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 6, 8, 1, 5", "ref": "0, 6, 8, 1, 5", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.001318, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 20, "M1": 45, "M2": 127}, "seed": 75305729602001118, "version": "po-v0", "tier": 2, "n_raw": 14, "small": 6, "valid_answers": ["0, 6, 8, 1, 5", "0, 7, 5, 4, 4", "0, 8, 2, 7, 3", "1, 3, 11, 0, 5", "1, 4, 8, 3, 4", "1, 5, 5, 6, 3", "1, 6, 2, 9, 2", "2, 1, 11, 2, 4", "2, 2, 8, 5, 3", "2, 3, 5, 8, 2", "2, 4, 2, 11, 1", "3, 0, 8, 7, 2", "3, 1, 5, 10, 1", "3, 2, 2, 13, 0"]}, "cell": "moment_weights/2", "stage": "C"}
+{"id": "po-moment_weights-t2-eval-000097", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 38 and sum of squares 114. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 38 and sum of squares 114. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 10, 2, 0, 6", "ref": "2, 10, 2, 0, 6", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.001035, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 20, "M1": 38, "M2": 114}, "seed": 75305729602000120, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 5, "valid_answers": ["2, 10, 2, 0, 6", "3, 8, 2, 2, 5", "4, 5, 5, 1, 5", "4, 6, 2, 4, 4", "5, 2, 8, 0, 5", "5, 3, 5, 3, 4", "5, 4, 2, 6, 3", "6, 0, 8, 2, 4", "6, 1, 5, 5, 3", "6, 2, 2, 8, 2", "7, 0, 2, 10, 1"]}, "cell": "moment_weights/2", "stage": "C"}
+{"id": "po-moment_weights-t2-eval-000470", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 58 and sum of squares 196. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 58 and sum of squares 196. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 3, 6, 1, 10", "ref": "0, 3, 6, 1, 10", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000847, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 20, "M1": 58, "M2": 196}, "seed": 75305729602001979, "version": "po-v0", "tier": 2, "n_raw": 9, "small": 8, "valid_answers": ["0, 3, 6, 1, 10", "0, 4, 3, 4, 9", "0, 5, 0, 7, 8", "1, 0, 9, 0, 10", "1, 1, 6, 3, 9", "1, 2, 3, 6, 8", "1, 3, 0, 9, 7", "2, 0, 3, 8, 7", "2, 1, 0, 11, 6"]}, "cell": "moment_weights/2", "stage": "C"}
+{"id": "po-moment_weights-t2-eval-000769", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 60 and sum of squares 208. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 60 and sum of squares 208. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 3, 5, 1, 11", "ref": "0, 3, 5, 1, 11", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000565, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 20, "M1": 60, "M2": 208}, "seed": 75305729602017667, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 8, "valid_answers": ["0, 3, 5, 1, 11", "0, 4, 2, 4, 10", "1, 0, 8, 0, 11", "1, 1, 5, 3, 10", "1, 2, 2, 6, 9", "2, 0, 2, 8, 8"]}, "cell": "moment_weights/2", "stage": "C"}
+{"id": "po-moment_weights-t2-eval-000120", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 36 and sum of squares 106. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 36 and sum of squares 106. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 9, 2, 1, 5", "ref": "3, 9, 2, 1, 5", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.000941, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 20, "M1": 36, "M2": 106}, "seed": 75305729602000152, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 5, "valid_answers": ["3, 9, 2, 1, 5", "4, 6, 5, 0, 5", "4, 7, 2, 3, 4", "5, 4, 5, 2, 4", "5, 5, 2, 5, 3", "6, 1, 8, 1, 4", "6, 2, 5, 4, 3", "6, 3, 2, 7, 2", "7, 0, 5, 6, 2", "7, 1, 2, 9, 1"]}, "cell": "moment_weights/2", "stage": "C"}
+{"id": "po-moment_weights-t2-eval-000042", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 35 and sum of squares 91. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 35 and sum of squares 91. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 14, 1, 1, 4", "ref": "0, 14, 1, 1, 4", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.001506, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 20, "M1": 35, "M2": 91}, "seed": 75305729602000046, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 6, "valid_answers": ["0, 14, 1, 1, 4", "1, 11, 4, 0, 4", "1, 12, 1, 3, 3", "2, 9, 4, 2, 3", "2, 10, 1, 5, 2", "3, 6, 7, 1, 3", "3, 7, 4, 4, 2", "3, 8, 1, 7, 1", "4, 3, 10, 0, 3", "4, 4, 7, 3, 2", "4, 5, 4, 6, 1", "4, 6, 1, 9, 0", "5, 1, 10, 2, 2", "5, 2, 7, 5, 1", "5, 3, 4, 8, 0", "6, 0, 7, 7, 0"]}, "cell": "moment_weights/2", "stage": "C"}
+{"id": "po-moment_weights-t2-eval-000637", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 19 and sum of squares 45. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 19 and sum of squares 45. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8, 9, 1, 0, 2", "ref": "8, 9, 1, 0, 2", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000565, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 20, "M1": 19, "M2": 45}, "seed": 75305729602006242, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 9, "valid_answers": ["8, 9, 1, 0, 2", "9, 7, 1, 2, 1", "10, 4, 4, 1, 1", "10, 5, 1, 4, 0", "11, 1, 7, 0, 1", "11, 2, 4, 3, 0"]}, "cell": "moment_weights/2", "stage": "C"}
+{"id": "po-moment_weights-t2-eval-000664", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 29 and sum of squares 91. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 20 numbers, each equal to 0, 1, 2, 3 or 4, has sum 29 and sum of squares 91. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 7, 1, 0, 5", "ref": "7, 7, 1, 0, 5", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000565, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 20, "M1": 29, "M2": 91}, "seed": 75305729602007672, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 7, "valid_answers": ["7, 7, 1, 0, 5", "8, 5, 1, 2, 4", "9, 2, 4, 1, 4", "9, 3, 1, 4, 3", "10, 0, 4, 3, 3", "10, 1, 1, 6, 2"]}, "cell": "moment_weights/2", "stage": "C"}
+{"id": "po-moment_weights-t3-eval-000923", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 70 and sum of squares 192. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 70 and sum of squares 192. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 4, 19, 0, 7", "ref": "0, 4, 19, 0, 7", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 23, "bucket": null, "guess_rate": 0.000496, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 30, "M1": 70, "M2": 192}, "seed": 75305729603008312, "version": "po-v0", "tier": 3, "n_raw": 23, "small": 10, "valid_answers": ["0, 4, 19, 0, 7", "0, 5, 16, 3, 6", "0, 6, 13, 6, 5", "0, 7, 10, 9, 4", "0, 8, 7, 12, 3", "0, 9, 4, 15, 2", "0, 10, 1, 18, 1", "1, 2, 19, 2, 6", "1, 3, 16, 5, 5", "1, 4, 13, 8, 4", "1, 5, 10, 11, 3", "1, 6, 7, 14, 2", "1, 7, 4, 17, 1", "1, 8, 1, 20, 0", "2, 0, 19, 4, 5", "2, 1, 16, 7, 4", "2, 2, 13, 10, 3", "2, 3, 10, 13, 2", "2, 4, 7, 16, 1", "2, 5, 4, 19, 0", "3, 0, 13, 12, 2", "3, 1, 10, 15, 1", "3, 2, 7, 18, 0"]}, "cell": "moment_weights/3", "stage": "C"}
+{"id": "po-moment_weights-t3-eval-000540", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 72 and sum of squares 242. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 72 and sum of squares 242. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 14, 1, 0, 14", "ref": "1, 14, 1, 0, 14", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 0.000388, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 30, "M1": 72, "M2": 242}, "seed": 75305729603001389, "version": "po-v0", "tier": 3, "n_raw": 18, "small": 9, "valid_answers": ["1, 14, 1, 0, 14", "2, 12, 1, 2, 13", "3, 9, 4, 1, 13", "3, 10, 1, 4, 12", "4, 6, 7, 0, 13", "4, 7, 4, 3, 12", "4, 8, 1, 6, 11", "5, 4, 7, 2, 12", "5, 5, 4, 5, 11", "5, 6, 1, 8, 10", "6, 1, 10, 1, 12", "6, 2, 7, 4, 11", "6, 3, 4, 7, 10", "6, 4, 1, 10, 9", "7, 0, 7, 6, 10", "7, 1, 4, 9, 9", "7, 2, 1, 12, 8", "8, 0, 1, 14, 7"]}, "cell": "moment_weights/3", "stage": "C"}
+{"id": "po-moment_weights-t3-eval-000052", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 69 and sum of squares 231. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 69 and sum of squares 231. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 14, 0, 1, 13", "ref": "2, 14, 0, 1, 13", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.000431, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 30, "M1": 69, "M2": 231}, "seed": 75305729603000056, "version": "po-v0", "tier": 3, "n_raw": 20, "small": 9, "valid_answers": ["2, 14, 0, 1, 13", "3, 11, 3, 0, 13", "3, 12, 0, 3, 12", "4, 9, 3, 2, 12", "4, 10, 0, 5, 11", "5, 6, 6, 1, 12", "5, 7, 3, 4, 11", "5, 8, 0, 7, 10", "6, 3, 9, 0, 12", "6, 4, 6, 3, 11", "6, 5, 3, 6, 10", "6, 6, 0, 9, 9", "7, 1, 9, 2, 11", "7, 2, 6, 5, 10", "7, 3, 3, 8, 9", "7, 4, 0, 11, 8", "8, 0, 6, 7, 9", "8, 1, 3, 10, 8", "8, 2, 0, 13, 7", "9, 0, 0, 15, 6"]}, "cell": "moment_weights/3", "stage": "C"}
+{"id": "po-moment_weights-t3-eval-001125", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 77 and sum of squares 221. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 77 and sum of squares 221. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, 21, 1, 8", "ref": "0, 0, 21, 1, 8", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 17, "bucket": "9-20", "guess_rate": 0.000367, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 30, "M1": 77, "M2": 221}, "seed": 75305729603022526, "version": "po-v0", "tier": 3, "n_raw": 17, "small": 12, "valid_answers": ["0, 0, 21, 1, 8", "0, 1, 18, 4, 7", "0, 2, 15, 7, 6", "0, 3, 12, 10, 5", "0, 4, 9, 13, 4", "0, 5, 6, 16, 3", "0, 6, 3, 19, 2", "0, 7, 0, 22, 1", "1, 0, 15, 9, 5", "1, 1, 12, 12, 4", "1, 2, 9, 15, 3", "1, 3, 6, 18, 2", "1, 4, 3, 21, 1", "1, 5, 0, 24, 0", "2, 0, 9, 17, 2", "2, 1, 6, 20, 1", "2, 2, 3, 23, 0"]}, "cell": "moment_weights/3", "stage": "C"}
+{"id": "po-moment_weights-t3-eval-001016", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 48 and sum of squares 108. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 48 and sum of squares 108. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 20, 6, 0, 4", "ref": "0, 20, 6, 0, 4", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 26, "bucket": null, "guess_rate": 0.000561, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 30, "M1": 48, "M2": 108}, "seed": 75305729603013089, "version": "po-v0", "tier": 3, "n_raw": 26, "small": 9, "valid_answers": ["0, 20, 6, 0, 4", "0, 21, 3, 3, 3", "0, 22, 0, 6, 2", "1, 18, 6, 2, 3", "1, 19, 3, 5, 2", "1, 20, 0, 8, 1", "2, 15, 9, 1, 3", "2, 16, 6, 4, 2", "2, 17, 3, 7, 1", "2, 18, 0, 10, 0", "3, 12, 12, 0, 3", "3, 13, 9, 3, 2", "3, 14, 6, 6, 1", "3, 15, 3, 9, 0", "4, 10, 12, 2, 2", "4, 11, 9, 5, 1", "4, 12, 6, 8, 0", "5, 7, 15, 1, 2", "5, 8, 12, 4, 1", "5, 9, 9, 7, 0", "6, 4, 18, 0, 2", "6, 5, 15, 3, 1", "6, 6, 12, 6, 0", "7, 2, 18, 2, 1", "7, 3, 15, 5, 0", "8, 0, 18, 4, 0"]}, "cell": "moment_weights/3", "stage": "C"}
+{"id": "po-moment_weights-t3-eval-000515", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 74 and sum of squares 246. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 74 and sum of squares 246. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 14, 2, 0, 14", "ref": "0, 14, 2, 0, 14", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 0.000388, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 30, "M1": 74, "M2": 246}, "seed": 75305729603001282, "version": "po-v0", "tier": 3, "n_raw": 18, "small": 9, "valid_answers": ["0, 14, 2, 0, 14", "1, 12, 2, 2, 13", "2, 9, 5, 1, 13", "2, 10, 2, 4, 12", "3, 6, 8, 0, 13", "3, 7, 5, 3, 12", "3, 8, 2, 6, 11", "4, 4, 8, 2, 12", "4, 5, 5, 5, 11", "4, 6, 2, 8, 10", "5, 1, 11, 1, 12", "5, 2, 8, 4, 11", "5, 3, 5, 7, 10", "5, 4, 2, 10, 9", "6, 0, 8, 6, 10", "6, 1, 5, 9, 9", "6, 2, 2, 12, 8", "7, 0, 2, 14, 7"]}, "cell": "moment_weights/3", "stage": "C"}
+{"id": "po-moment_weights-t3-eval-000600", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 47 and sum of squares 123. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 47 and sum of squares 123. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 19, 2, 0, 6", "ref": "3, 19, 2, 0, 6", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 25, "bucket": null, "guess_rate": 0.000539, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 30, "M1": 47, "M2": 123}, "seed": 75305729603001866, "version": "po-v0", "tier": 3, "n_raw": 25, "small": 8, "valid_answers": ["3, 19, 2, 0, 6", "4, 17, 2, 2, 5", "5, 14, 5, 1, 5", "5, 15, 2, 4, 4", "6, 11, 8, 0, 5", "6, 12, 5, 3, 4", "6, 13, 2, 6, 3", "7, 9, 8, 2, 4", "7, 10, 5, 5, 3", "7, 11, 2, 8, 2", "8, 6, 11, 1, 4", "8, 7, 8, 4, 3", "8, 8, 5, 7, 2", "8, 9, 2, 10, 1", "9, 3, 14, 0, 4", "9, 4, 11, 3, 3", "9, 5, 8, 6, 2", "9, 6, 5, 9, 1", "9, 7, 2, 12, 0", "10, 1, 14, 2, 3", "10, 2, 11, 5, 2", "10, 3, 8, 8, 1", "10, 4, 5, 11, 0", "11, 0, 11, 7, 1", "11, 1, 8, 10, 0"]}, "cell": "moment_weights/3", "stage": "C"}
+{"id": "po-moment_weights-t3-eval-000566", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 40 and sum of squares 98. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 40 and sum of squares 98. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 17, 2, 1, 4", "ref": "6, 17, 2, 1, 4", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.00041, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 30, "M1": 40, "M2": 98}, "seed": 75305729603001593, "version": "po-v0", "tier": 3, "n_raw": 19, "small": 9, "valid_answers": ["6, 17, 2, 1, 4", "7, 14, 5, 0, 4", "7, 15, 2, 3, 3", "8, 12, 5, 2, 3", "8, 13, 2, 5, 2", "9, 9, 8, 1, 3", "9, 10, 5, 4, 2", "9, 11, 2, 7, 1", "10, 6, 11, 0, 3", "10, 7, 8, 3, 2", "10, 8, 5, 6, 1", "10, 9, 2, 9, 0", "11, 4, 11, 2, 2", "11, 5, 8, 5, 1", "11, 6, 5, 8, 0", "12, 1, 14, 1, 2", "12, 2, 11, 4, 1", "12, 3, 8, 7, 0", "13, 0, 11, 6, 0"]}, "cell": "moment_weights/3", "stage": "C"}
+{"id": "po-moment_weights-t3-eval-000159", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 46 and sum of squares 140. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 46 and sum of squares 140. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9, 11, 2, 1, 7", "ref": "9, 11, 2, 1, 7", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.000302, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 30, "M1": 46, "M2": 140}, "seed": 75305729603000190, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 10, "valid_answers": ["9, 11, 2, 1, 7", "10, 8, 5, 0, 7", "10, 9, 2, 3, 6", "11, 6, 5, 2, 6", "11, 7, 2, 5, 5", "12, 3, 8, 1, 6", "12, 4, 5, 4, 5", "12, 5, 2, 7, 4", "13, 0, 11, 0, 6", "13, 1, 8, 3, 5", "13, 2, 5, 6, 4", "13, 3, 2, 9, 3", "14, 0, 5, 8, 3", "14, 1, 2, 11, 2"]}, "cell": "moment_weights/3", "stage": "C"}
+{"id": "po-moment_weights-t3-eval-000098", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 56 and sum of squares 156. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 56 and sum of squares 156. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 20, 2, 0, 8", "ref": "0, 20, 2, 0, 8", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 31, "bucket": null, "guess_rate": 0.000668, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 30, "M1": 56, "M2": 156}, "seed": 75305729603000104, "version": "po-v0", "tier": 3, "n_raw": 31, "small": 8, "valid_answers": ["0, 20, 2, 0, 8", "1, 18, 2, 2, 7", "2, 15, 5, 1, 7", "2, 16, 2, 4, 6", "3, 12, 8, 0, 7", "3, 13, 5, 3, 6", "3, 14, 2, 6, 5", "4, 10, 8, 2, 6", "4, 11, 5, 5, 5", "4, 12, 2, 8, 4", "5, 7, 11, 1, 6", "5, 8, 8, 4, 5", "5, 9, 5, 7, 4", "5, 10, 2, 10, 3", "6, 4, 14, 0, 6", "6, 5, 11, 3, 5", "6, 6, 8, 6, 4", "6, 7, 5, 9, 3", "6, 8, 2, 12, 2", "7, 2, 14, 2, 5", "7, 3, 11, 5, 4", "7, 4, 8, 8, 3", "7, 5, 5, 11, 2", "7, 6, 2, 14, 1", "8, 0, 14, 4, 4", "8, 1, 11, 7, 3", "8, 2, 8, 10, 2", "8, 3, 5, 13, 1", "8, 4, 2, 16, 0", "9, 0, 8, 12, 1", "9, 1, 5, 15, 0"]}, "cell": "moment_weights/3", "stage": "C"}
+{"id": "po-moment_weights-t3-eval-000092", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 60 and sum of squares 164. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 60 and sum of squares 164. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 15, 7, 1, 7", "ref": "0, 15, 7, 1, 7", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 35, "bucket": null, "guess_rate": 0.000755, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 30, "M1": 60, "M2": 164}, "seed": 75305729603000098, "version": "po-v0", "tier": 3, "n_raw": 35, "small": 9, "valid_answers": ["0, 15, 7, 1, 7", "0, 16, 4, 4, 6", "0, 17, 1, 7, 5", "1, 12, 10, 0, 7", "1, 13, 7, 3, 6", "1, 14, 4, 6, 5", "1, 15, 1, 9, 4", "2, 10, 10, 2, 6", "2, 11, 7, 5, 5", "2, 12, 4, 8, 4", "2, 13, 1, 11, 3", "3, 7, 13, 1, 6", "3, 8, 10, 4, 5", "3, 9, 7, 7, 4", "3, 10, 4, 10, 3", "3, 11, 1, 13, 2", "4, 4, 16, 0, 6", "4, 5, 13, 3, 5", "4, 6, 10, 6, 4", "4, 7, 7, 9, 3", "4, 8, 4, 12, 2", "4, 9, 1, 15, 1", "5, 2, 16, 2, 5", "5, 3, 13, 5, 4", "5, 4, 10, 8, 3", "5, 5, 7, 11, 2", "5, 6, 4, 14, 1", "5, 7, 1, 17, 0", "6, 0, 16, 4, 4", "6, 1, 13, 7, 3", "6, 2, 10, 10, 2", "6, 3, 7, 13, 1", "6, 4, 4, 16, 0", "7, 0, 10, 12, 1", "7, 1, 7, 15, 0"]}, "cell": "moment_weights/3", "stage": "C"}
+{"id": "po-moment_weights-t3-eval-000330", "task": "cx_po_moment_weights", "source": "cx_po_moment_weights", "split": "eval", "problem": "A list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 60 and sum of squares 176. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nA list of 30 numbers, each equal to 0, 1, 2, 3 or 4, has sum 60 and sum of squares 176. Find how many times each value can occur: nonnegative integers a, b, c, d, e (the numbers of 0s, 1s, 2s, 3s and 4s). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 19, 1, 1, 9", "ref": "0, 19, 1, 1, 9", "family": "po_moment_weights", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 33, "bucket": null, "guess_rate": 0.000712, "solvable": true, "holdout": false, "meta": {"family": "po_moment_weights", "rung": "witness", "params": {"W": 30, "M1": 60, "M2": 176}, "seed": 75305729603000537, "version": "po-v0", "tier": 3, "n_raw": 33, "small": 7, "valid_answers": ["0, 19, 1, 1, 9", "1, 16, 4, 0, 9", "1, 17, 1, 3, 8", "2, 14, 4, 2, 8", "2, 15, 1, 5, 7", "3, 11, 7, 1, 8", "3, 12, 4, 4, 7", "3, 13, 1, 7, 6", "4, 8, 10, 0, 8", "4, 9, 7, 3, 7", "4, 10, 4, 6, 6", "4, 11, 1, 9, 5", "5, 6, 10, 2, 7", "5, 7, 7, 5, 6", "5, 8, 4, 8, 5", "5, 9, 1, 11, 4", "6, 3, 13, 1, 7", "6, 4, 10, 4, 6", "6, 5, 7, 7, 5", "6, 6, 4, 10, 4", "6, 7, 1, 13, 3", "7, 0, 16, 0, 7", "7, 1, 13, 3, 6", "7, 2, 10, 6, 5", "7, 3, 7, 9, 4", "7, 4, 4, 12, 3", "7, 5, 1, 15, 2", "8, 0, 10, 8, 4", "8, 1, 7, 11, 3", "8, 2, 4, 14, 2", "8, 3, 1, 17, 1", "9, 0, 4, 16, 1", "9, 1, 1, 19, 0"]}, "cell": "moment_weights/3", "stage": "C"}
+{"id": "po-order_mod-t1-eval-000025", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 378 whose multiplicative order modulo 379 is 6, that is, x^6 \\equiv 1 \\pmod{379} and x^j \\not\\equiv 1 for 1 <= j < 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 378 whose multiplicative order modulo 379 is 6, that is, x^6 \\equiv 1 \\pmod{379} and x^j \\not\\equiv 1 for 1 <= j < 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "52", "ref": "52", "family": "po_order_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.005291, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 379, "k": 6}, "seed": 171199287101000079, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 52, "valid_answers": ["52", "328"]}, "cell": "order_mod/1", "stage": "C"}
+{"id": "po-order_mod-t1-eval-000034", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 568 whose multiplicative order modulo 569 is 8, that is, x^8 \\equiv 1 \\pmod{569} and x^j \\not\\equiv 1 for 1 <= j < 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 568 whose multiplicative order modulo 569 is 8, that is, x^8 \\equiv 1 \\pmod{569} and x^j \\not\\equiv 1 for 1 <= j < 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "76", "ref": "76", "family": "po_order_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.007042, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 569, "k": 8}, "seed": 171199287101000145, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 76, "valid_answers": ["76", "277", "292", "493"]}, "cell": "order_mod/1", "stage": "C"}
+{"id": "po-order_mod-t1-eval-000113", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 400 whose multiplicative order modulo 401 is 8, that is, x^8 \\equiv 1 \\pmod{401} and x^j \\not\\equiv 1 for 1 <= j < 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 400 whose multiplicative order modulo 401 is 8, that is, x^8 \\equiv 1 \\pmod{401} and x^j \\not\\equiv 1 for 1 <= j < 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "45", "ref": "45", "family": "po_order_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.01, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 401, "k": 8}, "seed": 171199287101000665, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 45, "valid_answers": ["45", "98", "303", "356"]}, "cell": "order_mod/1", "stage": "C"}
+{"id": "po-order_mod-t1-eval-000051", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 228 whose multiplicative order modulo 229 is 4, that is, x^4 \\equiv 1 \\pmod{229} and x^j \\not\\equiv 1 for 1 <= j < 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 228 whose multiplicative order modulo 229 is 4, that is, x^4 \\equiv 1 \\pmod{229} and x^j \\not\\equiv 1 for 1 <= j < 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "107", "ref": "107", "family": "po_order_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.008772, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 229, "k": 4}, "seed": 171199287101000239, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 107, "valid_answers": ["107", "122"]}, "cell": "order_mod/1", "stage": "C"}
+{"id": "po-order_mod-t1-eval-000123", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 570 whose multiplicative order modulo 571 is 15, that is, x^15 \\equiv 1 \\pmod{571} and x^j \\not\\equiv 1 for 1 <= j < 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 570 whose multiplicative order modulo 571 is 15, that is, x^15 \\equiv 1 \\pmod{571} and x^j \\not\\equiv 1 for 1 <= j < 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "134", "ref": "134", "family": "po_order_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.014035, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 571, "k": 15}, "seed": 171199287101000796, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 134, "valid_answers": ["134", "193", "255", "331", "468", "473", "500", "502"]}, "cell": "order_mod/1", "stage": "C"}
+{"id": "po-order_mod-t1-eval-000019", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 448 whose multiplicative order modulo 449 is 4, that is, x^4 \\equiv 1 \\pmod{449} and x^j \\not\\equiv 1 for 1 <= j < 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 448 whose multiplicative order modulo 449 is 4, that is, x^4 \\equiv 1 \\pmod{449} and x^j \\not\\equiv 1 for 1 <= j < 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "67", "ref": "67", "family": "po_order_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004464, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 449, "k": 4}, "seed": 171199287101000058, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 67, "valid_answers": ["67", "382"]}, "cell": "order_mod/1", "stage": "C"}
+{"id": "po-order_mod-t1-eval-000119", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 448 whose multiplicative order modulo 449 is 16, that is, x^16 \\equiv 1 \\pmod{449} and x^j \\not\\equiv 1 for 1 <= j < 16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 448 whose multiplicative order modulo 449 is 16, that is, x^16 \\equiv 1 \\pmod{449} and x^j \\not\\equiv 1 for 1 <= j < 16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "35", "ref": "35", "family": "po_order_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.017857, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 449, "k": 16}, "seed": 171199287101000739, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 35, "valid_answers": ["35", "77", "100", "220", "229", "349", "372", "414"]}, "cell": "order_mod/1", "stage": "C"}
+{"id": "po-order_mod-t1-eval-000154", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 490 whose multiplicative order modulo 491 is 7, that is, x^7 \\equiv 1 \\pmod{491} and x^j \\not\\equiv 1 for 1 <= j < 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 490 whose multiplicative order modulo 491 is 7, that is, x^7 \\equiv 1 \\pmod{491} and x^j \\not\\equiv 1 for 1 <= j < 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "138", "ref": "138", "family": "po_order_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.012245, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 491, "k": 7}, "seed": 171199287101001199, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 138, "valid_answers": ["138", "153", "223", "240", "332", "386"]}, "cell": "order_mod/1", "stage": "C"}
+{"id": "po-order_mod-t1-eval-000107", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 378 whose multiplicative order modulo 379 is 9, that is, x^9 \\equiv 1 \\pmod{379} and x^j \\not\\equiv 1 for 1 <= j < 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 378 whose multiplicative order modulo 379 is 9, that is, x^9 \\equiv 1 \\pmod{379} and x^j \\not\\equiv 1 for 1 <= j < 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "84", "ref": "84", "family": "po_order_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.015873, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 379, "k": 9}, "seed": 171199287101000615, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 84, "valid_answers": ["84", "115", "180", "185", "234", "339"]}, "cell": "order_mod/1", "stage": "C"}
+{"id": "po-order_mod-t1-eval-000014", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 540 whose multiplicative order modulo 541 is 30, that is, x^30 \\equiv 1 \\pmod{541} and x^j \\not\\equiv 1 for 1 <= j < 30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 540 whose multiplicative order modulo 541 is 30, that is, x^30 \\equiv 1 \\pmod{541} and x^j \\not\\equiv 1 for 1 <= j < 30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "234", "ref": "234", "family": "po_order_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.014815, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 541, "k": 30}, "seed": 171199287101000037, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 234, "valid_answers": ["234", "289", "300", "334", "343", "347", "426", "431"]}, "cell": "order_mod/1", "stage": "C"}
+{"id": "po-order_mod-t1-eval-000041", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 408 whose multiplicative order modulo 409 is 3, that is, x^3 \\equiv 1 \\pmod{409} and x^j \\not\\equiv 1 for 1 <= j < 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 408 whose multiplicative order modulo 409 is 3, that is, x^3 \\equiv 1 \\pmod{409} and x^j \\not\\equiv 1 for 1 <= j < 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "53", "ref": "53", "family": "po_order_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004902, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 409, "k": 3}, "seed": 171199287101000176, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 53, "valid_answers": ["53", "355"]}, "cell": "order_mod/1", "stage": "C"}
+{"id": "po-order_mod-t1-eval-000056", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 576 whose multiplicative order modulo 577 is 12, that is, x^12 \\equiv 1 \\pmod{577} and x^j \\not\\equiv 1 for 1 <= j < 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 576 whose multiplicative order modulo 577 is 12, that is, x^12 \\equiv 1 \\pmod{577} and x^j \\not\\equiv 1 for 1 <= j < 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "57", "ref": "57", "family": "po_order_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.006944, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 577, "k": 12}, "seed": 171199287101000264, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 57, "valid_answers": ["57", "81", "496", "520"]}, "cell": "order_mod/1", "stage": "C"}
+{"id": "po-order_mod-t2-eval-000632", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 1452 whose multiplicative order modulo 1453 is 6, that is, x^6 \\equiv 1 \\pmod{1453} and x^j \\not\\equiv 1 for 1 <= j < 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 1452 whose multiplicative order modulo 1453 is 6, that is, x^6 \\equiv 1 \\pmod{1453} and x^j \\not\\equiv 1 for 1 <= j < 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "694", "ref": "694", "family": "po_order_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001377, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 1453, "k": 6}, "seed": 171199287102007106, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 694, "valid_answers": ["694", "760"]}, "cell": "order_mod/2", "stage": "C"}
+{"id": "po-order_mod-t2-eval-000275", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 1482 whose multiplicative order modulo 1483 is 6, that is, x^6 \\equiv 1 \\pmod{1483} and x^j \\not\\equiv 1 for 1 <= j < 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 1482 whose multiplicative order modulo 1483 is 6, that is, x^6 \\equiv 1 \\pmod{1483} and x^j \\not\\equiv 1 for 1 <= j < 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "39", "ref": "39", "family": "po_order_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00135, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 1483, "k": 6}, "seed": 171199287102000849, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 39, "valid_answers": ["39", "1445"]}, "cell": "order_mod/2", "stage": "C"}
+{"id": "po-order_mod-t2-eval-000138", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 990 whose multiplicative order modulo 991 is 11, that is, x^11 \\equiv 1 \\pmod{991} and x^j \\not\\equiv 1 for 1 <= j < 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 990 whose multiplicative order modulo 991 is 11, that is, x^11 \\equiv 1 \\pmod{991} and x^j \\not\\equiv 1 for 1 <= j < 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "42", "ref": "42", "family": "po_order_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.010101, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 991, "k": 11}, "seed": 171199287102000351, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 42, "valid_answers": ["42", "50", "118", "134", "518", "673", "754", "773", "945", "947"]}, "cell": "order_mod/2", "stage": "C"}
+{"id": "po-order_mod-t2-eval-000332", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 700 whose multiplicative order modulo 701 is 20, that is, x^20 \\equiv 1 \\pmod{701} and x^j \\not\\equiv 1 for 1 <= j < 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 700 whose multiplicative order modulo 701 is 20, that is, x^20 \\equiv 1 \\pmod{701} and x^j \\not\\equiv 1 for 1 <= j < 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "93", "ref": "93", "family": "po_order_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.011429, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 701, "k": 20}, "seed": 171199287102001174, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 93, "valid_answers": ["93", "98", "251", "310", "391", "450", "603", "608"]}, "cell": "order_mod/2", "stage": "C"}
+{"id": "po-order_mod-t2-eval-000469", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 880 whose multiplicative order modulo 881 is 22, that is, x^22 \\equiv 1 \\pmod{881} and x^j \\not\\equiv 1 for 1 <= j < 22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 880 whose multiplicative order modulo 881 is 22, that is, x^22 \\equiv 1 \\pmod{881} and x^j \\not\\equiv 1 for 1 <= j < 22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "215", "ref": "215", "family": "po_order_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.011364, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 881, "k": 22}, "seed": 171199287102002218, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 215, "valid_answers": ["215", "345", "468", "644", "695", "710", "713", "738", "791", "849"]}, "cell": "order_mod/2", "stage": "C"}
+{"id": "po-order_mod-t2-eval-000373", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 1320 whose multiplicative order modulo 1321 is 10, that is, x^10 \\equiv 1 \\pmod{1321} and x^j \\not\\equiv 1 for 1 <= j < 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 1320 whose multiplicative order modulo 1321 is 10, that is, x^10 \\equiv 1 \\pmod{1321} and x^j \\not\\equiv 1 for 1 <= j < 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "64", "ref": "64", "family": "po_order_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00303, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 1321, "k": 10}, "seed": 171199287102001422, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 64, "valid_answers": ["64", "586", "805", "1188"]}, "cell": "order_mod/2", "stage": "C"}
+{"id": "po-order_mod-t2-eval-000557", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 690 whose multiplicative order modulo 691 is 15, that is, x^15 \\equiv 1 \\pmod{691} and x^j \\not\\equiv 1 for 1 <= j < 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 690 whose multiplicative order modulo 691 is 15, that is, x^15 \\equiv 1 \\pmod{691} and x^j \\not\\equiv 1 for 1 <= j < 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "113", "ref": "113", "family": "po_order_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.011594, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 691, "k": 15}, "seed": 171199287102003795, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 113, "valid_answers": ["113", "159", "197", "228", "258", "331", "383", "405"]}, "cell": "order_mod/2", "stage": "C"}
+{"id": "po-order_mod-t2-eval-000112", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 630 whose multiplicative order modulo 631 is 5, that is, x^5 \\equiv 1 \\pmod{631} and x^j \\not\\equiv 1 for 1 <= j < 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 630 whose multiplicative order modulo 631 is 5, that is, x^5 \\equiv 1 \\pmod{631} and x^j \\not\\equiv 1 for 1 <= j < 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "228", "ref": "228", "family": "po_order_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.006349, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 631, "k": 5}, "seed": 171199287102000283, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 228, "valid_answers": ["228", "242", "279", "512"]}, "cell": "order_mod/2", "stage": "C"}
+{"id": "po-order_mod-t2-eval-000342", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 1152 whose multiplicative order modulo 1153 is 6, that is, x^6 \\equiv 1 \\pmod{1153} and x^j \\not\\equiv 1 for 1 <= j < 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 1152 whose multiplicative order modulo 1153 is 6, that is, x^6 \\equiv 1 \\pmod{1153} and x^j \\not\\equiv 1 for 1 <= j < 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "503", "ref": "503", "family": "po_order_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001736, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 1153, "k": 6}, "seed": 171199287102001249, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 503, "valid_answers": ["503", "651"]}, "cell": "order_mod/2", "stage": "C"}
+{"id": "po-order_mod-t2-eval-000227", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 1470 whose multiplicative order modulo 1471 is 5, that is, x^5 \\equiv 1 \\pmod{1471} and x^j \\not\\equiv 1 for 1 <= j < 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 1470 whose multiplicative order modulo 1471 is 5, that is, x^5 \\equiv 1 \\pmod{1471} and x^j \\not\\equiv 1 for 1 <= j < 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "45", "ref": "45", "family": "po_order_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.002721, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 1471, "k": 5}, "seed": 171199287102000622, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 45, "valid_answers": ["45", "554", "948", "1394"]}, "cell": "order_mod/2", "stage": "C"}
+{"id": "po-order_mod-t2-eval-000358", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 1290 whose multiplicative order modulo 1291 is 10, that is, x^10 \\equiv 1 \\pmod{1291} and x^j \\not\\equiv 1 for 1 <= j < 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 1290 whose multiplicative order modulo 1291 is 10, that is, x^10 \\equiv 1 \\pmod{1291} and x^j \\not\\equiv 1 for 1 <= j < 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "228", "ref": "228", "family": "po_order_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.003101, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 1291, "k": 10}, "seed": 171199287102001346, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 228, "valid_answers": ["228", "436", "947", "972"]}, "cell": "order_mod/2", "stage": "C"}
+{"id": "po-order_mod-t2-eval-000566", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 1320 whose multiplicative order modulo 1321 is 4, that is, x^4 \\equiv 1 \\pmod{1321} and x^j \\not\\equiv 1 for 1 <= j < 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 1320 whose multiplicative order modulo 1321 is 4, that is, x^4 \\equiv 1 \\pmod{1321} and x^j \\not\\equiv 1 for 1 <= j < 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "257", "ref": "257", "family": "po_order_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001515, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 1321, "k": 4}, "seed": 171199287102004137, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 257, "valid_answers": ["257", "1064"]}, "cell": "order_mod/2", "stage": "C"}
+{"id": "po-order_mod-t3-eval-001360", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 3120 whose multiplicative order modulo 3121 is 104, that is, x^104 \\equiv 1 \\pmod{3121} and x^j \\not\\equiv 1 for 1 <= j < 104. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 3120 whose multiplicative order modulo 3121 is 104, that is, x^104 \\equiv 1 \\pmod{3121} and x^j \\not\\equiv 1 for 1 <= j < 104. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "106", "ref": "106", "family": "po_order_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 48, "bucket": null, "guess_rate": 0.015385, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 3121, "k": 104}, "seed": 171199287103005780, "version": "po-v0", "tier": 3, "n_raw": 48, "small": 106, "valid_answers": ["106", "114", "265", "357", "486", "542", "766", "767", "841", "848", "876", "898", "912", "931", "942", "989", "1001", "1054", "1206", "1215", "1294", "1355", "1451", "1477", "1644", "1670", "1766", "1827", "1906", "1915", "2067", "2120", "2132", "2179", "2190", "2209", "2223", "2245", "2273", "2280", "2354", "2355", "2579", "2635", "2764", "2856", "3007", "3015"]}, "cell": "order_mod/3", "stage": "C"}
+{"id": "po-order_mod-t3-eval-001609", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 1620 whose multiplicative order modulo 1621 is 45, that is, x^45 \\equiv 1 \\pmod{1621} and x^j \\not\\equiv 1 for 1 <= j < 45. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 1620 whose multiplicative order modulo 1621 is 45, that is, x^45 \\equiv 1 \\pmod{1621} and x^j \\not\\equiv 1 for 1 <= j < 45. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3", "ref": "3", "family": "po_order_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 24, "bucket": null, "guess_rate": 0.014815, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 1621, "k": 45}, "seed": 171199287103008425, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 3, "valid_answers": ["3", "9", "35", "77", "81", "315", "344", "400", "458", "552", "566", "655", "880", "921", "1019", "1066", "1081", "1142", "1183", "1200", "1225", "1441", "1577", "1601"]}, "cell": "order_mod/3", "stage": "C"}
+{"id": "po-order_mod-t3-eval-001227", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 3342 whose multiplicative order modulo 3343 is 3, that is, x^3 \\equiv 1 \\pmod{3343} and x^j \\not\\equiv 1 for 1 <= j < 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 3342 whose multiplicative order modulo 3343 is 3, that is, x^3 \\equiv 1 \\pmod{3343} and x^j \\not\\equiv 1 for 1 <= j < 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1424", "ref": "1424", "family": "po_order_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000598, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 3343, "k": 3}, "seed": 171199287103004640, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 1424, "valid_answers": ["1424", "1918"]}, "cell": "order_mod/3", "stage": "C"}
+{"id": "po-order_mod-t3-eval-001401", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 3570 whose multiplicative order modulo 3571 is 15, that is, x^15 \\equiv 1 \\pmod{3571} and x^j \\not\\equiv 1 for 1 <= j < 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 3570 whose multiplicative order modulo 3571 is 15, that is, x^15 \\equiv 1 \\pmod{3571} and x^j \\not\\equiv 1 for 1 <= j < 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "794", "ref": "794", "family": "po_order_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.002241, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 3571, "k": 15}, "seed": 171199287103006222, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 794, "valid_answers": ["794", "895", "1121", "1191", "1787", "1940", "3220", "3337"]}, "cell": "order_mod/3", "stage": "C"}
+{"id": "po-order_mod-t3-eval-001663", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 3760 whose multiplicative order modulo 3761 is 20, that is, x^20 \\equiv 1 \\pmod{3761} and x^j \\not\\equiv 1 for 1 <= j < 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 3760 whose multiplicative order modulo 3761 is 20, that is, x^20 \\equiv 1 \\pmod{3761} and x^j \\not\\equiv 1 for 1 <= j < 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "934", "ref": "934", "family": "po_order_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.002128, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 3761, "k": 20}, "seed": 171199287103009096, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 934, "valid_answers": ["934", "1225", "1655", "1793", "1968", "2106", "2536", "2827"]}, "cell": "order_mod/3", "stage": "C"}
+{"id": "po-order_mod-t3-eval-000695", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 2346 whose multiplicative order modulo 2347 is 46, that is, x^46 \\equiv 1 \\pmod{2347} and x^j \\not\\equiv 1 for 1 <= j < 46. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 2346 whose multiplicative order modulo 2347 is 46, that is, x^46 \\equiv 1 \\pmod{2347} and x^j \\not\\equiv 1 for 1 <= j < 46. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "13", "ref": "13", "family": "po_order_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 0.009378, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 2347, "k": 46}, "seed": 171199287103001977, "version": "po-v0", "tier": 3, "n_raw": 22, "small": 13, "valid_answers": ["13", "182", "201", "247", "467", "970", "1111", "1472", "1483", "1832", "1844", "1845", "1950", "1986", "1987", "2001", "2081", "2151", "2178", "2197", "2328", "2333"]}, "cell": "order_mod/3", "stage": "C"}
+{"id": "po-order_mod-t3-eval-000725", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 3726 whose multiplicative order modulo 3727 is 23, that is, x^23 \\equiv 1 \\pmod{3727} and x^j \\not\\equiv 1 for 1 <= j < 23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 3726 whose multiplicative order modulo 3727 is 23, that is, x^23 \\equiv 1 \\pmod{3727} and x^j \\not\\equiv 1 for 1 <= j < 23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "416", "ref": "416", "family": "po_order_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 0.005904, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 3727, "k": 23}, "seed": 171199287103002090, "version": "po-v0", "tier": 3, "n_raw": 22, "small": 416, "valid_answers": ["416", "439", "543", "557", "564", "638", "791", "797", "801", "908", "1080", "1301", "1513", "1614", "1619", "2040", "2268", "2611", "2644", "3272", "3550", "3576"]}, "cell": "order_mod/3", "stage": "C"}
+{"id": "po-order_mod-t3-eval-000608", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 3696 whose multiplicative order modulo 3697 is 21, that is, x^21 \\equiv 1 \\pmod{3697} and x^j \\not\\equiv 1 for 1 <= j < 21. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 3696 whose multiplicative order modulo 3697 is 21, that is, x^21 \\equiv 1 \\pmod{3697} and x^j \\not\\equiv 1 for 1 <= j < 21. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "51", "ref": "51", "family": "po_order_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.003247, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 3697, "k": 21}, "seed": 171199287103001683, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 51, "valid_answers": ["51", "106", "145", "335", "514", "1315", "1709", "2540", "2601", "2726", "3056", "3388"]}, "cell": "order_mod/3", "stage": "C"}
+{"id": "po-order_mod-t3-eval-000070", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 3360 whose multiplicative order modulo 3361 is 56, that is, x^56 \\equiv 1 \\pmod{3361} and x^j \\not\\equiv 1 for 1 <= j < 56. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 3360 whose multiplicative order modulo 3361 is 56, that is, x^56 \\equiv 1 \\pmod{3361} and x^j \\not\\equiv 1 for 1 <= j < 56. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8", "ref": "8", "family": "po_order_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 24, "bucket": null, "guess_rate": 0.007143, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 3361, "k": 56}, "seed": 171199287103000178, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 8, "valid_answers": ["8", "343", "420", "446", "478", "512", "842", "1441", "1477", "1568", "1575", "1656", "1705", "1786", "1793", "1884", "1920", "2519", "2849", "2883", "2915", "2941", "3018", "3353"]}, "cell": "order_mod/3", "stage": "C"}
+{"id": "po-order_mod-t3-eval-000334", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 1656 whose multiplicative order modulo 1657 is 12, that is, x^12 \\equiv 1 \\pmod{1657} and x^j \\not\\equiv 1 for 1 <= j < 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 1656 whose multiplicative order modulo 1657 is 12, that is, x^12 \\equiv 1 \\pmod{1657} and x^j \\not\\equiv 1 for 1 <= j < 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "129", "ref": "129", "family": "po_order_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.002415, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 1657, "k": 12}, "seed": 171199287103000892, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 129, "valid_answers": ["129", "745", "912", "1528"]}, "cell": "order_mod/3", "stage": "C"}
+{"id": "po-order_mod-t3-eval-001623", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 2860 whose multiplicative order modulo 2861 is 20, that is, x^20 \\equiv 1 \\pmod{2861} and x^j \\not\\equiv 1 for 1 <= j < 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 2860 whose multiplicative order modulo 2861 is 20, that is, x^20 \\equiv 1 \\pmod{2861} and x^j \\not\\equiv 1 for 1 <= j < 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "160", "ref": "160", "family": "po_order_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.002797, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 2861, "k": 20}, "seed": 171199287103008565, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 160, "valid_answers": ["160", "952", "1055", "1145", "1716", "1806", "1909", "2701"]}, "cell": "order_mod/3", "stage": "C"}
+{"id": "po-order_mod-t3-eval-002078", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 2376 whose multiplicative order modulo 2377 is 12, that is, x^12 \\equiv 1 \\pmod{2377} and x^j \\not\\equiv 1 for 1 <= j < 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 1 <= x <= 2376 whose multiplicative order modulo 2377 is 12, that is, x^12 \\equiv 1 \\pmod{2377} and x^j \\not\\equiv 1 for 1 <= j < 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "74", "ref": "74", "family": "po_order_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001684, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 2377, "k": 12}, "seed": 171199287103028941, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 74, "valid_answers": ["74", "1060", "1317", "2303"]}, "cell": "order_mod/3", "stage": "C"}
+{"id": "po-parallelogram-t0-eval-000517", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-5, -4), (5, 2) and (-2, 4) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-5, -4), (5, 2) and (-2, 4) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, -2", "ref": "-12, -2", "family": "po_parallelogram", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002755, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-5, -4], [5, 2], [-2, 4]]}, "seed": 421451545100000541, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 6, "valid_answers": ["-12, -2", "2, -6", "8, 10"]}, "cell": "parallelogram/0", "stage": "A"}
+{"id": "po-parallelogram-t0-eval-000955", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (3, -2), (1, 4) and (3, -4) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (3, -2), (1, 4) and (3, -4) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 2", "ref": "1, 2", "family": "po_parallelogram", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.004115, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[3, -2], [1, 4], [3, -4]]}, "seed": 421451545100001004, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["1, 2", "1, 6", "5, -10"]}, "cell": "parallelogram/0", "stage": "A"}
+{"id": "po-parallelogram-t0-eval-002078", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-4, -1), (-3, -1) and (1, 5) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-4, -1), (-3, -1) and (1, 5) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -7", "ref": "-8, -7", "family": "po_parallelogram", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002755, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-4, -1], [-3, -1], [1, 5]]}, "seed": 421451545100002195, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-8, -7", "0, 5", "2, 5"]}, "cell": "parallelogram/0", "stage": "A"}
+{"id": "po-parallelogram-t0-eval-001312", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (5, 4), (5, -1) and (2, -4) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (5, 4), (5, -1) and (2, -4) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -9", "ref": "2, -9", "family": "po_parallelogram", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002755, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[5, 4], [5, -1], [2, -4]]}, "seed": 421451545100001383, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["2, -9", "2, 1", "8, 7"]}, "cell": "parallelogram/0", "stage": "A"}
+{"id": "po-parallelogram-t0-eval-001160", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-1, 3), (1, 3) and (5, -4) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-1, 3), (1, 3) and (5, -4) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 10", "ref": "-5, 10", "family": "po_parallelogram", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002755, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-1, 3], [1, 3], [5, -4]]}, "seed": 421451545100001219, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-5, 10", "3, -4", "7, -4"]}, "cell": "parallelogram/0", "stage": "A"}
+{"id": "po-parallelogram-t0-eval-001483", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (2, -2), (-4, 1) and (-4, 0) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (2, -2), (-4, 1) and (-4, 0) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 3", "ref": "-10, 3", "family": "po_parallelogram", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.004115, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[2, -2], [-4, 1], [-4, 0]]}, "seed": 421451545100001566, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["-10, 3", "2, -3", "2, -1"]}, "cell": "parallelogram/0", "stage": "A"}
+{"id": "po-parallelogram-t0-eval-000030", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-3, 5), (-5, 0) and (-2, -5) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-3, 5), (-5, 0) and (-2, -5) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 10", "ref": "-6, 10", "family": "po_parallelogram", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002755, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-3, 5], [-5, 0], [-2, -5]]}, "seed": 421451545100000031, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 0, "valid_answers": ["-6, 10", "-4, -10", "0, 0"]}, "cell": "parallelogram/0", "stage": "A"}
+{"id": "po-parallelogram-t0-eval-002129", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (4, -4), (-1, 0) and (0, 4) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (4, -4), (-1, 0) and (0, 4) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 8", "ref": "-5, 8", "family": "po_parallelogram", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.004115, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[4, -4], [-1, 0], [0, 4]]}, "seed": 421451545100002248, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-5, 8", "3, -8", "5, 0"]}, "cell": "parallelogram/0", "stage": "A"}
+{"id": "po-parallelogram-t0-eval-000242", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (0, 4), (-3, 1) and (3, -4) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (0, 4), (-3, 1) and (3, -4) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 9", "ref": "-6, 9", "family": "po_parallelogram", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.004115, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[0, 4], [-3, 1], [3, -4]]}, "seed": 421451545100000253, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 6, "valid_answers": ["-6, 9", "0, -7", "6, -1"]}, "cell": "parallelogram/0", "stage": "A"}
+{"id": "po-parallelogram-t0-eval-000658", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (3, -1), (2, -5) and (-3, 3) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (3, -1), (2, -5) and (-3, 3) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -1", "ref": "-4, -1", "family": "po_parallelogram", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002755, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[3, -1], [2, -5], [-3, 3]]}, "seed": 421451545100000696, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-4, -1", "-2, 7", "8, -9"]}, "cell": "parallelogram/0", "stage": "A"}
+{"id": "po-parallelogram-t0-eval-001937", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-4, -3), (-1, -5) and (5, -1) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-4, -3), (-1, -5) and (5, -1) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -7", "ref": "-10, -7", "family": "po_parallelogram", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002755, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-4, -3], [-1, -5], [5, -1]]}, "seed": 421451545100002044, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["-10, -7", "2, 1", "8, -3"]}, "cell": "parallelogram/0", "stage": "A"}
+{"id": "po-parallelogram-t0-eval-001414", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-1, 4), (-2, 0) and (3, -3) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-1, 4), (-2, 0) and (3, -3) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 7", "ref": "-6, 7", "family": "po_parallelogram", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.004115, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-1, 4], [-2, 0], [3, -3]]}, "seed": 421451545100001492, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-6, 7", "2, -7", "4, 1"]}, "cell": "parallelogram/0", "stage": "A"}
+{"id": "po-parallelogram-t1-eval-000256", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-5, 2), (-6, -6) and (-1, 10) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-5, 2), (-6, -6) and (-1, 10) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -14", "ref": "-10, -14", "family": "po_parallelogram", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000756, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-5, 2], [-6, -6], [-1, 10]]}, "seed": 421451545101000258, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 2, "valid_answers": ["-10, -14", "-2, 2", "0, 18"]}, "cell": "parallelogram/1", "stage": "B"}
+{"id": "po-parallelogram-t1-eval-001943", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-3, 7), (7, 0) and (3, 12) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-3, 7), (7, 0) and (3, 12) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 19", "ref": "-7, 19", "family": "po_parallelogram", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000533, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-3, 7], [7, 0], [3, 12]]}, "seed": 421451545101001961, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 5, "valid_answers": ["-7, 19", "1, -5", "13, 5"]}, "cell": "parallelogram/1", "stage": "B"}
+{"id": "po-parallelogram-t1-eval-001437", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-10, -3), (7, -6) and (12, 3) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-10, -3), (7, -6) and (12, 3) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-15, -12", "ref": "-15, -12", "family": "po_parallelogram", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000533, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-10, -3], [7, -6], [12, 3]]}, "seed": 421451545101001451, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 6, "valid_answers": ["-15, -12", "-5, 6", "29, 0"]}, "cell": "parallelogram/1", "stage": "B"}
+{"id": "po-parallelogram-t1-eval-000823", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-1, 5), (-12, -11) and (12, -10) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-1, 5), (-12, -11) and (12, -10) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-25, 4", "ref": "-25, 4", "family": "po_parallelogram", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000533, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-1, 5], [-12, -11], [12, -10]]}, "seed": 421451545101000831, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 23, "valid_answers": ["-25, 4", "1, -26", "23, 6"]}, "cell": "parallelogram/1", "stage": "B"}
+{"id": "po-parallelogram-t1-eval-000496", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-11, -3), (0, -4) and (10, -6) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-11, -3), (0, -4) and (10, -6) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-21, -1", "ref": "-21, -1", "family": "po_parallelogram", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.00063, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-11, -3], [0, -4], [10, -6]]}, "seed": 421451545101000501, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 5, "valid_answers": ["-21, -1", "-1, -5", "21, -7"]}, "cell": "parallelogram/1", "stage": "B"}
+{"id": "po-parallelogram-t1-eval-001745", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-8, 3), (11, 0) and (-1, -5) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-8, 3), (11, 0) and (-1, -5) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-20, -2", "ref": "-20, -2", "family": "po_parallelogram", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.00063, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-8, 3], [11, 0], [-1, -5]]}, "seed": 421451545101001762, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 8, "valid_answers": ["-20, -2", "4, 8", "18, -8"]}, "cell": "parallelogram/1", "stage": "B"}
+{"id": "po-parallelogram-t1-eval-001897", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-1, -9), (-7, 12) and (-12, -9) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-1, -9), (-7, 12) and (-12, -9) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-18, 12", "ref": "-18, 12", "family": "po_parallelogram", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000533, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-1, -9], [-7, 12], [-12, -9]]}, "seed": 421451545101001915, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 12, "valid_answers": ["-18, 12", "-6, -30", "4, 12"]}, "cell": "parallelogram/1", "stage": "B"}
+{"id": "po-parallelogram-t1-eval-001855", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (3, -3), (0, 7) and (12, -3) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (3, -3), (0, 7) and (12, -3) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 7", "ref": "-9, 7", "family": "po_parallelogram", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000533, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[3, -3], [0, 7], [12, -3]]}, "seed": 421451545101001873, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 9, "valid_answers": ["-9, 7", "9, 7", "15, -13"]}, "cell": "parallelogram/1", "stage": "B"}
+{"id": "po-parallelogram-t1-eval-001357", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-3, -5), (-3, 0) and (8, -11) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-3, -5), (-3, 0) and (8, -11) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-14, 6", "ref": "-14, 6", "family": "po_parallelogram", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.00063, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-3, -5], [-3, 0], [8, -11]]}, "seed": 421451545101001371, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 8, "valid_answers": ["-14, 6", "8, -16", "8, -6"]}, "cell": "parallelogram/1", "stage": "B"}
+{"id": "po-parallelogram-t1-eval-001706", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-12, 8), (-3, -12) and (-1, -6) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-12, 8), (-3, -12) and (-1, -6) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-14, 2", "ref": "-14, 2", "family": "po_parallelogram", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000533, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-12, 8], [-3, -12], [-1, -6]]}, "seed": 421451545101001722, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 14, "valid_answers": ["-14, 2", "-10, 14", "8, -26"]}, "cell": "parallelogram/1", "stage": "B"}
+{"id": "po-parallelogram-t1-eval-000593", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-12, 1), (6, -4) and (10, -3) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-12, 1), (6, -4) and (10, -3) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-16, 0", "ref": "-16, 0", "family": "po_parallelogram", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000533, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[-12, 1], [6, -4], [10, -3]]}, "seed": 421451545101000599, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 8, "valid_answers": ["-16, 0", "-8, 2", "28, -8"]}, "cell": "parallelogram/1", "stage": "B"}
+{"id": "po-parallelogram-t1-eval-000476", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (3, -10), (-2, 1) and (10, 3) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (3, -10), (-2, 1) and (10, 3) are three vertices of a parallelogram. Find integers x, y such that (x, y) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -12", "ref": "-9, -12", "family": "po_parallelogram", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000756, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 2, "P": [[3, -10], [-2, 1], [10, 3]]}, "seed": 421451545101000481, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 12, "valid_answers": ["-9, -12", "5, 14", "15, -8"]}, "cell": "parallelogram/1", "stage": "B"}
+{"id": "po-parallelogram-t2-eval-001198", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (3, 4, -5), (1, 6, 4) and (1, 3, -6) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (3, 4, -5), (1, 6, 4) and (1, 3, -6) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 5, 3", "ref": "-1, 5, 3", "family": "po_parallelogram", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 5.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[3, 4, -5], [1, 6, 4], [1, 3, -6]]}, "seed": 421451545102001201, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 5, "valid_answers": ["-1, 5, 3", "3, 1, -15", "3, 7, 5"]}, "cell": "parallelogram/2", "stage": "B"}
+{"id": "po-parallelogram-t2-eval-000322", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-2, 1, 6), (3, -1, 0) and (-1, 6, 4) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-2, 1, 6), (3, -1, 0) and (-1, 6, 4) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 8, 10", "ref": "-6, 8, 10", "family": "po_parallelogram", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 5.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[-2, 1, 6], [3, -1, 0], [-1, 6, 4]]}, "seed": 421451545102000323, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 4, "valid_answers": ["-6, 8, 10", "2, -6, 2", "4, 4, -2"]}, "cell": "parallelogram/2", "stage": "B"}
+{"id": "po-parallelogram-t2-eval-001624", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-5, -4, -2), (-6, 5, 0) and (0, 2, 5) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-5, -4, -2), (-6, 5, 0) and (0, 2, 5) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-11, -1, -7", "ref": "-11, -1, -7", "family": "po_parallelogram", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 5.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[-5, -4, -2], [-6, 5, 0], [0, 2, 5]]}, "seed": 421451545102001628, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 7, "valid_answers": ["-11, -1, -7", "-1, 11, 7", "1, -7, 3"]}, "cell": "parallelogram/2", "stage": "B"}
+{"id": "po-parallelogram-t2-eval-001714", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-3, -3, -2), (-1, 2, -2) and (-4, 0, -2) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-3, -3, -2), (-1, 2, -2) and (-4, 0, -2) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -5, -2", "ref": "-6, -5, -2", "family": "po_parallelogram", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000152, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[-3, -3, -2], [-1, 2, -2], [-4, 0, -2]]}, "seed": 421451545102001718, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 2, "valid_answers": ["-6, -5, -2", "-2, 5, -2", "0, -1, -2"]}, "cell": "parallelogram/2", "stage": "B"}
+{"id": "po-parallelogram-t2-eval-001170", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (3, 5, -5), (5, -2, 1) and (1, 6, -3) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (3, 5, -5), (5, -2, 1) and (1, 6, -3) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 13, -9", "ref": "-1, 13, -9", "family": "po_parallelogram", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 5.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[3, 5, -5], [5, -2, 1], [1, 6, -3]]}, "seed": 421451545102001173, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 3, "valid_answers": ["-1, 13, -9", "3, -1, 3", "7, -3, -1"]}, "cell": "parallelogram/2", "stage": "B"}
+{"id": "po-parallelogram-t2-eval-001923", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (0, 4, 4), (-3, 2, 1) and (-6, -6, 4) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (0, 4, 4), (-3, 2, 1) and (-6, -6, 4) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -8, 1", "ref": "-9, -8, 1", "family": "po_parallelogram", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 5.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[0, 4, 4], [-3, 2, 1], [-6, -6, 4]]}, "seed": 421451545102001928, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 7, "valid_answers": ["-9, -8, 1", "-3, -4, 7", "3, 12, 1"]}, "cell": "parallelogram/2", "stage": "B"}
+{"id": "po-parallelogram-t2-eval-002424", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-1, -1, 4), (-1, 3, -3) and (3, -5, -3) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-1, -1, 4), (-1, 3, -3) and (3, -5, -3) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 7, 4", "ref": "-5, 7, 4", "family": "po_parallelogram", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 8.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[-1, -1, 4], [-1, 3, -3], [3, -5, -3]]}, "seed": 421451545102002429, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 7, "valid_answers": ["-5, 7, 4", "3, -9, 4", "3, -1, -10"]}, "cell": "parallelogram/2", "stage": "B"}
+{"id": "po-parallelogram-t2-eval-001482", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-4, 2, -5), (3, -1, 3) and (1, -1, -1) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-4, 2, -5), (3, -1, 3) and (1, -1, -1) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 2, -9", "ref": "-6, 2, -9", "family": "po_parallelogram", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 8.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[-4, 2, -5], [3, -1, 3], [1, -1, -1]]}, "seed": 421451545102001485, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 2, "valid_answers": ["-6, 2, -9", "-2, 2, -1", "8, -4, 7"]}, "cell": "parallelogram/2", "stage": "B"}
+{"id": "po-parallelogram-t2-eval-000892", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (5, 2, 6), (4, 6, 4) and (2, 0, 1) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (5, 2, 6), (4, 6, 4) and (2, 0, 1) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 4, -1", "ref": "1, 4, -1", "family": "po_parallelogram", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 5.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[5, 2, 6], [4, 6, 4], [2, 0, 1]]}, "seed": 421451545102000895, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 4, "valid_answers": ["1, 4, -1", "3, -4, 3", "7, 8, 9"]}, "cell": "parallelogram/2", "stage": "B"}
+{"id": "po-parallelogram-t2-eval-000722", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-1, 3, -1), (3, 3, -5) and (4, -4, -4) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-1, 3, -1), (3, 3, -5) and (4, -4, -4) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 10, -2", "ref": "-2, 10, -2", "family": "po_parallelogram", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 8.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[-1, 3, -1], [3, 3, -5], [4, -4, -4]]}, "seed": 421451545102000725, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 4, "valid_answers": ["-2, 10, -2", "0, -4, 0", "8, -4, -8"]}, "cell": "parallelogram/2", "stage": "B"}
+{"id": "po-parallelogram-t2-eval-000075", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (1, 2, 1), (6, 4, 4) and (1, 1, 6) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (1, 2, 1), (6, 4, 4) and (1, 1, 6) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -1, 3", "ref": "-4, -1, 3", "family": "po_parallelogram", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 5.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[1, 2, 1], [6, 4, 4], [1, 1, 6]]}, "seed": 421451545102000076, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 4, "valid_answers": ["-4, -1, 3", "6, 3, 9", "6, 5, -1"]}, "cell": "parallelogram/2", "stage": "B"}
+{"id": "po-parallelogram-t2-eval-001878", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (4, -2, -5), (6, 3, -5) and (-6, 3, -4) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (4, -2, -5), (6, 3, -5) and (-6, 3, -4) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -2, -4", "ref": "-8, -2, -4", "family": "po_parallelogram", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 5.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[4, -2, -5], [6, 3, -5], [-6, 3, -4]]}, "seed": 421451545102001883, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 8, "valid_answers": ["-8, -2, -4", "-4, 8, -4", "16, -2, -6"]}, "cell": "parallelogram/2", "stage": "B"}
+{"id": "po-parallelogram-t3-eval-001615", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (15, 4, -2), (7, 1, -9) and (4, -15, -3) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (15, 4, -2), (7, 1, -9) and (4, -15, -3) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -18, -10", "ref": "-4, -18, -10", "family": "po_parallelogram", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[15, 4, -2], [7, 1, -9], [4, -15, -3]]}, "seed": 421451545103001615, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 12, "valid_answers": ["-4, -18, -10", "12, -12, 4", "18, 20, -8"]}, "cell": "parallelogram/3", "stage": "B"}
+{"id": "po-parallelogram-t3-eval-001686", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-15, -9, -4), (6, -10, -12) and (-6, -9, -8) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-15, -9, -4), (6, -10, -12) and (-6, -9, -8) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-27, -8, 0", "ref": "-27, -8, 0", "family": "po_parallelogram", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[-15, -9, -4], [6, -10, -12], [-6, -9, -8]]}, "seed": 421451545103001686, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 10, "valid_answers": ["-27, -8, 0", "-3, -10, -8", "15, -10, -16"]}, "cell": "parallelogram/3", "stage": "B"}
+{"id": "po-parallelogram-t3-eval-000052", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-9, 0, 7), (-13, 5, 7) and (3, -7, 4) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-9, 0, 7), (-13, 5, 7) and (3, -7, 4) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-25, 12, 10", "ref": "-25, 12, 10", "family": "po_parallelogram", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 6e-06, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[-9, 0, 7], [-13, 5, 7], [3, -7, 4]]}, "seed": 421451545103000052, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 4, "valid_answers": ["-25, 12, 10", "-1, -2, 4", "7, -12, 4"]}, "cell": "parallelogram/3", "stage": "B"}
+{"id": "po-parallelogram-t3-eval-002055", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (10, 2, -4), (11, 14, 7) and (-15, -8, -12) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (10, 2, -4), (11, 14, 7) and (-15, -8, -12) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-16, -20, -23", "ref": "-16, -20, -23", "family": "po_parallelogram", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[10, 2, -4], [11, 14, 7], [-15, -8, -12]]}, "seed": 421451545103002055, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 14, "valid_answers": ["-16, -20, -23", "-14, 4, -1", "36, 24, 15"]}, "cell": "parallelogram/3", "stage": "B"}
+{"id": "po-parallelogram-t3-eval-000386", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (5, 12, 12), (-12, -6, -1) and (8, -12, -8) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (5, 12, 12), (-12, -6, -1) and (8, -12, -8) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-15, 18, 19", "ref": "-15, 18, 19", "family": "po_parallelogram", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 7e-06, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[5, 12, 12], [-12, -6, -1], [8, -12, -8]]}, "seed": 421451545103000386, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 19, "valid_answers": ["-15, 18, 19", "-9, -30, -21", "25, 6, 5"]}, "cell": "parallelogram/3", "stage": "B"}
+{"id": "po-parallelogram-t3-eval-002235", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (5, 13, -2), (-3, -14, -7) and (-8, 4, -6) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (5, 13, -2), (-3, -14, -7) and (-8, 4, -6) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-16, -23, -11", "ref": "-16, -23, -11", "family": "po_parallelogram", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[5, 13, -2], [-3, -14, -7], [-8, 4, -6]]}, "seed": 421451545103002235, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 10, "valid_answers": ["-16, -23, -11", "0, 31, -1", "10, -5, -3"]}, "cell": "parallelogram/3", "stage": "B"}
+{"id": "po-parallelogram-t3-eval-001064", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (12, -6, -10), (7, 9, -8) and (14, 11, -6) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (12, -6, -10), (7, 9, -8) and (14, 11, -6) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, -8, -12", "ref": "5, -8, -12", "family": "po_parallelogram", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[12, -6, -10], [7, 9, -8], [14, 11, -6]]}, "seed": 421451545103001064, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 12, "valid_answers": ["5, -8, -12", "9, 26, -4", "19, -4, -8"]}, "cell": "parallelogram/3", "stage": "B"}
+{"id": "po-parallelogram-t3-eval-000453", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (14, -15, 3), (12, -15, -9) and (0, -10, 9) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (14, -15, 3), (12, -15, -9) and (0, -10, 9) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -10, -3", "ref": "-2, -10, -3", "family": "po_parallelogram", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[14, -15, 3], [12, -15, -9], [0, -10, 9]]}, "seed": 421451545103000453, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 10, "valid_answers": ["-2, -10, -3", "2, -10, 21", "26, -20, -15"]}, "cell": "parallelogram/3", "stage": "B"}
+{"id": "po-parallelogram-t3-eval-000496", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (14, -1, -6), (8, 9, 4) and (-15, -3, 2) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (14, -1, -6), (8, 9, 4) and (-15, -3, 2) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-21, 7, 12", "ref": "-21, 7, 12", "family": "po_parallelogram", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[14, -1, -6], [8, 9, 4], [-15, -3, 2]]}, "seed": 421451545103000496, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 13, "valid_answers": ["-21, 7, 12", "-9, -13, -8", "37, 11, -4"]}, "cell": "parallelogram/3", "stage": "B"}
+{"id": "po-parallelogram-t3-eval-001370", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (1, 13, -14), (-10, -14, 5) and (8, 1, 11) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (1, 13, -14), (-10, -14, 5) and (8, 1, 11) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-17, -2, -20", "ref": "-17, -2, -20", "family": "po_parallelogram", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[1, 13, -14], [-10, -14, 5], [8, 1, 11]]}, "seed": 421451545103001370, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 20, "valid_answers": ["-17, -2, -20", "-3, -26, 30", "19, 28, -8"]}, "cell": "parallelogram/3", "stage": "B"}
+{"id": "po-parallelogram-t3-eval-000214", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-6, 2, -8), (6, 8, -10) and (10, 13, -12) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-6, 2, -8), (6, 8, -10) and (10, 13, -12) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -3, -6", "ref": "-10, -3, -6", "family": "po_parallelogram", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 6e-06, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[-6, 2, -8], [6, 8, -10], [10, 13, -12]]}, "seed": 421451545103000214, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 10, "valid_answers": ["-10, -3, -6", "-2, 7, -10", "22, 19, -14"]}, "cell": "parallelogram/3", "stage": "B"}
+{"id": "po-parallelogram-t3-eval-000394", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-8, -7, -14), (5, 14, 13) and (-6, -6, 15) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nThe points (-8, -7, -14), (5, 14, 13) and (-6, -6, 15) are three vertices of a parallelogram. Find integers x, y, z such that (x, y, z) is the fourth vertex. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-19, -27, -12", "ref": "-19, -27, -12", "family": "po_parallelogram", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_parallelogram", "rung": "witness", "params": {"dim": 3, "P": [[-8, -7, -14], [5, 14, 13], [-6, -6, 15]]}, "seed": 421451545103000394, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 16, "valid_answers": ["-19, -27, -12", "3, 13, -16", "7, 15, 42"]}, "cell": "parallelogram/3", "stage": "B"}
+{"id": "po-partition_sq-t0-eval-000115", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 45 and a^2 + b^2 + c^2 = 861. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 45 and a^2 + b^2 + c^2 = 861. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 19, 22", "ref": "4, 19, 22", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.013605, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 45, "Q": 861}, "seed": 301652278900000542, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 22, "valid_answers": ["4, 19, 22", "8, 11, 26"]}, "cell": "partition_sq/0", "stage": "C"}
+{"id": "po-partition_sq-t0-eval-000319", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 48 and a^2 + b^2 + c^2 = 1002. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 48 and a^2 + b^2 + c^2 = 1002. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 19, 25", "ref": "4, 19, 25", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 48, "Q": 1002}, "seed": 301652278900002313, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 25, "valid_answers": ["4, 19, 25", "7, 13, 28"]}, "cell": "partition_sq/0", "stage": "C"}
+{"id": "po-partition_sq-t0-eval-000382", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 44 and a^2 + b^2 + c^2 = 958. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 44 and a^2 + b^2 + c^2 = 958. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 15, 27", "ref": "2, 15, 27", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.014286, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 44, "Q": 958}, "seed": 301652278900003577, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 27, "valid_answers": ["2, 15, 27", "6, 9, 29"]}, "cell": "partition_sq/0", "stage": "C"}
+{"id": "po-partition_sq-t0-eval-000126", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 48 and a^2 + b^2 + c^2 = 894. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 48 and a^2 + b^2 + c^2 = 894. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 19, 22", "ref": "7, 19, 22", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 48, "Q": 894}, "seed": 301652278900000601, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 22, "valid_answers": ["7, 19, 22", "10, 13, 25"]}, "cell": "partition_sq/0", "stage": "C"}
+{"id": "po-partition_sq-t0-eval-000141", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 47 and a^2 + b^2 + c^2 = 881. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 47 and a^2 + b^2 + c^2 = 881. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 19, 22", "ref": "6, 19, 22", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.012422, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 47, "Q": 881}, "seed": 301652278900000669, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 22, "valid_answers": ["6, 19, 22", "7, 16, 24"]}, "cell": "partition_sq/0", "stage": "C"}
+{"id": "po-partition_sq-t0-eval-000077", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 60 and a^2 + b^2 + c^2 = 1304. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 60 and a^2 + b^2 + c^2 = 1304. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "12, 22, 26", "ref": "12, 22, 26", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00738, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 60, "Q": 1304}, "seed": 301652278900000369, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 26, "valid_answers": ["12, 22, 26", "14, 18, 28"]}, "cell": "partition_sq/0", "stage": "C"}
+{"id": "po-partition_sq-t0-eval-000138", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 54 and a^2 + b^2 + c^2 = 1154. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 54 and a^2 + b^2 + c^2 = 1154. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 23, 24", "ref": "7, 23, 24", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.018433, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 54, "Q": 1154}, "seed": 301652278900000648, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 24, "valid_answers": ["7, 23, 24", "8, 19, 27", "9, 17, 28", "12, 13, 29"]}, "cell": "partition_sq/0", "stage": "C"}
+{"id": "po-partition_sq-t0-eval-000230", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 53 and a^2 + b^2 + c^2 = 1025. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 53 and a^2 + b^2 + c^2 = 1025. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 21, 22", "ref": "10, 21, 22", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.009615, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 53, "Q": 1025}, "seed": 301652278900001309, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 22, "valid_answers": ["10, 21, 22", "12, 16, 25"]}, "cell": "partition_sq/0", "stage": "C"}
+{"id": "po-partition_sq-t0-eval-000217", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 57 and a^2 + b^2 + c^2 = 1205. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 57 and a^2 + b^2 + c^2 = 1205. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 23, 24", "ref": "10, 23, 24", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00823, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 57, "Q": 1205}, "seed": 301652278900001203, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 24, "valid_answers": ["10, 23, 24", "14, 15, 28"]}, "cell": "partition_sq/0", "stage": "C"}
+{"id": "po-partition_sq-t0-eval-000403", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 52 and a^2 + b^2 + c^2 = 990. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 52 and a^2 + b^2 + c^2 = 990. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 19, 23", "ref": "10, 19, 23", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.01, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 52, "Q": 990}, "seed": 301652278900004287, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 23, "valid_answers": ["10, 19, 23", "13, 14, 25"]}, "cell": "partition_sq/0", "stage": "C"}
+{"id": "po-partition_sq-t0-eval-000399", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 54 and a^2 + b^2 + c^2 = 1158. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 54 and a^2 + b^2 + c^2 = 1158. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 22, 25", "ref": "7, 22, 25", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.009217, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 54, "Q": 1158}, "seed": 301652278900004177, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 25, "valid_answers": ["7, 22, 25", "11, 14, 29"]}, "cell": "partition_sq/0", "stage": "C"}
+{"id": "po-partition_sq-t0-eval-000411", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 51 and a^2 + b^2 + c^2 = 1115. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 51 and a^2 + b^2 + c^2 = 1115. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 19, 27", "ref": "5, 19, 27", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.010417, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 51, "Q": 1115}, "seed": 301652278900004766, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 27, "valid_answers": ["5, 19, 27", "7, 15, 29"]}, "cell": "partition_sq/0", "stage": "C"}
+{"id": "po-partition_sq-t1-eval-001339", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 45 and a^2 + b^2 + c^2 = 717. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 45 and a^2 + b^2 + c^2 = 717. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 16, 19", "ref": "10, 16, 19", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.013605, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 45, "Q": 717}, "seed": 301652278901005999, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 19, "valid_answers": ["10, 16, 19", "11, 14, 20"]}, "cell": "partition_sq/1", "stage": "C"}
+{"id": "po-partition_sq-t1-eval-000262", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 39 and a^2 + b^2 + c^2 = 563. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 39 and a^2 + b^2 + c^2 = 563. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 15, 17", "ref": "7, 15, 17", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.018519, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 39, "Q": 563}, "seed": 301652278901000723, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 17, "valid_answers": ["7, 15, 17", "9, 11, 19"]}, "cell": "partition_sq/1", "stage": "C"}
+{"id": "po-partition_sq-t1-eval-001640", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 76 and a^2 + b^2 + c^2 = 2194. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 76 and a^2 + b^2 + c^2 = 2194. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "12, 31, 33", "ref": "12, 31, 33", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004505, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 76, "Q": 2194}, "seed": 301652278901009453, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 33, "valid_answers": ["12, 31, 33", "13, 27, 36"]}, "cell": "partition_sq/1", "stage": "C"}
+{"id": "po-partition_sq-t1-eval-000075", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 64 and a^2 + b^2 + c^2 = 1538. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 64 and a^2 + b^2 + c^2 = 1538. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 24, 29", "ref": "11, 24, 29", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.006452, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 64, "Q": 1538}, "seed": 301652278901000190, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 29, "valid_answers": ["11, 24, 29", "15, 17, 32"]}, "cell": "partition_sq/1", "stage": "C"}
+{"id": "po-partition_sq-t1-eval-001388", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 54 and a^2 + b^2 + c^2 = 1388. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 54 and a^2 + b^2 + c^2 = 1388. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 22, 30", "ref": "2, 22, 30", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.009217, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 54, "Q": 1388}, "seed": 301652278901006344, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 30, "valid_answers": ["2, 22, 30", "6, 14, 34"]}, "cell": "partition_sq/1", "stage": "C"}
+{"id": "po-partition_sq-t1-eval-001584", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 76 and a^2 + b^2 + c^2 = 2814. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 76 and a^2 + b^2 + c^2 = 2814. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 37, 38", "ref": "1, 37, 38", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004505, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 76, "Q": 2814}, "seed": 301652278901008779, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 38, "valid_answers": ["1, 37, 38", "2, 31, 43"]}, "cell": "partition_sq/1", "stage": "C"}
+{"id": "po-partition_sq-t1-eval-000913", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 71 and a^2 + b^2 + c^2 = 2453. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 71 and a^2 + b^2 + c^2 = 2453. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 34, 36", "ref": "1, 34, 36", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.005195, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 71, "Q": 2453}, "seed": 301652278901003361, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 36, "valid_answers": ["1, 34, 36", "9, 16, 46"]}, "cell": "partition_sq/1", "stage": "C"}
+{"id": "po-partition_sq-t1-eval-001310", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 53 and a^2 + b^2 + c^2 = 1481. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 53 and a^2 + b^2 + c^2 = 1481. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 18, 34", "ref": "1, 18, 34", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.009615, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 53, "Q": 1481}, "seed": 301652278901005806, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 34, "valid_answers": ["1, 18, 34", "4, 13, 36"]}, "cell": "partition_sq/1", "stage": "C"}
+{"id": "po-partition_sq-t1-eval-001233", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 82 and a^2 + b^2 + c^2 = 2302. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 82 and a^2 + b^2 + c^2 = 2302. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "21, 30, 31", "ref": "21, 30, 31", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.003846, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 82, "Q": 2302}, "seed": 301652278901005245, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 31, "valid_answers": ["21, 30, 31", "22, 27, 33"]}, "cell": "partition_sq/1", "stage": "C"}
+{"id": "po-partition_sq-t1-eval-000777", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 87 and a^2 + b^2 + c^2 = 2657. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 87 and a^2 + b^2 + c^2 = 2657. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "20, 31, 36", "ref": "20, 31, 36", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.003401, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 87, "Q": 2657}, "seed": 301652278901002621, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 36, "valid_answers": ["20, 31, 36", "22, 27, 38"]}, "cell": "partition_sq/1", "stage": "C"}
+{"id": "po-partition_sq-t1-eval-000017", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 60 and a^2 + b^2 + c^2 = 1352. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 60 and a^2 + b^2 + c^2 = 1352. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 24, 26", "ref": "10, 24, 26", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00738, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 60, "Q": 1352}, "seed": 301652278901000046, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 26, "valid_answers": ["10, 24, 26", "14, 16, 30"]}, "cell": "partition_sq/1", "stage": "C"}
+{"id": "po-partition_sq-t1-eval-000577", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c with a + b + c = 64 and a^2 + b^2 + c^2 = 1686. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c with a + b + c = 64 and a^2 + b^2 + c^2 = 1686. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 26, 31", "ref": "7, 26, 31", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.006452, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 3, "S": 64, "Q": 1686}, "seed": 301652278901001784, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 31, "valid_answers": ["7, 26, 31", "10, 19, 35"]}, "cell": "partition_sq/1", "stage": "C"}
+{"id": "po-partition_sq-t2-eval-001349", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 79 and a^2 + b^2 + c^2 + d^2 = 2145. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 79 and a^2 + b^2 + c^2 + d^2 = 2145. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 16, 27, 34", "ref": "2, 16, 27, 34", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.002849, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 79, "Q": 2145}, "seed": 301652278902001520, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 32, "valid_answers": ["2, 16, 27, 34", "2, 19, 22, 36", "3, 14, 28, 34", "4, 12, 31, 32", "4, 18, 19, 38", "6, 10, 28, 35", "8, 15, 16, 40", "10, 11, 18, 40"]}, "cell": "partition_sq/2", "stage": "C"}
+{"id": "po-partition_sq-t2-eval-001600", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 57 and a^2 + b^2 + c^2 + d^2 = 1185. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 57 and a^2 + b^2 + c^2 + d^2 = 1185. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 12, 16, 28", "ref": "1, 12, 16, 28", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.005144, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 57, "Q": 1185}, "seed": 301652278902001820, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 26, "valid_answers": ["1, 12, 16, 28", "2, 8, 21, 26", "2, 12, 14, 29", "4, 10, 13, 30", "5, 8, 14, 30"]}, "cell": "partition_sq/2", "stage": "C"}
+{"id": "po-partition_sq-t2-eval-000366", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 105 and a^2 + b^2 + c^2 + d^2 = 3111. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 105 and a^2 + b^2 + c^2 + d^2 = 3111. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 26, 33, 35", "ref": "11, 26, 33, 35", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000577, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 105, "Q": 3111}, "seed": 301652278902000398, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 35, "valid_answers": ["11, 26, 33, 35", "13, 22, 33, 37", "15, 19, 34, 37", "15, 23, 26, 41"]}, "cell": "partition_sq/2", "stage": "C"}
+{"id": "po-partition_sq-t2-eval-001619", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 96 and a^2 + b^2 + c^2 + d^2 = 3062. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 96 and a^2 + b^2 + c^2 + d^2 = 3062. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 26, 33, 36", "ref": "1, 26, 33, 36", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.001912, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 96, "Q": 3062}, "seed": 301652278902001846, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 36, "valid_answers": ["1, 26, 33, 36", "2, 24, 31, 39", "4, 21, 29, 42", "6, 16, 33, 41", "6, 19, 27, 44", "7, 14, 36, 39", "7, 15, 32, 42", "9, 12, 34, 41", "9, 17, 24, 46", "12, 15, 22, 47"]}, "cell": "partition_sq/2", "stage": "C"}
+{"id": "po-partition_sq-t2-eval-002243", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 60 and a^2 + b^2 + c^2 + d^2 = 1200. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 60 and a^2 + b^2 + c^2 + d^2 = 1200. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 14, 18, 26", "ref": "2, 14, 18, 26", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001733, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 60, "Q": 1200}, "seed": 301652278902002595, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 26, "valid_answers": ["2, 14, 18, 26", "4, 12, 16, 28"]}, "cell": "partition_sq/2", "stage": "C"}
+{"id": "po-partition_sq-t2-eval-001203", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 93 and a^2 + b^2 + c^2 + d^2 = 2717. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 93 and a^2 + b^2 + c^2 + d^2 = 2717. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 28, 30, 32", "ref": "3, 28, 30, 32", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.00254, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 93, "Q": 2717}, "seed": 301652278902001351, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 32, "valid_answers": ["3, 28, 30, 32", "4, 24, 30, 35", "4, 26, 27, 36", "6, 19, 32, 36", "6, 22, 26, 39", "7, 18, 30, 38", "8, 18, 27, 40", "10, 14, 30, 39", "10, 18, 23, 42", "11, 12, 34, 36", "11, 16, 24, 42", "12, 18, 20, 43"]}, "cell": "partition_sq/2", "stage": "C"}
+{"id": "po-partition_sq-t2-eval-002054", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 59 and a^2 + b^2 + c^2 + d^2 = 1525. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 59 and a^2 + b^2 + c^2 + d^2 = 1525. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 4, 22, 32", "ref": "1, 4, 22, 32", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001837, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 59, "Q": 1525}, "seed": 301652278902002363, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 32, "valid_answers": ["1, 4, 22, 32", "2, 9, 12, 36"]}, "cell": "partition_sq/2", "stage": "C"}
+{"id": "po-partition_sq-t2-eval-002446", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 98 and a^2 + b^2 + c^2 + d^2 = 2950. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 98 and a^2 + b^2 + c^2 + d^2 = 2950. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 26, 32, 35", "ref": "5, 26, 32, 35", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 0.002686, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 98, "Q": 2950}, "seed": 301652278902002843, "version": "po-v0", "tier": 2, "n_raw": 15, "small": 35, "valid_answers": ["5, 26, 32, 35", "5, 27, 30, 36", "6, 23, 33, 36", "7, 25, 26, 40", "8, 19, 34, 37", "8, 23, 26, 41", "9, 18, 32, 39", "9, 23, 24, 42", "10, 16, 35, 37", "10, 17, 31, 40", "12, 14, 33, 39", "12, 15, 30, 41", "13, 16, 26, 43", "13, 19, 22, 44", "14, 17, 23, 44"]}, "cell": "partition_sq/2", "stage": "C"}
+{"id": "po-partition_sq-t2-eval-001855", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 75 and a^2 + b^2 + c^2 + d^2 = 1683. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 75 and a^2 + b^2 + c^2 + d^2 = 1683. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 20, 23, 27", "ref": "5, 20, 23, 27", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000842, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 75, "Q": 1683}, "seed": 301652278902002133, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 27, "valid_answers": ["5, 20, 23, 27", "7, 15, 25, 28"]}, "cell": "partition_sq/2", "stage": "C"}
+{"id": "po-partition_sq-t2-eval-001180", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 118 and a^2 + b^2 + c^2 + d^2 = 3610. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 118 and a^2 + b^2 + c^2 + d^2 = 3610. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "20, 31, 32, 35", "ref": "20, 31, 32, 35", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000399, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 118, "Q": 3610}, "seed": 301652278902001328, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 35, "valid_answers": ["20, 31, 32, 35", "21, 28, 33, 36", "23, 26, 31, 38", "24, 27, 28, 39"]}, "cell": "partition_sq/2", "stage": "C"}
+{"id": "po-partition_sq-t2-eval-001326", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 85 and a^2 + b^2 + c^2 + d^2 = 2307. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 85 and a^2 + b^2 + c^2 + d^2 = 2307. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 19, 29, 33", "ref": "4, 19, 29, 33", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001127, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 85, "Q": 2307}, "seed": 301652278902001496, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 33, "valid_answers": ["4, 19, 29, 33", "4, 21, 25, 35", "5, 19, 25, 36", "11, 15, 19, 40"]}, "cell": "partition_sq/2", "stage": "C"}
+{"id": "po-partition_sq-t2-eval-000802", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 127 and a^2 + b^2 + c^2 + d^2 = 4251. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 127 and a^2 + b^2 + c^2 + d^2 = 4251. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "19, 35, 36, 37", "ref": "19, 35, 36, 37", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000476, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 127, "Q": 4251}, "seed": 301652278902000895, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 37, "valid_answers": ["19, 35, 36, 37", "20, 31, 37, 39", "21, 29, 37, 40", "23, 28, 33, 43", "24, 25, 37, 41", "25, 27, 31, 44"]}, "cell": "partition_sq/2", "stage": "C"}
+{"id": "po-partition_sq-t3-eval-000450", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 123 and a^2 + b^2 + c^2 + d^2 = 3989. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 123 and a^2 + b^2 + c^2 + d^2 = 3989. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "20, 30, 33, 40", "ref": "20, 30, 33, 40", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000263, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 123, "Q": 3989}, "seed": 301652278903000466, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 40, "valid_answers": ["20, 30, 33, 40", "22, 29, 30, 42", "24, 25, 32, 42"]}, "cell": "partition_sq/3", "stage": "C"}
+{"id": "po-partition_sq-t3-eval-001259", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 97 and a^2 + b^2 + c^2 + d^2 = 2451. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 97 and a^2 + b^2 + c^2 + d^2 = 2451. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "16, 25, 27, 29", "ref": "16, 25, 27, 29", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000741, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 97, "Q": 2451}, "seed": 301652278903001319, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 29, "valid_answers": ["16, 25, 27, 29", "17, 24, 25, 31", "19, 20, 27, 31", "19, 21, 25, 32"]}, "cell": "partition_sq/3", "stage": "C"}
+{"id": "po-partition_sq-t3-eval-000687", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 165 and a^2 + b^2 + c^2 + d^2 = 7189. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 165 and a^2 + b^2 + c^2 + d^2 = 7189. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "26, 40, 47, 52", "ref": "26, 40, 47, 52", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000176, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 165, "Q": 7189}, "seed": 301652278903000712, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 52, "valid_answers": ["26, 40, 47, 52", "28, 38, 44, 55", "31, 32, 50, 52", "31, 34, 44, 56", "34, 35, 38, 58"]}, "cell": "partition_sq/3", "stage": "C"}
+{"id": "po-partition_sq-t3-eval-001966", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 160 and a^2 + b^2 + c^2 + d^2 = 6586. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 160 and a^2 + b^2 + c^2 + d^2 = 6586. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "29, 40, 44, 47", "ref": "29, 40, 44, 47", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000232, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 160, "Q": 6586}, "seed": 301652278903002072, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 47, "valid_answers": ["29, 40, 44, 47", "30, 39, 42, 49", "31, 36, 45, 48", "31, 38, 41, 50", "32, 35, 44, 49", "33, 36, 40, 51"]}, "cell": "partition_sq/3", "stage": "C"}
+{"id": "po-partition_sq-t3-eval-000947", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 154 and a^2 + b^2 + c^2 + d^2 = 7310. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 154 and a^2 + b^2 + c^2 + d^2 = 7310. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8, 39, 53, 54", "ref": "8, 39, 53, 54", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000566, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 154, "Q": 7310}, "seed": 301652278903000985, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 54, "valid_answers": ["8, 39, 53, 54", "9, 38, 48, 59", "11, 32, 54, 57", "11, 33, 50, 60", "14, 27, 56, 57", "14, 29, 48, 63", "15, 26, 53, 60", "17, 24, 51, 62", "17, 27, 44, 66", "18, 29, 39, 68", "20, 21, 50, 63", "20, 23, 45, 66", "23, 24, 38, 69"]}, "cell": "partition_sq/3", "stage": "C"}
+{"id": "po-partition_sq-t3-eval-001540", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 116 and a^2 + b^2 + c^2 + d^2 = 4620. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 116 and a^2 + b^2 + c^2 + d^2 = 4620. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 27, 41, 47", "ref": "1, 27, 41, 47", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.001159, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 116, "Q": 4620}, "seed": 301652278903001625, "version": "po-v0", "tier": 3, "n_raw": 11, "small": 47, "valid_answers": ["1, 27, 41, 47", "3, 23, 41, 49", "3, 29, 31, 53", "5, 19, 45, 47", "5, 27, 29, 55", "7, 17, 41, 51", "9, 17, 35, 55", "11, 13, 39, 53", "11, 17, 31, 57", "13, 21, 23, 59", "15, 17, 25, 59"]}, "cell": "partition_sq/3", "stage": "C"}
+{"id": "po-partition_sq-t3-eval-001422", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 111 and a^2 + b^2 + c^2 + d^2 = 4009. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 111 and a^2 + b^2 + c^2 + d^2 = 4009. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 31, 38, 40", "ref": "2, 31, 38, 40", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000847, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 111, "Q": 4009}, "seed": 301652278903001493, "version": "po-v0", "tier": 3, "n_raw": 7, "small": 40, "valid_answers": ["2, 31, 38, 40", "7, 20, 38, 46", "7, 26, 28, 50", "8, 22, 31, 50", "10, 16, 38, 47", "10, 23, 26, 52", "11, 20, 28, 52"]}, "cell": "partition_sq/3", "stage": "C"}
+{"id": "po-partition_sq-t3-eval-001879", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 168 and a^2 + b^2 + c^2 + d^2 = 8550. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 168 and a^2 + b^2 + c^2 + d^2 = 8550. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9, 48, 54, 57", "ref": "9, 48, 54, 57", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 0.000732, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 168, "Q": 8550}, "seed": 301652278903001983, "version": "po-v0", "tier": 3, "n_raw": 22, "small": 57, "valid_answers": ["9, 48, 54, 57", "9, 49, 52, 58", "10, 45, 52, 61", "12, 39, 54, 63", "12, 41, 50, 65", "13, 36, 58, 61", "13, 37, 54, 64", "14, 39, 47, 68", "16, 31, 58, 63", "16, 37, 45, 70", "17, 30, 56, 65", "19, 27, 58, 64", "19, 28, 54, 67", "19, 34, 43, 72", "20, 26, 57, 65", "20, 30, 47, 71", "21, 26, 53, 68", "21, 30, 45, 72", "23, 26, 48, 71", "23, 32, 39, 74", "26, 32, 35, 75", "27, 30, 36, 75"]}, "cell": "partition_sq/3", "stage": "C"}
+{"id": "po-partition_sq-t3-eval-000350", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 131 and a^2 + b^2 + c^2 + d^2 = 5683. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 131 and a^2 + b^2 + c^2 + d^2 = 5683. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 40, 41, 49", "ref": "1, 40, 41, 49", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.000792, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 131, "Q": 5683}, "seed": 301652278903000360, "version": "po-v0", "tier": 3, "n_raw": 11, "small": 49, "valid_answers": ["1, 40, 41, 49", "4, 31, 41, 55", "5, 28, 43, 55", "5, 29, 41, 56", "7, 23, 49, 52", "7, 28, 37, 59", "9, 21, 45, 56", "11, 20, 41, 59", "11, 21, 39, 60", "12, 27, 29, 63", "15, 20, 33, 63"]}, "cell": "partition_sq/3", "stage": "C"}
+{"id": "po-partition_sq-t3-eval-001025", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 69 and a^2 + b^2 + c^2 + d^2 = 1255. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 69 and a^2 + b^2 + c^2 + d^2 = 1255. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 17, 19, 22", "ref": "11, 17, 19, 22", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001102, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 69, "Q": 1255}, "seed": 301652278903001067, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 22, "valid_answers": ["11, 17, 19, 22", "13, 14, 19, 23"]}, "cell": "partition_sq/3", "stage": "C"}
+{"id": "po-partition_sq-t3-eval-000233", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 104 and a^2 + b^2 + c^2 + d^2 = 3306. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 104 and a^2 + b^2 + c^2 + d^2 = 3306. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 30, 34, 35", "ref": "5, 30, 34, 35", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.001781, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 104, "Q": 3306}, "seed": 301652278903000238, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 35, "valid_answers": ["5, 30, 34, 35", "5, 31, 32, 36", "6, 26, 35, 37", "7, 26, 30, 41", "9, 20, 35, 40", "11, 17, 36, 40", "11, 22, 26, 45", "12, 16, 35, 41", "12, 17, 32, 43", "15, 17, 26, 46", "16, 20, 21, 47", "17, 18, 22, 47"]}, "cell": "partition_sq/3", "stage": "C"}
+{"id": "po-partition_sq-t3-eval-001537", "task": "cx_po_partition_sq", "source": "cx_po_partition_sq", "split": "eval", "problem": "Find distinct positive integers a, b, c, d with a + b + c + d = 87 and a^2 + b^2 + c^2 + d^2 = 2317. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind distinct positive integers a, b, c, d with a + b + c + d = 87 and a^2 + b^2 + c^2 + d^2 = 2317. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 26, 28, 29", "ref": "4, 26, 28, 29", "family": "po_partition_sq", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.001308, "solvable": true, "holdout": false, "meta": {"family": "po_partition_sq", "rung": "witness", "params": {"k": 4, "S": 87, "Q": 2317}, "seed": 301652278903001622, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 29, "valid_answers": ["4, 26, 28, 29", "5, 22, 28, 32", "8, 16, 29, 34", "8, 20, 22, 37", "10, 17, 22, 38"]}, "cell": "partition_sq/3", "stage": "C"}
+{"id": "po-perm_cycles-t0-eval-000017", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose cycle lengths are 4, 1, 1, with pi(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose cycle lengths are 4, 1, 1, with pi(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "2, 3, 4, 1, 5, 6", "ref": "2, 3, 4, 1, 5, 6", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.016667, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 6, "ct": [4, 1, 1], "f1": 2, "f2": null}, "seed": 36965189200000167, "version": "po-v0", "tier": 0, "n_raw": 12, "small": 6, "valid_answers": ["2, 3, 4, 1, 5, 6", "2, 3, 5, 4, 1, 6", "2, 3, 6, 4, 5, 1", "2, 4, 1, 3, 5, 6", "2, 4, 3, 5, 1, 6", "2, 4, 3, 6, 5, 1", "2, 5, 1, 4, 3, 6", "2, 5, 3, 1, 4, 6", "2, 5, 3, 4, 6, 1", "2, 6, 1, 4, 5, 3", "2, 6, 3, 1, 5, 4", "2, 6, 3, 4, 1, 5"]}, "cell": "perm_cycles/0", "stage": "C"}
+{"id": "po-perm_cycles-t0-eval-000019", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose cycle lengths are 2, 2, 2, with pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose cycle lengths are 2, 2, 2, with pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "4, 3, 2, 1, 6, 5", "ref": "4, 3, 2, 1, 6, 5", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.004167, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 6, "ct": [2, 2, 2], "f1": 4, "f2": null}, "seed": 36965189200000229, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 6, "valid_answers": ["4, 3, 2, 1, 6, 5", "4, 5, 6, 1, 2, 3", "4, 6, 5, 1, 3, 2"]}, "cell": "perm_cycles/0", "stage": "C"}
+{"id": "po-perm_cycles-t0-eval-000003", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose cycle lengths are 3, 3, with pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose cycle lengths are 3, 3, with pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "5, 1, 4, 6, 2, 3", "ref": "5, 1, 4, 6, 2, 3", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.011111, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 6, "ct": [3, 3], "f1": 5, "f2": null}, "seed": 36965189200000027, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 6, "valid_answers": ["5, 1, 4, 6, 2, 3", "5, 1, 6, 3, 2, 4", "5, 3, 4, 2, 6, 1", "5, 3, 6, 1, 4, 2", "5, 4, 1, 6, 3, 2", "5, 4, 2, 3, 6, 1", "5, 6, 1, 2, 3, 4", "5, 6, 2, 1, 4, 3"]}, "cell": "perm_cycles/0", "stage": "C"}
+{"id": "po-perm_cycles-t0-eval-000007", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose cycle lengths are 3, 3, with pi(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose cycle lengths are 3, 3, with pi(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "6, 1, 4, 5, 3, 2", "ref": "6, 1, 4, 5, 3, 2", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.011111, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 6, "ct": [3, 3], "f1": 6, "f2": null}, "seed": 36965189200000047, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 6, "valid_answers": ["6, 1, 4, 5, 3, 2", "6, 1, 5, 3, 4, 2", "6, 3, 4, 2, 1, 5", "6, 3, 5, 1, 2, 4", "6, 4, 1, 5, 2, 3", "6, 4, 2, 3, 1, 5", "6, 5, 1, 2, 4, 3", "6, 5, 2, 1, 3, 4"]}, "cell": "perm_cycles/0", "stage": "C"}
+{"id": "po-perm_cycles-t0-eval-000010", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose cycle lengths are 2, 2, 1, 1, with pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose cycle lengths are 2, 2, 1, 1, with pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "3, 2, 1, 4, 6, 5", "ref": "3, 2, 1, 4, 6, 5", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.008333, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 6, "ct": [2, 2, 1, 1], "f1": 3, "f2": null}, "seed": 36965189200000061, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 6, "valid_answers": ["3, 2, 1, 4, 6, 5", "3, 2, 1, 5, 4, 6", "3, 2, 1, 6, 5, 4", "3, 4, 1, 2, 5, 6", "3, 5, 1, 4, 2, 6", "3, 6, 1, 4, 5, 2"]}, "cell": "perm_cycles/0", "stage": "C"}
+{"id": "po-perm_cycles-t0-eval-000001", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose cycle lengths are 2, 2, 1, 1, with pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose cycle lengths are 2, 2, 1, 1, with pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "5, 2, 3, 6, 1, 4", "ref": "5, 2, 3, 6, 1, 4", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.008333, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 6, "ct": [2, 2, 1, 1], "f1": 5, "f2": null}, "seed": 36965189200000019, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 6, "valid_answers": ["5, 2, 3, 6, 1, 4", "5, 2, 4, 3, 1, 6", "5, 2, 6, 4, 1, 3", "5, 3, 2, 4, 1, 6", "5, 4, 3, 2, 1, 6", "5, 6, 3, 4, 1, 2"]}, "cell": "perm_cycles/0", "stage": "C"}
+{"id": "po-perm_cycles-t0-eval-000008", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose cycle lengths are 2, 1, 1, 1, 1, with pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose cycle lengths are 2, 1, 1, 1, 1, with pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "1, 2, 3, 4, 6, 5", "ref": "1, 2, 3, 4, 6, 5", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.013889, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 6, "ct": [2, 1, 1, 1, 1], "f1": 1, "f2": null}, "seed": 36965189200000048, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 6, "valid_answers": ["1, 2, 3, 4, 6, 5", "1, 2, 3, 5, 4, 6", "1, 2, 3, 6, 5, 4", "1, 2, 4, 3, 5, 6", "1, 2, 5, 4, 3, 6", "1, 2, 6, 4, 5, 3", "1, 3, 2, 4, 5, 6", "1, 4, 3, 2, 5, 6", "1, 5, 3, 4, 2, 6", "1, 6, 3, 4, 5, 2"]}, "cell": "perm_cycles/0", "stage": "C"}
+{"id": "po-perm_cycles-t1-eval-000040", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 4, 1, 1, 1, with pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 4, 1, 1, 1, with pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "5, 1, 2, 4, 3, 6, 7", "ref": "5, 1, 2, 4, 3, 6, 7", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.003968, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [4, 1, 1, 1], "f1": 5, "f2": null}, "seed": 36965189201000468, "version": "po-v0", "tier": 1, "n_raw": 20, "small": 7, "valid_answers": ["5, 1, 2, 4, 3, 6, 7", "5, 1, 3, 2, 4, 6, 7", "5, 1, 3, 4, 6, 2, 7", "5, 1, 3, 4, 7, 6, 2", "5, 2, 1, 3, 4, 6, 7", "5, 2, 1, 4, 6, 3, 7", "5, 2, 1, 4, 7, 6, 3", "5, 2, 3, 1, 6, 4, 7", "5, 2, 3, 1, 7, 6, 4", "5, 2, 3, 4, 6, 7, 1", "5, 2, 3, 4, 7, 1, 6", "5, 2, 3, 6, 4, 1, 7", "5, 2, 3, 7, 4, 6, 1", "5, 2, 4, 1, 3, 6, 7", "5, 2, 6, 4, 3, 1, 7", "5, 2, 7, 4, 3, 6, 1", "5, 3, 1, 4, 2, 6, 7", "5, 4, 3, 1, 2, 6, 7", "5, 6, 3, 4, 2, 1, 7", "5, 7, 3, 4, 2, 6, 1"]}, "cell": "perm_cycles/1", "stage": "C"}
+{"id": "po-perm_cycles-t1-eval-000050", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 2, 2, 1, 1, 1, with pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 2, 2, 1, 1, 1, with pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "4, 2, 3, 1, 5, 7, 6", "ref": "4, 2, 3, 1, 5, 7, 6", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.001984, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [2, 2, 1, 1, 1], "f1": 4, "f2": null}, "seed": 36965189201002020, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 7, "valid_answers": ["4, 2, 3, 1, 5, 7, 6", "4, 2, 3, 1, 6, 5, 7", "4, 2, 3, 1, 7, 6, 5", "4, 2, 5, 1, 3, 6, 7", "4, 2, 6, 1, 5, 3, 7", "4, 2, 7, 1, 5, 6, 3", "4, 3, 2, 1, 5, 6, 7", "4, 5, 3, 1, 2, 6, 7", "4, 6, 3, 1, 5, 2, 7", "4, 7, 3, 1, 5, 6, 2"]}, "cell": "perm_cycles/1", "stage": "C"}
+{"id": "po-perm_cycles-t1-eval-000011", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 2, 2, with pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 2, 2, with pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "5, 1, 4, 3, 2, 7, 6", "ref": "5, 1, 4, 3, 2, 7, 6", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 35, "bucket": null, "guess_rate": 0.006944, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [3, 2, 2], "f1": 5, "f2": null}, "seed": 36965189201000056, "version": "po-v0", "tier": 1, "n_raw": 35, "small": 7, "valid_answers": ["5, 1, 4, 3, 2, 7, 6", "5, 1, 6, 7, 2, 3, 4", "5, 1, 7, 6, 2, 4, 3", "5, 3, 2, 1, 4, 7, 6", "5, 3, 2, 6, 1, 7, 4", "5, 3, 2, 6, 7, 4, 1", "5, 3, 2, 7, 1, 4, 6", "5, 3, 2, 7, 6, 1, 4", "5, 3, 4, 2, 1, 7, 6", "5, 3, 6, 7, 1, 2, 4", "5, 3, 7, 6, 1, 4, 2", "5, 4, 1, 2, 3, 7, 6", "5, 4, 2, 3, 1, 7, 6", "5, 4, 6, 2, 1, 7, 3", "5, 4, 6, 2, 7, 3, 1", "5, 4, 6, 7, 1, 3, 2", "5, 4, 7, 2, 1, 3, 6", "5, 4, 7, 2, 6, 1, 3", "5, 4, 7, 6, 1, 2, 3", "5, 6, 1, 7, 3, 2, 4", "5, 6, 2, 7, 1, 3, 4", "5, 6, 4, 3, 1, 7, 2", "5, 6, 4, 3, 7, 2, 1", "5, 6, 4, 7, 1, 2, 3", "5, 6, 7, 1, 4, 2, 3", "5, 6, 7, 2, 1, 4, 3", "5, 6, 7, 3, 1, 2, 4", "5, 7, 1, 6, 3, 4, 2", "5, 7, 2, 6, 1, 4, 3", "5, 7, 4, 3, 1, 2, 6", "5, 7, 4, 3, 6, 1, 2", "5, 7, 4, 6, 1, 3, 2", "5, 7, 6, 1, 4, 3, 2", "5, 7, 6, 2, 1, 3, 4", "5, 7, 6, 3, 1, 4, 2"]}, "cell": "perm_cycles/1", "stage": "C"}
+{"id": "po-perm_cycles-t1-eval-000013", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 1, 1, 1, 1, with pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 1, 1, 1, 1, with pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "4, 1, 3, 2, 5, 6, 7", "ref": "4, 1, 3, 2, 5, 6, 7", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000992, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [3, 1, 1, 1, 1], "f1": 4, "f2": null}, "seed": 36965189201000065, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 7, "valid_answers": ["4, 1, 3, 2, 5, 6, 7", "4, 2, 1, 3, 5, 6, 7", "4, 2, 3, 5, 1, 6, 7", "4, 2, 3, 6, 5, 1, 7", "4, 2, 3, 7, 5, 6, 1"]}, "cell": "perm_cycles/1", "stage": "C"}
+{"id": "po-perm_cycles-t1-eval-000007", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 3, 1, with pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 3, 1, with pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 3, 4, 2, 6, 7, 5", "ref": "1, 3, 4, 2, 6, 7, 5", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 0.007937, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [3, 3, 1], "f1": 1, "f2": null}, "seed": 36965189201000024, "version": "po-v0", "tier": 1, "n_raw": 40, "small": 7, "valid_answers": ["1, 3, 4, 2, 6, 7, 5", "1, 3, 4, 2, 7, 5, 6", "1, 3, 5, 6, 2, 7, 4", "1, 3, 5, 7, 2, 4, 6", "1, 3, 6, 5, 7, 2, 4", "1, 3, 6, 7, 4, 2, 5", "1, 3, 7, 5, 6, 4, 2", "1, 3, 7, 6, 4, 5, 2", "1, 4, 2, 3, 6, 7, 5", "1, 4, 2, 3, 7, 5, 6", "1, 4, 5, 6, 7, 2, 3", "1, 4, 5, 7, 6, 3, 2", "1, 4, 6, 5, 2, 7, 3", "1, 4, 6, 7, 3, 5, 2", "1, 4, 7, 5, 2, 3, 6", "1, 4, 7, 6, 3, 2, 5", "1, 5, 2, 6, 3, 7, 4", "1, 5, 2, 7, 3, 4, 6", "1, 5, 4, 6, 7, 3, 2", "1, 5, 4, 7, 6, 2, 3", "1, 5, 6, 2, 4, 7, 3", "1, 5, 6, 3, 7, 4, 2", "1, 5, 7, 2, 4, 3, 6", "1, 5, 7, 3, 6, 2, 4", "1, 6, 2, 5, 7, 3, 4", "1, 6, 2, 7, 4, 3, 5", "1, 6, 4, 5, 3, 7, 2", "1, 6, 4, 7, 2, 5, 3", "1, 6, 5, 2, 7, 4, 3", "1, 6, 5, 3, 4, 7, 2", "1, 6, 7, 2, 3, 4, 5", "1, 6, 7, 3, 2, 5, 4", "1, 7, 2, 5, 6, 4, 3", "1, 7, 2, 6, 4, 5, 3", "1, 7, 4, 5, 3, 2, 6", "1, 7, 4, 6, 2, 3, 5", "1, 7, 5, 2, 6, 3, 4", "1, 7, 5, 3, 4, 2, 6", "1, 7, 6, 2, 3, 5, 4", "1, 7, 6, 3, 2, 4, 5"]}, "cell": "perm_cycles/1", "stage": "C"}
+{"id": "po-perm_cycles-t1-eval-000042", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 4, 1, 1, 1, with pi(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 4, 1, 1, 1, with pi(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "2, 3, 4, 1, 5, 6, 7", "ref": "2, 3, 4, 1, 5, 6, 7", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.003968, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [4, 1, 1, 1], "f1": 2, "f2": null}, "seed": 36965189201000592, "version": "po-v0", "tier": 1, "n_raw": 20, "small": 7, "valid_answers": ["2, 3, 4, 1, 5, 6, 7", "2, 3, 5, 4, 1, 6, 7", "2, 3, 6, 4, 5, 1, 7", "2, 3, 7, 4, 5, 6, 1", "2, 4, 1, 3, 5, 6, 7", "2, 4, 3, 5, 1, 6, 7", "2, 4, 3, 6, 5, 1, 7", "2, 4, 3, 7, 5, 6, 1", "2, 5, 1, 4, 3, 6, 7", "2, 5, 3, 1, 4, 6, 7", "2, 5, 3, 4, 6, 1, 7", "2, 5, 3, 4, 7, 6, 1", "2, 6, 1, 4, 5, 3, 7", "2, 6, 3, 1, 5, 4, 7", "2, 6, 3, 4, 1, 5, 7", "2, 6, 3, 4, 5, 7, 1", "2, 7, 1, 4, 5, 6, 3", "2, 7, 3, 1, 5, 6, 4", "2, 7, 3, 4, 1, 6, 5", "2, 7, 3, 4, 5, 1, 6"]}, "cell": "perm_cycles/1", "stage": "C"}
+{"id": "po-perm_cycles-t1-eval-000014", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 2, 1, 1, with pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 2, 1, 1, with pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "3, 1, 2, 4, 5, 7, 6", "ref": "3, 1, 2, 4, 5, 7, 6", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 50, "bucket": null, "guess_rate": 0.009921, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [3, 2, 1, 1], "f1": 3, "f2": null}, "seed": 36965189201000069, "version": "po-v0", "tier": 1, "n_raw": 50, "small": 7, "valid_answers": ["3, 1, 2, 4, 5, 7, 6", "3, 1, 2, 4, 6, 5, 7", "3, 1, 2, 4, 7, 6, 5", "3, 1, 2, 5, 4, 6, 7", "3, 1, 2, 6, 5, 4, 7", "3, 1, 2, 7, 5, 6, 4", "3, 2, 1, 4, 6, 7, 5", "3, 2, 1, 4, 7, 5, 6", "3, 2, 1, 5, 6, 4, 7", "3, 2, 1, 5, 7, 6, 4", "3, 2, 1, 6, 4, 5, 7", "3, 2, 1, 6, 5, 7, 4", "3, 2, 1, 7, 4, 6, 5", "3, 2, 1, 7, 5, 4, 6", "3, 2, 4, 1, 5, 7, 6", "3, 2, 4, 1, 6, 5, 7", "3, 2, 4, 1, 7, 6, 5", "3, 2, 5, 4, 1, 7, 6", "3, 2, 5, 6, 1, 4, 7", "3, 2, 5, 7, 1, 6, 4", "3, 2, 6, 4, 7, 1, 5", "3, 2, 6, 5, 4, 1, 7", "3, 2, 6, 7, 5, 1, 4", "3, 2, 7, 4, 6, 5, 1", "3, 2, 7, 5, 4, 6, 1", "3, 2, 7, 6, 5, 4, 1", "3, 4, 1, 5, 2, 6, 7", "3, 4, 1, 6, 5, 2, 7", "3, 4, 1, 7, 5, 6, 2", "3, 4, 5, 2, 1, 6, 7", "3, 4, 6, 2, 5, 1, 7", "3, 4, 7, 2, 5, 6, 1", "3, 5, 1, 2, 4, 6, 7", "3, 5, 1, 4, 6, 2, 7", "3, 5, 1, 4, 7, 6, 2", "3, 5, 4, 1, 2, 6, 7", "3, 5, 6, 4, 2, 1, 7", "3, 5, 7, 4, 2, 6, 1", "3, 6, 1, 2, 5, 4, 7", "3, 6, 1, 4, 2, 5, 7", "3, 6, 1, 4, 5, 7, 2", "3, 6, 4, 1, 5, 2, 7", "3, 6, 5, 4, 1, 2, 7", "3, 6, 7, 4, 5, 2, 1", "3, 7, 1, 2, 5, 6, 4", "3, 7, 1, 4, 2, 6, 5", "3, 7, 1, 4, 5, 2, 6", "3, 7, 4, 1, 5, 6, 2", "3, 7, 5, 4, 1, 6, 2", "3, 7, 6, 4, 5, 1, 2"]}, "cell": "perm_cycles/1", "stage": "C"}
+{"id": "po-perm_cycles-t1-eval-000018", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 2, 2, with pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 2, 2, with pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "4, 1, 5, 2, 3, 7, 6", "ref": "4, 1, 5, 2, 3, 7, 6", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 35, "bucket": null, "guess_rate": 0.006944, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [3, 2, 2], "f1": 4, "f2": null}, "seed": 36965189201000113, "version": "po-v0", "tier": 1, "n_raw": 35, "small": 7, "valid_answers": ["4, 1, 5, 2, 3, 7, 6", "4, 1, 6, 2, 7, 3, 5", "4, 1, 7, 2, 6, 5, 3", "4, 3, 2, 1, 6, 7, 5", "4, 3, 2, 1, 7, 5, 6", "4, 3, 2, 5, 1, 7, 6", "4, 3, 2, 6, 7, 1, 5", "4, 3, 2, 7, 6, 5, 1", "4, 3, 5, 1, 2, 7, 6", "4, 3, 6, 1, 7, 2, 5", "4, 3, 7, 1, 6, 5, 2", "4, 5, 1, 3, 2, 7, 6", "4, 5, 2, 1, 3, 7, 6", "4, 5, 6, 1, 2, 7, 3", "4, 5, 6, 1, 7, 3, 2", "4, 5, 6, 7, 2, 3, 1", "4, 5, 7, 1, 2, 3, 6", "4, 5, 7, 1, 6, 2, 3", "4, 5, 7, 6, 2, 1, 3", "4, 6, 1, 3, 7, 2, 5", "4, 6, 2, 1, 7, 3, 5", "4, 6, 5, 1, 3, 7, 2", "4, 6, 5, 1, 7, 2, 3", "4, 6, 5, 7, 3, 2, 1", "4, 6, 7, 1, 2, 5, 3", "4, 6, 7, 1, 3, 2, 5", "4, 6, 7, 5, 1, 2, 3", "4, 7, 1, 3, 6, 5, 2", "4, 7, 2, 1, 6, 5, 3", "4, 7, 5, 1, 3, 2, 6", "4, 7, 5, 1, 6, 3, 2", "4, 7, 5, 6, 3, 1, 2", "4, 7, 6, 1, 2, 3, 5", "4, 7, 6, 1, 3, 5, 2", "4, 7, 6, 5, 1, 3, 2"]}, "cell": "perm_cycles/1", "stage": "C"}
+{"id": "po-perm_cycles-t2-eval-000035", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 2, 2, 1, 1, 1, with pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 2, 2, 1, 1, 1, with pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "4, 2, 3, 1, 5, 7, 6", "ref": "4, 2, 3, 1, 5, 7, 6", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.001984, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [2, 2, 1, 1, 1], "f1": 4, "f2": null}, "seed": 36965189202000321, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 7, "valid_answers": ["4, 2, 3, 1, 5, 7, 6", "4, 2, 3, 1, 6, 5, 7", "4, 2, 3, 1, 7, 6, 5", "4, 2, 5, 1, 3, 6, 7", "4, 2, 6, 1, 5, 3, 7", "4, 2, 7, 1, 5, 6, 3", "4, 3, 2, 1, 5, 6, 7", "4, 5, 3, 1, 2, 6, 7", "4, 6, 3, 1, 5, 2, 7", "4, 7, 3, 1, 5, 6, 2"]}, "cell": "perm_cycles/2", "stage": "C"}
+{"id": "po-perm_cycles-t2-eval-000002", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 2, 1, 1, with pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 2, 1, 1, with pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "3, 1, 2, 4, 5, 7, 6", "ref": "3, 1, 2, 4, 5, 7, 6", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 50, "bucket": null, "guess_rate": 0.009921, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [3, 2, 1, 1], "f1": 3, "f2": null}, "seed": 36965189202000013, "version": "po-v0", "tier": 2, "n_raw": 50, "small": 7, "valid_answers": ["3, 1, 2, 4, 5, 7, 6", "3, 1, 2, 4, 6, 5, 7", "3, 1, 2, 4, 7, 6, 5", "3, 1, 2, 5, 4, 6, 7", "3, 1, 2, 6, 5, 4, 7", "3, 1, 2, 7, 5, 6, 4", "3, 2, 1, 4, 6, 7, 5", "3, 2, 1, 4, 7, 5, 6", "3, 2, 1, 5, 6, 4, 7", "3, 2, 1, 5, 7, 6, 4", "3, 2, 1, 6, 4, 5, 7", "3, 2, 1, 6, 5, 7, 4", "3, 2, 1, 7, 4, 6, 5", "3, 2, 1, 7, 5, 4, 6", "3, 2, 4, 1, 5, 7, 6", "3, 2, 4, 1, 6, 5, 7", "3, 2, 4, 1, 7, 6, 5", "3, 2, 5, 4, 1, 7, 6", "3, 2, 5, 6, 1, 4, 7", "3, 2, 5, 7, 1, 6, 4", "3, 2, 6, 4, 7, 1, 5", "3, 2, 6, 5, 4, 1, 7", "3, 2, 6, 7, 5, 1, 4", "3, 2, 7, 4, 6, 5, 1", "3, 2, 7, 5, 4, 6, 1", "3, 2, 7, 6, 5, 4, 1", "3, 4, 1, 5, 2, 6, 7", "3, 4, 1, 6, 5, 2, 7", "3, 4, 1, 7, 5, 6, 2", "3, 4, 5, 2, 1, 6, 7", "3, 4, 6, 2, 5, 1, 7", "3, 4, 7, 2, 5, 6, 1", "3, 5, 1, 2, 4, 6, 7", "3, 5, 1, 4, 6, 2, 7", "3, 5, 1, 4, 7, 6, 2", "3, 5, 4, 1, 2, 6, 7", "3, 5, 6, 4, 2, 1, 7", "3, 5, 7, 4, 2, 6, 1", "3, 6, 1, 2, 5, 4, 7", "3, 6, 1, 4, 2, 5, 7", "3, 6, 1, 4, 5, 7, 2", "3, 6, 4, 1, 5, 2, 7", "3, 6, 5, 4, 1, 2, 7", "3, 6, 7, 4, 5, 2, 1", "3, 7, 1, 2, 5, 6, 4", "3, 7, 1, 4, 2, 6, 5", "3, 7, 1, 4, 5, 2, 6", "3, 7, 4, 1, 5, 6, 2", "3, 7, 5, 4, 1, 6, 2", "3, 7, 6, 4, 5, 1, 2"]}, "cell": "perm_cycles/2", "stage": "C"}
+{"id": "po-perm_cycles-t2-eval-000030", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 4, 1, 1, 1, with pi(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 4, 1, 1, 1, with pi(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "2, 3, 4, 1, 5, 6, 7", "ref": "2, 3, 4, 1, 5, 6, 7", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.003968, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [4, 1, 1, 1], "f1": 2, "f2": null}, "seed": 36965189202000221, "version": "po-v0", "tier": 2, "n_raw": 20, "small": 7, "valid_answers": ["2, 3, 4, 1, 5, 6, 7", "2, 3, 5, 4, 1, 6, 7", "2, 3, 6, 4, 5, 1, 7", "2, 3, 7, 4, 5, 6, 1", "2, 4, 1, 3, 5, 6, 7", "2, 4, 3, 5, 1, 6, 7", "2, 4, 3, 6, 5, 1, 7", "2, 4, 3, 7, 5, 6, 1", "2, 5, 1, 4, 3, 6, 7", "2, 5, 3, 1, 4, 6, 7", "2, 5, 3, 4, 6, 1, 7", "2, 5, 3, 4, 7, 6, 1", "2, 6, 1, 4, 5, 3, 7", "2, 6, 3, 1, 5, 4, 7", "2, 6, 3, 4, 1, 5, 7", "2, 6, 3, 4, 5, 7, 1", "2, 7, 1, 4, 5, 6, 3", "2, 7, 3, 1, 5, 6, 4", "2, 7, 3, 4, 1, 6, 5", "2, 7, 3, 4, 5, 1, 6"]}, "cell": "perm_cycles/2", "stage": "C"}
+{"id": "po-perm_cycles-t2-eval-000041", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 4, 1, 1, 1, with pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 4, 1, 1, 1, with pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "5, 1, 2, 4, 3, 6, 7", "ref": "5, 1, 2, 4, 3, 6, 7", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.003968, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [4, 1, 1, 1], "f1": 5, "f2": null}, "seed": 36965189202000906, "version": "po-v0", "tier": 2, "n_raw": 20, "small": 7, "valid_answers": ["5, 1, 2, 4, 3, 6, 7", "5, 1, 3, 2, 4, 6, 7", "5, 1, 3, 4, 6, 2, 7", "5, 1, 3, 4, 7, 6, 2", "5, 2, 1, 3, 4, 6, 7", "5, 2, 1, 4, 6, 3, 7", "5, 2, 1, 4, 7, 6, 3", "5, 2, 3, 1, 6, 4, 7", "5, 2, 3, 1, 7, 6, 4", "5, 2, 3, 4, 6, 7, 1", "5, 2, 3, 4, 7, 1, 6", "5, 2, 3, 6, 4, 1, 7", "5, 2, 3, 7, 4, 6, 1", "5, 2, 4, 1, 3, 6, 7", "5, 2, 6, 4, 3, 1, 7", "5, 2, 7, 4, 3, 6, 1", "5, 3, 1, 4, 2, 6, 7", "5, 4, 3, 1, 2, 6, 7", "5, 6, 3, 4, 2, 1, 7", "5, 7, 3, 4, 2, 6, 1"]}, "cell": "perm_cycles/2", "stage": "C"}
+{"id": "po-perm_cycles-t2-eval-000034", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 3, 1, with pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 3, 1, with pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 3, 4, 2, 6, 7, 5", "ref": "1, 3, 4, 2, 6, 7, 5", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 0.007937, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [3, 3, 1], "f1": 1, "f2": null}, "seed": 36965189202000305, "version": "po-v0", "tier": 2, "n_raw": 40, "small": 7, "valid_answers": ["1, 3, 4, 2, 6, 7, 5", "1, 3, 4, 2, 7, 5, 6", "1, 3, 5, 6, 2, 7, 4", "1, 3, 5, 7, 2, 4, 6", "1, 3, 6, 5, 7, 2, 4", "1, 3, 6, 7, 4, 2, 5", "1, 3, 7, 5, 6, 4, 2", "1, 3, 7, 6, 4, 5, 2", "1, 4, 2, 3, 6, 7, 5", "1, 4, 2, 3, 7, 5, 6", "1, 4, 5, 6, 7, 2, 3", "1, 4, 5, 7, 6, 3, 2", "1, 4, 6, 5, 2, 7, 3", "1, 4, 6, 7, 3, 5, 2", "1, 4, 7, 5, 2, 3, 6", "1, 4, 7, 6, 3, 2, 5", "1, 5, 2, 6, 3, 7, 4", "1, 5, 2, 7, 3, 4, 6", "1, 5, 4, 6, 7, 3, 2", "1, 5, 4, 7, 6, 2, 3", "1, 5, 6, 2, 4, 7, 3", "1, 5, 6, 3, 7, 4, 2", "1, 5, 7, 2, 4, 3, 6", "1, 5, 7, 3, 6, 2, 4", "1, 6, 2, 5, 7, 3, 4", "1, 6, 2, 7, 4, 3, 5", "1, 6, 4, 5, 3, 7, 2", "1, 6, 4, 7, 2, 5, 3", "1, 6, 5, 2, 7, 4, 3", "1, 6, 5, 3, 4, 7, 2", "1, 6, 7, 2, 3, 4, 5", "1, 6, 7, 3, 2, 5, 4", "1, 7, 2, 5, 6, 4, 3", "1, 7, 2, 6, 4, 5, 3", "1, 7, 4, 5, 3, 2, 6", "1, 7, 4, 6, 2, 3, 5", "1, 7, 5, 2, 6, 3, 4", "1, 7, 5, 3, 4, 2, 6", "1, 7, 6, 2, 3, 5, 4", "1, 7, 6, 3, 2, 4, 5"]}, "cell": "perm_cycles/2", "stage": "C"}
+{"id": "po-perm_cycles-t2-eval-000044", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 1, 1, 1, 1, with pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 1, 1, 1, 1, with pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "4, 1, 3, 2, 5, 6, 7", "ref": "4, 1, 3, 2, 5, 6, 7", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000992, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [3, 1, 1, 1, 1], "f1": 4, "f2": null}, "seed": 36965189202001092, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 7, "valid_answers": ["4, 1, 3, 2, 5, 6, 7", "4, 2, 1, 3, 5, 6, 7", "4, 2, 3, 5, 1, 6, 7", "4, 2, 3, 6, 5, 1, 7", "4, 2, 3, 7, 5, 6, 1"]}, "cell": "perm_cycles/2", "stage": "C"}
+{"id": "po-perm_cycles-t2-eval-000013", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 2, 2, with pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 2, 2, with pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "5, 1, 4, 3, 2, 7, 6", "ref": "5, 1, 4, 3, 2, 7, 6", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 35, "bucket": null, "guess_rate": 0.006944, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [3, 2, 2], "f1": 5, "f2": null}, "seed": 36965189202000042, "version": "po-v0", "tier": 2, "n_raw": 35, "small": 7, "valid_answers": ["5, 1, 4, 3, 2, 7, 6", "5, 1, 6, 7, 2, 3, 4", "5, 1, 7, 6, 2, 4, 3", "5, 3, 2, 1, 4, 7, 6", "5, 3, 2, 6, 1, 7, 4", "5, 3, 2, 6, 7, 4, 1", "5, 3, 2, 7, 1, 4, 6", "5, 3, 2, 7, 6, 1, 4", "5, 3, 4, 2, 1, 7, 6", "5, 3, 6, 7, 1, 2, 4", "5, 3, 7, 6, 1, 4, 2", "5, 4, 1, 2, 3, 7, 6", "5, 4, 2, 3, 1, 7, 6", "5, 4, 6, 2, 1, 7, 3", "5, 4, 6, 2, 7, 3, 1", "5, 4, 6, 7, 1, 3, 2", "5, 4, 7, 2, 1, 3, 6", "5, 4, 7, 2, 6, 1, 3", "5, 4, 7, 6, 1, 2, 3", "5, 6, 1, 7, 3, 2, 4", "5, 6, 2, 7, 1, 3, 4", "5, 6, 4, 3, 1, 7, 2", "5, 6, 4, 3, 7, 2, 1", "5, 6, 4, 7, 1, 2, 3", "5, 6, 7, 1, 4, 2, 3", "5, 6, 7, 2, 1, 4, 3", "5, 6, 7, 3, 1, 2, 4", "5, 7, 1, 6, 3, 4, 2", "5, 7, 2, 6, 1, 4, 3", "5, 7, 4, 3, 1, 2, 6", "5, 7, 4, 3, 6, 1, 2", "5, 7, 4, 6, 1, 3, 2", "5, 7, 6, 1, 4, 3, 2", "5, 7, 6, 2, 1, 3, 4", "5, 7, 6, 3, 1, 4, 2"]}, "cell": "perm_cycles/2", "stage": "C"}
+{"id": "po-perm_cycles-t2-eval-000004", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 2, 2, with pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose cycle lengths are 3, 2, 2, with pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "4, 1, 5, 2, 3, 7, 6", "ref": "4, 1, 5, 2, 3, 7, 6", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 35, "bucket": null, "guess_rate": 0.006944, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 7, "ct": [3, 2, 2], "f1": 4, "f2": null}, "seed": 36965189202000016, "version": "po-v0", "tier": 2, "n_raw": 35, "small": 7, "valid_answers": ["4, 1, 5, 2, 3, 7, 6", "4, 1, 6, 2, 7, 3, 5", "4, 1, 7, 2, 6, 5, 3", "4, 3, 2, 1, 6, 7, 5", "4, 3, 2, 1, 7, 5, 6", "4, 3, 2, 5, 1, 7, 6", "4, 3, 2, 6, 7, 1, 5", "4, 3, 2, 7, 6, 5, 1", "4, 3, 5, 1, 2, 7, 6", "4, 3, 6, 1, 7, 2, 5", "4, 3, 7, 1, 6, 5, 2", "4, 5, 1, 3, 2, 7, 6", "4, 5, 2, 1, 3, 7, 6", "4, 5, 6, 1, 2, 7, 3", "4, 5, 6, 1, 7, 3, 2", "4, 5, 6, 7, 2, 3, 1", "4, 5, 7, 1, 2, 3, 6", "4, 5, 7, 1, 6, 2, 3", "4, 5, 7, 6, 2, 1, 3", "4, 6, 1, 3, 7, 2, 5", "4, 6, 2, 1, 7, 3, 5", "4, 6, 5, 1, 3, 7, 2", "4, 6, 5, 1, 7, 2, 3", "4, 6, 5, 7, 3, 2, 1", "4, 6, 7, 1, 2, 5, 3", "4, 6, 7, 1, 3, 2, 5", "4, 6, 7, 5, 1, 2, 3", "4, 7, 1, 3, 6, 5, 2", "4, 7, 2, 1, 6, 5, 3", "4, 7, 5, 1, 3, 2, 6", "4, 7, 5, 1, 6, 3, 2", "4, 7, 5, 6, 3, 1, 2", "4, 7, 6, 1, 2, 3, 5", "4, 7, 6, 1, 3, 5, 2", "4, 7, 6, 5, 1, 3, 2"]}, "cell": "perm_cycles/2", "stage": "C"}
+{"id": "po-perm_cycles-t3-eval-000310", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose cycle lengths are 3, 2, 2, 1, with pi(1) = 2 and pi(2) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose cycle lengths are 3, 2, 2, 1, with pi(1) = 2 and pi(2) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "2, 8, 3, 5, 4, 7, 6, 1", "ref": "2, 8, 3, 5, 4, 7, 6, 1", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 0.000372, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 8, "ct": [3, 2, 2, 1], "f1": 2, "f2": 8}, "seed": 36965189203001116, "version": "po-v0", "tier": 3, "n_raw": 15, "small": 8, "valid_answers": ["2, 8, 3, 5, 4, 7, 6, 1", "2, 8, 3, 6, 7, 4, 5, 1", "2, 8, 3, 7, 6, 5, 4, 1", "2, 8, 4, 3, 5, 7, 6, 1", "2, 8, 4, 3, 6, 5, 7, 1", "2, 8, 4, 3, 7, 6, 5, 1", "2, 8, 5, 4, 3, 7, 6, 1", "2, 8, 5, 6, 3, 4, 7, 1", "2, 8, 5, 7, 3, 6, 4, 1", "2, 8, 6, 4, 7, 3, 5, 1", "2, 8, 6, 5, 4, 3, 7, 1", "2, 8, 6, 7, 5, 3, 4, 1", "2, 8, 7, 4, 6, 5, 3, 1", "2, 8, 7, 5, 4, 6, 3, 1", "2, 8, 7, 6, 5, 4, 3, 1"]}, "cell": "perm_cycles/3", "stage": "C"}
+{"id": "po-perm_cycles-t3-eval-000429", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose cycle lengths are 2, 2, 2, 2, with pi(1) = 4 and pi(2) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose cycle lengths are 2, 2, 2, 2, with pi(1) = 4 and pi(2) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "4, 8, 5, 1, 3, 7, 6, 2", "ref": "4, 8, 5, 1, 3, 7, 6, 2", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 7.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 8, "ct": [2, 2, 2, 2], "f1": 4, "f2": 8}, "seed": 36965189203002074, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 8, "valid_answers": ["4, 8, 5, 1, 3, 7, 6, 2", "4, 8, 6, 1, 7, 3, 5, 2", "4, 8, 7, 1, 6, 5, 3, 2"]}, "cell": "perm_cycles/3", "stage": "C"}
+{"id": "po-perm_cycles-t3-eval-000098", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose cycle lengths are 4, 3, 1, with pi(1) = 8 and pi(2) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose cycle lengths are 4, 3, 1, with pi(1) = 8 and pi(2) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "8, 7, 1, 2, 3, 6, 4, 5", "ref": "8, 7, 1, 2, 3, 6, 4, 5", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 56, "bucket": null, "guess_rate": 0.001389, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 8, "ct": [4, 3, 1], "f1": 8, "f2": 7}, "seed": 36965189203000213, "version": "po-v0", "tier": 3, "n_raw": 56, "small": 8, "valid_answers": ["8, 7, 1, 2, 3, 6, 4, 5", "8, 7, 1, 2, 4, 6, 5, 3", "8, 7, 1, 2, 5, 3, 4, 6", "8, 7, 1, 2, 5, 4, 6, 3", "8, 7, 1, 3, 2, 6, 5, 4", "8, 7, 1, 3, 5, 2, 6, 4", "8, 7, 1, 4, 2, 3, 5, 6", "8, 7, 1, 4, 2, 5, 6, 3", "8, 7, 1, 4, 3, 2, 6, 5", "8, 7, 1, 4, 6, 2, 5, 3", "8, 7, 1, 5, 2, 6, 4, 3", "8, 7, 1, 6, 5, 2, 4, 3", "8, 7, 2, 1, 3, 6, 5, 4", "8, 7, 2, 1, 4, 6, 3, 5", "8, 7, 2, 1, 5, 3, 6, 4", "8, 7, 2, 1, 5, 4, 3, 6", "8, 7, 2, 3, 1, 6, 4, 5", "8, 7, 2, 3, 5, 1, 4, 6", "8, 7, 2, 4, 1, 3, 6, 5", "8, 7, 2, 4, 1, 5, 3, 6", "8, 7, 2, 4, 3, 1, 5, 6", "8, 7, 2, 4, 6, 1, 3, 5", "8, 7, 2, 5, 1, 6, 3, 4", "8, 7, 2, 6, 5, 1, 3, 4", "8, 7, 3, 1, 2, 4, 5, 6", "8, 7, 3, 1, 2, 5, 6, 4", "8, 7, 3, 1, 4, 2, 6, 5", "8, 7, 3, 1, 6, 2, 5, 4", "8, 7, 3, 2, 1, 4, 6, 5", "8, 7, 3, 2, 1, 5, 4, 6", "8, 7, 3, 2, 4, 1, 5, 6", "8, 7, 3, 2, 6, 1, 4, 5", "8, 7, 3, 5, 1, 2, 6, 4", "8, 7, 3, 5, 2, 1, 4, 6", "8, 7, 3, 5, 6, 4, 1, 2", "8, 7, 3, 6, 1, 2, 4, 5", "8, 7, 3, 6, 2, 1, 5, 4", "8, 7, 3, 6, 4, 5, 1, 2", "8, 7, 4, 1, 2, 6, 5, 3", "8, 7, 4, 1, 5, 2, 6, 3", "8, 7, 4, 2, 1, 6, 3, 5", "8, 7, 4, 2, 5, 1, 3, 6", "8, 7, 4, 5, 3, 6, 1, 2", "8, 7, 4, 6, 5, 3, 1, 2", "8, 7, 5, 1, 2, 6, 3, 4", "8, 7, 5, 2, 1, 6, 4, 3", "8, 7, 5, 3, 4, 6, 1, 2", "8, 7, 5, 4, 1, 2, 6, 3", "8, 7, 5, 4, 2, 1, 3, 6", "8, 7, 5, 4, 6, 3, 1, 2", "8, 7, 6, 1, 5, 2, 3, 4", "8, 7, 6, 2, 5, 1, 4, 3", "8, 7, 6, 3, 5, 4, 1, 2", "8, 7, 6, 4, 1, 2, 3, 5", "8, 7, 6, 4, 2, 1, 5, 3", "8, 7, 6, 4, 3, 5, 1, 2"]}, "cell": "perm_cycles/3", "stage": "C"}
+{"id": "po-perm_cycles-t3-eval-000088", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose cycle lengths are 4, 2, 2, with pi(1) = 5 and pi(2) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose cycle lengths are 4, 2, 2, with pi(1) = 5 and pi(2) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "5, 3, 1, 6, 2, 4, 8, 7", "ref": "5, 3, 1, 6, 2, 4, 8, 7", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 33, "bucket": null, "guess_rate": 0.000818, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 8, "ct": [4, 2, 2], "f1": 5, "f2": 3}, "seed": 36965189203000172, "version": "po-v0", "tier": 3, "n_raw": 33, "small": 8, "valid_answers": ["5, 3, 1, 6, 2, 4, 8, 7", "5, 3, 1, 7, 2, 8, 4, 6", "5, 3, 1, 8, 2, 7, 6, 4", "5, 3, 2, 1, 6, 4, 8, 7", "5, 3, 2, 1, 7, 8, 4, 6", "5, 3, 2, 1, 8, 7, 6, 4", "5, 3, 2, 6, 1, 7, 8, 4", "5, 3, 2, 6, 1, 8, 4, 7", "5, 3, 2, 6, 4, 1, 8, 7", "5, 3, 2, 6, 7, 4, 8, 1", "5, 3, 2, 6, 8, 4, 1, 7", "5, 3, 2, 7, 1, 4, 8, 6", "5, 3, 2, 7, 1, 8, 6, 4", "5, 3, 2, 7, 4, 8, 1, 6", "5, 3, 2, 7, 6, 8, 4, 1", "5, 3, 2, 7, 8, 1, 4, 6", "5, 3, 2, 8, 1, 4, 6, 7", "5, 3, 2, 8, 1, 7, 4, 6", "5, 3, 2, 8, 4, 7, 6, 1", "5, 3, 2, 8, 6, 7, 1, 4", "5, 3, 2, 8, 7, 1, 6, 4", "5, 3, 4, 6, 1, 2, 8, 7", "5, 3, 4, 7, 1, 8, 2, 6", "5, 3, 4, 8, 1, 7, 6, 2", "5, 3, 6, 2, 1, 4, 8, 7", "5, 3, 6, 7, 1, 8, 4, 2", "5, 3, 6, 8, 1, 7, 2, 4", "5, 3, 7, 2, 1, 8, 4, 6", "5, 3, 7, 6, 1, 4, 8, 2", "5, 3, 7, 8, 1, 2, 6, 4", "5, 3, 8, 2, 1, 7, 6, 4", "5, 3, 8, 6, 1, 4, 2, 7", "5, 3, 8, 7, 1, 2, 4, 6"]}, "cell": "perm_cycles/3", "stage": "C"}
+{"id": "po-perm_cycles-t3-eval-000450", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose cycle lengths are 3, 2, 2, 1, with pi(1) = 6 and pi(2) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose cycle lengths are 3, 2, 2, 1, with pi(1) = 6 and pi(2) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "6, 8, 1, 4, 7, 3, 5, 2", "ref": "6, 8, 1, 4, 7, 3, 5, 2", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 32, "bucket": null, "guess_rate": 0.000794, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 8, "ct": [3, 2, 2, 1], "f1": 6, "f2": 8}, "seed": 36965189203002321, "version": "po-v0", "tier": 3, "n_raw": 32, "small": 8, "valid_answers": ["6, 8, 1, 4, 7, 3, 5, 2", "6, 8, 1, 5, 4, 3, 7, 2", "6, 8, 1, 7, 5, 3, 4, 2", "6, 8, 2, 4, 7, 1, 5, 3", "6, 8, 2, 5, 4, 1, 7, 3", "6, 8, 2, 7, 5, 1, 4, 3", "6, 8, 3, 1, 7, 4, 5, 2", "6, 8, 3, 2, 7, 1, 5, 4", "6, 8, 3, 5, 4, 1, 2, 7", "6, 8, 3, 5, 4, 7, 1, 2", "6, 8, 3, 5, 7, 1, 4, 2", "6, 8, 3, 7, 1, 5, 4, 2", "6, 8, 3, 7, 2, 1, 4, 5", "6, 8, 3, 7, 4, 1, 5, 2", "6, 8, 4, 3, 1, 5, 7, 2", "6, 8, 4, 3, 2, 1, 7, 5", "6, 8, 4, 3, 5, 1, 2, 7", "6, 8, 4, 3, 5, 7, 1, 2", "6, 8, 4, 5, 3, 1, 7, 2", "6, 8, 4, 7, 5, 1, 3, 2", "6, 8, 5, 1, 3, 4, 7, 2", "6, 8, 5, 2, 3, 1, 7, 4", "6, 8, 5, 3, 4, 1, 7, 2", "6, 8, 5, 4, 3, 1, 2, 7", "6, 8, 5, 4, 3, 7, 1, 2", "6, 8, 5, 4, 7, 1, 3, 2", "6, 8, 7, 1, 5, 4, 3, 2", "6, 8, 7, 2, 5, 1, 3, 4", "6, 8, 7, 3, 5, 1, 4, 2", "6, 8, 7, 4, 1, 5, 3, 2", "6, 8, 7, 4, 2, 1, 3, 5", "6, 8, 7, 4, 3, 1, 5, 2"]}, "cell": "perm_cycles/3", "stage": "C"}
+{"id": "po-perm_cycles-t3-eval-000543", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose cycle lengths are 4, 2, 2, with pi(1) = 4 and pi(2) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose cycle lengths are 4, 2, 2, with pi(1) = 4 and pi(2) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "4, 6, 1, 5, 3, 2, 8, 7", "ref": "4, 6, 1, 5, 3, 2, 8, 7", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 33, "bucket": null, "guess_rate": 0.000818, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 8, "ct": [4, 2, 2], "f1": 4, "f2": 6}, "seed": 36965189203004392, "version": "po-v0", "tier": 3, "n_raw": 33, "small": 8, "valid_answers": ["4, 6, 1, 5, 3, 2, 8, 7", "4, 6, 1, 7, 8, 2, 3, 5", "4, 6, 1, 8, 7, 2, 5, 3", "4, 6, 2, 1, 3, 5, 8, 7", "4, 6, 2, 1, 7, 8, 5, 3", "4, 6, 2, 1, 8, 7, 3, 5", "4, 6, 5, 1, 2, 3, 8, 7", "4, 6, 5, 1, 3, 7, 8, 2", "4, 6, 5, 1, 3, 8, 2, 7", "4, 6, 5, 1, 7, 2, 8, 3", "4, 6, 5, 1, 8, 2, 3, 7", "4, 6, 5, 2, 3, 1, 8, 7", "4, 6, 5, 3, 1, 2, 8, 7", "4, 6, 5, 7, 3, 2, 8, 1", "4, 6, 5, 8, 3, 2, 1, 7", "4, 6, 7, 1, 2, 8, 3, 5", "4, 6, 7, 1, 3, 2, 8, 5", "4, 6, 7, 1, 8, 2, 5, 3", "4, 6, 7, 1, 8, 3, 2, 5", "4, 6, 7, 1, 8, 5, 3, 2", "4, 6, 7, 2, 8, 1, 3, 5", "4, 6, 7, 3, 8, 2, 1, 5", "4, 6, 7, 5, 8, 2, 3, 1", "4, 6, 7, 8, 1, 2, 3, 5", "4, 6, 8, 1, 2, 7, 5, 3", "4, 6, 8, 1, 3, 2, 5, 7", "4, 6, 8, 1, 7, 2, 3, 5", "4, 6, 8, 1, 7, 3, 5, 2", "4, 6, 8, 1, 7, 5, 2, 3", "4, 6, 8, 2, 7, 1, 5, 3", "4, 6, 8, 3, 7, 2, 5, 1", "4, 6, 8, 5, 7, 2, 1, 3", "4, 6, 8, 7, 1, 2, 5, 3"]}, "cell": "perm_cycles/3", "stage": "C"}
+{"id": "po-perm_cycles-t3-eval-000186", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose cycle lengths are 3, 2, 2, 1, with pi(1) = 1 and pi(2) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose cycle lengths are 3, 2, 2, 1, with pi(1) = 1 and pi(2) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "1, 8, 2, 5, 4, 7, 6, 3", "ref": "1, 8, 2, 5, 4, 7, 6, 3", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 35, "bucket": null, "guess_rate": 0.000868, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 8, "ct": [3, 2, 2, 1], "f1": 1, "f2": 8}, "seed": 36965189203000494, "version": "po-v0", "tier": 3, "n_raw": 35, "small": 8, "valid_answers": ["1, 8, 2, 5, 4, 7, 6, 3", "1, 8, 2, 6, 7, 4, 5, 3", "1, 8, 2, 7, 6, 5, 4, 3", "1, 8, 4, 3, 2, 7, 6, 5", "1, 8, 4, 3, 6, 5, 2, 7", "1, 8, 4, 3, 6, 7, 5, 2", "1, 8, 4, 3, 7, 2, 5, 6", "1, 8, 4, 3, 7, 5, 6, 2", "1, 8, 4, 5, 3, 7, 6, 2", "1, 8, 4, 6, 7, 3, 5, 2", "1, 8, 4, 7, 6, 5, 3, 2", "1, 8, 5, 2, 3, 7, 6, 4", "1, 8, 5, 3, 4, 7, 6, 2", "1, 8, 5, 6, 3, 4, 2, 7", "1, 8, 5, 6, 3, 7, 4, 2", "1, 8, 5, 6, 7, 4, 3, 2", "1, 8, 5, 7, 3, 2, 4, 6", "1, 8, 5, 7, 3, 4, 6, 2", "1, 8, 5, 7, 6, 3, 4, 2", "1, 8, 6, 2, 7, 3, 5, 4", "1, 8, 6, 3, 7, 4, 5, 2", "1, 8, 6, 5, 4, 3, 2, 7", "1, 8, 6, 5, 4, 7, 3, 2", "1, 8, 6, 5, 7, 3, 4, 2", "1, 8, 6, 7, 2, 3, 4, 5", "1, 8, 6, 7, 3, 5, 4, 2", "1, 8, 6, 7, 4, 3, 5, 2", "1, 8, 7, 2, 6, 5, 3, 4", "1, 8, 7, 3, 6, 5, 4, 2", "1, 8, 7, 5, 4, 2, 3, 6", "1, 8, 7, 5, 4, 3, 6, 2", "1, 8, 7, 5, 6, 4, 3, 2", "1, 8, 7, 6, 2, 4, 3, 5", "1, 8, 7, 6, 3, 4, 5, 2", "1, 8, 7, 6, 4, 5, 3, 2"]}, "cell": "perm_cycles/3", "stage": "C"}
+{"id": "po-perm_cycles-t3-eval-000306", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose cycle lengths are 3, 2, 2, 1, with pi(1) = 3 and pi(2) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose cycle lengths are 3, 2, 2, 1, with pi(1) = 3 and pi(2) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "3, 2, 1, 5, 4, 7, 8, 6", "ref": "3, 2, 1, 5, 4, 7, 8, 6", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 35, "bucket": null, "guess_rate": 0.000868, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 8, "ct": [3, 2, 2, 1], "f1": 3, "f2": 2}, "seed": 36965189203001074, "version": "po-v0", "tier": 3, "n_raw": 35, "small": 8, "valid_answers": ["3, 2, 1, 5, 4, 7, 8, 6", "3, 2, 1, 5, 4, 8, 6, 7", "3, 2, 1, 5, 6, 4, 8, 7", "3, 2, 1, 5, 7, 8, 4, 6", "3, 2, 1, 5, 8, 7, 6, 4", "3, 2, 1, 6, 4, 5, 8, 7", "3, 2, 1, 6, 7, 4, 8, 5", "3, 2, 1, 6, 7, 8, 5, 4", "3, 2, 1, 6, 8, 4, 5, 7", "3, 2, 1, 6, 8, 7, 4, 5", "3, 2, 1, 7, 4, 8, 5, 6", "3, 2, 1, 7, 6, 5, 8, 4", "3, 2, 1, 7, 6, 8, 4, 5", "3, 2, 1, 7, 8, 4, 6, 5", "3, 2, 1, 7, 8, 5, 4, 6", "3, 2, 1, 8, 4, 7, 6, 5", "3, 2, 1, 8, 6, 5, 4, 7", "3, 2, 1, 8, 6, 7, 5, 4", "3, 2, 1, 8, 7, 4, 5, 6", "3, 2, 1, 8, 7, 5, 6, 4", "3, 2, 4, 1, 6, 5, 8, 7", "3, 2, 4, 1, 7, 8, 5, 6", "3, 2, 4, 1, 8, 7, 6, 5", "3, 2, 5, 6, 1, 4, 8, 7", "3, 2, 5, 7, 1, 8, 4, 6", "3, 2, 5, 8, 1, 7, 6, 4", "3, 2, 6, 5, 4, 1, 8, 7", "3, 2, 6, 7, 8, 1, 4, 5", "3, 2, 6, 8, 7, 1, 5, 4", "3, 2, 7, 5, 4, 8, 1, 6", "3, 2, 7, 6, 8, 4, 1, 5", "3, 2, 7, 8, 6, 5, 1, 4", "3, 2, 8, 5, 4, 7, 6, 1", "3, 2, 8, 6, 7, 4, 5, 1", "3, 2, 8, 7, 6, 5, 4, 1"]}, "cell": "perm_cycles/3", "stage": "C"}
+{"id": "po-perm_cycles-t3-eval-000596", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose cycle lengths are 2, 2, 2, 2, with pi(1) = 5 and pi(2) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose cycle lengths are 2, 2, 2, 2, with pi(1) = 5 and pi(2) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "5, 6, 4, 3, 1, 2, 8, 7", "ref": "5, 6, 4, 3, 1, 2, 8, 7", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 7.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 8, "ct": [2, 2, 2, 2], "f1": 5, "f2": 6}, "seed": 36965189203008292, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 8, "valid_answers": ["5, 6, 4, 3, 1, 2, 8, 7", "5, 6, 7, 8, 1, 2, 3, 4", "5, 6, 8, 7, 1, 2, 4, 3"]}, "cell": "perm_cycles/3", "stage": "C"}
+{"id": "po-perm_cycles-t3-eval-000564", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose cycle lengths are 2, 2, 2, 1, 1, with pi(1) = 4 and pi(2) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose cycle lengths are 2, 2, 2, 1, 1, with pi(1) = 4 and pi(2) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "4, 6, 3, 1, 5, 2, 8, 7", "ref": "4, 6, 3, 1, 5, 2, 8, 7", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000149, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 8, "ct": [2, 2, 2, 1, 1], "f1": 4, "f2": 6}, "seed": 36965189203005636, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 8, "valid_answers": ["4, 6, 3, 1, 5, 2, 8, 7", "4, 6, 3, 1, 7, 2, 5, 8", "4, 6, 3, 1, 8, 2, 7, 5", "4, 6, 5, 1, 3, 2, 7, 8", "4, 6, 7, 1, 5, 2, 3, 8", "4, 6, 8, 1, 5, 2, 7, 3"]}, "cell": "perm_cycles/3", "stage": "C"}
+{"id": "po-perm_cycles-t3-eval-000195", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose cycle lengths are 5, 3, with pi(1) = 8 and pi(2) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose cycle lengths are 5, 3, with pi(1) = 8 and pi(2) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "8, 1, 2, 3, 6, 7, 5, 4", "ref": "8, 1, 2, 3, 6, 7, 5, 4", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 64, "bucket": null, "guess_rate": 0.001587, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 8, "ct": [5, 3], "f1": 8, "f2": 1}, "seed": 36965189203000524, "version": "po-v0", "tier": 3, "n_raw": 64, "small": 8, "valid_answers": ["8, 1, 2, 3, 6, 7, 5, 4", "8, 1, 2, 3, 7, 5, 6, 4", "8, 1, 2, 5, 6, 4, 3, 7", "8, 1, 2, 5, 7, 3, 4, 6", "8, 1, 2, 6, 3, 7, 4, 5", "8, 1, 2, 6, 4, 5, 3, 7", "8, 1, 2, 7, 3, 4, 6, 5", "8, 1, 2, 7, 4, 3, 5, 6", "8, 1, 4, 2, 6, 7, 5, 3", "8, 1, 4, 2, 7, 5, 6, 3", "8, 1, 4, 5, 3, 2, 6, 7", "8, 1, 4, 5, 3, 7, 2, 6", "8, 1, 4, 5, 6, 7, 3, 2", "8, 1, 4, 5, 7, 3, 6, 2", "8, 1, 4, 6, 2, 3, 5, 7", "8, 1, 4, 6, 3, 7, 5, 2", "8, 1, 4, 6, 7, 3, 2, 5", "8, 1, 4, 6, 7, 5, 3, 2", "8, 1, 4, 7, 2, 5, 3, 6", "8, 1, 4, 7, 3, 5, 6, 2", "8, 1, 4, 7, 6, 2, 3, 5", "8, 1, 4, 7, 6, 3, 5, 2", "8, 1, 5, 2, 6, 3, 4, 7", "8, 1, 5, 2, 7, 4, 3, 6", "8, 1, 5, 3, 4, 2, 6, 7", "8, 1, 5, 3, 4, 7, 2, 6", "8, 1, 5, 3, 6, 7, 4, 2", "8, 1, 5, 3, 7, 4, 6, 2", "8, 1, 5, 6, 2, 7, 4, 3", "8, 1, 5, 6, 4, 7, 3, 2", "8, 1, 5, 6, 7, 2, 3, 4", "8, 1, 5, 6, 7, 3, 4, 2", "8, 1, 5, 7, 2, 4, 6, 3", "8, 1, 5, 7, 4, 3, 6, 2", "8, 1, 5, 7, 6, 3, 2, 4", "8, 1, 5, 7, 6, 4, 3, 2", "8, 1, 6, 2, 3, 5, 4, 7", "8, 1, 6, 2, 4, 7, 3, 5", "8, 1, 6, 3, 2, 4, 5, 7", "8, 1, 6, 3, 4, 7, 5, 2", "8, 1, 6, 3, 7, 4, 2, 5", "8, 1, 6, 3, 7, 5, 4, 2", "8, 1, 6, 5, 2, 7, 3, 4", "8, 1, 6, 5, 3, 7, 4, 2", "8, 1, 6, 5, 7, 2, 4, 3", "8, 1, 6, 5, 7, 4, 3, 2", "8, 1, 6, 7, 3, 4, 5, 2", "8, 1, 6, 7, 3, 5, 2, 4", "8, 1, 6, 7, 4, 2, 5, 3", "8, 1, 6, 7, 4, 5, 3, 2", "8, 1, 7, 2, 3, 4, 5, 6", "8, 1, 7, 2, 4, 3, 6, 5", "8, 1, 7, 3, 2, 5, 4, 6", "8, 1, 7, 3, 4, 5, 6, 2", "8, 1, 7, 3, 6, 2, 4, 5", "8, 1, 7, 3, 6, 4, 5, 2", "8, 1, 7, 5, 2, 3, 6, 4", "8, 1, 7, 5, 3, 4, 6, 2", "8, 1, 7, 5, 6, 3, 4, 2", "8, 1, 7, 5, 6, 4, 2, 3", "8, 1, 7, 6, 3, 2, 5, 4", "8, 1, 7, 6, 3, 5, 4, 2", "8, 1, 7, 6, 4, 3, 5, 2", "8, 1, 7, 6, 4, 5, 2, 3"]}, "cell": "perm_cycles/3", "stage": "C"}
+{"id": "po-perm_cycles-t3-eval-000476", "task": "cx_po_perm_cycles", "source": "cx_po_perm_cycles", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose cycle lengths are 3, 3, 1, 1, with pi(1) = 1 and pi(2) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose cycle lengths are 3, 3, 1, 1, with pi(1) = 1 and pi(2) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "1, 4, 2, 3, 5, 7, 8, 6", "ref": "1, 4, 2, 3, 5, 7, 8, 6", "family": "po_perm_cycles", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 0.000992, "solvable": true, "holdout": false, "meta": {"family": "po_perm_cycles", "rung": "witness", "params": {"n": 8, "ct": [3, 3, 1, 1], "f1": 1, "f2": 4}, "seed": 36965189203002791, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 8, "valid_answers": ["1, 4, 2, 3, 5, 7, 8, 6", "1, 4, 2, 3, 5, 8, 6, 7", "1, 4, 2, 3, 6, 7, 5, 8", "1, 4, 2, 3, 6, 8, 7, 5", "1, 4, 2, 3, 7, 5, 6, 8", "1, 4, 2, 3, 7, 6, 8, 5", "1, 4, 2, 3, 8, 5, 7, 6", "1, 4, 2, 3, 8, 6, 5, 7", "1, 4, 3, 5, 2, 7, 8, 6", "1, 4, 3, 5, 2, 8, 6, 7", "1, 4, 3, 6, 7, 2, 8, 5", "1, 4, 3, 6, 8, 2, 5, 7", "1, 4, 3, 7, 6, 8, 2, 5", "1, 4, 3, 7, 8, 5, 2, 6", "1, 4, 3, 8, 6, 7, 5, 2", "1, 4, 3, 8, 7, 5, 6, 2", "1, 4, 5, 6, 7, 2, 3, 8", "1, 4, 5, 6, 8, 2, 7, 3", "1, 4, 5, 7, 6, 3, 2, 8", "1, 4, 5, 7, 8, 6, 2, 3", "1, 4, 5, 8, 6, 3, 7, 2", "1, 4, 5, 8, 7, 6, 3, 2", "1, 4, 6, 5, 2, 7, 3, 8", "1, 4, 6, 5, 2, 8, 7, 3", "1, 4, 6, 7, 3, 5, 2, 8", "1, 4, 6, 7, 5, 8, 2, 3", "1, 4, 6, 8, 3, 5, 7, 2", "1, 4, 6, 8, 5, 7, 3, 2", "1, 4, 7, 5, 2, 3, 6, 8", "1, 4, 7, 5, 2, 6, 8, 3", "1, 4, 7, 6, 3, 2, 5, 8", "1, 4, 7, 6, 5, 2, 8, 3", "1, 4, 7, 8, 3, 6, 5, 2", "1, 4, 7, 8, 5, 3, 6, 2", "1, 4, 8, 5, 2, 3, 7, 6", "1, 4, 8, 5, 2, 6, 3, 7", "1, 4, 8, 6, 3, 2, 7, 5", "1, 4, 8, 6, 5, 2, 3, 7", "1, 4, 8, 7, 3, 6, 2, 5", "1, 4, 8, 7, 5, 3, 2, 6"]}, "cell": "perm_cycles/3", "stage": "C"}
+{"id": "po-perm_descents-t0-eval-000059", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 4} and with pi(6) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 4} and with pi(6) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "5, 4, 1, 6, 2, 3", "ref": "5, 4, 1, 6, 2, 3", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.008333, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 6, "D": [1, 2, 4], "last": 3}, "seed": 167197441500000125, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 6, "valid_answers": ["5, 4, 1, 6, 2, 3", "5, 4, 2, 6, 1, 3", "6, 4, 1, 5, 2, 3", "6, 4, 2, 5, 1, 3", "6, 5, 1, 4, 2, 3", "6, 5, 2, 4, 1, 3"]}, "cell": "perm_descents/0", "stage": "C"}
+{"id": "po-perm_descents-t0-eval-000050", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 5} and with pi(6) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 5} and with pi(6) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "2, 1, 3, 5, 6, 4", "ref": "2, 1, 3, 5, 6, 4", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005556, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 6, "D": [1, 5], "last": 4}, "seed": 167197441500000102, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 6, "valid_answers": ["2, 1, 3, 5, 6, 4", "3, 1, 2, 5, 6, 4", "5, 1, 2, 3, 6, 4", "6, 1, 2, 3, 5, 4"]}, "cell": "perm_descents/0", "stage": "C"}
+{"id": "po-perm_descents-t0-eval-000089", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 3, 5} and with pi(6) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 3, 5} and with pi(6) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "5, 3, 2, 1, 6, 4", "ref": "5, 3, 2, 1, 6, 4", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002778, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 6, "D": [1, 2, 3, 5], "last": 4}, "seed": 167197441500000289, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 6, "valid_answers": ["5, 3, 2, 1, 6, 4", "6, 3, 2, 1, 5, 4"]}, "cell": "perm_descents/0", "stage": "C"}
+{"id": "po-perm_descents-t0-eval-000060", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 5} and with pi(6) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 5} and with pi(6) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "2, 1, 4, 5, 6, 3", "ref": "2, 1, 4, 5, 6, 3", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005556, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 6, "D": [1, 5], "last": 3}, "seed": 167197441500000127, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 6, "valid_answers": ["2, 1, 4, 5, 6, 3", "4, 1, 2, 5, 6, 3", "5, 1, 2, 4, 6, 3", "6, 1, 2, 4, 5, 3"]}, "cell": "perm_descents/0", "stage": "C"}
+{"id": "po-perm_descents-t0-eval-000066", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 3, 5} and with pi(6) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 3, 5} and with pi(6) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "5, 4, 3, 1, 6, 2", "ref": "5, 4, 3, 1, 6, 2", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005556, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 6, "D": [1, 2, 3, 5], "last": 2}, "seed": 167197441500000153, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 6, "valid_answers": ["5, 4, 3, 1, 6, 2", "6, 4, 3, 1, 5, 2", "6, 5, 3, 1, 4, 2", "6, 5, 4, 1, 3, 2"]}, "cell": "perm_descents/0", "stage": "C"}
+{"id": "po-perm_descents-t0-eval-000065", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {4} and with pi(6) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {4} and with pi(6) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "1, 2, 3, 5, 4, 6", "ref": "1, 2, 3, 5, 4, 6", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005556, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 6, "D": [4], "last": 6}, "seed": 167197441500000152, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 6, "valid_answers": ["1, 2, 3, 5, 4, 6", "1, 2, 4, 5, 3, 6", "1, 3, 4, 5, 2, 6", "2, 3, 4, 5, 1, 6"]}, "cell": "perm_descents/0", "stage": "C"}
+{"id": "po-perm_descents-t0-eval-000004", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {2} and with pi(6) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {2} and with pi(6) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "1, 3, 2, 4, 5, 6", "ref": "1, 3, 2, 4, 5, 6", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.0125, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 6, "D": [2], "last": 6}, "seed": 167197441500000008, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 6, "valid_answers": ["1, 3, 2, 4, 5, 6", "1, 4, 2, 3, 5, 6", "1, 5, 2, 3, 4, 6", "2, 3, 1, 4, 5, 6", "2, 4, 1, 3, 5, 6", "2, 5, 1, 3, 4, 6", "3, 4, 1, 2, 5, 6", "3, 5, 1, 2, 4, 6", "4, 5, 1, 2, 3, 6"]}, "cell": "perm_descents/0", "stage": "C"}
+{"id": "po-perm_descents-t0-eval-000088", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {2} and with pi(6) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {2} and with pi(6) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "1, 6, 2, 3, 4, 5", "ref": "1, 6, 2, 3, 4, 5", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005556, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 6, "D": [2], "last": 5}, "seed": 167197441500000286, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 6, "valid_answers": ["1, 6, 2, 3, 4, 5", "2, 6, 1, 3, 4, 5", "3, 6, 1, 2, 4, 5", "4, 6, 1, 2, 3, 5"]}, "cell": "perm_descents/0", "stage": "C"}
+{"id": "po-perm_descents-t0-eval-000057", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3, 5} and with pi(6) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3, 5} and with pi(6) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "1, 5, 4, 3, 6, 2", "ref": "1, 5, 4, 3, 6, 2", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.015278, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 6, "D": [2, 3, 5], "last": 2}, "seed": 167197441500000119, "version": "po-v0", "tier": 0, "n_raw": 11, "small": 6, "valid_answers": ["1, 5, 4, 3, 6, 2", "1, 6, 4, 3, 5, 2", "1, 6, 5, 3, 4, 2", "3, 5, 4, 1, 6, 2", "3, 6, 4, 1, 5, 2", "3, 6, 5, 1, 4, 2", "4, 5, 3, 1, 6, 2", "4, 6, 3, 1, 5, 2", "4, 6, 5, 1, 3, 2", "5, 6, 3, 1, 4, 2", "5, 6, 4, 1, 3, 2"]}, "cell": "perm_descents/0", "stage": "C"}
+{"id": "po-perm_descents-t0-eval-000075", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3, 4} and with pi(6) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3, 4} and with pi(6) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "1, 6, 4, 3, 2, 5", "ref": "1, 6, 4, 3, 2, 5", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005556, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 6, "D": [2, 3, 4], "last": 5}, "seed": 167197441500000174, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 6, "valid_answers": ["1, 6, 4, 3, 2, 5", "2, 6, 4, 3, 1, 5", "3, 6, 4, 2, 1, 5", "4, 6, 3, 2, 1, 5"]}, "cell": "perm_descents/0", "stage": "C"}
+{"id": "po-perm_descents-t0-eval-000068", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 5} and with pi(6) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 5} and with pi(6) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "2, 1, 3, 4, 6, 5", "ref": "2, 1, 3, 4, 6, 5", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.004167, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 6, "D": [1, 5], "last": 5}, "seed": 167197441500000156, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 6, "valid_answers": ["2, 1, 3, 4, 6, 5", "3, 1, 2, 4, 6, 5", "4, 1, 2, 3, 6, 5"]}, "cell": "perm_descents/0", "stage": "C"}
+{"id": "po-perm_descents-t0-eval-000086", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 3, 4} and with pi(6) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 3, 4} and with pi(6) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "4, 3, 6, 5, 1, 2", "ref": "4, 3, 6, 5, 1, 2", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.006944, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 6, "D": [1, 3, 4], "last": 2}, "seed": 167197441500000256, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 6, "valid_answers": ["4, 3, 6, 5, 1, 2", "5, 3, 6, 4, 1, 2", "5, 4, 6, 3, 1, 2", "6, 3, 5, 4, 1, 2", "6, 4, 5, 3, 1, 2"]}, "cell": "perm_descents/0", "stage": "C"}
+{"id": "po-perm_descents-t1-eval-000291", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 5} and with pi(7) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 5} and with pi(7) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "5, 4, 3, 6, 7, 1, 2", "ref": "5, 4, 3, 6, 7, 1, 2", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.00119, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [1, 2, 5], "last": 2}, "seed": 167197441501005219, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 7, "valid_answers": ["5, 4, 3, 6, 7, 1, 2", "6, 4, 3, 5, 7, 1, 2", "6, 5, 3, 4, 7, 1, 2", "7, 4, 3, 5, 6, 1, 2", "7, 5, 3, 4, 6, 1, 2", "7, 6, 3, 4, 5, 1, 2"]}, "cell": "perm_descents/1", "stage": "C"}
+{"id": "po-perm_descents-t1-eval-000175", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 5, 6} and with pi(7) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 5, 6} and with pi(7) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 3, 2, 5, 7, 6, 4", "ref": "1, 3, 2, 5, 7, 6, 4", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.002579, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [2, 5, 6], "last": 4}, "seed": 167197441501000402, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 7, "valid_answers": ["1, 3, 2, 5, 7, 6, 4", "1, 3, 2, 6, 7, 5, 4", "1, 5, 2, 3, 7, 6, 4", "1, 6, 2, 3, 7, 5, 4", "1, 7, 2, 3, 6, 5, 4", "2, 3, 1, 5, 7, 6, 4", "2, 3, 1, 6, 7, 5, 4", "2, 5, 1, 3, 7, 6, 4", "2, 6, 1, 3, 7, 5, 4", "2, 7, 1, 3, 6, 5, 4", "3, 5, 1, 2, 7, 6, 4", "3, 6, 1, 2, 7, 5, 4", "3, 7, 1, 2, 6, 5, 4"]}, "cell": "perm_descents/1", "stage": "C"}
+{"id": "po-perm_descents-t1-eval-000027", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {5, 6} and with pi(7) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {5, 6} and with pi(7) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 2, 4, 5, 7, 6, 3", "ref": "1, 2, 4, 5, 7, 6, 3", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000595, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [5, 6], "last": 3}, "seed": 167197441501000032, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 7, "valid_answers": ["1, 2, 4, 5, 7, 6, 3", "1, 2, 4, 6, 7, 5, 3", "1, 2, 5, 6, 7, 4, 3"]}, "cell": "perm_descents/1", "stage": "C"}
+{"id": "po-perm_descents-t1-eval-000229", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 3} and with pi(7) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 3} and with pi(7) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "7, 3, 2, 1, 4, 5, 6", "ref": "7, 3, 2, 1, 4, 5, 6", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.00119, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [1, 2, 3], "last": 6}, "seed": 167197441501000841, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 7, "valid_answers": ["7, 3, 2, 1, 4, 5, 6", "7, 4, 2, 1, 3, 5, 6", "7, 4, 3, 1, 2, 5, 6", "7, 5, 2, 1, 3, 4, 6", "7, 5, 3, 1, 2, 4, 6", "7, 5, 4, 1, 2, 3, 6"]}, "cell": "perm_descents/1", "stage": "C"}
+{"id": "po-perm_descents-t1-eval-000050", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 4} and with pi(7) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 4} and with pi(7) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "6, 2, 1, 7, 3, 4, 5", "ref": "6, 2, 1, 7, 3, 4, 5", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 17, "bucket": "9-20", "guess_rate": 0.003373, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [1, 2, 4], "last": 5}, "seed": 167197441501000064, "version": "po-v0", "tier": 1, "n_raw": 17, "small": 7, "valid_answers": ["6, 2, 1, 7, 3, 4, 5", "6, 3, 1, 7, 2, 4, 5", "6, 3, 2, 7, 1, 4, 5", "6, 4, 1, 7, 2, 3, 5", "6, 4, 2, 7, 1, 3, 5", "6, 4, 3, 7, 1, 2, 5", "7, 2, 1, 6, 3, 4, 5", "7, 3, 1, 6, 2, 4, 5", "7, 3, 2, 6, 1, 4, 5", "7, 4, 1, 6, 2, 3, 5", "7, 4, 2, 6, 1, 3, 5", "7, 4, 3, 6, 1, 2, 5", "7, 6, 1, 3, 2, 4, 5", "7, 6, 1, 4, 2, 3, 5", "7, 6, 2, 3, 1, 4, 5", "7, 6, 2, 4, 1, 3, 5", "7, 6, 3, 4, 1, 2, 5"]}, "cell": "perm_descents/1", "stage": "C"}
+{"id": "po-perm_descents-t1-eval-000162", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3, 5} and with pi(7) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3, 5} and with pi(7) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 4, 3, 2, 6, 5, 7", "ref": "1, 4, 3, 2, 6, 5, 7", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 0.007937, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [2, 3, 5], "last": 7}, "seed": 167197441501000342, "version": "po-v0", "tier": 1, "n_raw": 40, "small": 7, "valid_answers": ["1, 4, 3, 2, 6, 5, 7", "1, 5, 3, 2, 6, 4, 7", "1, 5, 4, 2, 6, 3, 7", "1, 5, 4, 3, 6, 2, 7", "1, 6, 3, 2, 5, 4, 7", "1, 6, 4, 2, 5, 3, 7", "1, 6, 4, 3, 5, 2, 7", "1, 6, 5, 2, 4, 3, 7", "1, 6, 5, 3, 4, 2, 7", "2, 4, 3, 1, 6, 5, 7", "2, 5, 3, 1, 6, 4, 7", "2, 5, 4, 1, 6, 3, 7", "2, 5, 4, 3, 6, 1, 7", "2, 6, 3, 1, 5, 4, 7", "2, 6, 4, 1, 5, 3, 7", "2, 6, 4, 3, 5, 1, 7", "2, 6, 5, 1, 4, 3, 7", "2, 6, 5, 3, 4, 1, 7", "3, 4, 2, 1, 6, 5, 7", "3, 5, 2, 1, 6, 4, 7", "3, 5, 4, 1, 6, 2, 7", "3, 5, 4, 2, 6, 1, 7", "3, 6, 2, 1, 5, 4, 7", "3, 6, 4, 1, 5, 2, 7", "3, 6, 4, 2, 5, 1, 7", "3, 6, 5, 1, 4, 2, 7", "3, 6, 5, 2, 4, 1, 7", "4, 5, 2, 1, 6, 3, 7", "4, 5, 3, 1, 6, 2, 7", "4, 5, 3, 2, 6, 1, 7", "4, 6, 2, 1, 5, 3, 7", "4, 6, 3, 1, 5, 2, 7", "4, 6, 3, 2, 5, 1, 7", "4, 6, 5, 1, 3, 2, 7", "4, 6, 5, 2, 3, 1, 7", "5, 6, 2, 1, 4, 3, 7", "5, 6, 3, 1, 4, 2, 7", "5, 6, 3, 2, 4, 1, 7", "5, 6, 4, 1, 3, 2, 7", "5, 6, 4, 2, 3, 1, 7"]}, "cell": "perm_descents/1", "stage": "C"}
+{"id": "po-perm_descents-t1-eval-000144", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3, 5, 6} and with pi(7) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3, 5, 6} and with pi(7) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "2, 5, 4, 3, 7, 6, 1", "ref": "2, 5, 4, 3, 7, 6, 1", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 0.007937, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [2, 3, 5, 6], "last": 1}, "seed": 167197441501000286, "version": "po-v0", "tier": 1, "n_raw": 40, "small": 7, "valid_answers": ["2, 5, 4, 3, 7, 6, 1", "2, 6, 4, 3, 7, 5, 1", "2, 6, 5, 3, 7, 4, 1", "2, 6, 5, 4, 7, 3, 1", "2, 7, 4, 3, 6, 5, 1", "2, 7, 5, 3, 6, 4, 1", "2, 7, 5, 4, 6, 3, 1", "2, 7, 6, 3, 5, 4, 1", "2, 7, 6, 4, 5, 3, 1", "3, 5, 4, 2, 7, 6, 1", "3, 6, 4, 2, 7, 5, 1", "3, 6, 5, 2, 7, 4, 1", "3, 6, 5, 4, 7, 2, 1", "3, 7, 4, 2, 6, 5, 1", "3, 7, 5, 2, 6, 4, 1", "3, 7, 5, 4, 6, 2, 1", "3, 7, 6, 2, 5, 4, 1", "3, 7, 6, 4, 5, 2, 1", "4, 5, 3, 2, 7, 6, 1", "4, 6, 3, 2, 7, 5, 1", "4, 6, 5, 2, 7, 3, 1", "4, 6, 5, 3, 7, 2, 1", "4, 7, 3, 2, 6, 5, 1", "4, 7, 5, 2, 6, 3, 1", "4, 7, 5, 3, 6, 2, 1", "4, 7, 6, 2, 5, 3, 1", "4, 7, 6, 3, 5, 2, 1", "5, 6, 3, 2, 7, 4, 1", "5, 6, 4, 2, 7, 3, 1", "5, 6, 4, 3, 7, 2, 1", "5, 7, 3, 2, 6, 4, 1", "5, 7, 4, 2, 6, 3, 1", "5, 7, 4, 3, 6, 2, 1", "5, 7, 6, 2, 4, 3, 1", "5, 7, 6, 3, 4, 2, 1", "6, 7, 3, 2, 5, 4, 1", "6, 7, 4, 2, 5, 3, 1", "6, 7, 4, 3, 5, 2, 1", "6, 7, 5, 2, 4, 3, 1", "6, 7, 5, 3, 4, 2, 1"]}, "cell": "perm_descents/1", "stage": "C"}
+{"id": "po-perm_descents-t1-eval-000065", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 6} and with pi(7) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 6} and with pi(7) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "4, 3, 1, 5, 6, 7, 2", "ref": "4, 3, 1, 5, 6, 7, 2", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.001984, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [1, 2, 6], "last": 2}, "seed": 167197441501000087, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 7, "valid_answers": ["4, 3, 1, 5, 6, 7, 2", "5, 3, 1, 4, 6, 7, 2", "5, 4, 1, 3, 6, 7, 2", "6, 3, 1, 4, 5, 7, 2", "6, 4, 1, 3, 5, 7, 2", "6, 5, 1, 3, 4, 7, 2", "7, 3, 1, 4, 5, 6, 2", "7, 4, 1, 3, 5, 6, 2", "7, 5, 1, 3, 4, 6, 2", "7, 6, 1, 3, 4, 5, 2"]}, "cell": "perm_descents/1", "stage": "C"}
+{"id": "po-perm_descents-t1-eval-000010", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2} and with pi(7) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2} and with pi(7) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "7, 2, 1, 3, 4, 5, 6", "ref": "7, 2, 1, 3, 4, 5, 6", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000794, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [1, 2], "last": 6}, "seed": 167197441501000011, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 7, "valid_answers": ["7, 2, 1, 3, 4, 5, 6", "7, 3, 1, 2, 4, 5, 6", "7, 4, 1, 2, 3, 5, 6", "7, 5, 1, 2, 3, 4, 6"]}, "cell": "perm_descents/1", "stage": "C"}
+{"id": "po-perm_descents-t1-eval-000201", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3} and with pi(7) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3} and with pi(7) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 4, 3, 2, 5, 6, 7", "ref": "1, 4, 3, 2, 5, 6, 7", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 26, "bucket": null, "guess_rate": 0.005159, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [2, 3], "last": 7}, "seed": 167197441501000531, "version": "po-v0", "tier": 1, "n_raw": 26, "small": 7, "valid_answers": ["1, 4, 3, 2, 5, 6, 7", "1, 5, 3, 2, 4, 6, 7", "1, 5, 4, 2, 3, 6, 7", "1, 6, 3, 2, 4, 5, 7", "1, 6, 4, 2, 3, 5, 7", "1, 6, 5, 2, 3, 4, 7", "2, 4, 3, 1, 5, 6, 7", "2, 5, 3, 1, 4, 6, 7", "2, 5, 4, 1, 3, 6, 7", "2, 6, 3, 1, 4, 5, 7", "2, 6, 4, 1, 3, 5, 7", "2, 6, 5, 1, 3, 4, 7", "3, 4, 2, 1, 5, 6, 7", "3, 5, 2, 1, 4, 6, 7", "3, 5, 4, 1, 2, 6, 7", "3, 6, 2, 1, 4, 5, 7", "3, 6, 4, 1, 2, 5, 7", "3, 6, 5, 1, 2, 4, 7", "4, 5, 2, 1, 3, 6, 7", "4, 5, 3, 1, 2, 6, 7", "4, 6, 2, 1, 3, 5, 7", "4, 6, 3, 1, 2, 5, 7", "4, 6, 5, 1, 2, 3, 7", "5, 6, 2, 1, 3, 4, 7", "5, 6, 3, 1, 2, 4, 7", "5, 6, 4, 1, 2, 3, 7"]}, "cell": "perm_descents/1", "stage": "C"}
+{"id": "po-perm_descents-t1-eval-000052", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 6} and with pi(7) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 6} and with pi(7) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 4, 2, 5, 6, 7, 3", "ref": "1, 4, 2, 5, 6, 7, 3", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.002778, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [2, 6], "last": 3}, "seed": 167197441501000067, "version": "po-v0", "tier": 1, "n_raw": 14, "small": 7, "valid_answers": ["1, 4, 2, 5, 6, 7, 3", "1, 5, 2, 4, 6, 7, 3", "1, 6, 2, 4, 5, 7, 3", "1, 7, 2, 4, 5, 6, 3", "2, 4, 1, 5, 6, 7, 3", "2, 5, 1, 4, 6, 7, 3", "2, 6, 1, 4, 5, 7, 3", "2, 7, 1, 4, 5, 6, 3", "4, 5, 1, 2, 6, 7, 3", "4, 6, 1, 2, 5, 7, 3", "4, 7, 1, 2, 5, 6, 3", "5, 6, 1, 2, 4, 7, 3", "5, 7, 1, 2, 4, 6, 3", "6, 7, 1, 2, 4, 5, 3"]}, "cell": "perm_descents/1", "stage": "C"}
+{"id": "po-perm_descents-t1-eval-000220", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 5, 6} and with pi(7) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 5, 6} and with pi(7) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "2, 1, 3, 5, 7, 6, 4", "ref": "2, 1, 3, 5, 7, 6, 4", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.001389, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [1, 5, 6], "last": 4}, "seed": 167197441501000728, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 7, "valid_answers": ["2, 1, 3, 5, 7, 6, 4", "2, 1, 3, 6, 7, 5, 4", "3, 1, 2, 5, 7, 6, 4", "3, 1, 2, 6, 7, 5, 4", "5, 1, 2, 3, 7, 6, 4", "6, 1, 2, 3, 7, 5, 4", "7, 1, 2, 3, 6, 5, 4"]}, "cell": "perm_descents/1", "stage": "C"}
+{"id": "po-perm_descents-t2-eval-000069", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3, 5, 6} and with pi(7) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3, 5, 6} and with pi(7) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "2, 5, 4, 3, 7, 6, 1", "ref": "2, 5, 4, 3, 7, 6, 1", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 0.007937, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [2, 3, 5, 6], "last": 1}, "seed": 167197441502000082, "version": "po-v0", "tier": 2, "n_raw": 40, "small": 7, "valid_answers": ["2, 5, 4, 3, 7, 6, 1", "2, 6, 4, 3, 7, 5, 1", "2, 6, 5, 3, 7, 4, 1", "2, 6, 5, 4, 7, 3, 1", "2, 7, 4, 3, 6, 5, 1", "2, 7, 5, 3, 6, 4, 1", "2, 7, 5, 4, 6, 3, 1", "2, 7, 6, 3, 5, 4, 1", "2, 7, 6, 4, 5, 3, 1", "3, 5, 4, 2, 7, 6, 1", "3, 6, 4, 2, 7, 5, 1", "3, 6, 5, 2, 7, 4, 1", "3, 6, 5, 4, 7, 2, 1", "3, 7, 4, 2, 6, 5, 1", "3, 7, 5, 2, 6, 4, 1", "3, 7, 5, 4, 6, 2, 1", "3, 7, 6, 2, 5, 4, 1", "3, 7, 6, 4, 5, 2, 1", "4, 5, 3, 2, 7, 6, 1", "4, 6, 3, 2, 7, 5, 1", "4, 6, 5, 2, 7, 3, 1", "4, 6, 5, 3, 7, 2, 1", "4, 7, 3, 2, 6, 5, 1", "4, 7, 5, 2, 6, 3, 1", "4, 7, 5, 3, 6, 2, 1", "4, 7, 6, 2, 5, 3, 1", "4, 7, 6, 3, 5, 2, 1", "5, 6, 3, 2, 7, 4, 1", "5, 6, 4, 2, 7, 3, 1", "5, 6, 4, 3, 7, 2, 1", "5, 7, 3, 2, 6, 4, 1", "5, 7, 4, 2, 6, 3, 1", "5, 7, 4, 3, 6, 2, 1", "5, 7, 6, 2, 4, 3, 1", "5, 7, 6, 3, 4, 2, 1", "6, 7, 3, 2, 5, 4, 1", "6, 7, 4, 2, 5, 3, 1", "6, 7, 4, 3, 5, 2, 1", "6, 7, 5, 2, 4, 3, 1", "6, 7, 5, 3, 4, 2, 1"]}, "cell": "perm_descents/2", "stage": "C"}
+{"id": "po-perm_descents-t2-eval-000043", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 4, 5} and with pi(7) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 4, 5} and with pi(7) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "4, 1, 5, 7, 6, 2, 3", "ref": "4, 1, 5, 7, 6, 2, 3", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.00377, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [1, 4, 5], "last": 3}, "seed": 167197441502000047, "version": "po-v0", "tier": 2, "n_raw": 19, "small": 7, "valid_answers": ["4, 1, 5, 7, 6, 2, 3", "4, 1, 6, 7, 5, 2, 3", "4, 2, 5, 7, 6, 1, 3", "4, 2, 6, 7, 5, 1, 3", "5, 1, 4, 7, 6, 2, 3", "5, 1, 6, 7, 4, 2, 3", "5, 2, 4, 7, 6, 1, 3", "5, 2, 6, 7, 4, 1, 3", "5, 4, 6, 7, 2, 1, 3", "6, 1, 4, 7, 5, 2, 3", "6, 1, 5, 7, 4, 2, 3", "6, 2, 4, 7, 5, 1, 3", "6, 2, 5, 7, 4, 1, 3", "6, 4, 5, 7, 2, 1, 3", "7, 1, 4, 6, 5, 2, 3", "7, 1, 5, 6, 4, 2, 3", "7, 2, 4, 6, 5, 1, 3", "7, 2, 5, 6, 4, 1, 3", "7, 4, 5, 6, 2, 1, 3"]}, "cell": "perm_descents/2", "stage": "C"}
+{"id": "po-perm_descents-t2-eval-000120", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 3, 6} and with pi(7) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 3, 6} and with pi(7) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "2, 1, 4, 3, 5, 7, 6", "ref": "2, 1, 4, 3, 5, 7, 6", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.003175, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [1, 3, 6], "last": 6}, "seed": 167197441502000189, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 7, "valid_answers": ["2, 1, 4, 3, 5, 7, 6", "2, 1, 5, 3, 4, 7, 6", "3, 1, 4, 2, 5, 7, 6", "3, 1, 5, 2, 4, 7, 6", "3, 2, 4, 1, 5, 7, 6", "3, 2, 5, 1, 4, 7, 6", "4, 1, 3, 2, 5, 7, 6", "4, 1, 5, 2, 3, 7, 6", "4, 2, 3, 1, 5, 7, 6", "4, 2, 5, 1, 3, 7, 6", "4, 3, 5, 1, 2, 7, 6", "5, 1, 3, 2, 4, 7, 6", "5, 1, 4, 2, 3, 7, 6", "5, 2, 3, 1, 4, 7, 6", "5, 2, 4, 1, 3, 7, 6", "5, 3, 4, 1, 2, 7, 6"]}, "cell": "perm_descents/2", "stage": "C"}
+{"id": "po-perm_descents-t2-eval-000093", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {3, 4} and with pi(7) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {3, 4} and with pi(7) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 2, 5, 4, 3, 6, 7", "ref": "1, 2, 5, 4, 3, 6, 7", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 26, "bucket": null, "guess_rate": 0.005159, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [3, 4], "last": 7}, "seed": 167197441502000122, "version": "po-v0", "tier": 2, "n_raw": 26, "small": 7, "valid_answers": ["1, 2, 5, 4, 3, 6, 7", "1, 2, 6, 4, 3, 5, 7", "1, 2, 6, 5, 3, 4, 7", "1, 3, 5, 4, 2, 6, 7", "1, 3, 6, 4, 2, 5, 7", "1, 3, 6, 5, 2, 4, 7", "1, 4, 5, 3, 2, 6, 7", "1, 4, 6, 3, 2, 5, 7", "1, 4, 6, 5, 2, 3, 7", "1, 5, 6, 3, 2, 4, 7", "1, 5, 6, 4, 2, 3, 7", "2, 3, 5, 4, 1, 6, 7", "2, 3, 6, 4, 1, 5, 7", "2, 3, 6, 5, 1, 4, 7", "2, 4, 5, 3, 1, 6, 7", "2, 4, 6, 3, 1, 5, 7", "2, 4, 6, 5, 1, 3, 7", "2, 5, 6, 3, 1, 4, 7", "2, 5, 6, 4, 1, 3, 7", "3, 4, 5, 2, 1, 6, 7", "3, 4, 6, 2, 1, 5, 7", "3, 4, 6, 5, 1, 2, 7", "3, 5, 6, 2, 1, 4, 7", "3, 5, 6, 4, 1, 2, 7", "4, 5, 6, 2, 1, 3, 7", "4, 5, 6, 3, 1, 2, 7"]}, "cell": "perm_descents/2", "stage": "C"}
+{"id": "po-perm_descents-t2-eval-000075", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {3, 5, 6} and with pi(7) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {3, 5, 6} and with pi(7) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 2, 5, 4, 7, 6, 3", "ref": "1, 2, 5, 4, 7, 6, 3", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 17, "bucket": "9-20", "guess_rate": 0.003373, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [3, 5, 6], "last": 3}, "seed": 167197441502000092, "version": "po-v0", "tier": 2, "n_raw": 17, "small": 7, "valid_answers": ["1, 2, 5, 4, 7, 6, 3", "1, 2, 6, 4, 7, 5, 3", "1, 2, 6, 5, 7, 4, 3", "1, 2, 7, 4, 6, 5, 3", "1, 2, 7, 5, 6, 4, 3", "1, 4, 5, 2, 7, 6, 3", "1, 4, 6, 2, 7, 5, 3", "1, 4, 7, 2, 6, 5, 3", "1, 5, 6, 2, 7, 4, 3", "1, 5, 7, 2, 6, 4, 3", "1, 6, 7, 2, 5, 4, 3", "2, 4, 5, 1, 7, 6, 3", "2, 4, 6, 1, 7, 5, 3", "2, 4, 7, 1, 6, 5, 3", "2, 5, 6, 1, 7, 4, 3", "2, 5, 7, 1, 6, 4, 3", "2, 6, 7, 1, 5, 4, 3"]}, "cell": "perm_descents/2", "stage": "C"}
+{"id": "po-perm_descents-t2-eval-000266", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3, 4, 5} and with pi(7) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3, 4, 5} and with pi(7) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "3, 7, 6, 5, 4, 1, 2", "ref": "3, 7, 6, 5, 4, 1, 2", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000794, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [2, 3, 4, 5], "last": 2}, "seed": 167197441502001263, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 7, "valid_answers": ["3, 7, 6, 5, 4, 1, 2", "4, 7, 6, 5, 3, 1, 2", "5, 7, 6, 4, 3, 1, 2", "6, 7, 5, 4, 3, 1, 2"]}, "cell": "perm_descents/2", "stage": "C"}
+{"id": "po-perm_descents-t2-eval-000016", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 6} and with pi(7) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 6} and with pi(7) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 4, 2, 5, 6, 7, 3", "ref": "1, 4, 2, 5, 6, 7, 3", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.002778, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [2, 6], "last": 3}, "seed": 167197441502000016, "version": "po-v0", "tier": 2, "n_raw": 14, "small": 7, "valid_answers": ["1, 4, 2, 5, 6, 7, 3", "1, 5, 2, 4, 6, 7, 3", "1, 6, 2, 4, 5, 7, 3", "1, 7, 2, 4, 5, 6, 3", "2, 4, 1, 5, 6, 7, 3", "2, 5, 1, 4, 6, 7, 3", "2, 6, 1, 4, 5, 7, 3", "2, 7, 1, 4, 5, 6, 3", "4, 5, 1, 2, 6, 7, 3", "4, 6, 1, 2, 5, 7, 3", "4, 7, 1, 2, 5, 6, 3", "5, 6, 1, 2, 4, 7, 3", "5, 7, 1, 2, 4, 6, 3", "6, 7, 1, 2, 4, 5, 3"]}, "cell": "perm_descents/2", "stage": "C"}
+{"id": "po-perm_descents-t2-eval-000073", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {3, 6} and with pi(7) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {3, 6} and with pi(7) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 2, 5, 3, 6, 7, 4", "ref": "1, 2, 5, 3, 6, 7, 4", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 0.003571, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [3, 6], "last": 4}, "seed": 167197441502000087, "version": "po-v0", "tier": 2, "n_raw": 18, "small": 7, "valid_answers": ["1, 2, 5, 3, 6, 7, 4", "1, 2, 6, 3, 5, 7, 4", "1, 2, 7, 3, 5, 6, 4", "1, 3, 5, 2, 6, 7, 4", "1, 3, 6, 2, 5, 7, 4", "1, 3, 7, 2, 5, 6, 4", "1, 5, 6, 2, 3, 7, 4", "1, 5, 7, 2, 3, 6, 4", "1, 6, 7, 2, 3, 5, 4", "2, 3, 5, 1, 6, 7, 4", "2, 3, 6, 1, 5, 7, 4", "2, 3, 7, 1, 5, 6, 4", "2, 5, 6, 1, 3, 7, 4", "2, 5, 7, 1, 3, 6, 4", "2, 6, 7, 1, 3, 5, 4", "3, 5, 6, 1, 2, 7, 4", "3, 5, 7, 1, 2, 6, 4", "3, 6, 7, 1, 2, 5, 4"]}, "cell": "perm_descents/2", "stage": "C"}
+{"id": "po-perm_descents-t2-eval-000207", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 5, 6} and with pi(7) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 5, 6} and with pi(7) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "2, 1, 3, 5, 7, 6, 4", "ref": "2, 1, 3, 5, 7, 6, 4", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.001389, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [1, 5, 6], "last": 4}, "seed": 167197441502000583, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 7, "valid_answers": ["2, 1, 3, 5, 7, 6, 4", "2, 1, 3, 6, 7, 5, 4", "3, 1, 2, 5, 7, 6, 4", "3, 1, 2, 6, 7, 5, 4", "5, 1, 2, 3, 7, 6, 4", "6, 1, 2, 3, 7, 5, 4", "7, 1, 2, 3, 6, 5, 4"]}, "cell": "perm_descents/2", "stage": "C"}
+{"id": "po-perm_descents-t2-eval-000210", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 5} and with pi(7) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 5} and with pi(7) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "2, 1, 3, 4, 7, 5, 6", "ref": "2, 1, 3, 4, 7, 5, 6", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.00377, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [1, 5], "last": 6}, "seed": 167197441502000613, "version": "po-v0", "tier": 2, "n_raw": 19, "small": 7, "valid_answers": ["2, 1, 3, 4, 7, 5, 6", "2, 1, 3, 5, 7, 4, 6", "2, 1, 4, 5, 7, 3, 6", "3, 1, 2, 4, 7, 5, 6", "3, 1, 2, 5, 7, 4, 6", "3, 1, 4, 5, 7, 2, 6", "3, 2, 4, 5, 7, 1, 6", "4, 1, 2, 3, 7, 5, 6", "4, 1, 2, 5, 7, 3, 6", "4, 1, 3, 5, 7, 2, 6", "4, 2, 3, 5, 7, 1, 6", "5, 1, 2, 3, 7, 4, 6", "5, 1, 2, 4, 7, 3, 6", "5, 1, 3, 4, 7, 2, 6", "5, 2, 3, 4, 7, 1, 6", "7, 1, 2, 3, 5, 4, 6", "7, 1, 2, 4, 5, 3, 6", "7, 1, 3, 4, 5, 2, 6", "7, 2, 3, 4, 5, 1, 6"]}, "cell": "perm_descents/2", "stage": "C"}
+{"id": "po-perm_descents-t2-eval-000274", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {5, 6} and with pi(7) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {5, 6} and with pi(7) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 2, 4, 5, 7, 6, 3", "ref": "1, 2, 4, 5, 7, 6, 3", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000595, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [5, 6], "last": 3}, "seed": 167197441502001503, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 7, "valid_answers": ["1, 2, 4, 5, 7, 6, 3", "1, 2, 4, 6, 7, 5, 3", "1, 2, 5, 6, 7, 4, 3"]}, "cell": "perm_descents/2", "stage": "C"}
+{"id": "po-perm_descents-t2-eval-000151", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 4} and with pi(7) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 4} and with pi(7) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "2, 1, 6, 7, 3, 4, 5", "ref": "2, 1, 6, 7, 3, 4, 5", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 0.003571, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 7, "D": [1, 4], "last": 5}, "seed": 167197441502000289, "version": "po-v0", "tier": 2, "n_raw": 18, "small": 7, "valid_answers": ["2, 1, 6, 7, 3, 4, 5", "3, 1, 6, 7, 2, 4, 5", "3, 2, 6, 7, 1, 4, 5", "4, 1, 6, 7, 2, 3, 5", "4, 2, 6, 7, 1, 3, 5", "4, 3, 6, 7, 1, 2, 5", "6, 1, 2, 7, 3, 4, 5", "6, 1, 3, 7, 2, 4, 5", "6, 1, 4, 7, 2, 3, 5", "6, 2, 3, 7, 1, 4, 5", "6, 2, 4, 7, 1, 3, 5", "6, 3, 4, 7, 1, 2, 5", "7, 1, 2, 6, 3, 4, 5", "7, 1, 3, 6, 2, 4, 5", "7, 1, 4, 6, 2, 3, 5", "7, 2, 3, 6, 1, 4, 5", "7, 2, 4, 6, 1, 3, 5", "7, 3, 4, 6, 1, 2, 5"]}, "cell": "perm_descents/2", "stage": "C"}
+{"id": "po-perm_descents-t3-eval-000469", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {4, 5, 6, 7} and with pi(8) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {4, 5, 6, 7} and with pi(8) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "1, 2, 4, 8, 7, 6, 5, 3", "ref": "1, 2, 4, 8, 7, 6, 5, 3", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 9.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 8, "D": [4, 5, 6, 7], "last": 3}, "seed": 167197441503010568, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 8, "valid_answers": ["1, 2, 4, 8, 7, 6, 5, 3", "1, 2, 5, 8, 7, 6, 4, 3", "1, 2, 6, 8, 7, 5, 4, 3", "1, 2, 7, 8, 6, 5, 4, 3"]}, "cell": "perm_descents/3", "stage": "C"}
+{"id": "po-perm_descents-t3-eval-000087", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {3, 4, 5, 6} and with pi(8) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {3, 4, 5, 6} and with pi(8) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "1, 2, 8, 7, 6, 4, 3, 5", "ref": "1, 2, 8, 7, 6, 4, 3, 5", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 0.000372, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 8, "D": [3, 4, 5, 6], "last": 5}, "seed": 167197441503000314, "version": "po-v0", "tier": 3, "n_raw": 15, "small": 8, "valid_answers": ["1, 2, 8, 7, 6, 4, 3, 5", "1, 3, 8, 7, 6, 4, 2, 5", "1, 4, 8, 7, 6, 3, 2, 5", "1, 6, 8, 7, 4, 3, 2, 5", "1, 7, 8, 6, 4, 3, 2, 5", "2, 3, 8, 7, 6, 4, 1, 5", "2, 4, 8, 7, 6, 3, 1, 5", "2, 6, 8, 7, 4, 3, 1, 5", "2, 7, 8, 6, 4, 3, 1, 5", "3, 4, 8, 7, 6, 2, 1, 5", "3, 6, 8, 7, 4, 2, 1, 5", "3, 7, 8, 6, 4, 2, 1, 5", "4, 6, 8, 7, 3, 2, 1, 5", "4, 7, 8, 6, 3, 2, 1, 5", "6, 7, 8, 4, 3, 2, 1, 5"]}, "cell": "perm_descents/3", "stage": "C"}
+{"id": "po-perm_descents-t3-eval-000346", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {4, 5, 7} and with pi(8) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {4, 5, 7} and with pi(8) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "1, 2, 3, 7, 6, 5, 8, 4", "ref": "1, 2, 3, 7, 6, 5, 8, 4", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 39, "bucket": null, "guess_rate": 0.000967, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 8, "D": [4, 5, 7], "last": 4}, "seed": 167197441503002732, "version": "po-v0", "tier": 3, "n_raw": 39, "small": 8, "valid_answers": ["1, 2, 3, 7, 6, 5, 8, 4", "1, 2, 3, 8, 6, 5, 7, 4", "1, 2, 3, 8, 7, 5, 6, 4", "1, 2, 5, 7, 6, 3, 8, 4", "1, 2, 5, 8, 6, 3, 7, 4", "1, 2, 5, 8, 7, 3, 6, 4", "1, 2, 6, 7, 5, 3, 8, 4", "1, 2, 6, 8, 5, 3, 7, 4", "1, 2, 6, 8, 7, 3, 5, 4", "1, 2, 7, 8, 5, 3, 6, 4", "1, 2, 7, 8, 6, 3, 5, 4", "1, 3, 5, 7, 6, 2, 8, 4", "1, 3, 5, 8, 6, 2, 7, 4", "1, 3, 5, 8, 7, 2, 6, 4", "1, 3, 6, 7, 5, 2, 8, 4", "1, 3, 6, 8, 5, 2, 7, 4", "1, 3, 6, 8, 7, 2, 5, 4", "1, 3, 7, 8, 5, 2, 6, 4", "1, 3, 7, 8, 6, 2, 5, 4", "1, 5, 6, 7, 3, 2, 8, 4", "1, 5, 6, 8, 3, 2, 7, 4", "1, 5, 7, 8, 3, 2, 6, 4", "1, 6, 7, 8, 3, 2, 5, 4", "2, 3, 5, 7, 6, 1, 8, 4", "2, 3, 5, 8, 6, 1, 7, 4", "2, 3, 5, 8, 7, 1, 6, 4", "2, 3, 6, 7, 5, 1, 8, 4", "2, 3, 6, 8, 5, 1, 7, 4", "2, 3, 6, 8, 7, 1, 5, 4", "2, 3, 7, 8, 5, 1, 6, 4", "2, 3, 7, 8, 6, 1, 5, 4", "2, 5, 6, 7, 3, 1, 8, 4", "2, 5, 6, 8, 3, 1, 7, 4", "2, 5, 7, 8, 3, 1, 6, 4", "2, 6, 7, 8, 3, 1, 5, 4", "3, 5, 6, 7, 2, 1, 8, 4", "3, 5, 6, 8, 2, 1, 7, 4", "3, 5, 7, 8, 2, 1, 6, 4", "3, 6, 7, 8, 2, 1, 5, 4"]}, "cell": "perm_descents/3", "stage": "C"}
+{"id": "po-perm_descents-t3-eval-000107", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3, 4, 5, 7} and with pi(8) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {2, 3, 4, 5, 7} and with pi(8) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "1, 7, 6, 5, 4, 3, 8, 2", "ref": "1, 7, 6, 5, 4, 3, 8, 2", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 29, "bucket": null, "guess_rate": 0.000719, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 8, "D": [2, 3, 4, 5, 7], "last": 2}, "seed": 167197441503000379, "version": "po-v0", "tier": 3, "n_raw": 29, "small": 8, "valid_answers": ["1, 7, 6, 5, 4, 3, 8, 2", "1, 8, 6, 5, 4, 3, 7, 2", "1, 8, 7, 5, 4, 3, 6, 2", "1, 8, 7, 6, 4, 3, 5, 2", "1, 8, 7, 6, 5, 3, 4, 2", "3, 7, 6, 5, 4, 1, 8, 2", "3, 8, 6, 5, 4, 1, 7, 2", "3, 8, 7, 5, 4, 1, 6, 2", "3, 8, 7, 6, 4, 1, 5, 2", "3, 8, 7, 6, 5, 1, 4, 2", "4, 7, 6, 5, 3, 1, 8, 2", "4, 8, 6, 5, 3, 1, 7, 2", "4, 8, 7, 5, 3, 1, 6, 2", "4, 8, 7, 6, 3, 1, 5, 2", "4, 8, 7, 6, 5, 1, 3, 2", "5, 7, 6, 4, 3, 1, 8, 2", "5, 8, 6, 4, 3, 1, 7, 2", "5, 8, 7, 4, 3, 1, 6, 2", "5, 8, 7, 6, 3, 1, 4, 2", "5, 8, 7, 6, 4, 1, 3, 2", "6, 7, 5, 4, 3, 1, 8, 2", "6, 8, 5, 4, 3, 1, 7, 2", "6, 8, 7, 4, 3, 1, 5, 2", "6, 8, 7, 5, 3, 1, 4, 2", "6, 8, 7, 5, 4, 1, 3, 2", "7, 8, 5, 4, 3, 1, 6, 2", "7, 8, 6, 4, 3, 1, 5, 2", "7, 8, 6, 5, 3, 1, 4, 2", "7, 8, 6, 5, 4, 1, 3, 2"]}, "cell": "perm_descents/3", "stage": "C"}
+{"id": "po-perm_descents-t3-eval-000025", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 3, 6, 7} and with pi(8) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 3, 6, 7} and with pi(8) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "5, 4, 3, 2, 6, 8, 7, 1", "ref": "5, 4, 3, 2, 6, 8, 7, 1", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 50, "bucket": null, "guess_rate": 0.00124, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 8, "D": [1, 2, 3, 6, 7], "last": 1}, "seed": 167197441503000068, "version": "po-v0", "tier": 3, "n_raw": 50, "small": 8, "valid_answers": ["5, 4, 3, 2, 6, 8, 7, 1", "5, 4, 3, 2, 7, 8, 6, 1", "6, 4, 3, 2, 5, 8, 7, 1", "6, 4, 3, 2, 7, 8, 5, 1", "6, 5, 3, 2, 4, 8, 7, 1", "6, 5, 3, 2, 7, 8, 4, 1", "6, 5, 4, 2, 3, 8, 7, 1", "6, 5, 4, 2, 7, 8, 3, 1", "6, 5, 4, 3, 7, 8, 2, 1", "7, 4, 3, 2, 5, 8, 6, 1", "7, 4, 3, 2, 6, 8, 5, 1", "7, 5, 3, 2, 4, 8, 6, 1", "7, 5, 3, 2, 6, 8, 4, 1", "7, 5, 4, 2, 3, 8, 6, 1", "7, 5, 4, 2, 6, 8, 3, 1", "7, 5, 4, 3, 6, 8, 2, 1", "7, 6, 3, 2, 4, 8, 5, 1", "7, 6, 3, 2, 5, 8, 4, 1", "7, 6, 4, 2, 3, 8, 5, 1", "7, 6, 4, 2, 5, 8, 3, 1", "7, 6, 4, 3, 5, 8, 2, 1", "7, 6, 5, 2, 3, 8, 4, 1", "7, 6, 5, 2, 4, 8, 3, 1", "7, 6, 5, 3, 4, 8, 2, 1", "8, 4, 3, 2, 5, 7, 6, 1", "8, 4, 3, 2, 6, 7, 5, 1", "8, 5, 3, 2, 4, 7, 6, 1", "8, 5, 3, 2, 6, 7, 4, 1", "8, 5, 4, 2, 3, 7, 6, 1", "8, 5, 4, 2, 6, 7, 3, 1", "8, 5, 4, 3, 6, 7, 2, 1", "8, 6, 3, 2, 4, 7, 5, 1", "8, 6, 3, 2, 5, 7, 4, 1", "8, 6, 4, 2, 3, 7, 5, 1", "8, 6, 4, 2, 5, 7, 3, 1", "8, 6, 4, 3, 5, 7, 2, 1", "8, 6, 5, 2, 3, 7, 4, 1", "8, 6, 5, 2, 4, 7, 3, 1", "8, 6, 5, 3, 4, 7, 2, 1", "8, 7, 3, 2, 4, 6, 5, 1", "8, 7, 3, 2, 5, 6, 4, 1", "8, 7, 4, 2, 3, 6, 5, 1", "8, 7, 4, 2, 5, 6, 3, 1", "8, 7, 4, 3, 5, 6, 2, 1", "8, 7, 5, 2, 3, 6, 4, 1", "8, 7, 5, 2, 4, 6, 3, 1", "8, 7, 5, 3, 4, 6, 2, 1", "8, 7, 6, 2, 3, 5, 4, 1", "8, 7, 6, 2, 4, 5, 3, 1", "8, 7, 6, 3, 4, 5, 2, 1"]}, "cell": "perm_descents/3", "stage": "C"}
+{"id": "po-perm_descents-t3-eval-000172", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 4, 5, 7} and with pi(8) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 4, 5, 7} and with pi(8) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "3, 2, 1, 6, 5, 4, 8, 7", "ref": "3, 2, 1, 6, 5, 4, 8, 7", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.000471, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 8, "D": [1, 2, 4, 5, 7], "last": 7}, "seed": 167197441503000706, "version": "po-v0", "tier": 3, "n_raw": 19, "small": 8, "valid_answers": ["3, 2, 1, 6, 5, 4, 8, 7", "4, 2, 1, 6, 5, 3, 8, 7", "4, 3, 1, 6, 5, 2, 8, 7", "4, 3, 2, 6, 5, 1, 8, 7", "5, 2, 1, 6, 4, 3, 8, 7", "5, 3, 1, 6, 4, 2, 8, 7", "5, 3, 2, 6, 4, 1, 8, 7", "5, 4, 1, 6, 3, 2, 8, 7", "5, 4, 2, 6, 3, 1, 8, 7", "5, 4, 3, 6, 2, 1, 8, 7", "6, 2, 1, 5, 4, 3, 8, 7", "6, 3, 1, 5, 4, 2, 8, 7", "6, 3, 2, 5, 4, 1, 8, 7", "6, 4, 1, 5, 3, 2, 8, 7", "6, 4, 2, 5, 3, 1, 8, 7", "6, 4, 3, 5, 2, 1, 8, 7", "6, 5, 1, 4, 3, 2, 8, 7", "6, 5, 2, 4, 3, 1, 8, 7", "6, 5, 3, 4, 2, 1, 8, 7"]}, "cell": "perm_descents/3", "stage": "C"}
+{"id": "po-perm_descents-t3-eval-000242", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {3, 4, 5, 7} and with pi(8) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {3, 4, 5, 7} and with pi(8) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "1, 2, 7, 5, 4, 3, 8, 6", "ref": "1, 2, 7, 5, 4, 3, 8, 6", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.000496, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 8, "D": [3, 4, 5, 7], "last": 6}, "seed": 167197441503001146, "version": "po-v0", "tier": 3, "n_raw": 20, "small": 8, "valid_answers": ["1, 2, 7, 5, 4, 3, 8, 6", "1, 2, 8, 5, 4, 3, 7, 6", "1, 3, 7, 5, 4, 2, 8, 6", "1, 3, 8, 5, 4, 2, 7, 6", "1, 4, 7, 5, 3, 2, 8, 6", "1, 4, 8, 5, 3, 2, 7, 6", "1, 5, 7, 4, 3, 2, 8, 6", "1, 5, 8, 4, 3, 2, 7, 6", "2, 3, 7, 5, 4, 1, 8, 6", "2, 3, 8, 5, 4, 1, 7, 6", "2, 4, 7, 5, 3, 1, 8, 6", "2, 4, 8, 5, 3, 1, 7, 6", "2, 5, 7, 4, 3, 1, 8, 6", "2, 5, 8, 4, 3, 1, 7, 6", "3, 4, 7, 5, 2, 1, 8, 6", "3, 4, 8, 5, 2, 1, 7, 6", "3, 5, 7, 4, 2, 1, 8, 6", "3, 5, 8, 4, 2, 1, 7, 6", "4, 5, 7, 3, 2, 1, 8, 6", "4, 5, 8, 3, 2, 1, 7, 6"]}, "cell": "perm_descents/3", "stage": "C"}
+{"id": "po-perm_descents-t3-eval-000288", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 3, 5, 6, 7} and with pi(8) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 3, 5, 6, 7} and with pi(8) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "5, 4, 3, 2, 8, 7, 6, 1", "ref": "5, 4, 3, 2, 8, 7, 6, 1", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 34, "bucket": null, "guess_rate": 0.000843, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 8, "D": [1, 2, 3, 5, 6, 7], "last": 1}, "seed": 167197441503001539, "version": "po-v0", "tier": 3, "n_raw": 34, "small": 8, "valid_answers": ["5, 4, 3, 2, 8, 7, 6, 1", "6, 4, 3, 2, 8, 7, 5, 1", "6, 5, 3, 2, 8, 7, 4, 1", "6, 5, 4, 2, 8, 7, 3, 1", "6, 5, 4, 3, 8, 7, 2, 1", "7, 4, 3, 2, 8, 6, 5, 1", "7, 5, 3, 2, 8, 6, 4, 1", "7, 5, 4, 2, 8, 6, 3, 1", "7, 5, 4, 3, 8, 6, 2, 1", "7, 6, 3, 2, 8, 5, 4, 1", "7, 6, 4, 2, 8, 5, 3, 1", "7, 6, 4, 3, 8, 5, 2, 1", "7, 6, 5, 2, 8, 4, 3, 1", "7, 6, 5, 3, 8, 4, 2, 1", "7, 6, 5, 4, 8, 3, 2, 1", "8, 4, 3, 2, 7, 6, 5, 1", "8, 5, 3, 2, 7, 6, 4, 1", "8, 5, 4, 2, 7, 6, 3, 1", "8, 5, 4, 3, 7, 6, 2, 1", "8, 6, 3, 2, 7, 5, 4, 1", "8, 6, 4, 2, 7, 5, 3, 1", "8, 6, 4, 3, 7, 5, 2, 1", "8, 6, 5, 2, 7, 4, 3, 1", "8, 6, 5, 3, 7, 4, 2, 1", "8, 6, 5, 4, 7, 3, 2, 1", "8, 7, 3, 2, 6, 5, 4, 1", "8, 7, 4, 2, 6, 5, 3, 1", "8, 7, 4, 3, 6, 5, 2, 1", "8, 7, 5, 2, 6, 4, 3, 1", "8, 7, 5, 3, 6, 4, 2, 1", "8, 7, 5, 4, 6, 3, 2, 1", "8, 7, 6, 2, 5, 4, 3, 1", "8, 7, 6, 3, 5, 4, 2, 1", "8, 7, 6, 4, 5, 3, 2, 1"]}, "cell": "perm_descents/3", "stage": "C"}
+{"id": "po-perm_descents-t3-eval-000012", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 4, 5} and with pi(8) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 4, 5} and with pi(8) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "6, 5, 4, 8, 7, 1, 2, 3", "ref": "6, 5, 4, 8, 7, 1, 2, 3", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000223, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 8, "D": [1, 2, 4, 5], "last": 3}, "seed": 167197441503000038, "version": "po-v0", "tier": 3, "n_raw": 9, "small": 8, "valid_answers": ["6, 5, 4, 8, 7, 1, 2, 3", "7, 5, 4, 8, 6, 1, 2, 3", "7, 6, 4, 8, 5, 1, 2, 3", "7, 6, 5, 8, 4, 1, 2, 3", "8, 5, 4, 7, 6, 1, 2, 3", "8, 6, 4, 7, 5, 1, 2, 3", "8, 6, 5, 7, 4, 1, 2, 3", "8, 7, 4, 6, 5, 1, 2, 3", "8, 7, 5, 6, 4, 1, 2, 3"]}, "cell": "perm_descents/3", "stage": "C"}
+{"id": "po-perm_descents-t3-eval-000018", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {3, 5, 6} and with pi(8) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {3, 5, 6} and with pi(8) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "1, 4, 6, 5, 8, 7, 2, 3", "ref": "1, 4, 6, 5, 8, 7, 2, 3", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 61, "bucket": null, "guess_rate": 0.001513, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 8, "D": [3, 5, 6], "last": 3}, "seed": 167197441503000051, "version": "po-v0", "tier": 3, "n_raw": 61, "small": 8, "valid_answers": ["1, 4, 6, 5, 8, 7, 2, 3", "1, 4, 7, 5, 8, 6, 2, 3", "1, 4, 7, 6, 8, 5, 2, 3", "1, 4, 8, 5, 7, 6, 2, 3", "1, 4, 8, 6, 7, 5, 2, 3", "1, 5, 6, 4, 8, 7, 2, 3", "1, 5, 7, 4, 8, 6, 2, 3", "1, 5, 7, 6, 8, 4, 2, 3", "1, 5, 8, 4, 7, 6, 2, 3", "1, 5, 8, 6, 7, 4, 2, 3", "1, 6, 7, 4, 8, 5, 2, 3", "1, 6, 7, 5, 8, 4, 2, 3", "1, 6, 8, 4, 7, 5, 2, 3", "1, 6, 8, 5, 7, 4, 2, 3", "1, 7, 8, 4, 6, 5, 2, 3", "1, 7, 8, 5, 6, 4, 2, 3", "2, 4, 6, 5, 8, 7, 1, 3", "2, 4, 7, 5, 8, 6, 1, 3", "2, 4, 7, 6, 8, 5, 1, 3", "2, 4, 8, 5, 7, 6, 1, 3", "2, 4, 8, 6, 7, 5, 1, 3", "2, 5, 6, 4, 8, 7, 1, 3", "2, 5, 7, 4, 8, 6, 1, 3", "2, 5, 7, 6, 8, 4, 1, 3", "2, 5, 8, 4, 7, 6, 1, 3", "2, 5, 8, 6, 7, 4, 1, 3", "2, 6, 7, 4, 8, 5, 1, 3", "2, 6, 7, 5, 8, 4, 1, 3", "2, 6, 8, 4, 7, 5, 1, 3", "2, 6, 8, 5, 7, 4, 1, 3", "2, 7, 8, 4, 6, 5, 1, 3", "2, 7, 8, 5, 6, 4, 1, 3", "4, 5, 6, 1, 8, 7, 2, 3", "4, 5, 6, 2, 8, 7, 1, 3", "4, 5, 7, 1, 8, 6, 2, 3", "4, 5, 7, 2, 8, 6, 1, 3", "4, 5, 7, 6, 8, 2, 1, 3", "4, 5, 8, 1, 7, 6, 2, 3", "4, 5, 8, 2, 7, 6, 1, 3", "4, 5, 8, 6, 7, 2, 1, 3", "4, 6, 7, 1, 8, 5, 2, 3", "4, 6, 7, 2, 8, 5, 1, 3", "4, 6, 7, 5, 8, 2, 1, 3", "4, 6, 8, 1, 7, 5, 2, 3", "4, 6, 8, 2, 7, 5, 1, 3", "4, 6, 8, 5, 7, 2, 1, 3", "4, 7, 8, 1, 6, 5, 2, 3", "4, 7, 8, 2, 6, 5, 1, 3", "4, 7, 8, 5, 6, 2, 1, 3", "5, 6, 7, 1, 8, 4, 2, 3", "5, 6, 7, 2, 8, 4, 1, 3", "5, 6, 7, 4, 8, 2, 1, 3", "5, 6, 8, 1, 7, 4, 2, 3", "5, 6, 8, 2, 7, 4, 1, 3", "5, 6, 8, 4, 7, 2, 1, 3", "5, 7, 8, 1, 6, 4, 2, 3", "5, 7, 8, 2, 6, 4, 1, 3", "5, 7, 8, 4, 6, 2, 1, 3", "6, 7, 8, 1, 5, 4, 2, 3", "6, 7, 8, 2, 5, 4, 1, 3", "6, 7, 8, 4, 5, 2, 1, 3"]}, "cell": "perm_descents/3", "stage": "C"}
+{"id": "po-perm_descents-t3-eval-000389", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 3, 7} and with pi(8) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 2, 3, 7} and with pi(8) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "5, 3, 2, 1, 6, 7, 8, 4", "ref": "5, 3, 2, 1, 6, 7, 8, 4", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.000496, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 8, "D": [1, 2, 3, 7], "last": 4}, "seed": 167197441503003660, "version": "po-v0", "tier": 3, "n_raw": 20, "small": 8, "valid_answers": ["5, 3, 2, 1, 6, 7, 8, 4", "6, 3, 2, 1, 5, 7, 8, 4", "6, 5, 2, 1, 3, 7, 8, 4", "6, 5, 3, 1, 2, 7, 8, 4", "7, 3, 2, 1, 5, 6, 8, 4", "7, 5, 2, 1, 3, 6, 8, 4", "7, 5, 3, 1, 2, 6, 8, 4", "7, 6, 2, 1, 3, 5, 8, 4", "7, 6, 3, 1, 2, 5, 8, 4", "7, 6, 5, 1, 2, 3, 8, 4", "8, 3, 2, 1, 5, 6, 7, 4", "8, 5, 2, 1, 3, 6, 7, 4", "8, 5, 3, 1, 2, 6, 7, 4", "8, 6, 2, 1, 3, 5, 7, 4", "8, 6, 3, 1, 2, 5, 7, 4", "8, 6, 5, 1, 2, 3, 7, 4", "8, 7, 2, 1, 3, 5, 6, 4", "8, 7, 3, 1, 2, 5, 6, 4", "8, 7, 5, 1, 2, 3, 6, 4", "8, 7, 6, 1, 2, 3, 5, 4"]}, "cell": "perm_descents/3", "stage": "C"}
+{"id": "po-perm_descents-t3-eval-000080", "task": "cx_po_perm_descents", "source": "cx_po_perm_descents", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 5, 6} and with pi(8) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 whose descent set (the positions i with pi(i) > pi(i+1)) is {1, 5, 6} and with pi(8) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "2, 1, 3, 4, 8, 6, 5, 7", "ref": "2, 1, 3, 4, 8, 6, 5, 7", "family": "po_perm_descents", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 55, "bucket": null, "guess_rate": 0.001364, "solvable": true, "holdout": false, "meta": {"family": "po_perm_descents", "rung": "witness", "params": {"n": 8, "D": [1, 5, 6], "last": 7}, "seed": 167197441503000275, "version": "po-v0", "tier": 3, "n_raw": 55, "small": 8, "valid_answers": ["2, 1, 3, 4, 8, 6, 5, 7", "2, 1, 3, 5, 8, 6, 4, 7", "2, 1, 3, 6, 8, 5, 4, 7", "2, 1, 4, 5, 8, 6, 3, 7", "2, 1, 4, 6, 8, 5, 3, 7", "2, 1, 5, 6, 8, 4, 3, 7", "3, 1, 2, 4, 8, 6, 5, 7", "3, 1, 2, 5, 8, 6, 4, 7", "3, 1, 2, 6, 8, 5, 4, 7", "3, 1, 4, 5, 8, 6, 2, 7", "3, 1, 4, 6, 8, 5, 2, 7", "3, 1, 5, 6, 8, 4, 2, 7", "3, 2, 4, 5, 8, 6, 1, 7", "3, 2, 4, 6, 8, 5, 1, 7", "3, 2, 5, 6, 8, 4, 1, 7", "4, 1, 2, 3, 8, 6, 5, 7", "4, 1, 2, 5, 8, 6, 3, 7", "4, 1, 2, 6, 8, 5, 3, 7", "4, 1, 3, 5, 8, 6, 2, 7", "4, 1, 3, 6, 8, 5, 2, 7", "4, 1, 5, 6, 8, 3, 2, 7", "4, 2, 3, 5, 8, 6, 1, 7", "4, 2, 3, 6, 8, 5, 1, 7", "4, 2, 5, 6, 8, 3, 1, 7", "4, 3, 5, 6, 8, 2, 1, 7", "5, 1, 2, 3, 8, 6, 4, 7", "5, 1, 2, 4, 8, 6, 3, 7", "5, 1, 2, 6, 8, 4, 3, 7", "5, 1, 3, 4, 8, 6, 2, 7", "5, 1, 3, 6, 8, 4, 2, 7", "5, 1, 4, 6, 8, 3, 2, 7", "5, 2, 3, 4, 8, 6, 1, 7", "5, 2, 3, 6, 8, 4, 1, 7", "5, 2, 4, 6, 8, 3, 1, 7", "5, 3, 4, 6, 8, 2, 1, 7", "6, 1, 2, 3, 8, 5, 4, 7", "6, 1, 2, 4, 8, 5, 3, 7", "6, 1, 2, 5, 8, 4, 3, 7", "6, 1, 3, 4, 8, 5, 2, 7", "6, 1, 3, 5, 8, 4, 2, 7", "6, 1, 4, 5, 8, 3, 2, 7", "6, 2, 3, 4, 8, 5, 1, 7", "6, 2, 3, 5, 8, 4, 1, 7", "6, 2, 4, 5, 8, 3, 1, 7", "6, 3, 4, 5, 8, 2, 1, 7", "8, 1, 2, 3, 6, 5, 4, 7", "8, 1, 2, 4, 6, 5, 3, 7", "8, 1, 2, 5, 6, 4, 3, 7", "8, 1, 3, 4, 6, 5, 2, 7", "8, 1, 3, 5, 6, 4, 2, 7", "8, 1, 4, 5, 6, 3, 2, 7", "8, 2, 3, 4, 6, 5, 1, 7", "8, 2, 3, 5, 6, 4, 1, 7", "8, 2, 4, 5, 6, 3, 1, 7", "8, 3, 4, 5, 6, 2, 1, 7"]}, "cell": "perm_descents/3", "stage": "C"}
+{"id": "po-perm_inversions-t0-eval-000013", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 with exactly 3 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 with exactly 3 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "2, 1, 3, 5, 6, 4", "ref": "2, 1, 3, 5, 6, 4", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.0125, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 6, "inv": 3, "first": 2, "last": null}, "seed": 28913166900000103, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 6, "valid_answers": ["2, 1, 3, 5, 6, 4", "2, 1, 3, 6, 4, 5", "2, 1, 4, 3, 6, 5", "2, 1, 4, 5, 3, 6", "2, 1, 5, 3, 4, 6", "2, 3, 1, 4, 6, 5", "2, 3, 1, 5, 4, 6", "2, 3, 4, 1, 5, 6", "2, 4, 1, 3, 5, 6"]}, "cell": "perm_inversions/0", "stage": "C"}
+{"id": "po-perm_inversions-t0-eval-000010", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 with exactly 13 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 with exactly 13 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "6, 3, 5, 4, 2, 1", "ref": "6, 3, 5, 4, 2, 1", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.0125, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 6, "inv": 13, "first": 6, "last": null}, "seed": 28913166900000072, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 6, "valid_answers": ["6, 3, 5, 4, 2, 1", "6, 4, 3, 5, 2, 1", "6, 4, 5, 2, 3, 1", "6, 4, 5, 3, 1, 2", "6, 5, 2, 4, 3, 1", "6, 5, 3, 2, 4, 1", "6, 5, 3, 4, 1, 2", "6, 5, 4, 1, 3, 2", "6, 5, 4, 2, 1, 3"]}, "cell": "perm_inversions/0", "stage": "C"}
+{"id": "po-perm_inversions-t0-eval-000022", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 with exactly 5 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 with exactly 5 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "5, 1, 2, 3, 6, 4", "ref": "5, 1, 2, 3, 6, 4", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005556, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 6, "inv": 5, "first": 5, "last": null}, "seed": 28913166900000623, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 6, "valid_answers": ["5, 1, 2, 3, 6, 4", "5, 1, 2, 4, 3, 6", "5, 1, 3, 2, 4, 6", "5, 2, 1, 3, 4, 6"]}, "cell": "perm_inversions/0", "stage": "C"}
+{"id": "po-perm_inversions-t0-eval-000001", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 with exactly 12 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 with exactly 12 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "4, 5, 6, 3, 2, 1", "ref": "4, 5, 6, 3, 2, 1", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005556, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 6, "inv": 12, "first": 4, "last": null}, "seed": 28913166900000008, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 6, "valid_answers": ["4, 5, 6, 3, 2, 1", "4, 6, 3, 5, 2, 1", "4, 6, 5, 2, 3, 1", "4, 6, 5, 3, 1, 2"]}, "cell": "perm_inversions/0", "stage": "C"}
+{"id": "po-perm_inversions-t0-eval-000009", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 with exactly 4 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 with exactly 4 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "3, 1, 2, 5, 6, 4", "ref": "3, 1, 2, 5, 6, 4", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.0125, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 6, "inv": 4, "first": 3, "last": null}, "seed": 28913166900000067, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 6, "valid_answers": ["3, 1, 2, 5, 6, 4", "3, 1, 2, 6, 4, 5", "3, 1, 4, 2, 6, 5", "3, 1, 4, 5, 2, 6", "3, 1, 5, 2, 4, 6", "3, 2, 1, 4, 6, 5", "3, 2, 1, 5, 4, 6", "3, 2, 4, 1, 5, 6", "3, 4, 1, 2, 5, 6"]}, "cell": "perm_inversions/0", "stage": "C"}
+{"id": "po-perm_inversions-t0-eval-000014", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 with exactly 1 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 with exactly 1 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "1, 2, 3, 4, 6, 5", "ref": "1, 2, 3, 4, 6, 5", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005556, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 6, "inv": 1, "first": 1, "last": null}, "seed": 28913166900000112, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 6, "valid_answers": ["1, 2, 3, 4, 6, 5", "1, 2, 3, 5, 4, 6", "1, 2, 4, 3, 5, 6", "1, 3, 2, 4, 5, 6"]}, "cell": "perm_inversions/0", "stage": "C"}
+{"id": "po-perm_inversions-t0-eval-000004", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 with exactly 2 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 with exactly 2 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "2, 1, 3, 4, 6, 5", "ref": "2, 1, 3, 4, 6, 5", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005556, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 6, "inv": 2, "first": 2, "last": null}, "seed": 28913166900000028, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 6, "valid_answers": ["2, 1, 3, 4, 6, 5", "2, 1, 3, 5, 4, 6", "2, 1, 4, 3, 5, 6", "2, 3, 1, 4, 5, 6"]}, "cell": "perm_inversions/0", "stage": "C"}
+{"id": "po-perm_inversions-t0-eval-000019", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 with exactly 9 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 with exactly 9 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "1, 5, 6, 4, 3, 2", "ref": "1, 5, 6, 4, 3, 2", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005556, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 6, "inv": 9, "first": 1, "last": null}, "seed": 28913166900000190, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 6, "valid_answers": ["1, 5, 6, 4, 3, 2", "1, 6, 4, 5, 3, 2", "1, 6, 5, 3, 4, 2", "1, 6, 5, 4, 2, 3"]}, "cell": "perm_inversions/0", "stage": "C"}
+{"id": "po-perm_inversions-t0-eval-000000", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 6 with exactly 10 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 6 with exactly 10 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(6).\n\n<think>\n", "answer": "3, 4, 6, 5, 2, 1", "ref": "3, 4, 6, 5, 2, 1", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.0125, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 6, "inv": 10, "first": 3, "last": null}, "seed": 28913166900000002, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 6, "valid_answers": ["3, 4, 6, 5, 2, 1", "3, 5, 4, 6, 2, 1", "3, 5, 6, 2, 4, 1", "3, 5, 6, 4, 1, 2", "3, 6, 2, 5, 4, 1", "3, 6, 4, 2, 5, 1", "3, 6, 4, 5, 1, 2", "3, 6, 5, 1, 4, 2", "3, 6, 5, 2, 1, 4"]}, "cell": "perm_inversions/0", "stage": "C"}
+{"id": "po-perm_inversions-t1-eval-000045", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 3 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 3 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "3, 1, 2, 4, 5, 7, 6", "ref": "3, 1, 2, 4, 5, 7, 6", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000992, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 3, "first": 3, "last": null}, "seed": 28913166901000676, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 7, "valid_answers": ["3, 1, 2, 4, 5, 7, 6", "3, 1, 2, 4, 6, 5, 7", "3, 1, 2, 5, 4, 6, 7", "3, 1, 4, 2, 5, 6, 7", "3, 2, 1, 4, 5, 6, 7"]}, "cell": "perm_inversions/1", "stage": "C"}
+{"id": "po-perm_inversions-t1-eval-000011", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 16 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 16 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "6, 2, 7, 5, 4, 3, 1", "ref": "6, 2, 7, 5, 4, 3, 1", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 49, "bucket": null, "guess_rate": 0.009722, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 16, "first": 6, "last": null}, "seed": 28913166901000064, "version": "po-v0", "tier": 1, "n_raw": 49, "small": 7, "valid_answers": ["6, 2, 7, 5, 4, 3, 1", "6, 3, 5, 7, 4, 2, 1", "6, 3, 7, 4, 5, 2, 1", "6, 3, 7, 5, 2, 4, 1", "6, 3, 7, 5, 4, 1, 2", "6, 4, 3, 7, 5, 2, 1", "6, 4, 5, 3, 7, 2, 1", "6, 4, 5, 7, 2, 3, 1", "6, 4, 5, 7, 3, 1, 2", "6, 4, 7, 2, 5, 3, 1", "6, 4, 7, 3, 2, 5, 1", "6, 4, 7, 3, 5, 1, 2", "6, 4, 7, 5, 1, 3, 2", "6, 4, 7, 5, 2, 1, 3", "6, 5, 2, 7, 4, 3, 1", "6, 5, 3, 4, 7, 2, 1", "6, 5, 3, 7, 2, 4, 1", "6, 5, 3, 7, 4, 1, 2", "6, 5, 4, 2, 7, 3, 1", "6, 5, 4, 3, 2, 7, 1", "6, 5, 4, 3, 7, 1, 2", "6, 5, 4, 7, 1, 3, 2", "6, 5, 4, 7, 2, 1, 3", "6, 5, 7, 1, 4, 3, 2", "6, 5, 7, 2, 3, 4, 1", "6, 5, 7, 2, 4, 1, 3", "6, 5, 7, 3, 1, 4, 2", "6, 5, 7, 3, 2, 1, 4", "6, 5, 7, 4, 1, 2, 3", "6, 7, 1, 5, 4, 3, 2", "6, 7, 2, 4, 5, 3, 1", "6, 7, 2, 5, 3, 4, 1", "6, 7, 2, 5, 4, 1, 3", "6, 7, 3, 2, 5, 4, 1", "6, 7, 3, 4, 2, 5, 1", "6, 7, 3, 4, 5, 1, 2", "6, 7, 3, 5, 1, 4, 2", "6, 7, 3, 5, 2, 1, 4", "6, 7, 4, 1, 5, 3, 2", "6, 7, 4, 2, 3, 5, 1", "6, 7, 4, 2, 5, 1, 3", "6, 7, 4, 3, 1, 5, 2", "6, 7, 4, 3, 2, 1, 5", "6, 7, 4, 5, 1, 2, 3", "6, 7, 5, 1, 3, 4, 2", "6, 7, 5, 1, 4, 2, 3", "6, 7, 5, 2, 1, 4, 3", "6, 7, 5, 2, 3, 1, 4", "6, 7, 5, 3, 1, 2, 4"]}, "cell": "perm_inversions/1", "stage": "C"}
+{"id": "po-perm_inversions-t1-eval-000008", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 7 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 7 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "4, 1, 2, 5, 7, 6, 3", "ref": "4, 1, 2, 5, 7, 6, 3", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 49, "bucket": null, "guess_rate": 0.009722, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 7, "first": 4, "last": null}, "seed": 28913166901000040, "version": "po-v0", "tier": 1, "n_raw": 49, "small": 7, "valid_answers": ["4, 1, 2, 5, 7, 6, 3", "4, 1, 2, 6, 5, 7, 3", "4, 1, 2, 6, 7, 3, 5", "4, 1, 2, 7, 3, 6, 5", "4, 1, 2, 7, 5, 3, 6", "4, 1, 3, 2, 7, 6, 5", "4, 1, 3, 5, 6, 7, 2", "4, 1, 3, 5, 7, 2, 6", "4, 1, 3, 6, 2, 7, 5", "4, 1, 3, 6, 5, 2, 7", "4, 1, 3, 7, 2, 5, 6", "4, 1, 5, 2, 6, 7, 3", "4, 1, 5, 2, 7, 3, 6", "4, 1, 5, 3, 2, 7, 6", "4, 1, 5, 3, 6, 2, 7", "4, 1, 5, 6, 2, 3, 7", "4, 1, 6, 2, 3, 7, 5", "4, 1, 6, 2, 5, 3, 7", "4, 1, 6, 3, 2, 5, 7", "4, 1, 7, 2, 3, 5, 6", "4, 2, 1, 3, 7, 6, 5", "4, 2, 1, 5, 6, 7, 3", "4, 2, 1, 5, 7, 3, 6", "4, 2, 1, 6, 3, 7, 5", "4, 2, 1, 6, 5, 3, 7", "4, 2, 1, 7, 3, 5, 6", "4, 2, 3, 1, 6, 7, 5", "4, 2, 3, 1, 7, 5, 6", "4, 2, 3, 5, 1, 7, 6", "4, 2, 3, 5, 6, 1, 7", "4, 2, 3, 6, 1, 5, 7", "4, 2, 5, 1, 3, 7, 6", "4, 2, 5, 1, 6, 3, 7", "4, 2, 5, 3, 1, 6, 7", "4, 2, 6, 1, 3, 5, 7", "4, 3, 1, 2, 6, 7, 5", "4, 3, 1, 2, 7, 5, 6", "4, 3, 1, 5, 2, 7, 6", "4, 3, 1, 5, 6, 2, 7", "4, 3, 1, 6, 2, 5, 7", "4, 3, 2, 1, 5, 7, 6", "4, 3, 2, 1, 6, 5, 7", "4, 3, 2, 5, 1, 6, 7", "4, 3, 5, 1, 2, 6, 7", "4, 5, 1, 2, 3, 7, 6", "4, 5, 1, 2, 6, 3, 7", "4, 5, 1, 3, 2, 6, 7", "4, 5, 2, 1, 3, 6, 7", "4, 6, 1, 2, 3, 5, 7"]}, "cell": "perm_inversions/1", "stage": "C"}
+{"id": "po-perm_inversions-t1-eval-000055", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 14 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 14 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 6, 7, 5, 4, 3, 2", "ref": "1, 6, 7, 5, 4, 3, 2", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000992, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 14, "first": 1, "last": null}, "seed": 28913166901002728, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 7, "valid_answers": ["1, 6, 7, 5, 4, 3, 2", "1, 7, 5, 6, 4, 3, 2", "1, 7, 6, 4, 5, 3, 2", "1, 7, 6, 5, 3, 4, 2", "1, 7, 6, 5, 4, 2, 3"]}, "cell": "perm_inversions/1", "stage": "C"}
+{"id": "po-perm_inversions-t1-eval-000020", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 19 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 19 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "7, 4, 6, 5, 3, 2, 1", "ref": "7, 4, 6, 5, 3, 2, 1", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.002778, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 19, "first": 7, "last": null}, "seed": 28913166901000132, "version": "po-v0", "tier": 1, "n_raw": 14, "small": 7, "valid_answers": ["7, 4, 6, 5, 3, 2, 1", "7, 5, 4, 6, 3, 2, 1", "7, 5, 6, 3, 4, 2, 1", "7, 5, 6, 4, 2, 3, 1", "7, 5, 6, 4, 3, 1, 2", "7, 6, 3, 5, 4, 2, 1", "7, 6, 4, 3, 5, 2, 1", "7, 6, 4, 5, 2, 3, 1", "7, 6, 4, 5, 3, 1, 2", "7, 6, 5, 2, 4, 3, 1", "7, 6, 5, 3, 2, 4, 1", "7, 6, 5, 3, 4, 1, 2", "7, 6, 5, 4, 1, 3, 2", "7, 6, 5, 4, 2, 1, 3"]}, "cell": "perm_inversions/1", "stage": "C"}
+{"id": "po-perm_inversions-t1-eval-000013", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 10 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 10 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "7, 1, 2, 4, 6, 5, 3", "ref": "7, 1, 2, 4, 6, 5, 3", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 49, "bucket": null, "guess_rate": 0.009722, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 10, "first": 7, "last": null}, "seed": 28913166901000079, "version": "po-v0", "tier": 1, "n_raw": 49, "small": 7, "valid_answers": ["7, 1, 2, 4, 6, 5, 3", "7, 1, 2, 5, 4, 6, 3", "7, 1, 2, 5, 6, 3, 4", "7, 1, 2, 6, 3, 5, 4", "7, 1, 2, 6, 4, 3, 5", "7, 1, 3, 2, 6, 5, 4", "7, 1, 3, 4, 5, 6, 2", "7, 1, 3, 4, 6, 2, 5", "7, 1, 3, 5, 2, 6, 4", "7, 1, 3, 5, 4, 2, 6", "7, 1, 3, 6, 2, 4, 5", "7, 1, 4, 2, 5, 6, 3", "7, 1, 4, 2, 6, 3, 5", "7, 1, 4, 3, 2, 6, 5", "7, 1, 4, 3, 5, 2, 6", "7, 1, 4, 5, 2, 3, 6", "7, 1, 5, 2, 3, 6, 4", "7, 1, 5, 2, 4, 3, 6", "7, 1, 5, 3, 2, 4, 6", "7, 1, 6, 2, 3, 4, 5", "7, 2, 1, 3, 6, 5, 4", "7, 2, 1, 4, 5, 6, 3", "7, 2, 1, 4, 6, 3, 5", "7, 2, 1, 5, 3, 6, 4", "7, 2, 1, 5, 4, 3, 6", "7, 2, 1, 6, 3, 4, 5", "7, 2, 3, 1, 5, 6, 4", "7, 2, 3, 1, 6, 4, 5", "7, 2, 3, 4, 1, 6, 5", "7, 2, 3, 4, 5, 1, 6", "7, 2, 3, 5, 1, 4, 6", "7, 2, 4, 1, 3, 6, 5", "7, 2, 4, 1, 5, 3, 6", "7, 2, 4, 3, 1, 5, 6", "7, 2, 5, 1, 3, 4, 6", "7, 3, 1, 2, 5, 6, 4", "7, 3, 1, 2, 6, 4, 5", "7, 3, 1, 4, 2, 6, 5", "7, 3, 1, 4, 5, 2, 6", "7, 3, 1, 5, 2, 4, 6", "7, 3, 2, 1, 4, 6, 5", "7, 3, 2, 1, 5, 4, 6", "7, 3, 2, 4, 1, 5, 6", "7, 3, 4, 1, 2, 5, 6", "7, 4, 1, 2, 3, 6, 5", "7, 4, 1, 2, 5, 3, 6", "7, 4, 1, 3, 2, 5, 6", "7, 4, 2, 1, 3, 5, 6", "7, 5, 1, 2, 3, 4, 6"]}, "cell": "perm_inversions/1", "stage": "C"}
+{"id": "po-perm_inversions-t1-eval-000000", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 5 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 5 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "2, 1, 3, 5, 7, 6, 4", "ref": "2, 1, 3, 5, 7, 6, 4", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 49, "bucket": null, "guess_rate": 0.009722, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 5, "first": 2, "last": null}, "seed": 28913166901000000, "version": "po-v0", "tier": 1, "n_raw": 49, "small": 7, "valid_answers": ["2, 1, 3, 5, 7, 6, 4", "2, 1, 3, 6, 5, 7, 4", "2, 1, 3, 6, 7, 4, 5", "2, 1, 3, 7, 4, 6, 5", "2, 1, 3, 7, 5, 4, 6", "2, 1, 4, 3, 7, 6, 5", "2, 1, 4, 5, 6, 7, 3", "2, 1, 4, 5, 7, 3, 6", "2, 1, 4, 6, 3, 7, 5", "2, 1, 4, 6, 5, 3, 7", "2, 1, 4, 7, 3, 5, 6", "2, 1, 5, 3, 6, 7, 4", "2, 1, 5, 3, 7, 4, 6", "2, 1, 5, 4, 3, 7, 6", "2, 1, 5, 4, 6, 3, 7", "2, 1, 5, 6, 3, 4, 7", "2, 1, 6, 3, 4, 7, 5", "2, 1, 6, 3, 5, 4, 7", "2, 1, 6, 4, 3, 5, 7", "2, 1, 7, 3, 4, 5, 6", "2, 3, 1, 4, 7, 6, 5", "2, 3, 1, 5, 6, 7, 4", "2, 3, 1, 5, 7, 4, 6", "2, 3, 1, 6, 4, 7, 5", "2, 3, 1, 6, 5, 4, 7", "2, 3, 1, 7, 4, 5, 6", "2, 3, 4, 1, 6, 7, 5", "2, 3, 4, 1, 7, 5, 6", "2, 3, 4, 5, 1, 7, 6", "2, 3, 4, 5, 6, 1, 7", "2, 3, 4, 6, 1, 5, 7", "2, 3, 5, 1, 4, 7, 6", "2, 3, 5, 1, 6, 4, 7", "2, 3, 5, 4, 1, 6, 7", "2, 3, 6, 1, 4, 5, 7", "2, 4, 1, 3, 6, 7, 5", "2, 4, 1, 3, 7, 5, 6", "2, 4, 1, 5, 3, 7, 6", "2, 4, 1, 5, 6, 3, 7", "2, 4, 1, 6, 3, 5, 7", "2, 4, 3, 1, 5, 7, 6", "2, 4, 3, 1, 6, 5, 7", "2, 4, 3, 5, 1, 6, 7", "2, 4, 5, 1, 3, 6, 7", "2, 5, 1, 3, 4, 7, 6", "2, 5, 1, 3, 6, 4, 7", "2, 5, 1, 4, 3, 6, 7", "2, 5, 3, 1, 4, 6, 7", "2, 6, 1, 3, 4, 5, 7"]}, "cell": "perm_inversions/1", "stage": "C"}
+{"id": "po-perm_inversions-t1-eval-000051", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 19 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 19 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "6, 5, 7, 4, 3, 2, 1", "ref": "6, 5, 7, 4, 3, 2, 1", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000992, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 19, "first": 6, "last": null}, "seed": 28913166901001096, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 7, "valid_answers": ["6, 5, 7, 4, 3, 2, 1", "6, 7, 4, 5, 3, 2, 1", "6, 7, 5, 3, 4, 2, 1", "6, 7, 5, 4, 2, 3, 1", "6, 7, 5, 4, 3, 1, 2"]}, "cell": "perm_inversions/1", "stage": "C"}
+{"id": "po-perm_inversions-t1-eval-000024", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 16 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 16 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "4, 5, 7, 6, 3, 2, 1", "ref": "4, 5, 7, 6, 3, 2, 1", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.002778, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 16, "first": 4, "last": null}, "seed": 28913166901000151, "version": "po-v0", "tier": 1, "n_raw": 14, "small": 7, "valid_answers": ["4, 5, 7, 6, 3, 2, 1", "4, 6, 5, 7, 3, 2, 1", "4, 6, 7, 3, 5, 2, 1", "4, 6, 7, 5, 2, 3, 1", "4, 6, 7, 5, 3, 1, 2", "4, 7, 3, 6, 5, 2, 1", "4, 7, 5, 3, 6, 2, 1", "4, 7, 5, 6, 2, 3, 1", "4, 7, 5, 6, 3, 1, 2", "4, 7, 6, 2, 5, 3, 1", "4, 7, 6, 3, 2, 5, 1", "4, 7, 6, 3, 5, 1, 2", "4, 7, 6, 5, 1, 3, 2", "4, 7, 6, 5, 2, 1, 3"]}, "cell": "perm_inversions/1", "stage": "C"}
+{"id": "po-perm_inversions-t1-eval-000042", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 14 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 14 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "3, 4, 7, 6, 5, 2, 1", "ref": "3, 4, 7, 6, 5, 2, 1", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 29, "bucket": null, "guess_rate": 0.005754, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 14, "first": 3, "last": null}, "seed": 28913166901000408, "version": "po-v0", "tier": 1, "n_raw": 29, "small": 7, "valid_answers": ["3, 4, 7, 6, 5, 2, 1", "3, 5, 6, 7, 4, 2, 1", "3, 5, 7, 4, 6, 2, 1", "3, 5, 7, 6, 2, 4, 1", "3, 5, 7, 6, 4, 1, 2", "3, 6, 4, 7, 5, 2, 1", "3, 6, 5, 4, 7, 2, 1", "3, 6, 5, 7, 2, 4, 1", "3, 6, 5, 7, 4, 1, 2", "3, 6, 7, 2, 5, 4, 1", "3, 6, 7, 4, 2, 5, 1", "3, 6, 7, 4, 5, 1, 2", "3, 6, 7, 5, 1, 4, 2", "3, 6, 7, 5, 2, 1, 4", "3, 7, 2, 6, 5, 4, 1", "3, 7, 4, 5, 6, 2, 1", "3, 7, 4, 6, 2, 5, 1", "3, 7, 4, 6, 5, 1, 2", "3, 7, 5, 2, 6, 4, 1", "3, 7, 5, 4, 2, 6, 1", "3, 7, 5, 4, 6, 1, 2", "3, 7, 5, 6, 1, 4, 2", "3, 7, 5, 6, 2, 1, 4", "3, 7, 6, 1, 5, 4, 2", "3, 7, 6, 2, 4, 5, 1", "3, 7, 6, 2, 5, 1, 4", "3, 7, 6, 4, 1, 5, 2", "3, 7, 6, 4, 2, 1, 5", "3, 7, 6, 5, 1, 2, 4"]}, "cell": "perm_inversions/1", "stage": "C"}
+{"id": "po-perm_inversions-t1-eval-000005", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 8 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 8 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "7, 1, 2, 3, 5, 6, 4", "ref": "7, 1, 2, 3, 5, 6, 4", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.002778, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 8, "first": 7, "last": null}, "seed": 28913166901000021, "version": "po-v0", "tier": 1, "n_raw": 14, "small": 7, "valid_answers": ["7, 1, 2, 3, 5, 6, 4", "7, 1, 2, 3, 6, 4, 5", "7, 1, 2, 4, 3, 6, 5", "7, 1, 2, 4, 5, 3, 6", "7, 1, 2, 5, 3, 4, 6", "7, 1, 3, 2, 4, 6, 5", "7, 1, 3, 2, 5, 4, 6", "7, 1, 3, 4, 2, 5, 6", "7, 1, 4, 2, 3, 5, 6", "7, 2, 1, 3, 4, 6, 5", "7, 2, 1, 3, 5, 4, 6", "7, 2, 1, 4, 3, 5, 6", "7, 2, 3, 1, 4, 5, 6", "7, 3, 1, 2, 4, 5, 6"]}, "cell": "perm_inversions/1", "stage": "C"}
+{"id": "po-perm_inversions-t1-eval-000025", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 11 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 11 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 3, 7, 6, 5, 4, 2", "ref": "1, 3, 7, 6, 5, 4, 2", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 49, "bucket": null, "guess_rate": 0.009722, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 11, "first": 1, "last": null}, "seed": 28913166901000154, "version": "po-v0", "tier": 1, "n_raw": 49, "small": 7, "valid_answers": ["1, 3, 7, 6, 5, 4, 2", "1, 4, 6, 7, 5, 3, 2", "1, 4, 7, 5, 6, 3, 2", "1, 4, 7, 6, 3, 5, 2", "1, 4, 7, 6, 5, 2, 3", "1, 5, 4, 7, 6, 3, 2", "1, 5, 6, 4, 7, 3, 2", "1, 5, 6, 7, 3, 4, 2", "1, 5, 6, 7, 4, 2, 3", "1, 5, 7, 3, 6, 4, 2", "1, 5, 7, 4, 3, 6, 2", "1, 5, 7, 4, 6, 2, 3", "1, 5, 7, 6, 2, 4, 3", "1, 5, 7, 6, 3, 2, 4", "1, 6, 3, 7, 5, 4, 2", "1, 6, 4, 5, 7, 3, 2", "1, 6, 4, 7, 3, 5, 2", "1, 6, 4, 7, 5, 2, 3", "1, 6, 5, 3, 7, 4, 2", "1, 6, 5, 4, 3, 7, 2", "1, 6, 5, 4, 7, 2, 3", "1, 6, 5, 7, 2, 4, 3", "1, 6, 5, 7, 3, 2, 4", "1, 6, 7, 2, 5, 4, 3", "1, 6, 7, 3, 4, 5, 2", "1, 6, 7, 3, 5, 2, 4", "1, 6, 7, 4, 2, 5, 3", "1, 6, 7, 4, 3, 2, 5", "1, 6, 7, 5, 2, 3, 4", "1, 7, 2, 6, 5, 4, 3", "1, 7, 3, 5, 6, 4, 2", "1, 7, 3, 6, 4, 5, 2", "1, 7, 3, 6, 5, 2, 4", "1, 7, 4, 3, 6, 5, 2", "1, 7, 4, 5, 3, 6, 2", "1, 7, 4, 5, 6, 2, 3", "1, 7, 4, 6, 2, 5, 3", "1, 7, 4, 6, 3, 2, 5", "1, 7, 5, 2, 6, 4, 3", "1, 7, 5, 3, 4, 6, 2", "1, 7, 5, 3, 6, 2, 4", "1, 7, 5, 4, 2, 6, 3", "1, 7, 5, 4, 3, 2, 6", "1, 7, 5, 6, 2, 3, 4", "1, 7, 6, 2, 4, 5, 3", "1, 7, 6, 2, 5, 3, 4", "1, 7, 6, 3, 2, 5, 4", "1, 7, 6, 3, 4, 2, 5", "1, 7, 6, 4, 2, 3, 5"]}, "cell": "perm_inversions/1", "stage": "C"}
+{"id": "po-perm_inversions-t2-eval-000019", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 8 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 8 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "7, 1, 2, 3, 5, 6, 4", "ref": "7, 1, 2, 3, 5, 6, 4", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.002778, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 8, "first": 7, "last": null}, "seed": 28913166902000106, "version": "po-v0", "tier": 2, "n_raw": 14, "small": 7, "valid_answers": ["7, 1, 2, 3, 5, 6, 4", "7, 1, 2, 3, 6, 4, 5", "7, 1, 2, 4, 3, 6, 5", "7, 1, 2, 4, 5, 3, 6", "7, 1, 2, 5, 3, 4, 6", "7, 1, 3, 2, 4, 6, 5", "7, 1, 3, 2, 5, 4, 6", "7, 1, 3, 4, 2, 5, 6", "7, 1, 4, 2, 3, 5, 6", "7, 2, 1, 3, 4, 6, 5", "7, 2, 1, 3, 5, 4, 6", "7, 2, 1, 4, 3, 5, 6", "7, 2, 3, 1, 4, 5, 6", "7, 3, 1, 2, 4, 5, 6"]}, "cell": "perm_inversions/2", "stage": "C"}
+{"id": "po-perm_inversions-t2-eval-000053", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 3 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 3 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "3, 1, 2, 4, 5, 7, 6", "ref": "3, 1, 2, 4, 5, 7, 6", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000992, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 3, "first": 3, "last": null}, "seed": 28913166902001816, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 7, "valid_answers": ["3, 1, 2, 4, 5, 7, 6", "3, 1, 2, 4, 6, 5, 7", "3, 1, 2, 5, 4, 6, 7", "3, 1, 4, 2, 5, 6, 7", "3, 2, 1, 4, 5, 6, 7"]}, "cell": "perm_inversions/2", "stage": "C"}
+{"id": "po-perm_inversions-t2-eval-000021", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 19 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 19 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "6, 5, 7, 4, 3, 2, 1", "ref": "6, 5, 7, 4, 3, 2, 1", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000992, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 19, "first": 6, "last": null}, "seed": 28913166902000138, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 7, "valid_answers": ["6, 5, 7, 4, 3, 2, 1", "6, 7, 4, 5, 3, 2, 1", "6, 7, 5, 3, 4, 2, 1", "6, 7, 5, 4, 2, 3, 1", "6, 7, 5, 4, 3, 1, 2"]}, "cell": "perm_inversions/2", "stage": "C"}
+{"id": "po-perm_inversions-t2-eval-000001", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 19 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 19 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "7, 4, 6, 5, 3, 2, 1", "ref": "7, 4, 6, 5, 3, 2, 1", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.002778, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 19, "first": 7, "last": null}, "seed": 28913166902000014, "version": "po-v0", "tier": 2, "n_raw": 14, "small": 7, "valid_answers": ["7, 4, 6, 5, 3, 2, 1", "7, 5, 4, 6, 3, 2, 1", "7, 5, 6, 3, 4, 2, 1", "7, 5, 6, 4, 2, 3, 1", "7, 5, 6, 4, 3, 1, 2", "7, 6, 3, 5, 4, 2, 1", "7, 6, 4, 3, 5, 2, 1", "7, 6, 4, 5, 2, 3, 1", "7, 6, 4, 5, 3, 1, 2", "7, 6, 5, 2, 4, 3, 1", "7, 6, 5, 3, 2, 4, 1", "7, 6, 5, 3, 4, 1, 2", "7, 6, 5, 4, 1, 3, 2", "7, 6, 5, 4, 2, 1, 3"]}, "cell": "perm_inversions/2", "stage": "C"}
+{"id": "po-perm_inversions-t2-eval-000017", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 6 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 6 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "5, 1, 2, 3, 6, 7, 4", "ref": "5, 1, 2, 3, 6, 7, 4", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.002778, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 6, "first": 5, "last": null}, "seed": 28913166902000099, "version": "po-v0", "tier": 2, "n_raw": 14, "small": 7, "valid_answers": ["5, 1, 2, 3, 6, 7, 4", "5, 1, 2, 3, 7, 4, 6", "5, 1, 2, 4, 3, 7, 6", "5, 1, 2, 4, 6, 3, 7", "5, 1, 2, 6, 3, 4, 7", "5, 1, 3, 2, 4, 7, 6", "5, 1, 3, 2, 6, 4, 7", "5, 1, 3, 4, 2, 6, 7", "5, 1, 4, 2, 3, 6, 7", "5, 2, 1, 3, 4, 7, 6", "5, 2, 1, 3, 6, 4, 7", "5, 2, 1, 4, 3, 6, 7", "5, 2, 3, 1, 4, 6, 7", "5, 3, 1, 2, 4, 6, 7"]}, "cell": "perm_inversions/2", "stage": "C"}
+{"id": "po-perm_inversions-t2-eval-000015", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 11 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 11 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 3, 7, 6, 5, 4, 2", "ref": "1, 3, 7, 6, 5, 4, 2", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 49, "bucket": null, "guess_rate": 0.009722, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 11, "first": 1, "last": null}, "seed": 28913166902000075, "version": "po-v0", "tier": 2, "n_raw": 49, "small": 7, "valid_answers": ["1, 3, 7, 6, 5, 4, 2", "1, 4, 6, 7, 5, 3, 2", "1, 4, 7, 5, 6, 3, 2", "1, 4, 7, 6, 3, 5, 2", "1, 4, 7, 6, 5, 2, 3", "1, 5, 4, 7, 6, 3, 2", "1, 5, 6, 4, 7, 3, 2", "1, 5, 6, 7, 3, 4, 2", "1, 5, 6, 7, 4, 2, 3", "1, 5, 7, 3, 6, 4, 2", "1, 5, 7, 4, 3, 6, 2", "1, 5, 7, 4, 6, 2, 3", "1, 5, 7, 6, 2, 4, 3", "1, 5, 7, 6, 3, 2, 4", "1, 6, 3, 7, 5, 4, 2", "1, 6, 4, 5, 7, 3, 2", "1, 6, 4, 7, 3, 5, 2", "1, 6, 4, 7, 5, 2, 3", "1, 6, 5, 3, 7, 4, 2", "1, 6, 5, 4, 3, 7, 2", "1, 6, 5, 4, 7, 2, 3", "1, 6, 5, 7, 2, 4, 3", "1, 6, 5, 7, 3, 2, 4", "1, 6, 7, 2, 5, 4, 3", "1, 6, 7, 3, 4, 5, 2", "1, 6, 7, 3, 5, 2, 4", "1, 6, 7, 4, 2, 5, 3", "1, 6, 7, 4, 3, 2, 5", "1, 6, 7, 5, 2, 3, 4", "1, 7, 2, 6, 5, 4, 3", "1, 7, 3, 5, 6, 4, 2", "1, 7, 3, 6, 4, 5, 2", "1, 7, 3, 6, 5, 2, 4", "1, 7, 4, 3, 6, 5, 2", "1, 7, 4, 5, 3, 6, 2", "1, 7, 4, 5, 6, 2, 3", "1, 7, 4, 6, 2, 5, 3", "1, 7, 4, 6, 3, 2, 5", "1, 7, 5, 2, 6, 4, 3", "1, 7, 5, 3, 4, 6, 2", "1, 7, 5, 3, 6, 2, 4", "1, 7, 5, 4, 2, 6, 3", "1, 7, 5, 4, 3, 2, 6", "1, 7, 5, 6, 2, 3, 4", "1, 7, 6, 2, 4, 5, 3", "1, 7, 6, 2, 5, 3, 4", "1, 7, 6, 3, 2, 5, 4", "1, 7, 6, 3, 4, 2, 5", "1, 7, 6, 4, 2, 3, 5"]}, "cell": "perm_inversions/2", "stage": "C"}
+{"id": "po-perm_inversions-t2-eval-000004", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 10 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 10 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "7, 1, 2, 4, 6, 5, 3", "ref": "7, 1, 2, 4, 6, 5, 3", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 49, "bucket": null, "guess_rate": 0.009722, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 10, "first": 7, "last": null}, "seed": 28913166902000026, "version": "po-v0", "tier": 2, "n_raw": 49, "small": 7, "valid_answers": ["7, 1, 2, 4, 6, 5, 3", "7, 1, 2, 5, 4, 6, 3", "7, 1, 2, 5, 6, 3, 4", "7, 1, 2, 6, 3, 5, 4", "7, 1, 2, 6, 4, 3, 5", "7, 1, 3, 2, 6, 5, 4", "7, 1, 3, 4, 5, 6, 2", "7, 1, 3, 4, 6, 2, 5", "7, 1, 3, 5, 2, 6, 4", "7, 1, 3, 5, 4, 2, 6", "7, 1, 3, 6, 2, 4, 5", "7, 1, 4, 2, 5, 6, 3", "7, 1, 4, 2, 6, 3, 5", "7, 1, 4, 3, 2, 6, 5", "7, 1, 4, 3, 5, 2, 6", "7, 1, 4, 5, 2, 3, 6", "7, 1, 5, 2, 3, 6, 4", "7, 1, 5, 2, 4, 3, 6", "7, 1, 5, 3, 2, 4, 6", "7, 1, 6, 2, 3, 4, 5", "7, 2, 1, 3, 6, 5, 4", "7, 2, 1, 4, 5, 6, 3", "7, 2, 1, 4, 6, 3, 5", "7, 2, 1, 5, 3, 6, 4", "7, 2, 1, 5, 4, 3, 6", "7, 2, 1, 6, 3, 4, 5", "7, 2, 3, 1, 5, 6, 4", "7, 2, 3, 1, 6, 4, 5", "7, 2, 3, 4, 1, 6, 5", "7, 2, 3, 4, 5, 1, 6", "7, 2, 3, 5, 1, 4, 6", "7, 2, 4, 1, 3, 6, 5", "7, 2, 4, 1, 5, 3, 6", "7, 2, 4, 3, 1, 5, 6", "7, 2, 5, 1, 3, 4, 6", "7, 3, 1, 2, 5, 6, 4", "7, 3, 1, 2, 6, 4, 5", "7, 3, 1, 4, 2, 6, 5", "7, 3, 1, 4, 5, 2, 6", "7, 3, 1, 5, 2, 4, 6", "7, 3, 2, 1, 4, 6, 5", "7, 3, 2, 1, 5, 4, 6", "7, 3, 2, 4, 1, 5, 6", "7, 3, 4, 1, 2, 5, 6", "7, 4, 1, 2, 3, 6, 5", "7, 4, 1, 2, 5, 3, 6", "7, 4, 1, 3, 2, 5, 6", "7, 4, 2, 1, 3, 5, 6", "7, 5, 1, 2, 3, 4, 6"]}, "cell": "perm_inversions/2", "stage": "C"}
+{"id": "po-perm_inversions-t2-eval-000042", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 5 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 5 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "4, 1, 2, 3, 6, 7, 5", "ref": "4, 1, 2, 3, 6, 7, 5", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.002778, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 5, "first": 4, "last": null}, "seed": 28913166902000443, "version": "po-v0", "tier": 2, "n_raw": 14, "small": 7, "valid_answers": ["4, 1, 2, 3, 6, 7, 5", "4, 1, 2, 3, 7, 5, 6", "4, 1, 2, 5, 3, 7, 6", "4, 1, 2, 5, 6, 3, 7", "4, 1, 2, 6, 3, 5, 7", "4, 1, 3, 2, 5, 7, 6", "4, 1, 3, 2, 6, 5, 7", "4, 1, 3, 5, 2, 6, 7", "4, 1, 5, 2, 3, 6, 7", "4, 2, 1, 3, 5, 7, 6", "4, 2, 1, 3, 6, 5, 7", "4, 2, 1, 5, 3, 6, 7", "4, 2, 3, 1, 5, 6, 7", "4, 3, 1, 2, 5, 6, 7"]}, "cell": "perm_inversions/2", "stage": "C"}
+{"id": "po-perm_inversions-t2-eval-000013", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 5 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 5 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "2, 1, 3, 5, 7, 6, 4", "ref": "2, 1, 3, 5, 7, 6, 4", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 49, "bucket": null, "guess_rate": 0.009722, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 5, "first": 2, "last": null}, "seed": 28913166902000060, "version": "po-v0", "tier": 2, "n_raw": 49, "small": 7, "valid_answers": ["2, 1, 3, 5, 7, 6, 4", "2, 1, 3, 6, 5, 7, 4", "2, 1, 3, 6, 7, 4, 5", "2, 1, 3, 7, 4, 6, 5", "2, 1, 3, 7, 5, 4, 6", "2, 1, 4, 3, 7, 6, 5", "2, 1, 4, 5, 6, 7, 3", "2, 1, 4, 5, 7, 3, 6", "2, 1, 4, 6, 3, 7, 5", "2, 1, 4, 6, 5, 3, 7", "2, 1, 4, 7, 3, 5, 6", "2, 1, 5, 3, 6, 7, 4", "2, 1, 5, 3, 7, 4, 6", "2, 1, 5, 4, 3, 7, 6", "2, 1, 5, 4, 6, 3, 7", "2, 1, 5, 6, 3, 4, 7", "2, 1, 6, 3, 4, 7, 5", "2, 1, 6, 3, 5, 4, 7", "2, 1, 6, 4, 3, 5, 7", "2, 1, 7, 3, 4, 5, 6", "2, 3, 1, 4, 7, 6, 5", "2, 3, 1, 5, 6, 7, 4", "2, 3, 1, 5, 7, 4, 6", "2, 3, 1, 6, 4, 7, 5", "2, 3, 1, 6, 5, 4, 7", "2, 3, 1, 7, 4, 5, 6", "2, 3, 4, 1, 6, 7, 5", "2, 3, 4, 1, 7, 5, 6", "2, 3, 4, 5, 1, 7, 6", "2, 3, 4, 5, 6, 1, 7", "2, 3, 4, 6, 1, 5, 7", "2, 3, 5, 1, 4, 7, 6", "2, 3, 5, 1, 6, 4, 7", "2, 3, 5, 4, 1, 6, 7", "2, 3, 6, 1, 4, 5, 7", "2, 4, 1, 3, 6, 7, 5", "2, 4, 1, 3, 7, 5, 6", "2, 4, 1, 5, 3, 7, 6", "2, 4, 1, 5, 6, 3, 7", "2, 4, 1, 6, 3, 5, 7", "2, 4, 3, 1, 5, 7, 6", "2, 4, 3, 1, 6, 5, 7", "2, 4, 3, 5, 1, 6, 7", "2, 4, 5, 1, 3, 6, 7", "2, 5, 1, 3, 4, 7, 6", "2, 5, 1, 3, 6, 4, 7", "2, 5, 1, 4, 3, 6, 7", "2, 5, 3, 1, 4, 6, 7", "2, 6, 1, 3, 4, 5, 7"]}, "cell": "perm_inversions/2", "stage": "C"}
+{"id": "po-perm_inversions-t2-eval-000011", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 7 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 7 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "4, 1, 2, 5, 7, 6, 3", "ref": "4, 1, 2, 5, 7, 6, 3", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 49, "bucket": null, "guess_rate": 0.009722, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 7, "first": 4, "last": null}, "seed": 28913166902000055, "version": "po-v0", "tier": 2, "n_raw": 49, "small": 7, "valid_answers": ["4, 1, 2, 5, 7, 6, 3", "4, 1, 2, 6, 5, 7, 3", "4, 1, 2, 6, 7, 3, 5", "4, 1, 2, 7, 3, 6, 5", "4, 1, 2, 7, 5, 3, 6", "4, 1, 3, 2, 7, 6, 5", "4, 1, 3, 5, 6, 7, 2", "4, 1, 3, 5, 7, 2, 6", "4, 1, 3, 6, 2, 7, 5", "4, 1, 3, 6, 5, 2, 7", "4, 1, 3, 7, 2, 5, 6", "4, 1, 5, 2, 6, 7, 3", "4, 1, 5, 2, 7, 3, 6", "4, 1, 5, 3, 2, 7, 6", "4, 1, 5, 3, 6, 2, 7", "4, 1, 5, 6, 2, 3, 7", "4, 1, 6, 2, 3, 7, 5", "4, 1, 6, 2, 5, 3, 7", "4, 1, 6, 3, 2, 5, 7", "4, 1, 7, 2, 3, 5, 6", "4, 2, 1, 3, 7, 6, 5", "4, 2, 1, 5, 6, 7, 3", "4, 2, 1, 5, 7, 3, 6", "4, 2, 1, 6, 3, 7, 5", "4, 2, 1, 6, 5, 3, 7", "4, 2, 1, 7, 3, 5, 6", "4, 2, 3, 1, 6, 7, 5", "4, 2, 3, 1, 7, 5, 6", "4, 2, 3, 5, 1, 7, 6", "4, 2, 3, 5, 6, 1, 7", "4, 2, 3, 6, 1, 5, 7", "4, 2, 5, 1, 3, 7, 6", "4, 2, 5, 1, 6, 3, 7", "4, 2, 5, 3, 1, 6, 7", "4, 2, 6, 1, 3, 5, 7", "4, 3, 1, 2, 6, 7, 5", "4, 3, 1, 2, 7, 5, 6", "4, 3, 1, 5, 2, 7, 6", "4, 3, 1, 5, 6, 2, 7", "4, 3, 1, 6, 2, 5, 7", "4, 3, 2, 1, 5, 7, 6", "4, 3, 2, 1, 6, 5, 7", "4, 3, 2, 5, 1, 6, 7", "4, 3, 5, 1, 2, 6, 7", "4, 5, 1, 2, 3, 7, 6", "4, 5, 1, 2, 6, 3, 7", "4, 5, 1, 3, 2, 6, 7", "4, 5, 2, 1, 3, 6, 7", "4, 6, 1, 2, 3, 5, 7"]}, "cell": "perm_inversions/2", "stage": "C"}
+{"id": "po-perm_inversions-t2-eval-000035", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 16 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 16 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "4, 5, 7, 6, 3, 2, 1", "ref": "4, 5, 7, 6, 3, 2, 1", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.002778, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 16, "first": 4, "last": null}, "seed": 28913166902000330, "version": "po-v0", "tier": 2, "n_raw": 14, "small": 7, "valid_answers": ["4, 5, 7, 6, 3, 2, 1", "4, 6, 5, 7, 3, 2, 1", "4, 6, 7, 3, 5, 2, 1", "4, 6, 7, 5, 2, 3, 1", "4, 6, 7, 5, 3, 1, 2", "4, 7, 3, 6, 5, 2, 1", "4, 7, 5, 3, 6, 2, 1", "4, 7, 5, 6, 2, 3, 1", "4, 7, 5, 6, 3, 1, 2", "4, 7, 6, 2, 5, 3, 1", "4, 7, 6, 3, 2, 5, 1", "4, 7, 6, 3, 5, 1, 2", "4, 7, 6, 5, 1, 3, 2", "4, 7, 6, 5, 2, 1, 3"]}, "cell": "perm_inversions/2", "stage": "C"}
+{"id": "po-perm_inversions-t2-eval-000029", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 7 with exactly 14 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 7 with exactly 14 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(7).\n\n<think>\n", "answer": "1, 6, 7, 5, 4, 3, 2", "ref": "1, 6, 7, 5, 4, 3, 2", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000992, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 7, "inv": 14, "first": 1, "last": null}, "seed": 28913166902000223, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 7, "valid_answers": ["1, 6, 7, 5, 4, 3, 2", "1, 7, 5, 6, 4, 3, 2", "1, 7, 6, 4, 5, 3, 2", "1, 7, 6, 5, 3, 4, 2", "1, 7, 6, 5, 4, 2, 3"]}, "cell": "perm_inversions/2", "stage": "C"}
+{"id": "po-perm_inversions-t3-eval-000151", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 with exactly 19 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 3 and pi(8) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 with exactly 19 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 3 and pi(8) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "3, 4, 8, 7, 6, 5, 2, 1", "ref": "3, 4, 8, 7, 6, 5, 2, 1", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 49, "bucket": null, "guess_rate": 0.001215, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 8, "inv": 19, "first": 3, "last": 1}, "seed": 28913166903000745, "version": "po-v0", "tier": 3, "n_raw": 49, "small": 8, "valid_answers": ["3, 4, 8, 7, 6, 5, 2, 1", "3, 5, 7, 8, 6, 4, 2, 1", "3, 5, 8, 6, 7, 4, 2, 1", "3, 5, 8, 7, 4, 6, 2, 1", "3, 5, 8, 7, 6, 2, 4, 1", "3, 6, 5, 8, 7, 4, 2, 1", "3, 6, 7, 5, 8, 4, 2, 1", "3, 6, 7, 8, 4, 5, 2, 1", "3, 6, 7, 8, 5, 2, 4, 1", "3, 6, 8, 4, 7, 5, 2, 1", "3, 6, 8, 5, 4, 7, 2, 1", "3, 6, 8, 5, 7, 2, 4, 1", "3, 6, 8, 7, 2, 5, 4, 1", "3, 6, 8, 7, 4, 2, 5, 1", "3, 7, 4, 8, 6, 5, 2, 1", "3, 7, 5, 6, 8, 4, 2, 1", "3, 7, 5, 8, 4, 6, 2, 1", "3, 7, 5, 8, 6, 2, 4, 1", "3, 7, 6, 4, 8, 5, 2, 1", "3, 7, 6, 5, 4, 8, 2, 1", "3, 7, 6, 5, 8, 2, 4, 1", "3, 7, 6, 8, 2, 5, 4, 1", "3, 7, 6, 8, 4, 2, 5, 1", "3, 7, 8, 2, 6, 5, 4, 1", "3, 7, 8, 4, 5, 6, 2, 1", "3, 7, 8, 4, 6, 2, 5, 1", "3, 7, 8, 5, 2, 6, 4, 1", "3, 7, 8, 5, 4, 2, 6, 1", "3, 7, 8, 6, 2, 4, 5, 1", "3, 8, 2, 7, 6, 5, 4, 1", "3, 8, 4, 6, 7, 5, 2, 1", "3, 8, 4, 7, 5, 6, 2, 1", "3, 8, 4, 7, 6, 2, 5, 1", "3, 8, 5, 4, 7, 6, 2, 1", "3, 8, 5, 6, 4, 7, 2, 1", "3, 8, 5, 6, 7, 2, 4, 1", "3, 8, 5, 7, 2, 6, 4, 1", "3, 8, 5, 7, 4, 2, 6, 1", "3, 8, 6, 2, 7, 5, 4, 1", "3, 8, 6, 4, 5, 7, 2, 1", "3, 8, 6, 4, 7, 2, 5, 1", "3, 8, 6, 5, 2, 7, 4, 1", "3, 8, 6, 5, 4, 2, 7, 1", "3, 8, 6, 7, 2, 4, 5, 1", "3, 8, 7, 2, 5, 6, 4, 1", "3, 8, 7, 2, 6, 4, 5, 1", "3, 8, 7, 4, 2, 6, 5, 1", "3, 8, 7, 4, 5, 2, 6, 1", "3, 8, 7, 5, 2, 4, 6, 1"]}, "cell": "perm_inversions/3", "stage": "C"}
+{"id": "po-perm_inversions-t3-eval-000021", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 with exactly 13 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 8 and pi(8) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 with exactly 13 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 8 and pi(8) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "8, 1, 2, 4, 7, 6, 3, 5", "ref": "8, 1, 2, 4, 7, 6, 3, 5", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 49, "bucket": null, "guess_rate": 0.001215, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 8, "inv": 13, "first": 8, "last": 5}, "seed": 28913166903000072, "version": "po-v0", "tier": 3, "n_raw": 49, "small": 8, "valid_answers": ["8, 1, 2, 4, 7, 6, 3, 5", "8, 1, 2, 6, 4, 7, 3, 5", "8, 1, 2, 6, 7, 3, 4, 5", "8, 1, 2, 7, 3, 6, 4, 5", "8, 1, 2, 7, 4, 3, 6, 5", "8, 1, 3, 2, 7, 6, 4, 5", "8, 1, 3, 4, 6, 7, 2, 5", "8, 1, 3, 4, 7, 2, 6, 5", "8, 1, 3, 6, 2, 7, 4, 5", "8, 1, 3, 6, 4, 2, 7, 5", "8, 1, 3, 7, 2, 4, 6, 5", "8, 1, 4, 2, 6, 7, 3, 5", "8, 1, 4, 2, 7, 3, 6, 5", "8, 1, 4, 3, 2, 7, 6, 5", "8, 1, 4, 3, 6, 2, 7, 5", "8, 1, 4, 6, 2, 3, 7, 5", "8, 1, 6, 2, 3, 7, 4, 5", "8, 1, 6, 2, 4, 3, 7, 5", "8, 1, 6, 3, 2, 4, 7, 5", "8, 1, 7, 2, 3, 4, 6, 5", "8, 2, 1, 3, 7, 6, 4, 5", "8, 2, 1, 4, 6, 7, 3, 5", "8, 2, 1, 4, 7, 3, 6, 5", "8, 2, 1, 6, 3, 7, 4, 5", "8, 2, 1, 6, 4, 3, 7, 5", "8, 2, 1, 7, 3, 4, 6, 5", "8, 2, 3, 1, 6, 7, 4, 5", "8, 2, 3, 1, 7, 4, 6, 5", "8, 2, 3, 4, 1, 7, 6, 5", "8, 2, 3, 4, 6, 1, 7, 5", "8, 2, 3, 6, 1, 4, 7, 5", "8, 2, 4, 1, 3, 7, 6, 5", "8, 2, 4, 1, 6, 3, 7, 5", "8, 2, 4, 3, 1, 6, 7, 5", "8, 2, 6, 1, 3, 4, 7, 5", "8, 3, 1, 2, 6, 7, 4, 5", "8, 3, 1, 2, 7, 4, 6, 5", "8, 3, 1, 4, 2, 7, 6, 5", "8, 3, 1, 4, 6, 2, 7, 5", "8, 3, 1, 6, 2, 4, 7, 5", "8, 3, 2, 1, 4, 7, 6, 5", "8, 3, 2, 1, 6, 4, 7, 5", "8, 3, 2, 4, 1, 6, 7, 5", "8, 3, 4, 1, 2, 6, 7, 5", "8, 4, 1, 2, 3, 7, 6, 5", "8, 4, 1, 2, 6, 3, 7, 5", "8, 4, 1, 3, 2, 6, 7, 5", "8, 4, 2, 1, 3, 6, 7, 5", "8, 6, 1, 2, 3, 4, 7, 5"]}, "cell": "perm_inversions/3", "stage": "C"}
+{"id": "po-perm_inversions-t3-eval-000147", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 with exactly 19 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4 and pi(8) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 with exactly 19 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4 and pi(8) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "4, 5, 8, 7, 6, 2, 1, 3", "ref": "4, 5, 8, 7, 6, 2, 1, 3", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 29, "bucket": null, "guess_rate": 0.000719, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 8, "inv": 19, "first": 4, "last": 3}, "seed": 28913166903000717, "version": "po-v0", "tier": 3, "n_raw": 29, "small": 8, "valid_answers": ["4, 5, 8, 7, 6, 2, 1, 3", "4, 6, 7, 8, 5, 2, 1, 3", "4, 6, 8, 5, 7, 2, 1, 3", "4, 6, 8, 7, 2, 5, 1, 3", "4, 6, 8, 7, 5, 1, 2, 3", "4, 7, 5, 8, 6, 2, 1, 3", "4, 7, 6, 5, 8, 2, 1, 3", "4, 7, 6, 8, 2, 5, 1, 3", "4, 7, 6, 8, 5, 1, 2, 3", "4, 7, 8, 2, 6, 5, 1, 3", "4, 7, 8, 5, 2, 6, 1, 3", "4, 7, 8, 5, 6, 1, 2, 3", "4, 7, 8, 6, 1, 5, 2, 3", "4, 7, 8, 6, 2, 1, 5, 3", "4, 8, 2, 7, 6, 5, 1, 3", "4, 8, 5, 6, 7, 2, 1, 3", "4, 8, 5, 7, 2, 6, 1, 3", "4, 8, 5, 7, 6, 1, 2, 3", "4, 8, 6, 2, 7, 5, 1, 3", "4, 8, 6, 5, 2, 7, 1, 3", "4, 8, 6, 5, 7, 1, 2, 3", "4, 8, 6, 7, 1, 5, 2, 3", "4, 8, 6, 7, 2, 1, 5, 3", "4, 8, 7, 1, 6, 5, 2, 3", "4, 8, 7, 2, 5, 6, 1, 3", "4, 8, 7, 2, 6, 1, 5, 3", "4, 8, 7, 5, 1, 6, 2, 3", "4, 8, 7, 5, 2, 1, 6, 3", "4, 8, 7, 6, 1, 2, 5, 3"]}, "cell": "perm_inversions/3", "stage": "C"}
+{"id": "po-perm_inversions-t3-eval-000430", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 with exactly 1 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 1 and pi(8) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 with exactly 1 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 1 and pi(8) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "1, 2, 3, 4, 5, 7, 6, 8", "ref": "1, 2, 3, 4, 5, 7, 6, 8", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000124, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 8, "inv": 1, "first": 1, "last": 8}, "seed": 28913166903013303, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 8, "valid_answers": ["1, 2, 3, 4, 5, 7, 6, 8", "1, 2, 3, 4, 6, 5, 7, 8", "1, 2, 3, 5, 4, 6, 7, 8", "1, 2, 4, 3, 5, 6, 7, 8", "1, 3, 2, 4, 5, 6, 7, 8"]}, "cell": "perm_inversions/3", "stage": "C"}
+{"id": "po-perm_inversions-t3-eval-000367", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 with exactly 7 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 6 and pi(8) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 with exactly 7 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 6 and pi(8) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "6, 1, 2, 3, 5, 7, 4, 8", "ref": "6, 1, 2, 3, 5, 7, 4, 8", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.000347, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 8, "inv": 7, "first": 6, "last": 8}, "seed": 28913166903004616, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 8, "valid_answers": ["6, 1, 2, 3, 5, 7, 4, 8", "6, 1, 2, 3, 7, 4, 5, 8", "6, 1, 2, 4, 3, 7, 5, 8", "6, 1, 2, 4, 5, 3, 7, 8", "6, 1, 2, 5, 3, 4, 7, 8", "6, 1, 3, 2, 4, 7, 5, 8", "6, 1, 3, 2, 5, 4, 7, 8", "6, 1, 3, 4, 2, 5, 7, 8", "6, 1, 4, 2, 3, 5, 7, 8", "6, 2, 1, 3, 4, 7, 5, 8", "6, 2, 1, 3, 5, 4, 7, 8", "6, 2, 1, 4, 3, 5, 7, 8", "6, 2, 3, 1, 4, 5, 7, 8", "6, 3, 1, 2, 4, 5, 7, 8"]}, "cell": "perm_inversions/3", "stage": "C"}
+{"id": "po-perm_inversions-t3-eval-000434", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 with exactly 20 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4 and pi(8) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 with exactly 20 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4 and pi(8) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "4, 7, 8, 6, 3, 2, 1, 5", "ref": "4, 7, 8, 6, 3, 2, 1, 5", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000124, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 8, "inv": 20, "first": 4, "last": 5}, "seed": 28913166903014339, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 8, "valid_answers": ["4, 7, 8, 6, 3, 2, 1, 5", "4, 8, 6, 7, 3, 2, 1, 5", "4, 8, 7, 3, 6, 2, 1, 5", "4, 8, 7, 6, 2, 3, 1, 5", "4, 8, 7, 6, 3, 1, 2, 5"]}, "cell": "perm_inversions/3", "stage": "C"}
+{"id": "po-perm_inversions-t3-eval-000166", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 with exactly 17 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 6 and pi(8) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 with exactly 17 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 6 and pi(8) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "6, 3, 7, 5, 4, 2, 1, 8", "ref": "6, 3, 7, 5, 4, 2, 1, 8", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 29, "bucket": null, "guess_rate": 0.000719, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 8, "inv": 17, "first": 6, "last": 8}, "seed": 28913166903000828, "version": "po-v0", "tier": 3, "n_raw": 29, "small": 8, "valid_answers": ["6, 3, 7, 5, 4, 2, 1, 8", "6, 4, 5, 7, 3, 2, 1, 8", "6, 4, 7, 3, 5, 2, 1, 8", "6, 4, 7, 5, 2, 3, 1, 8", "6, 4, 7, 5, 3, 1, 2, 8", "6, 5, 3, 7, 4, 2, 1, 8", "6, 5, 4, 3, 7, 2, 1, 8", "6, 5, 4, 7, 2, 3, 1, 8", "6, 5, 4, 7, 3, 1, 2, 8", "6, 5, 7, 2, 4, 3, 1, 8", "6, 5, 7, 3, 2, 4, 1, 8", "6, 5, 7, 3, 4, 1, 2, 8", "6, 5, 7, 4, 1, 3, 2, 8", "6, 5, 7, 4, 2, 1, 3, 8", "6, 7, 2, 5, 4, 3, 1, 8", "6, 7, 3, 4, 5, 2, 1, 8", "6, 7, 3, 5, 2, 4, 1, 8", "6, 7, 3, 5, 4, 1, 2, 8", "6, 7, 4, 2, 5, 3, 1, 8", "6, 7, 4, 3, 2, 5, 1, 8", "6, 7, 4, 3, 5, 1, 2, 8", "6, 7, 4, 5, 1, 3, 2, 8", "6, 7, 4, 5, 2, 1, 3, 8", "6, 7, 5, 1, 4, 3, 2, 8", "6, 7, 5, 2, 3, 4, 1, 8", "6, 7, 5, 2, 4, 1, 3, 8", "6, 7, 5, 3, 1, 4, 2, 8", "6, 7, 5, 3, 2, 1, 4, 8", "6, 7, 5, 4, 1, 2, 3, 8"]}, "cell": "perm_inversions/3", "stage": "C"}
+{"id": "po-perm_inversions-t3-eval-000110", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 with exactly 9 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 5 and pi(8) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 with exactly 9 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 5 and pi(8) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "5, 1, 2, 3, 7, 8, 6, 4", "ref": "5, 1, 2, 3, 7, 8, 6, 4", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.000347, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 8, "inv": 9, "first": 5, "last": 4}, "seed": 28913166903000497, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 8, "valid_answers": ["5, 1, 2, 3, 7, 8, 6, 4", "5, 1, 2, 3, 8, 6, 7, 4", "5, 1, 2, 6, 3, 8, 7, 4", "5, 1, 2, 6, 7, 3, 8, 4", "5, 1, 2, 7, 3, 6, 8, 4", "5, 1, 3, 2, 6, 8, 7, 4", "5, 1, 3, 2, 7, 6, 8, 4", "5, 1, 3, 6, 2, 7, 8, 4", "5, 1, 6, 2, 3, 7, 8, 4", "5, 2, 1, 3, 6, 8, 7, 4", "5, 2, 1, 3, 7, 6, 8, 4", "5, 2, 1, 6, 3, 7, 8, 4", "5, 2, 3, 1, 6, 7, 8, 4", "5, 3, 1, 2, 6, 7, 8, 4"]}, "cell": "perm_inversions/3", "stage": "C"}
+{"id": "po-perm_inversions-t3-eval-000335", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 with exactly 11 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4 and pi(8) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 with exactly 11 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4 and pi(8) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "4, 2, 3, 5, 7, 8, 6, 1", "ref": "4, 2, 3, 5, 7, 8, 6, 1", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.000347, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 8, "inv": 11, "first": 4, "last": 1}, "seed": 28913166903003525, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 8, "valid_answers": ["4, 2, 3, 5, 7, 8, 6, 1", "4, 2, 3, 5, 8, 6, 7, 1", "4, 2, 3, 6, 5, 8, 7, 1", "4, 2, 3, 6, 7, 5, 8, 1", "4, 2, 3, 7, 5, 6, 8, 1", "4, 2, 5, 3, 6, 8, 7, 1", "4, 2, 5, 3, 7, 6, 8, 1", "4, 2, 5, 6, 3, 7, 8, 1", "4, 2, 6, 3, 5, 7, 8, 1", "4, 3, 2, 5, 6, 8, 7, 1", "4, 3, 2, 5, 7, 6, 8, 1", "4, 3, 2, 6, 5, 7, 8, 1", "4, 3, 5, 2, 6, 7, 8, 1", "4, 5, 2, 3, 6, 7, 8, 1"]}, "cell": "perm_inversions/3", "stage": "C"}
+{"id": "po-perm_inversions-t3-eval-000323", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 with exactly 15 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 6 and pi(8) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 with exactly 15 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 6 and pi(8) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "6, 2, 3, 5, 8, 7, 4, 1", "ref": "6, 2, 3, 5, 8, 7, 4, 1", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 49, "bucket": null, "guess_rate": 0.001215, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 8, "inv": 15, "first": 6, "last": 1}, "seed": 28913166903002915, "version": "po-v0", "tier": 3, "n_raw": 49, "small": 8, "valid_answers": ["6, 2, 3, 5, 8, 7, 4, 1", "6, 2, 3, 7, 5, 8, 4, 1", "6, 2, 3, 7, 8, 4, 5, 1", "6, 2, 3, 8, 4, 7, 5, 1", "6, 2, 3, 8, 5, 4, 7, 1", "6, 2, 4, 3, 8, 7, 5, 1", "6, 2, 4, 5, 7, 8, 3, 1", "6, 2, 4, 5, 8, 3, 7, 1", "6, 2, 4, 7, 3, 8, 5, 1", "6, 2, 4, 7, 5, 3, 8, 1", "6, 2, 4, 8, 3, 5, 7, 1", "6, 2, 5, 3, 7, 8, 4, 1", "6, 2, 5, 3, 8, 4, 7, 1", "6, 2, 5, 4, 3, 8, 7, 1", "6, 2, 5, 4, 7, 3, 8, 1", "6, 2, 5, 7, 3, 4, 8, 1", "6, 2, 7, 3, 4, 8, 5, 1", "6, 2, 7, 3, 5, 4, 8, 1", "6, 2, 7, 4, 3, 5, 8, 1", "6, 2, 8, 3, 4, 5, 7, 1", "6, 3, 2, 4, 8, 7, 5, 1", "6, 3, 2, 5, 7, 8, 4, 1", "6, 3, 2, 5, 8, 4, 7, 1", "6, 3, 2, 7, 4, 8, 5, 1", "6, 3, 2, 7, 5, 4, 8, 1", "6, 3, 2, 8, 4, 5, 7, 1", "6, 3, 4, 2, 7, 8, 5, 1", "6, 3, 4, 2, 8, 5, 7, 1", "6, 3, 4, 5, 2, 8, 7, 1", "6, 3, 4, 5, 7, 2, 8, 1", "6, 3, 4, 7, 2, 5, 8, 1", "6, 3, 5, 2, 4, 8, 7, 1", "6, 3, 5, 2, 7, 4, 8, 1", "6, 3, 5, 4, 2, 7, 8, 1", "6, 3, 7, 2, 4, 5, 8, 1", "6, 4, 2, 3, 7, 8, 5, 1", "6, 4, 2, 3, 8, 5, 7, 1", "6, 4, 2, 5, 3, 8, 7, 1", "6, 4, 2, 5, 7, 3, 8, 1", "6, 4, 2, 7, 3, 5, 8, 1", "6, 4, 3, 2, 5, 8, 7, 1", "6, 4, 3, 2, 7, 5, 8, 1", "6, 4, 3, 5, 2, 7, 8, 1", "6, 4, 5, 2, 3, 7, 8, 1", "6, 5, 2, 3, 4, 8, 7, 1", "6, 5, 2, 3, 7, 4, 8, 1", "6, 5, 2, 4, 3, 7, 8, 1", "6, 5, 3, 2, 4, 7, 8, 1", "6, 7, 2, 3, 4, 5, 8, 1"]}, "cell": "perm_inversions/3", "stage": "C"}
+{"id": "po-perm_inversions-t3-eval-000209", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 with exactly 21 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4 and pi(8) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 with exactly 21 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4 and pi(8) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "4, 5, 8, 7, 6, 3, 2, 1", "ref": "4, 5, 8, 7, 6, 3, 2, 1", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 29, "bucket": null, "guess_rate": 0.000719, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 8, "inv": 21, "first": 4, "last": 1}, "seed": 28913166903001174, "version": "po-v0", "tier": 3, "n_raw": 29, "small": 8, "valid_answers": ["4, 5, 8, 7, 6, 3, 2, 1", "4, 6, 7, 8, 5, 3, 2, 1", "4, 6, 8, 5, 7, 3, 2, 1", "4, 6, 8, 7, 3, 5, 2, 1", "4, 6, 8, 7, 5, 2, 3, 1", "4, 7, 5, 8, 6, 3, 2, 1", "4, 7, 6, 5, 8, 3, 2, 1", "4, 7, 6, 8, 3, 5, 2, 1", "4, 7, 6, 8, 5, 2, 3, 1", "4, 7, 8, 3, 6, 5, 2, 1", "4, 7, 8, 5, 3, 6, 2, 1", "4, 7, 8, 5, 6, 2, 3, 1", "4, 7, 8, 6, 2, 5, 3, 1", "4, 7, 8, 6, 3, 2, 5, 1", "4, 8, 3, 7, 6, 5, 2, 1", "4, 8, 5, 6, 7, 3, 2, 1", "4, 8, 5, 7, 3, 6, 2, 1", "4, 8, 5, 7, 6, 2, 3, 1", "4, 8, 6, 3, 7, 5, 2, 1", "4, 8, 6, 5, 3, 7, 2, 1", "4, 8, 6, 5, 7, 2, 3, 1", "4, 8, 6, 7, 2, 5, 3, 1", "4, 8, 6, 7, 3, 2, 5, 1", "4, 8, 7, 2, 6, 5, 3, 1", "4, 8, 7, 3, 5, 6, 2, 1", "4, 8, 7, 3, 6, 2, 5, 1", "4, 8, 7, 5, 2, 6, 3, 1", "4, 8, 7, 5, 3, 2, 6, 1", "4, 8, 7, 6, 2, 3, 5, 1"]}, "cell": "perm_inversions/3", "stage": "C"}
+{"id": "po-perm_inversions-t3-eval-000248", "task": "cx_po_perm_inversions", "source": "cx_po_perm_inversions", "split": "eval", "problem": "Find a permutation pi of 1, 2, ..., 8 with exactly 8 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4 and pi(8) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).", "prompt": "Problem:\nFind a permutation pi of 1, 2, ..., 8 with exactly 8 inversions (pairs i < j with pi(i) > pi(j)), pi(1) = 4 and pi(8) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. Write the final answer in \\boxed{} as pi(1), pi(2), ..., pi(8).\n\n<think>\n", "answer": "4, 1, 2, 3, 8, 7, 5, 6", "ref": "4, 1, 2, 3, 8, 7, 5, 6", "family": "po_perm_inversions", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 29, "bucket": null, "guess_rate": 0.000719, "solvable": true, "holdout": false, "meta": {"family": "po_perm_inversions", "rung": "witness", "params": {"n": 8, "inv": 8, "first": 4, "last": 6}, "seed": 28913166903001633, "version": "po-v0", "tier": 3, "n_raw": 29, "small": 8, "valid_answers": ["4, 1, 2, 3, 8, 7, 5, 6", "4, 1, 2, 5, 7, 8, 3, 6", "4, 1, 2, 5, 8, 3, 7, 6", "4, 1, 2, 7, 3, 8, 5, 6", "4, 1, 2, 7, 5, 3, 8, 6", "4, 1, 2, 8, 3, 5, 7, 6", "4, 1, 3, 2, 7, 8, 5, 6", "4, 1, 3, 2, 8, 5, 7, 6", "4, 1, 3, 5, 2, 8, 7, 6", "4, 1, 3, 5, 7, 2, 8, 6", "4, 1, 3, 7, 2, 5, 8, 6", "4, 1, 5, 2, 3, 8, 7, 6", "4, 1, 5, 2, 7, 3, 8, 6", "4, 1, 5, 3, 2, 7, 8, 6", "4, 1, 7, 2, 3, 5, 8, 6", "4, 2, 1, 3, 7, 8, 5, 6", "4, 2, 1, 3, 8, 5, 7, 6", "4, 2, 1, 5, 3, 8, 7, 6", "4, 2, 1, 5, 7, 3, 8, 6", "4, 2, 1, 7, 3, 5, 8, 6", "4, 2, 3, 1, 5, 8, 7, 6", "4, 2, 3, 1, 7, 5, 8, 6", "4, 2, 3, 5, 1, 7, 8, 6", "4, 2, 5, 1, 3, 7, 8, 6", "4, 3, 1, 2, 5, 8, 7, 6", "4, 3, 1, 2, 7, 5, 8, 6", "4, 3, 1, 5, 2, 7, 8, 6", "4, 3, 2, 1, 5, 7, 8, 6", "4, 5, 1, 2, 3, 7, 8, 6"]}, "cell": "perm_inversions/3", "stage": "C"}
+{"id": "po-poly_critical-t0-eval-000276", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 0 and p(-1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 0 and p(-1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, -6, -3", "ref": "4, -6, -3", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001365, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "val", "x": -1, "v": "7"}]}, "seed": 207778151700001081, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["4, -6, -3", "5, -4, -2", "6, -2, -1"]}, "cell": "poly_critical/0", "stage": "A"}
+{"id": "po-poly_critical-t0-eval-000062", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 0 and p(0) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 0 and p(0) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -6, -3", "ref": "2, -6, -3", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.003186, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "val", "x": 0, "v": "2"}]}, "seed": 207778151700000082, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 2, "valid_answers": ["2, -6, -3", "2, -4, -2", "2, -2, -1", "2, 0, 0", "2, 2, 1", "2, 4, 2", "2, 6, 3"]}, "cell": "poly_critical/0", "stage": "A"}
+{"id": "po-poly_critical-t0-eval-000056", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = 0 and p(2) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = 0 and p(2) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, -6, 3", "ref": "6, -6, 3", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.003186, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "0"}, {"t": "val", "x": 2, "v": "6"}]}, "seed": 207778151700000071, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 6, "valid_answers": ["6, -6, 3", "6, -4, 2", "6, -2, 1", "6, 0, 0", "6, 2, -1", "6, 4, -2", "6, 6, -3"]}, "cell": "poly_critical/0", "stage": "A"}
+{"id": "po-poly_critical-t0-eval-000040", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = 0 and p(1) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = 0 and p(1) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 0, -2", "ref": "-6, 0, -2", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002276, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": 1, "v": "-8"}]}, "seed": 207778151700000048, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 4, "valid_answers": ["-6, 0, -2", "-5, 0, -3", "-4, 0, -4", "-3, 0, -5", "-2, 0, -6"]}, "cell": "poly_critical/0", "stage": "A"}
+{"id": "po-poly_critical-t0-eval-000138", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = 0 and p(-1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = 0 and p(-1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 0, 6", "ref": "1, 0, 6", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.002731, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": -1, "v": "7"}]}, "seed": 207778151700000204, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 4, "valid_answers": ["1, 0, 6", "2, 0, 5", "3, 0, 4", "4, 0, 3", "5, 0, 2", "6, 0, 1"]}, "cell": "poly_critical/0", "stage": "A"}
+{"id": "po-poly_critical-t0-eval-000075", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 0 and p(0) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 0 and p(0) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -6, -3", "ref": "-1, -6, -3", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.003186, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "val", "x": 0, "v": "-1"}]}, "seed": 207778151700000103, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 1, "valid_answers": ["-1, -6, -3", "-1, -4, -2", "-1, -2, -1", "-1, 0, 0", "-1, 2, 1", "-1, 4, 2", "-1, 6, 3"]}, "cell": "poly_critical/0", "stage": "A"}
+{"id": "po-poly_critical-t0-eval-000058", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = 0 and p(2) = 22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = 0 and p(2) = 22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 0, 6", "ref": "-2, 0, 6", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001365, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": 2, "v": "22"}]}, "seed": 207778151700000074, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-2, 0, 6", "2, 0, 5", "6, 0, 4"]}, "cell": "poly_critical/0", "stage": "A"}
+{"id": "po-poly_critical-t0-eval-000265", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = 0 and p(1) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = 0 and p(1) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 0, 0", "ref": "-6, 0, 0", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.003186, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": 1, "v": "-6"}]}, "seed": 207778151700000779, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 3, "valid_answers": ["-6, 0, 0", "-5, 0, -1", "-4, 0, -2", "-3, 0, -3", "-2, 0, -4", "-1, 0, -5", "0, 0, -6"]}, "cell": "poly_critical/0", "stage": "A"}
+{"id": "po-poly_critical-t0-eval-000193", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 0 and p(2) = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 0 and p(2) = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -4, -2", "ref": "-2, -4, -2", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00091, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "val", "x": 2, "v": "-18"}]}, "seed": 207778151700000369, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-2, -4, -2", "6, -6, -3"]}, "cell": "poly_critical/0", "stage": "A"}
+{"id": "po-poly_critical-t0-eval-000137", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 0 and p(1) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 0 and p(1) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 2, 1", "ref": "-6, 2, 1", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002276, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "val", "x": 1, "v": "-3"}]}, "seed": 207778151700000203, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 2, "valid_answers": ["-6, 2, 1", "-3, 0, 0", "0, -2, -1", "3, -4, -2", "6, -6, -3"]}, "cell": "poly_critical/0", "stage": "A"}
+{"id": "po-poly_critical-t0-eval-000096", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(2) = 0 and p(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(2) = 0 and p(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 4, -1", "ref": "-1, 4, -1", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001365, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": 2, "m": 1, "v": "0"}, {"t": "val", "x": 1, "v": "2"}]}, "seed": 207778151700000135, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["-1, 4, -1", "2, 0, 0", "5, -4, 1"]}, "cell": "poly_critical/0", "stage": "A"}
+{"id": "po-poly_critical-t0-eval-000045", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = 0 and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = 0 and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 0, -6", "ref": "1, 0, -6", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.005917, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": 0, "v": "1"}]}, "seed": 207778151700000054, "version": "po-v0", "tier": 0, "n_raw": 13, "small": 1, "valid_answers": ["1, 0, -6", "1, 0, -5", "1, 0, -4", "1, 0, -3", "1, 0, -2", "1, 0, -1", "1, 0, 0", "1, 0, 1", "1, 0, 2", "1, 0, 3", "1, 0, 4", "1, 0, 5", "1, 0, 6"]}, "cell": "poly_critical/0", "stage": "A"}
+{"id": "po-poly_critical-t1-eval-002362", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(-1) = -9 and p(2) = 42. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(-1) = -9 and p(2) = 42. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 0, 2, 5", "ref": "-6, 0, 2, 5", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": -1, "v": "-9"}, {"t": "val", "x": 2, "v": "42"}]}, "seed": 207778151701004470, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 6, "valid_answers": ["-6, 0, 2, 5", "-2, 0, -1, 6"]}, "cell": "poly_critical/1", "stage": "B"}
+{"id": "po-poly_critical-t1-eval-002254", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(0) = 4 and p(2) = 48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(0) = 4 and p(2) = 48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 0, -1, 6", "ref": "4, 0, -1, 6", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00014, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": 0, "v": "4"}, {"t": "val", "x": 2, "v": "48"}]}, "seed": 207778151701004219, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 4, "valid_answers": ["4, 0, -1, 6", "4, 0, 1, 5", "4, 0, 3, 4", "4, 0, 5, 3"]}, "cell": "poly_critical/1", "stage": "B"}
+{"id": "po-poly_critical-t1-eval-002333", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(2) = -54 and p(-1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(2) = -54 and p(-1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 0, -1, -6", "ref": "-2, 0, -1, -6", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": 2, "v": "-54"}, {"t": "val", "x": -1, "v": "3"}]}, "seed": 207778151701004405, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 5, "valid_answers": ["-2, 0, -1, -6", "2, 0, -4, -5"]}, "cell": "poly_critical/1", "stage": "B"}
+{"id": "po-poly_critical-t1-eval-000376", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(-1) = 0, p(0) = 2 and p(-1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(-1) = 0, p(0) = 2 and p(-1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -3, -6, -3", "ref": "2, -3, -6, -3", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000245, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "val", "x": 0, "v": "2"}, {"t": "val", "x": -1, "v": "2"}]}, "seed": 207778151701000587, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 2, "valid_answers": ["2, -3, -6, -3", "2, -2, -4, -2", "2, -1, -2, -1", "2, 0, 0, 0", "2, 1, 2, 1", "2, 2, 4, 2", "2, 3, 6, 3"]}, "cell": "poly_critical/1", "stage": "B"}
+{"id": "po-poly_critical-t1-eval-000795", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(-1) = 0, p(-1) = -5 and p(2) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(-1) = 0, p(-1) = -5 and p(2) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -1, 1, 1", "ref": "-6, -1, 1, 1", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "val", "x": -1, "v": "-5"}, {"t": "val", "x": 2, "v": "4"}]}, "seed": 207778151701001292, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 4, "valid_answers": ["-6, -1, 1, 1", "-4, 2, 1, 0", "-2, 5, 1, -1"]}, "cell": "poly_critical/1", "stage": "B"}
+{"id": "po-poly_critical-t1-eval-002199", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(-1) = 0, p(2) = -49 and p(-1) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(-1) = 0, p(2) = -49 and p(-1) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -4, -5, -2", "ref": "-5, -4, -5, -2", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00014, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "val", "x": 2, "v": "-49"}, {"t": "val", "x": -1, "v": "-4"}]}, "seed": 207778151701004081, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 5, "valid_answers": ["-5, -4, -5, -2", "-3, -1, -5, -3", "-1, 2, -5, -4", "1, 5, -5, -5"]}, "cell": "poly_critical/1", "stage": "B"}
+{"id": "po-poly_critical-t1-eval-001965", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(-1) = 5 and p(1) = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(-1) = 5 and p(1) = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 6, 3, -4", "ref": "4, 6, 3, -4", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "0"}, {"t": "val", "x": -1, "v": "5"}, {"t": "val", "x": 1, "v": "9"}]}, "seed": 207778151701003562, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 5, "valid_answers": ["4, 6, 3, -4", "5, 5, 2, -3", "6, 4, 1, -2"]}, "cell": "poly_critical/1", "stage": "B"}
+{"id": "po-poly_critical-t1-eval-002145", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(-1) = 0, p(0) = -4 and p(-1) = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(-1) = 0, p(0) = -4 and p(-1) = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 2, -5, -4", "ref": "-4, 2, -5, -4", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000175, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "val", "x": 0, "v": "-4"}, {"t": "val", "x": -1, "v": "-7"}]}, "seed": 207778151701003964, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 4, "valid_answers": ["-4, 2, -5, -4", "-4, 3, -3, -3", "-4, 4, -1, -2", "-4, 5, 1, -1", "-4, 6, 3, 0"]}, "cell": "poly_critical/1", "stage": "B"}
+{"id": "po-poly_critical-t1-eval-000026", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(2) = 5 and p(-1) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(2) = 5 and p(-1) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -6, -3, 4", "ref": "-3, -6, -3, 4", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000175, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "0"}, {"t": "val", "x": 2, "v": "5"}, {"t": "val", "x": -1, "v": "-4"}]}, "seed": 207778151701000036, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 3, "valid_answers": ["-3, -6, -3, 4", "-1, -3, -3, 3", "1, 0, -3, 2", "3, 3, -3, 1", "5, 6, -3, 0"]}, "cell": "poly_critical/1", "stage": "B"}
+{"id": "po-poly_critical-t1-eval-001525", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(2) = 8 and p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(2) = 8 and p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -1, -4, 3", "ref": "2, -1, -4, 3", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "0"}, {"t": "val", "x": 2, "v": "8"}, {"t": "val", "x": 1, "v": "0"}]}, "seed": 207778151701002691, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["2, -1, -4, 3", "4, -6, 0, 2"]}, "cell": "poly_critical/1", "stage": "B"}
+{"id": "po-poly_critical-t1-eval-002134", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(1) = 4 and p(-1) = 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(1) = 4 and p(-1) = 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 0, 6, -3", "ref": "1, 0, 6, -3", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.00021, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": 1, "v": "4"}, {"t": "val", "x": -1, "v": "10"}]}, "seed": 207778151701003940, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 4, "valid_answers": ["1, 0, 6, -3", "2, 0, 5, -3", "3, 0, 4, -3", "4, 0, 3, -3", "5, 0, 2, -3", "6, 0, 1, -3"]}, "cell": "poly_critical/1", "stage": "B"}
+{"id": "po-poly_critical-t1-eval-000128", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(1) = -5 and p(-1) = -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(1) = -5 and p(-1) = -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 1, 1, -1", "ref": "-6, 1, 1, -1", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.00028, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "0"}, {"t": "val", "x": 1, "v": "-5"}, {"t": "val", "x": -1, "v": "-5"}]}, "seed": 207778151701000199, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 3, "valid_answers": ["-6, 1, 1, -1", "-5, 0, 0, 0", "-4, -1, -1, 1", "-3, -2, -2, 2", "-2, -3, -3, 3", "-1, -4, -4, 4", "0, -5, -5, 5", "1, -6, -6, 6"]}, "cell": "poly_critical/1", "stage": "B"}
+{"id": "po-poly_critical-t2-eval-001379", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(0) = 6 and p''(-2) = -90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(0) = 6 and p''(-2) = -90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 0, -9, 6", "ref": "6, 0, -9, 6", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 3.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": 0, "v": "6"}, {"t": "der", "x": -2, "m": 2, "v": "-90"}]}, "seed": 207778151702003545, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 7, "valid_answers": ["6, 0, -9, 6", "6, 0, -3, 7", "6, 0, 3, 8", "6, 0, 9, 9"]}, "cell": "poly_critical/2", "stage": "B"}
+{"id": "po-poly_critical-t2-eval-000385", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(-1) = 16 and p''(3) = -144. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(-1) = 16 and p''(3) = -144. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 0, 9, -9", "ref": "-2, 0, 9, -9", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": -1, "v": "16"}, {"t": "der", "x": 3, "m": 2, "v": "-144"}]}, "seed": 207778151702000963, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 8, "valid_answers": ["-2, 0, 9, -9", "8, 0, 0, -8"]}, "cell": "poly_critical/2", "stage": "B"}
+{"id": "po-poly_critical-t2-eval-002420", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(3) = 68 and p(0) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(3) = 68 and p(0) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -8, -2, 4", "ref": "2, -8, -2, 4", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 1, "m": 1, "v": "0"}, {"t": "val", "x": 3, "v": "68"}, {"t": "val", "x": 0, "v": "2"}]}, "seed": 207778151702006150, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 8, "valid_answers": ["2, -8, -2, 4", "2, 1, -8, 5"]}, "cell": "poly_critical/2", "stage": "B"}
+{"id": "po-poly_critical-t2-eval-000837", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(1) = -12 and p(0) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(1) = -12 and p(0) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 0, -9, 3", "ref": "-6, 0, -9, 3", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.0001, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": 1, "v": "-12"}, {"t": "val", "x": 0, "v": "-6"}]}, "seed": 207778151702002114, "version": "po-v0", "tier": 2, "n_raw": 13, "small": 6, "valid_answers": ["-6, 0, -9, 3", "-6, 0, -8, 2", "-6, 0, -7, 1", "-6, 0, -6, 0", "-6, 0, -5, -1", "-6, 0, -4, -2", "-6, 0, -3, -3", "-6, 0, -2, -4", "-6, 0, -1, -5", "-6, 0, 0, -6", "-6, 0, 1, -7", "-6, 0, 2, -8", "-6, 0, 3, -9"]}, "cell": "poly_critical/2", "stage": "B"}
+{"id": "po-poly_critical-t2-eval-000891", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(-1) = -21 and p(0) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(-1) = -21 and p(0) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -3, -9, 7", "ref": "-8, -3, -9, 7", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 2.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 1, "m": 1, "v": "0"}, {"t": "val", "x": -1, "v": "-21"}, {"t": "val", "x": 0, "v": "-8"}]}, "seed": 207778151702002229, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 8, "valid_answers": ["-8, -3, -9, 7", "-8, 2, -7, 4", "-8, 7, -5, 1"]}, "cell": "poly_critical/2", "stage": "B"}
+{"id": "po-poly_critical-t2-eval-000690", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(-2) = -70 and p(0) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(-2) = -70 and p(0) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -9, -6, 7", "ref": "-8, -9, -6, 7", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 1, "m": 1, "v": "0"}, {"t": "val", "x": -2, "v": "-70"}, {"t": "val", "x": 0, "v": "-8"}]}, "seed": 207778151702001728, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 8, "valid_answers": ["-8, -9, -6, 7", "-8, 5, -7, 3"]}, "cell": "poly_critical/2", "stage": "B"}
+{"id": "po-poly_critical-t2-eval-001541", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(1) = 2 and p(-2) = 98. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = 0, p(1) = 2 and p(-2) = 98. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 0, 8, -8", "ref": "2, 0, 8, -8", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": 1, "v": "2"}, {"t": "val", "x": -2, "v": "98"}]}, "seed": 207778151702003959, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 8, "valid_answers": ["2, 0, 8, -8", "6, 0, 5, -9"]}, "cell": "poly_critical/2", "stage": "B"}
+{"id": "po-poly_critical-t2-eval-000693", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(2) = 0, p(3) = -2 and p(0) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(2) = 0, p(3) = -2 and p(0) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 0, -9, 3", "ref": "-2, 0, -9, 3", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 5.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 2, "m": 1, "v": "0"}, {"t": "val", "x": 3, "v": "-2"}, {"t": "val", "x": 0, "v": "-2"}]}, "seed": 207778151702001740, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 2, "valid_answers": ["-2, 0, -9, 3", "-2, 0, -6, 2", "-2, 0, -3, 1", "-2, 0, 0, 0", "-2, 0, 3, -1", "-2, 0, 6, -2", "-2, 0, 9, -3"]}, "cell": "poly_critical/2", "stage": "B"}
+{"id": "po-poly_critical-t2-eval-000774", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(1) = 7 and p''(-1) = 32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(1) = 7 and p''(-1) = 32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 4, 4, -4", "ref": "3, 4, 4, -4", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 1, "m": 1, "v": "0"}, {"t": "val", "x": 1, "v": "7"}, {"t": "der", "x": -1, "m": 2, "v": "32"}]}, "seed": 207778151702001952, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 4, "valid_answers": ["3, 4, 4, -4", "8, -5, 7, -3"]}, "cell": "poly_critical/2", "stage": "B"}
+{"id": "po-poly_critical-t2-eval-001232", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(-2) = 0, p(-1) = 11 and p(0) = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(-2) = 0, p(-1) = 11 and p(0) = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9, -8, -8, -2", "ref": "9, -8, -8, -2", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": -2, "m": 1, "v": "0"}, {"t": "val", "x": -1, "v": "11"}, {"t": "val", "x": 0, "v": "9"}]}, "seed": 207778151702003135, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 9, "valid_answers": ["9, -8, -8, -2", "9, 0, 3, 1"]}, "cell": "poly_critical/2", "stage": "B"}
+{"id": "po-poly_critical-t2-eval-001956", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(-2) = 0, p(3) = -99 and p(-2) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(-2) = 0, p(3) = -99 and p(-2) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -8, -5, -1", "ref": "-3, -8, -5, -1", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": -2, "m": 1, "v": "0"}, {"t": "val", "x": 3, "v": "-99"}, {"t": "val", "x": -2, "v": "1"}]}, "seed": 207778151702004992, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 8, "valid_answers": ["-3, -8, -5, -1", "9, 0, -6, -2"]}, "cell": "poly_critical/2", "stage": "B"}
+{"id": "po-poly_critical-t2-eval-000981", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(3) = 75 and p''(1) = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 0, p(3) = 75 and p''(1) = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -2, -8, 6", "ref": "-9, -2, -8, 6", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 2.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 1, "m": 1, "v": "0"}, {"t": "val", "x": 3, "v": "75"}, {"t": "der", "x": 1, "m": 2, "v": "20"}]}, "seed": 207778151702002460, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 5, "valid_answers": ["-9, -2, -8, 6", "0, -5, -5, 5", "9, -8, -2, 4"]}, "cell": "poly_critical/2", "stage": "B"}
+{"id": "po-poly_critical-t3-eval-001547", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-1) = 0, p'(2) = 0, p(2) = -5 and p(-1) = -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-1) = 0, p'(2) = 0, p(2) = -5 and p(-1) = -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 0, 0, 0, 0", "ref": "-5, 0, 0, 0, 0", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "der", "x": 2, "m": 1, "v": "0"}, {"t": "val", "x": 2, "v": "-5"}, {"t": "val", "x": -1, "v": "-5"}]}, "seed": 207778151703018472, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 4, "valid_answers": ["-5, 0, 0, 0, 0", "-1, 4, -3, -2, 1"]}, "cell": "poly_critical/3", "stage": "B"}
+{"id": "po-poly_critical-t3-eval-001048", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(1) = 0, p'(0) = 0, p(0) = 1 and p''(1) = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(1) = 0, p'(0) = 0, p(0) = 1 and p''(1) = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 0, -6, 4, 0", "ref": "1, 0, -6, 4, 0", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "0"}, {"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": 0, "v": "1"}, {"t": "der", "x": 1, "m": 2, "v": "12"}]}, "seed": 207778151703011788, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 4, "valid_answers": ["1, 0, -6, 4, 0", "1, 0, 0, -4, 3"]}, "cell": "poly_critical/3", "stage": "B"}
+{"id": "po-poly_critical-t3-eval-000390", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(0) = 0, p'(2) = 0, p(2) = 9 and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(0) = 0, p'(2) = 0, p(2) = 9 and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 0, -2, 6, -2", "ref": "1, 0, -2, 6, -2", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 8e-06, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "der", "x": 2, "m": 1, "v": "0"}, {"t": "val", "x": 2, "v": "9"}, {"t": "val", "x": 0, "v": "1"}]}, "seed": 207778151703004405, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 2, "valid_answers": ["1, 0, -2, 6, -2", "1, 0, 2, 2, -1", "1, 0, 6, -2, 0"]}, "cell": "poly_critical/3", "stage": "B"}
+{"id": "po-poly_critical-t3-eval-000002", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(2) = 0, p'(0) = 0, p(3) = -1 and p''(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(2) = 0, p'(0) = 0, p(3) = -1 and p''(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, -6, 2, 0", "ref": "-1, 0, -6, 2, 0", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 1.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": 2, "m": 1, "v": "0"}, {"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": 3, "v": "-1"}, {"t": "der", "x": 1, "m": 2, "v": "0"}]}, "seed": 207778151703000033, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 1, "valid_answers": ["-1, 0, -6, 2, 0", "-1, 0, -3, 1, 0", "-1, 0, 0, 0, 0", "-1, 0, 3, -1, 0", "-1, 0, 6, -2, 0"]}, "cell": "poly_critical/3", "stage": "B"}
+{"id": "po-poly_critical-t3-eval-002053", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(2) = 0, p'(0) = 0, p(2) = 9 and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(2) = 0, p'(0) = 0, p(2) = 9 and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 0, -2, 6, -2", "ref": "1, 0, -2, 6, -2", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 8e-06, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": 2, "m": 1, "v": "0"}, {"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": 2, "v": "9"}, {"t": "val", "x": 0, "v": "1"}]}, "seed": 207778151703024795, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 2, "valid_answers": ["1, 0, -2, 6, -2", "1, 0, 2, 2, -1", "1, 0, 6, -2, 0"]}, "cell": "poly_critical/3", "stage": "B"}
+{"id": "po-poly_critical-t3-eval-000324", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-1) = 0, p'(1) = 0, p(-1) = -2 and p''(0) = -12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-1) = 0, p'(1) = 0, p(-1) = -2 and p''(0) = -12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -6, -6, 2, 3", "ref": "-3, -6, -6, 2, 3", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 1.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "der", "x": 1, "m": 1, "v": "0"}, {"t": "val", "x": -1, "v": "-2"}, {"t": "der", "x": 0, "m": 2, "v": "-12"}]}, "seed": 207778151703003648, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 6, "valid_answers": ["-3, -6, -6, 2, 3", "-1, -3, -6, 1, 3", "1, 0, -6, 0, 3", "3, 3, -6, -1, 3", "5, 6, -6, -2, 3"]}, "cell": "poly_critical/3", "stage": "B"}
+{"id": "po-poly_critical-t3-eval-001464", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-1) = 0, p'(0) = 0, p(2) = -20 and p(-2) = -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-1) = 0, p'(0) = 0, p(2) = -20 and p(-2) = -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 0, 4, 0, -2", "ref": "-4, 0, 4, 0, -2", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": 2, "v": "-20"}, {"t": "val", "x": -2, "v": "-20"}]}, "seed": 207778151703017276, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 4, "valid_answers": ["-4, 0, 4, 0, -2", "4, 0, 6, 0, -3"]}, "cell": "poly_critical/3", "stage": "B"}
+{"id": "po-poly_critical-t3-eval-000125", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(1) = 0, p'(2) = 0, p(2) = -4 and p(0) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(1) = 0, p'(2) = 0, p(2) = -4 and p(0) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 0, -4, 4, -1", "ref": "-4, 0, -4, 4, -1", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 8e-06, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "0"}, {"t": "der", "x": 2, "m": 1, "v": "0"}, {"t": "val", "x": 2, "v": "-4"}, {"t": "val", "x": 0, "v": "-4"}]}, "seed": 207778151703001486, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 4, "valid_answers": ["-4, 0, -4, 4, -1", "-4, 0, 0, 0, 0", "-4, 0, 4, -4, 1"]}, "cell": "poly_critical/3", "stage": "B"}
+{"id": "po-poly_critical-t3-eval-001085", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-1) = 0, p'(2) = 0, p(-2) = 6 and p(3) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-1) = 0, p'(2) = 0, p(-2) = 6 and p(3) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 4, -3, -2, 1", "ref": "-6, 4, -3, -2, 1", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "der", "x": 2, "m": 1, "v": "0"}, {"t": "val", "x": -2, "v": "6"}, {"t": "val", "x": 3, "v": "6"}]}, "seed": 207778151703012368, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 6, "valid_answers": ["-6, 4, -3, -2, 1", "6, 0, 0, 0, 0"]}, "cell": "poly_critical/3", "stage": "B"}
+{"id": "po-poly_critical-t3-eval-000283", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-1) = 0, p'(0) = 0, p(-1) = -2 and p(0) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-1) = 0, p'(0) = 0, p(-1) = -2 and p(0) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, -5, -6, -2", "ref": "-1, 0, -5, -6, -2", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 1.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": -1, "v": "-2"}, {"t": "val", "x": 0, "v": "-1"}]}, "seed": 207778151703003174, "version": "po-v0", "tier": 3, "n_raw": 7, "small": 2, "valid_answers": ["-1, 0, -5, -6, -2", "-1, 0, -4, -4, -1", "-1, 0, -3, -2, 0", "-1, 0, -2, 0, 1", "-1, 0, -1, 2, 2", "-1, 0, 0, 4, 3", "-1, 0, 1, 6, 4"]}, "cell": "poly_critical/3", "stage": "B"}
+{"id": "po-poly_critical-t3-eval-000824", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(0) = 0, p'(1) = 0, p(1) = 5 and p''(0) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(0) = 0, p'(1) = 0, p(1) = 5 and p''(0) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 0, 4, 4, -5", "ref": "2, 0, 4, 4, -5", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 8e-06, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "der", "x": 1, "m": 1, "v": "0"}, {"t": "val", "x": 1, "v": "5"}, {"t": "der", "x": 0, "m": 2, "v": "8"}]}, "seed": 207778151703009403, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 4, "valid_answers": ["2, 0, 4, 4, -5", "3, 0, 4, 0, -2", "4, 0, 4, -4, 1"]}, "cell": "poly_critical/3", "stage": "B"}
+{"id": "po-poly_critical-t3-eval-002415", "task": "cx_po_poly_critical", "source": "cx_po_poly_critical", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-1) = 0, p'(1) = 0, p(0) = 5 and p(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-1) = 0, p'(1) = 0, p(0) = 5 and p(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, -3, 6, 1, -3", "ref": "5, -3, 6, 1, -3", "family": "po_poly_critical", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_critical", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "der", "x": 1, "m": 1, "v": "0"}, {"t": "val", "x": 0, "v": "5"}, {"t": "val", "x": 1, "v": "6"}]}, "seed": 207778151703029951, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 5, "valid_answers": ["5, -3, 6, 1, -3", "5, 0, 2, 0, -1", "5, 3, -2, -1, 1", "5, 6, -6, -2, 3"]}, "cell": "poly_critical/3", "stage": "B"}
+{"id": "po-poly_integral-t0-eval-000017", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{1}^{2} p(x)\\,dx = 6 and p(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{1}^{2} p(x)\\,dx = 6 and p(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 4, 0", "ref": "0, 4, 0", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001503, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "int", "lo": 1, "hi": 2, "v": "6"}, {"t": "val", "x": 1, "v": "4"}]}, "seed": 143586649400000024, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["0, 4, 0", "5, -4, 3"]}, "cell": "poly_integral/0", "stage": "A"}
+{"id": "po-poly_integral-t0-eval-001957", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{1} p(x)\\,dx = 5 and p(1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{1} p(x)\\,dx = 5 and p(1) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 4, 0", "ref": "3, 4, 0", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001503, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "int", "lo": 0, "hi": 1, "v": "5"}, {"t": "val", "x": 1, "v": "7"}]}, "seed": 143586649400006698, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["3, 4, 0", "4, 0, 3"]}, "cell": "poly_integral/0", "stage": "A"}
+{"id": "po-poly_integral-t0-eval-001353", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{1} p(x)\\,dx = \\frac{5}{2} and p(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{1} p(x)\\,dx = \\frac{5}{2} and p(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 5, 0", "ref": "0, 5, 0", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001503, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "int", "lo": 0, "hi": 1, "v": "5/2"}, {"t": "val", "x": 1, "v": "5"}]}, "seed": 143586649400003277, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["0, 5, 0", "1, 1, 3"]}, "cell": "poly_integral/0", "stage": "A"}
+{"id": "po-poly_integral-t0-eval-002416", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{1} p(x)\\,dx = -\\frac{1}{6} and p(1) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{1} p(x)\\,dx = -\\frac{1}{6} and p(1) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, -5", "ref": "1, 1, -5", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001503, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "int", "lo": 0, "hi": 1, "v": "-1/6"}, {"t": "val", "x": 1, "v": "-3"}]}, "seed": 143586649400012622, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["1, 1, -5", "2, -3, -2"]}, "cell": "poly_integral/0", "stage": "A"}
+{"id": "po-poly_integral-t0-eval-001800", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = -6 and p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = -6 and p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 1, 3", "ref": "-4, 1, 3", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002254, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "-6"}, {"t": "val", "x": 1, "v": "0"}]}, "seed": 143586649400005486, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["-4, 1, 3", "-3, 3, 0", "-2, 5, -3"]}, "cell": "poly_integral/0", "stage": "A"}
+{"id": "po-poly_integral-t0-eval-002220", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{0} p(x)\\,dx = \\frac{2}{3} and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{0} p(x)\\,dx = \\frac{2}{3} and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -2, -4", "ref": "1, -2, -4", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.003005, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "int", "lo": -1, "hi": 0, "v": "2/3"}, {"t": "val", "x": 0, "v": "1"}]}, "seed": 143586649400009219, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 1, "valid_answers": ["1, -2, -4", "1, 0, -1", "1, 2, 2", "1, 4, 5"]}, "cell": "poly_integral/0", "stage": "A"}
+{"id": "po-poly_integral-t0-eval-001788", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{1} p(x)\\,dx = -\\frac{20}{3} and p(1) = -10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{1} p(x)\\,dx = -\\frac{20}{3} and p(1) = -10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 0, -5", "ref": "-5, 0, -5", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001503, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "int", "lo": 0, "hi": 1, "v": "-20/3"}, {"t": "val", "x": 1, "v": "-10"}]}, "seed": 143586649400005400, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-5, 0, -5", "-4, -4, -2"]}, "cell": "poly_integral/0", "stage": "A"}
+{"id": "po-poly_integral-t0-eval-002107", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{2} p(x)\\,dx = 20 and p(0) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{2} p(x)\\,dx = 20 and p(0) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 1, 3", "ref": "5, 1, 3", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001503, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "20"}, {"t": "val", "x": 0, "v": "5"}]}, "seed": 143586649400007948, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["5, 1, 3", "5, 5, 0"]}, "cell": "poly_integral/0", "stage": "A"}
+{"id": "po-poly_integral-t0-eval-000788", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{2} p(x)\\,dx = \\frac{10}{3} and p(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{2} p(x)\\,dx = \\frac{10}{3} and p(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 5, -1", "ref": "-2, 5, -1", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.006011, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "10/3"}, {"t": "val", "x": 1, "v": "2"}]}, "seed": 143586649400001607, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 2, "valid_answers": ["-2, 5, -1", "-1, 4, -1", "0, 3, -1", "1, 2, -1", "2, 1, -1", "3, 0, -1", "4, -1, -1", "5, -2, -1"]}, "cell": "poly_integral/0", "stage": "A"}
+{"id": "po-poly_integral-t0-eval-002027", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{1} p(x)\\,dx = 1 and p(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{1} p(x)\\,dx = 1 and p(1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 4, -3", "ref": "0, 4, -3", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002254, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "int", "lo": 0, "hi": 1, "v": "1"}, {"t": "val", "x": 1, "v": "1"}]}, "seed": 143586649400007270, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 1, "valid_answers": ["0, 4, -3", "1, 0, 0", "2, -4, 3"]}, "cell": "poly_integral/0", "stage": "A"}
+{"id": "po-poly_integral-t0-eval-001674", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{2} p(x)\\,dx = -24 and p(1) = -11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{2} p(x)\\,dx = -24 and p(1) = -11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -3, -3", "ref": "-5, -3, -3", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002254, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "-24"}, {"t": "val", "x": 1, "v": "-11"}]}, "seed": 143586649400004726, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-5, -3, -3", "-4, -4, -3", "-3, -5, -3"]}, "cell": "poly_integral/0", "stage": "A"}
+{"id": "po-poly_integral-t0-eval-000596", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{0} p(x)\\,dx = 5 and p(0) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{0} p(x)\\,dx = 5 and p(0) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -4, 0", "ref": "3, -4, 0", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001503, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "int", "lo": -1, "hi": 0, "v": "5"}, {"t": "val", "x": 0, "v": "3"}]}, "seed": 143586649400001075, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["3, -4, 0", "3, -2, 3"]}, "cell": "poly_integral/0", "stage": "A"}
+{"id": "po-poly_integral-t1-eval-000154", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = 12 and p(2) = -23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = 12 and p(2) = -23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, -9, -3", "ref": "7, -9, -3", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000292, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "12"}, {"t": "val", "x": 2, "v": "-23"}]}, "seed": 143586649401000204, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 9, "valid_answers": ["7, -9, -3", "9, 2, -9"]}, "cell": "poly_integral/1", "stage": "B"}
+{"id": "po-poly_integral-t1-eval-001494", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = -8 and \\int_{0}^{1} p(x)\\,dx = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = -8 and \\int_{0}^{1} p(x)\\,dx = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -8, 9", "ref": "-7, -8, 9", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.001021, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "-8"}, {"t": "int", "lo": 0, "hi": 1, "v": "-8"}]}, "seed": 143586649401002034, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 8, "valid_answers": ["-7, -8, 9", "-6, -8, 6", "-5, -8, 3", "-4, -8, 0", "-3, -8, -3", "-2, -8, -6", "-1, -8, -9"]}, "cell": "poly_integral/1", "stage": "B"}
+{"id": "po-poly_integral-t1-eval-002405", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{1}^{2} p(x)\\,dx = -\\frac{26}{3} and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{1}^{2} p(x)\\,dx = -\\frac{26}{3} and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -8, 1", "ref": "1, -8, 1", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000292, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "int", "lo": 1, "hi": 2, "v": "-26/3"}, {"t": "val", "x": 0, "v": "1"}]}, "seed": 143586649401003326, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 8, "valid_answers": ["1, -8, 1", "1, 6, -8"]}, "cell": "poly_integral/1", "stage": "B"}
+{"id": "po-poly_integral-t1-eval-001175", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = \\frac{20}{3} and \\int_{-1}^{0} p(x)\\,dx = \\frac{41}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = \\frac{20}{3} and \\int_{-1}^{0} p(x)\\,dx = \\frac{41}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -7, 7", "ref": "1, -7, 7", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000875, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "20/3"}, {"t": "int", "lo": -1, "hi": 0, "v": "41/6"}]}, "seed": 143586649401001605, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 7, "valid_answers": ["1, -7, 7", "2, -7, 4", "3, -7, 1", "4, -7, -2", "5, -7, -5", "6, -7, -8"]}, "cell": "poly_integral/1", "stage": "B"}
+{"id": "po-poly_integral-t1-eval-000717", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = \\frac{58}{3} and \\int_{0}^{2} p(x)\\,dx = \\frac{70}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = \\frac{58}{3} and \\int_{0}^{2} p(x)\\,dx = \\frac{70}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, -6, 8", "ref": "7, -6, 8", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000437, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "58/3"}, {"t": "int", "lo": 0, "hi": 2, "v": "70/3"}]}, "seed": 143586649401000962, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 8, "valid_answers": ["7, -6, 8", "8, -3, 5", "9, 0, 2"]}, "cell": "poly_integral/1", "stage": "B"}
+{"id": "po-poly_integral-t1-eval-002454", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{0} p(x)\\,dx = -\\frac{3}{2} and \\int_{0}^{1} p(x)\\,dx = -\\frac{5}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{0} p(x)\\,dx = -\\frac{3}{2} and \\int_{0}^{1} p(x)\\,dx = -\\frac{5}{2}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -1, 9", "ref": "-5, -1, 9", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.001021, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 0, "v": "-3/2"}, {"t": "int", "lo": 0, "hi": 1, "v": "-5/2"}]}, "seed": 143586649401003394, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 2, "valid_answers": ["-5, -1, 9", "-4, -1, 6", "-3, -1, 3", "-2, -1, 0", "-1, -1, -3", "0, -1, -6", "1, -1, -9"]}, "cell": "poly_integral/1", "stage": "B"}
+{"id": "po-poly_integral-t1-eval-002184", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{0} p(x)\\,dx = -\\frac{1}{6} and p(-1) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{0} p(x)\\,dx = -\\frac{1}{6} and p(-1) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -3, -8", "ref": "1, -3, -8", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000583, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 0, "v": "-1/6"}, {"t": "val", "x": -1, "v": "-4"}]}, "seed": 143586649401003026, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 5, "valid_answers": ["1, -3, -8", "2, 1, -5", "3, 5, -2", "4, 9, 1"]}, "cell": "poly_integral/1", "stage": "B"}
+{"id": "po-poly_integral-t1-eval-000468", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{2} p(x)\\,dx = \\frac{8}{3} and p(-1) = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{2} p(x)\\,dx = \\frac{8}{3} and p(-1) = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -5, 7", "ref": "-3, -5, 7", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000292, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "8/3"}, {"t": "val", "x": -1, "v": "9"}]}, "seed": 143586649401000623, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["-3, -5, 7", "4, -4, 1"]}, "cell": "poly_integral/1", "stage": "B"}
+{"id": "po-poly_integral-t1-eval-001463", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = -\\frac{14}{3} and p(-1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = -\\frac{14}{3} and p(-1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -1, 8", "ref": "-5, -1, 8", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000729, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "-14/3"}, {"t": "val", "x": -1, "v": "4"}]}, "seed": 143586649401001994, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 5, "valid_answers": ["-5, -1, 8", "-4, -3, 5", "-3, -5, 2", "-2, -7, -1", "-1, -9, -4"]}, "cell": "poly_integral/1", "stage": "B"}
+{"id": "po-poly_integral-t1-eval-000423", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{0} p(x)\\,dx = -2 and p(0) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{0} p(x)\\,dx = -2 and p(0) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -6, -9", "ref": "-2, -6, -9", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.001021, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 0, "v": "-2"}, {"t": "val", "x": 0, "v": "-2"}]}, "seed": 143586649401000552, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 2, "valid_answers": ["-2, -6, -9", "-2, -4, -6", "-2, -2, -3", "-2, 0, 0", "-2, 2, 3", "-2, 4, 6", "-2, 6, 9"]}, "cell": "poly_integral/1", "stage": "B"}
+{"id": "po-poly_integral-t1-eval-000138", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{2} p(x)\\,dx = \\frac{34}{3} and p(1) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{2} p(x)\\,dx = \\frac{34}{3} and p(1) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 9, -7", "ref": "6, 9, -7", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000583, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "34/3"}, {"t": "val", "x": 1, "v": "8"}]}, "seed": 143586649401000183, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 8, "valid_answers": ["6, 9, -7", "7, 8, -7", "8, 7, -7", "9, 6, -7"]}, "cell": "poly_integral/1", "stage": "B"}
+{"id": "po-poly_integral-t1-eval-001765", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = \\frac{2}{3} and \\int_{-1}^{0} p(x)\\,dx = \\frac{7}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = \\frac{2}{3} and \\int_{-1}^{0} p(x)\\,dx = \\frac{7}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -4, 7", "ref": "-2, -4, 7", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000875, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "2/3"}, {"t": "int", "lo": -1, "hi": 0, "v": "7/3"}]}, "seed": 143586649401002420, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 4, "valid_answers": ["-2, -4, 7", "-1, -4, 4", "0, -4, 1", "1, -4, -2", "2, -4, -5", "3, -4, -8"]}, "cell": "poly_integral/1", "stage": "B"}
+{"id": "po-poly_integral-t2-eval-001765", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = -28, p(2) = -40 and p(0) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = -28, p(2) = -40 and p(0) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -6, 6, -6", "ref": "-4, -6, 6, -6", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000175, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "-28"}, {"t": "val", "x": 2, "v": "-40"}, {"t": "val", "x": 0, "v": "-4"}]}, "seed": 143586649402003623, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 4, "valid_answers": ["-4, -6, 6, -6", "-4, -4, 3, -5", "-4, -2, 0, -4", "-4, 0, -3, -3", "-4, 2, -6, -2"]}, "cell": "poly_integral/2", "stage": "B"}
+{"id": "po-poly_integral-t2-eval-000453", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{0} p(x)\\,dx = \\frac{1}{3}, p(1) = 2 and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{0} p(x)\\,dx = \\frac{1}{3}, p(1) = 2 and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -3, -2, 6", "ref": "1, -3, -2, 6", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "int", "lo": -1, "hi": 0, "v": "1/3"}, {"t": "val", "x": 1, "v": "2"}, {"t": "val", "x": 0, "v": "1"}]}, "seed": 143586649402000900, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 4, "valid_answers": ["1, -3, -2, 6", "1, 4, 1, -4"]}, "cell": "poly_integral/2", "stage": "B"}
+{"id": "po-poly_integral-t2-eval-001634", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = -4, \\int_{0}^{1} p(x)\\,dx = \\frac{1}{4} and p(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = -4, \\int_{0}^{1} p(x)\\,dx = \\frac{1}{4} and p(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 5, 0, -1", "ref": "-2, 5, 0, -1", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "-4"}, {"t": "int", "lo": 0, "hi": 1, "v": "1/4"}, {"t": "val", "x": 1, "v": "2"}]}, "seed": 143586649402003313, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 3, "valid_answers": ["-2, 5, 0, -1", "-1, 3, -3, 3"]}, "cell": "poly_integral/2", "stage": "B"}
+{"id": "po-poly_integral-t2-eval-000316", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = -\\frac{98}{3}, \\int_{-1}^{1} p(x)\\,dx = -\\frac{38}{3} and \\int_{1}^{2} p(x)\\,dx = -\\frac{307}{12}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = -\\frac{98}{3}, \\int_{-1}^{1} p(x)\\,dx = -\\frac{38}{3} and \\int_{1}^{2} p(x)\\,dx = -\\frac{307}{12}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 1, -1, -5", "ref": "-6, 1, -1, -5", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "-98/3"}, {"t": "int", "lo": -1, "hi": 1, "v": "-38/3"}, {"t": "int", "lo": 1, "hi": 2, "v": "-307/12"}]}, "seed": 143586649402000620, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 5, "valid_answers": ["-6, 1, -1, -5", "-5, 0, -4, -3"]}, "cell": "poly_integral/2", "stage": "B"}
+{"id": "po-poly_integral-t2-eval-002487", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = -\\frac{22}{3}, \\int_{-1}^{1} p(x)\\,dx = \\frac{20}{3} and \\int_{1}^{2} p(x)\\,dx = -\\frac{67}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = -\\frac{22}{3}, \\int_{-1}^{1} p(x)\\,dx = \\frac{20}{3} and \\int_{1}^{2} p(x)\\,dx = -\\frac{67}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 4, 1, -6", "ref": "3, 4, 1, -6", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "-22/3"}, {"t": "int", "lo": -1, "hi": 1, "v": "20/3"}, {"t": "int", "lo": 1, "hi": 2, "v": "-67/6"}]}, "seed": 143586649402005156, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 4, "valid_answers": ["3, 4, 1, -6", "4, 3, -2, -4", "5, 2, -5, -2"]}, "cell": "poly_integral/2", "stage": "B"}
+{"id": "po-poly_integral-t2-eval-001456", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{1} p(x)\\,dx = -\\frac{5}{6}, p(1) = 5 and \\int_{-1}^{1} p(x)\\,dx = -\\frac{14}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{1} p(x)\\,dx = -\\frac{5}{6}, p(1) = 5 and \\int_{-1}^{1} p(x)\\,dx = -\\frac{14}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 2, 5, 2", "ref": "-4, 2, 5, 2", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "int", "lo": 0, "hi": 1, "v": "-5/6"}, {"t": "val", "x": 1, "v": "5"}, {"t": "int", "lo": -1, "hi": 1, "v": "-14/3"}]}, "seed": 143586649402002924, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 5, "valid_answers": ["-4, 2, 5, 2", "-3, 0, 2, 6"]}, "cell": "poly_integral/2", "stage": "B"}
+{"id": "po-poly_integral-t2-eval-001755", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = 8, p(1) = 3 and \\int_{1}^{2} p(x)\\,dx = \\frac{29}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = 8, p(1) = 3 and \\int_{1}^{2} p(x)\\,dx = \\frac{29}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 3, 0, 1", "ref": "-1, 3, 0, 1", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "8"}, {"t": "val", "x": 1, "v": "3"}, {"t": "int", "lo": 1, "hi": 2, "v": "29/4"}]}, "seed": 143586649402003598, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 3, "valid_answers": ["-1, 3, 0, 1", "0, -2, 6, -1"]}, "cell": "poly_integral/2", "stage": "B"}
+{"id": "po-poly_integral-t2-eval-002144", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{0} p(x)\\,dx = -\\frac{7}{4}, \\int_{0}^{2} p(x)\\,dx = 16 and \\int_{-1}^{1} p(x)\\,dx = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{0} p(x)\\,dx = -\\frac{7}{4}, \\int_{0}^{2} p(x)\\,dx = 16 and \\int_{-1}^{1} p(x)\\,dx = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 1, 6, 1", "ref": "-3, 1, 6, 1", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "int", "lo": -1, "hi": 0, "v": "-7/4"}, {"t": "int", "lo": 0, "hi": 2, "v": "16"}, {"t": "int", "lo": -1, "hi": 1, "v": "-2"}]}, "seed": 143586649402004407, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 3, "valid_answers": ["-3, 1, 6, 1", "-2, 0, 3, 3", "-1, -1, 0, 5"]}, "cell": "poly_integral/2", "stage": "B"}
+{"id": "po-poly_integral-t2-eval-001542", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = 0, p(-1) = 9 and \\int_{-1}^{0} p(x)\\,dx = \\frac{9}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = 0, p(-1) = 9 and \\int_{-1}^{0} p(x)\\,dx = \\frac{9}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -4, 6, -1", "ref": "-2, -4, 6, -1", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "0"}, {"t": "val", "x": -1, "v": "9"}, {"t": "int", "lo": -1, "hi": 0, "v": "9/4"}]}, "seed": 143586649402003144, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 6, "valid_answers": ["-2, -4, 6, -1", "-1, -1, 6, -3", "0, 2, 6, -5"]}, "cell": "poly_integral/2", "stage": "B"}
+{"id": "po-poly_integral-t2-eval-001673", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = -\\frac{34}{3}, p(1) = -15 and \\int_{0}^{2} p(x)\\,dx = -\\frac{106}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = -\\frac{34}{3}, p(1) = -15 and \\int_{0}^{2} p(x)\\,dx = -\\frac{106}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -6, -2, -2", "ref": "-5, -6, -2, -2", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "-34/3"}, {"t": "val", "x": 1, "v": "-15"}, {"t": "int", "lo": 0, "hi": 2, "v": "-106/3"}]}, "seed": 143586649402003411, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 5, "valid_answers": ["-5, -6, -2, -2", "-4, -5, -5, -1"]}, "cell": "poly_integral/2", "stage": "B"}
+{"id": "po-poly_integral-t2-eval-001077", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = \\frac{10}{3}, p(1) = -3 and p(2) = 25. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = \\frac{10}{3}, p(1) = -3 and p(2) = 25. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -6, 2, 4", "ref": "-3, -6, 2, 4", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "10/3"}, {"t": "val", "x": 1, "v": "-3"}, {"t": "val", "x": 2, "v": "25"}]}, "seed": 143586649402002177, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 5, "valid_answers": ["-3, -6, 2, 4", "-3, -4, -1, 5", "-3, -2, -4, 6"]}, "cell": "poly_integral/2", "stage": "B"}
+{"id": "po-poly_integral-t2-eval-002053", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = \\frac{98}{3}, \\int_{1}^{2} p(x)\\,dx = \\frac{155}{6} and \\int_{0}^{1} p(x)\\,dx = \\frac{41}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = \\frac{98}{3}, \\int_{1}^{2} p(x)\\,dx = \\frac{155}{6} and \\int_{0}^{1} p(x)\\,dx = \\frac{41}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 4, 4, 2", "ref": "3, 4, 4, 2", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00014, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "98/3"}, {"t": "int", "lo": 1, "hi": 2, "v": "155/6"}, {"t": "int", "lo": 0, "hi": 1, "v": "41/6"}]}, "seed": 143586649402004216, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 4, "valid_answers": ["3, 4, 4, 2", "4, 3, 1, 4", "5, 2, -2, 6", "6, -3, 4, 4"]}, "cell": "poly_integral/2", "stage": "B"}
+{"id": "po-poly_integral-t3-eval-001093", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{1} p(x)\\,dx = -\\frac{33}{4}, \\int_{-1}^{0} p(x)\\,dx = -\\frac{55}{4} and \\int_{1}^{2} p(x)\\,dx = -\\frac{71}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{1} p(x)\\,dx = -\\frac{33}{4}, \\int_{-1}^{0} p(x)\\,dx = -\\frac{55}{4} and \\int_{1}^{2} p(x)\\,dx = -\\frac{71}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 6, -6, -1", "ref": "-9, 6, -6, -1", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": 0, "hi": 1, "v": "-33/4"}, {"t": "int", "lo": -1, "hi": 0, "v": "-55/4"}, {"t": "int", "lo": 1, "hi": 2, "v": "-71/4"}]}, "seed": 143586649403002178, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 9, "valid_answers": ["-9, 6, -6, -1", "-8, 5, -9, 1"]}, "cell": "poly_integral/3", "stage": "C"}
+{"id": "po-poly_integral-t3-eval-000628", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = 12, \\int_{0}^{1} p(x)\\,dx = \\frac{9}{2} and p(-1) = 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = 12, \\int_{0}^{1} p(x)\\,dx = \\frac{9}{2} and p(-1) = 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -7, 9, 8", "ref": "3, -7, 9, 8", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 3.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "12"}, {"t": "int", "lo": 0, "hi": 1, "v": "9/2"}, {"t": "val", "x": -1, "v": "11"}]}, "seed": 143586649403001254, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 5, "valid_answers": ["3, -7, 9, 8", "4, -5, 6, 4", "5, -3, 3, 0", "6, -1, 0, -4", "7, 1, -3, -8"]}, "cell": "poly_integral/3", "stage": "C"}
+{"id": "po-poly_integral-t3-eval-000730", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = -\\frac{82}{3}, \\int_{-1}^{1} p(x)\\,dx = \\frac{8}{3} and p(-1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = -\\frac{82}{3}, \\int_{-1}^{1} p(x)\\,dx = \\frac{8}{3} and p(-1) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 4, -5, -7", "ref": "3, 4, -5, -7", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "-82/3"}, {"t": "int", "lo": -1, "hi": 1, "v": "8/3"}, {"t": "val", "x": -1, "v": "1"}]}, "seed": 143586649403001460, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 7, "valid_answers": ["3, 4, -5, -7", "4, -3, -8, -2"]}, "cell": "poly_integral/3", "stage": "C"}
+{"id": "po-poly_integral-t3-eval-000678", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{1} p(x)\\,dx = \\frac{33}{4}, \\int_{1}^{2} p(x)\\,dx = \\frac{47}{4} and \\int_{-1}^{0} p(x)\\,dx = \\frac{7}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{1} p(x)\\,dx = \\frac{33}{4}, \\int_{1}^{2} p(x)\\,dx = \\frac{47}{4} and \\int_{-1}^{0} p(x)\\,dx = \\frac{7}{4}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 9, 6, -5", "ref": "3, 9, 6, -5", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 4.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": 0, "hi": 1, "v": "33/4"}, {"t": "int", "lo": 1, "hi": 2, "v": "47/4"}, {"t": "int", "lo": -1, "hi": 0, "v": "7/4"}]}, "seed": 143586649403001357, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 6, "valid_answers": ["3, 9, 6, -5", "4, 8, 3, -3", "5, 7, 0, -1", "6, 6, -3, 1", "7, 5, -6, 3", "8, 4, -9, 5"]}, "cell": "poly_integral/3", "stage": "C"}
+{"id": "po-poly_integral-t3-eval-000020", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = \\frac{44}{3}, \\int_{0}^{1} p(x)\\,dx = \\frac{73}{12} and p'(0) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = \\frac{44}{3}, \\int_{0}^{1} p(x)\\,dx = \\frac{73}{12} and p'(0) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 0, 7, -5", "ref": "5, 0, 7, -5", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 3.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "44/3"}, {"t": "int", "lo": 0, "hi": 1, "v": "73/12"}, {"t": "der", "x": 0, "m": 1, "v": "0"}]}, "seed": 143586649403000039, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 6, "valid_answers": ["5, 0, 7, -5", "6, 0, 4, -5", "7, 0, 1, -5", "8, 0, -2, -5", "9, 0, -5, -5"]}, "cell": "poly_integral/3", "stage": "C"}
+{"id": "po-poly_integral-t3-eval-001071", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{1} p(x)\\,dx = 0, \\int_{0}^{2} p(x)\\,dx = -48 and p'(1) = -44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{1} p(x)\\,dx = 0, \\int_{0}^{2} p(x)\\,dx = -48 and p'(1) = -44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, -2, -9, -8", "ref": "6, -2, -9, -8", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": 0, "hi": 1, "v": "0"}, {"t": "int", "lo": 0, "hi": 2, "v": "-48"}, {"t": "der", "x": 1, "m": 1, "v": "-44"}]}, "seed": 143586649403002132, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 8, "valid_answers": ["6, -2, -9, -8", "8, -8, -6, -8"]}, "cell": "poly_integral/3", "stage": "C"}
+{"id": "po-poly_integral-t3-eval-002019", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{0} p(x)\\,dx = -\\frac{53}{12}, \\int_{-1}^{1} p(x)\\,dx = -\\frac{58}{3} and p(1) = -28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{0} p(x)\\,dx = -\\frac{53}{12}, \\int_{-1}^{1} p(x)\\,dx = -\\frac{58}{3} and p(1) = -28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -6, -5, -9", "ref": "-8, -6, -5, -9", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 0, "v": "-53/12"}, {"t": "int", "lo": -1, "hi": 1, "v": "-58/3"}, {"t": "val", "x": 1, "v": "-28"}]}, "seed": 143586649403004029, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 8, "valid_answers": ["-8, -6, -5, -9", "-7, -8, -8, -5"]}, "cell": "poly_integral/3", "stage": "C"}
+{"id": "po-poly_integral-t3-eval-002134", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = \\frac{26}{3}, \\int_{0}^{1} p(x)\\,dx = \\frac{37}{12} and p(-1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = \\frac{26}{3}, \\int_{0}^{1} p(x)\\,dx = \\frac{37}{12} and p(-1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, -6, -2, 7", "ref": "5, -6, -2, 7", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 2.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "26/3"}, {"t": "int", "lo": 0, "hi": 1, "v": "37/12"}, {"t": "val", "x": -1, "v": "2"}]}, "seed": 143586649403004233, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 6, "valid_answers": ["5, -6, -2, 7", "6, -4, -5, 3", "7, -2, -8, -1"]}, "cell": "poly_integral/3", "stage": "C"}
+{"id": "po-poly_integral-t3-eval-001810", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = \\frac{14}{3}, \\int_{1}^{2} p(x)\\,dx = \\frac{179}{6} and p(-1) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = \\frac{14}{3}, \\int_{1}^{2} p(x)\\,dx = \\frac{179}{6} and p(-1) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 9, 7, 0", "ref": "0, 9, 7, 0", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 2.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "14/3"}, {"t": "int", "lo": 1, "hi": 2, "v": "179/6"}, {"t": "val", "x": -1, "v": "-2"}]}, "seed": 143586649403003595, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 4, "valid_answers": ["0, 9, 7, 0", "1, 3, 4, 4", "2, -3, 1, 8"]}, "cell": "poly_integral/3", "stage": "C"}
+{"id": "po-poly_integral-t3-eval-000671", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{0} p(x)\\,dx = \\frac{11}{6}, \\int_{0}^{2} p(x)\\,dx = \\frac{44}{3} and \\int_{1}^{2} p(x)\\,dx = \\frac{71}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{0} p(x)\\,dx = \\frac{11}{6}, \\int_{0}^{2} p(x)\\,dx = \\frac{44}{3} and \\int_{1}^{2} p(x)\\,dx = \\frac{71}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 2, 7, -2", "ref": "0, 2, 7, -2", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 4.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 0, "v": "11/6"}, {"t": "int", "lo": 0, "hi": 2, "v": "44/3"}, {"t": "int", "lo": 1, "hi": 2, "v": "71/6"}]}, "seed": 143586649403001331, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 2, "valid_answers": ["0, 2, 7, -2", "1, 1, 4, 0", "2, 0, 1, 2", "3, -1, -2, 4", "4, -2, -5, 6", "5, -3, -8, 8"]}, "cell": "poly_integral/3", "stage": "C"}
+{"id": "po-poly_integral-t3-eval-002440", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{0} p(x)\\,dx = -\\frac{11}{2}, \\int_{0}^{1} p(x)\\,dx = -\\frac{21}{2} and p'(0) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{0} p(x)\\,dx = -\\frac{11}{2}, \\int_{0}^{1} p(x)\\,dx = -\\frac{21}{2} and p'(0) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -3, 3, -4", "ref": "-9, -3, 3, -4", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 3.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 0, "v": "-11/2"}, {"t": "int", "lo": 0, "hi": 1, "v": "-21/2"}, {"t": "der", "x": 0, "m": 1, "v": "-3"}]}, "seed": 143586649403004829, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 6, "valid_answers": ["-9, -3, 3, -4", "-8, -3, 0, -4", "-7, -3, -3, -4", "-6, -3, -6, -4", "-5, -3, -9, -4"]}, "cell": "poly_integral/3", "stage": "C"}
+{"id": "po-poly_integral-t3-eval-000613", "task": "cx_po_poly_integral", "source": "cx_po_poly_integral", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = \\frac{52}{3}, \\int_{-1}^{0} p(x)\\,dx = \\frac{37}{6} and \\int_{1}^{2} p(x)\\,dx = \\frac{121}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = \\frac{52}{3}, \\int_{-1}^{0} p(x)\\,dx = \\frac{37}{6} and \\int_{1}^{2} p(x)\\,dx = \\frac{121}{6}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 7, 8, -4", "ref": "6, 7, 8, -4", "family": "po_poly_integral", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 3.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_integral", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "52/3"}, {"t": "int", "lo": -1, "hi": 0, "v": "37/6"}, {"t": "int", "lo": 1, "hi": 2, "v": "121/6"}]}, "seed": 143586649403001226, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 7, "valid_answers": ["6, 7, 8, -4", "7, 6, 5, -2", "8, 5, 2, 0", "9, 4, -1, 2"]}, "cell": "poly_integral/3", "stage": "C"}
+{"id": "po-poly_mixed-t0-eval-001334", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(-1) = 13 and p(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(-1) = 13 and p(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -4, 6", "ref": "3, -4, 6", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001821, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "val", "x": -1, "v": "13"}, {"t": "val", "x": 1, "v": "5"}]}, "seed": 157317360100001789, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 5, "valid_answers": ["3, -4, 6", "4, -4, 5", "5, -4, 4", "6, -4, 3"]}, "cell": "poly_mixed/0", "stage": "B"}
+{"id": "po-poly_mixed-t0-eval-001312", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(2) = -12 and p(2) - p(0) = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(2) = -12 and p(2) - p(0) = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, -5, -2", "ref": "6, -5, -2", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002276, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "val", "x": 2, "v": "-12"}, {"t": "diff", "x": 2, "y": 0, "v": "-18"}]}, "seed": 157317360100001761, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 6, "valid_answers": ["6, -5, -2", "6, -3, -3", "6, -1, -4", "6, 1, -5", "6, 3, -6"]}, "cell": "poly_mixed/0", "stage": "B"}
+{"id": "po-poly_mixed-t0-eval-001392", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = 6 and p(2) - p(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = 6 and p(2) - p(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 6, 0", "ref": "0, 6, 0", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001821, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "val", "x": 1, "v": "6"}, {"t": "diff", "x": 2, "y": 1, "v": "6"}]}, "seed": 157317360100001868, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["0, 6, 0", "2, 3, 1", "4, 0, 2", "6, -3, 3"]}, "cell": "poly_mixed/0", "stage": "B"}
+{"id": "po-poly_mixed-t0-eval-000820", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -3 and p'(0) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -3 and p'(0) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -3, -6", "ref": "-3, -3, -6", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.005917, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "val", "x": 0, "v": "-3"}, {"t": "der", "x": 0, "m": 1, "v": "-3"}]}, "seed": 157317360100001079, "version": "po-v0", "tier": 0, "n_raw": 13, "small": 3, "valid_answers": ["-3, -3, -6", "-3, -3, -5", "-3, -3, -4", "-3, -3, -3", "-3, -3, -2", "-3, -3, -1", "-3, -3, 0", "-3, -3, 1", "-3, -3, 2", "-3, -3, 3", "-3, -3, 4", "-3, -3, 5", "-3, -3, 6"]}, "cell": "poly_mixed/0", "stage": "B"}
+{"id": "po-poly_mixed-t0-eval-000858", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(-1) = -2 and p'(0) = -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(-1) = -2 and p'(0) = -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -5, -1", "ref": "-6, -5, -1", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.002731, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "val", "x": -1, "v": "-2"}, {"t": "der", "x": 0, "m": 1, "v": "-5"}]}, "seed": 157317360100001129, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 5, "valid_answers": ["-6, -5, -1", "-5, -5, -2", "-4, -5, -3", "-3, -5, -4", "-2, -5, -5", "-1, -5, -6"]}, "cell": "poly_mixed/0", "stage": "B"}
+{"id": "po-poly_mixed-t0-eval-000492", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) - p(2) = -6 and p(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) - p(2) = -6 and p(1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 5, -1", "ref": "1, 5, -1", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.002731, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "diff", "x": 0, "y": 2, "v": "-6"}, {"t": "val", "x": 1, "v": "5"}]}, "seed": 157317360100000645, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 3, "valid_answers": ["1, 5, -1", "2, 3, 0", "3, 1, 1", "4, -1, 2", "5, -3, 3", "6, -5, 4"]}, "cell": "poly_mixed/0", "stage": "B"}
+{"id": "po-poly_mixed-t0-eval-001814", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = 2 and p(0) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = 2 and p(0) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, -6, 4", "ref": "4, -6, 4", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.005007, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "val", "x": 1, "v": "2"}, {"t": "val", "x": 0, "v": "4"}]}, "seed": 157317360100002420, "version": "po-v0", "tier": 0, "n_raw": 11, "small": 4, "valid_answers": ["4, -6, 4", "4, -5, 3", "4, -4, 2", "4, -3, 1", "4, -2, 0", "4, -1, -1", "4, 0, -2", "4, 1, -3", "4, 2, -4", "4, 3, -5", "4, 4, -6"]}, "cell": "poly_mixed/0", "stage": "B"}
+{"id": "po-poly_mixed-t0-eval-002022", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(2) = 18 and p(2) - p(1) = 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(2) = 18 and p(2) - p(1) = 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -2, 5", "ref": "-6, -2, 5", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.005917, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": 2, "m": 1, "v": "18"}, {"t": "diff", "x": 2, "y": 1, "v": "13"}]}, "seed": 157317360100002690, "version": "po-v0", "tier": 0, "n_raw": 13, "small": 5, "valid_answers": ["-6, -2, 5", "-5, -2, 5", "-4, -2, 5", "-3, -2, 5", "-2, -2, 5", "-1, -2, 5", "0, -2, 5", "1, -2, 5", "2, -2, 5", "3, -2, 5", "4, -2, 5", "5, -2, 5", "6, -2, 5"]}, "cell": "poly_mixed/0", "stage": "B"}
+{"id": "po-poly_mixed-t0-eval-001104", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) - p(2) = -8 and \\int_{1}^{2} p(x)\\,dx = \\frac{20}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) - p(2) = -8 and \\int_{1}^{2} p(x)\\,dx = \\frac{20}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 6, -1", "ref": "0, 6, -1", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001365, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "diff", "x": 0, "y": 2, "v": "-8"}, {"t": "int", "lo": 1, "hi": 2, "v": "20/3"}]}, "seed": 157317360100001466, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["0, 6, -1", "2, 0, 2", "4, -6, 5"]}, "cell": "poly_mixed/0", "stage": "B"}
+{"id": "po-poly_mixed-t0-eval-002043", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) - p(1) = -3 and p'(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) - p(1) = -3 and p'(1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 2, 1", "ref": "-6, 2, 1", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.005917, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "diff", "x": 0, "y": 1, "v": "-3"}, {"t": "der", "x": 1, "m": 1, "v": "4"}]}, "seed": 157317360100002719, "version": "po-v0", "tier": 0, "n_raw": 13, "small": 2, "valid_answers": ["-6, 2, 1", "-5, 2, 1", "-4, 2, 1", "-3, 2, 1", "-2, 2, 1", "-1, 2, 1", "0, 2, 1", "1, 2, 1", "2, 2, 1", "3, 2, 1", "4, 2, 1", "5, 2, 1", "6, 2, 1"]}, "cell": "poly_mixed/0", "stage": "B"}
+{"id": "po-poly_mixed-t0-eval-000358", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = -\\frac{8}{3} and p(0) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{1} p(x)\\,dx = -\\frac{8}{3} and p(0) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -6, -1", "ref": "-1, -6, -1", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.005917, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "-8/3"}, {"t": "val", "x": 0, "v": "-1"}]}, "seed": 157317360100000471, "version": "po-v0", "tier": 0, "n_raw": 13, "small": 1, "valid_answers": ["-1, -6, -1", "-1, -5, -1", "-1, -4, -1", "-1, -3, -1", "-1, -2, -1", "-1, -1, -1", "-1, 0, -1", "-1, 1, -1", "-1, 2, -1", "-1, 3, -1", "-1, 4, -1", "-1, 5, -1", "-1, 6, -1"]}, "cell": "poly_mixed/0", "stage": "B"}
+{"id": "po-poly_mixed-t0-eval-002208", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = -10 and \\int_{-1}^{1} p(x)\\,dx = \\frac{2}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = -10 and \\int_{-1}^{1} p(x)\\,dx = \\frac{2}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -6, -2", "ref": "1, -6, -2", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00091, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "-10"}, {"t": "int", "lo": -1, "hi": 1, "v": "2/3"}]}, "seed": 157317360100002936, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["1, -6, -2", "2, 0, -5"]}, "cell": "poly_mixed/0", "stage": "B"}
+{"id": "po-poly_mixed-t1-eval-000010", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{0} p(x)\\,dx = -6, \\int_{0}^{2} p(x)\\,dx = -24 and \\int_{-1}^{1} p(x)\\,dx = -12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{0} p(x)\\,dx = -6, \\int_{0}^{2} p(x)\\,dx = -24 and \\int_{-1}^{1} p(x)\\,dx = -12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 2, 0, -4", "ref": "-6, 2, 0, -4", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "int", "lo": -1, "hi": 0, "v": "-6"}, {"t": "int", "lo": 0, "hi": 2, "v": "-24"}, {"t": "int", "lo": -1, "hi": 1, "v": "-12"}]}, "seed": 157317360101000020, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 5, "valid_answers": ["-6, 2, 0, -4", "-5, 1, -3, -2", "-4, 0, -6, 0"]}, "cell": "poly_mixed/1", "stage": "B"}
+{"id": "po-poly_mixed-t1-eval-001947", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(1) = -12, \\int_{-1}^{1} p(x)\\,dx = 4 and p(0) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(1) = -12, \\int_{-1}^{1} p(x)\\,dx = 4 and p(0) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -6, 0, -2", "ref": "2, -6, 0, -2", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 1, "m": 2, "v": "-12"}, {"t": "int", "lo": -1, "hi": 1, "v": "4"}, {"t": "val", "x": 0, "v": "2"}]}, "seed": 157317360101003549, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 2, "valid_answers": ["2, -6, 0, -2", "2, -5, 0, -2", "2, -4, 0, -2", "2, -3, 0, -2", "2, -2, 0, -2", "2, -1, 0, -2", "2, 0, 0, -2", "2, 1, 0, -2", "2, 2, 0, -2", "2, 3, 0, -2", "2, 4, 0, -2", "2, 5, 0, -2", "2, 6, 0, -2"]}, "cell": "poly_mixed/1", "stage": "B"}
+{"id": "po-poly_mixed-t1-eval-001378", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(0) - p(2) = -48, p''(-1) = -20 and p''(1) = 40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(0) - p(2) = -48, p''(-1) = -20 and p''(1) = 40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -6, 5, 5", "ref": "-6, -6, 5, 5", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "diff", "x": 0, "y": 2, "v": "-48"}, {"t": "der", "x": -1, "m": 2, "v": "-20"}, {"t": "der", "x": 1, "m": 2, "v": "40"}]}, "seed": 157317360101002514, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 6, "valid_answers": ["-6, -6, 5, 5", "-5, -6, 5, 5", "-4, -6, 5, 5", "-3, -6, 5, 5", "-2, -6, 5, 5", "-1, -6, 5, 5", "0, -6, 5, 5", "1, -6, 5, 5", "2, -6, 5, 5", "3, -6, 5, 5", "4, -6, 5, 5", "5, -6, 5, 5", "6, -6, 5, 5"]}, "cell": "poly_mixed/1", "stage": "B"}
+{"id": "po-poly_mixed-t1-eval-000409", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(1) = -28, p''(0) = -4 and p'(1) = -19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(1) = -28, p''(0) = -4 and p'(1) = -19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -3, -2, -4", "ref": "-6, -3, -2, -4", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 1, "m": 2, "v": "-28"}, {"t": "der", "x": 0, "m": 2, "v": "-4"}, {"t": "der", "x": 1, "m": 1, "v": "-19"}]}, "seed": 157317360101000717, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 4, "valid_answers": ["-6, -3, -2, -4", "-5, -3, -2, -4", "-4, -3, -2, -4", "-3, -3, -2, -4", "-2, -3, -2, -4", "-1, -3, -2, -4", "0, -3, -2, -4", "1, -3, -2, -4", "2, -3, -2, -4", "3, -3, -2, -4", "4, -3, -2, -4", "5, -3, -2, -4", "6, -3, -2, -4"]}, "cell": "poly_mixed/1", "stage": "B"}
+{"id": "po-poly_mixed-t1-eval-002070", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(0) = 4, p''(1) = -26 and p(2) - p(-1) = -27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(0) = 4, p''(1) = -26 and p(2) - p(-1) = -27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 4, 2, -5", "ref": "-6, 4, 2, -5", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 0, "m": 2, "v": "4"}, {"t": "der", "x": 1, "m": 2, "v": "-26"}, {"t": "diff", "x": 2, "y": -1, "v": "-27"}]}, "seed": 157317360101003751, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 5, "valid_answers": ["-6, 4, 2, -5", "-5, 4, 2, -5", "-4, 4, 2, -5", "-3, 4, 2, -5", "-2, 4, 2, -5", "-1, 4, 2, -5", "0, 4, 2, -5", "1, 4, 2, -5", "2, 4, 2, -5", "3, 4, 2, -5", "4, 4, 2, -5", "5, 4, 2, -5", "6, 4, 2, -5"]}, "cell": "poly_mixed/1", "stage": "B"}
+{"id": "po-poly_mixed-t1-eval-001665", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = -22, \\int_{0}^{1} p(x)\\,dx = -\\frac{31}{6} and p'(-1) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = -22, \\int_{0}^{1} p(x)\\,dx = -\\frac{31}{6} and p'(-1) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 6, -5, -6", "ref": "-5, 6, -5, -6", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "-22"}, {"t": "int", "lo": 0, "hi": 1, "v": "-31/6"}, {"t": "der", "x": -1, "m": 1, "v": "-2"}]}, "seed": 157317360101003030, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 6, "valid_answers": ["-5, 6, -5, -6", "0, -6, -5, -2"]}, "cell": "poly_mixed/1", "stage": "B"}
+{"id": "po-poly_mixed-t1-eval-001799", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(0) = 0, p''(2) = -24 and p'(2) = -23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(0) = 0, p''(2) = -24 and p'(2) = -23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 1, 0, -2", "ref": "-6, 1, 0, -2", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 0, "m": 2, "v": "0"}, {"t": "der", "x": 2, "m": 2, "v": "-24"}, {"t": "der", "x": 2, "m": 1, "v": "-23"}]}, "seed": 157317360101003276, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 2, "valid_answers": ["-6, 1, 0, -2", "-5, 1, 0, -2", "-4, 1, 0, -2", "-3, 1, 0, -2", "-2, 1, 0, -2", "-1, 1, 0, -2", "0, 1, 0, -2", "1, 1, 0, -2", "2, 1, 0, -2", "3, 1, 0, -2", "4, 1, 0, -2", "5, 1, 0, -2", "6, 1, 0, -2"]}, "cell": "poly_mixed/1", "stage": "B"}
+{"id": "po-poly_mixed-t1-eval-002458", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(2) = 41, p(-1) - p(2) = -33 and p''(0) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(2) = 41, p(-1) - p(2) = -33 and p''(0) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 1, 4, 2", "ref": "-6, 1, 4, 2", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 2, "m": 1, "v": "41"}, {"t": "diff", "x": -1, "y": 2, "v": "-33"}, {"t": "der", "x": 0, "m": 2, "v": "8"}]}, "seed": 157317360101004467, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 4, "valid_answers": ["-6, 1, 4, 2", "-5, 1, 4, 2", "-4, 1, 4, 2", "-3, 1, 4, 2", "-2, 1, 4, 2", "-1, 1, 4, 2", "0, 1, 4, 2", "1, 1, 4, 2", "2, 1, 4, 2", "3, 1, 4, 2", "4, 1, 4, 2", "5, 1, 4, 2", "6, 1, 4, 2"]}, "cell": "poly_mixed/1", "stage": "B"}
+{"id": "po-poly_mixed-t1-eval-002069", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(-1) = 25, p(-1) - p(2) = -30 and p''(2) = 38. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(-1) = 25, p(-1) - p(2) = -30 and p''(2) = 38. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 3, -5, 4", "ref": "-6, 3, -5, 4", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "25"}, {"t": "diff", "x": -1, "y": 2, "v": "-30"}, {"t": "der", "x": 2, "m": 2, "v": "38"}]}, "seed": 157317360101003750, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 5, "valid_answers": ["-6, 3, -5, 4", "-5, 3, -5, 4", "-4, 3, -5, 4", "-3, 3, -5, 4", "-2, 3, -5, 4", "-1, 3, -5, 4", "0, 3, -5, 4", "1, 3, -5, 4", "2, 3, -5, 4", "3, 3, -5, 4", "4, 3, -5, 4", "5, 3, -5, 4", "6, 3, -5, 4"]}, "cell": "poly_mixed/1", "stage": "B"}
+{"id": "po-poly_mixed-t1-eval-001477", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = -32, p'(-1) = -16 and p''(2) = -80. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = -32, p'(-1) = -16 and p''(2) = -80. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -6, -4, -6", "ref": "-6, -6, -4, -6", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "-32"}, {"t": "der", "x": -1, "m": 1, "v": "-16"}, {"t": "der", "x": 2, "m": 2, "v": "-80"}]}, "seed": 157317360101002683, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 6, "valid_answers": ["-6, -6, -4, -6", "-5, -6, -4, -6", "-4, -6, -4, -6", "-3, -6, -4, -6", "-2, -6, -4, -6", "-1, -6, -4, -6", "0, -6, -4, -6", "1, -6, -4, -6", "2, -6, -4, -6", "3, -6, -4, -6", "4, -6, -4, -6", "5, -6, -4, -6", "6, -6, -4, -6"]}, "cell": "poly_mixed/1", "stage": "B"}
+{"id": "po-poly_mixed-t1-eval-001093", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(0) = -8, p''(2) = -44 and p'(0) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(0) = -8, p''(2) = -44 and p'(0) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -6, -4, -3", "ref": "-6, -6, -4, -3", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 0, "m": 2, "v": "-8"}, {"t": "der", "x": 2, "m": 2, "v": "-44"}, {"t": "der", "x": 0, "m": 1, "v": "-6"}]}, "seed": 157317360101001992, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 6, "valid_answers": ["-6, -6, -4, -3", "-5, -6, -4, -3", "-4, -6, -4, -3", "-3, -6, -4, -3", "-2, -6, -4, -3", "-1, -6, -4, -3", "0, -6, -4, -3", "1, -6, -4, -3", "2, -6, -4, -3", "3, -6, -4, -3", "4, -6, -4, -3", "5, -6, -4, -3", "6, -6, -4, -3"]}, "cell": "poly_mixed/1", "stage": "B"}
+{"id": "po-poly_mixed-t1-eval-001039", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = -3, p(0) = 5 and p''(2) = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = -3, p(0) = 5 and p''(2) = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, -3, 3, -2", "ref": "5, -3, 3, -2", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "-3"}, {"t": "val", "x": 0, "v": "5"}, {"t": "der", "x": 2, "m": 2, "v": "-18"}]}, "seed": 157317360101001892, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 5, "valid_answers": ["5, -3, 3, -2", "5, 6, -3, -1"]}, "cell": "poly_mixed/1", "stage": "B"}
+{"id": "po-poly_mixed-t2-eval-002323", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(1) - p(2) = -34, p'(3) = 111 and p(2) - p(0) = 46. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(1) - p(2) = -34, p'(3) = 111 and p(2) - p(0) = 46. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 9, -1, 4", "ref": "-9, 9, -1, 4", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.000146, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "diff", "x": 1, "y": 2, "v": "-34"}, {"t": "der", "x": 3, "m": 1, "v": "111"}, {"t": "diff", "x": 2, "y": 0, "v": "46"}]}, "seed": 157317360102004813, "version": "po-v0", "tier": 2, "n_raw": 19, "small": 9, "valid_answers": ["-9, 9, -1, 4", "-8, 9, -1, 4", "-7, 9, -1, 4", "-6, 9, -1, 4", "-5, 9, -1, 4", "-4, 9, -1, 4", "-3, 9, -1, 4", "-2, 9, -1, 4", "-1, 9, -1, 4", "0, 9, -1, 4", "1, 9, -1, 4", "2, 9, -1, 4", "3, 9, -1, 4", "4, 9, -1, 4", "5, 9, -1, 4", "6, 9, -1, 4", "7, 9, -1, 4", "8, 9, -1, 4", "9, 9, -1, 4"]}, "cell": "poly_mixed/2", "stage": "C"}
+{"id": "po-poly_mixed-t2-eval-000218", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = \\frac{98}{3}, p(2) = 63 and p(0) - p(1) = -15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = \\frac{98}{3}, p(2) = 63 and p(0) - p(1) = -15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 4, 7, 4", "ref": "-5, 4, 7, 4", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 2.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "98/3"}, {"t": "val", "x": 2, "v": "63"}, {"t": "diff", "x": 0, "y": 1, "v": "-15"}]}, "seed": 157317360102000411, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 6, "valid_answers": ["-5, 4, 7, 4", "-5, 6, 4, 5", "-5, 8, 1, 6"]}, "cell": "poly_mixed/2", "stage": "C"}
+{"id": "po-poly_mixed-t2-eval-002098", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-1) = 1, p(-1) - p(2) = 9 and p(0) - p(-2) = 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-1) = 1, p(-1) - p(2) = 9 and p(0) - p(-2) = 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -9, -3, 3", "ref": "-2, -9, -3, 3", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "val", "x": -1, "v": "1"}, {"t": "diff", "x": -1, "y": 2, "v": "9"}, {"t": "diff", "x": 0, "y": -2, "v": "18"}]}, "seed": 157317360102004361, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 6, "valid_answers": ["-2, -9, -3, 3", "6, 1, -4, 0"]}, "cell": "poly_mixed/2", "stage": "C"}
+{"id": "po-poly_mixed-t2-eval-001194", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{1} p(x)\\,dx = -\\frac{25}{3}, \\int_{0}^{2} p(x)\\,dx = -\\frac{68}{3} and p(1) = -10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{1} p(x)\\,dx = -\\frac{25}{3}, \\int_{0}^{2} p(x)\\,dx = -\\frac{68}{3} and p(1) = -10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 2, -4, 0", "ref": "-8, 2, -4, 0", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 2.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": 0, "hi": 1, "v": "-25/3"}, {"t": "int", "lo": 0, "hi": 2, "v": "-68/3"}, {"t": "val", "x": 1, "v": "-10"}]}, "seed": 157317360102002451, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 7, "valid_answers": ["-8, 2, -4, 0", "-7, -3, 2, -2", "-6, -8, 8, -4"]}, "cell": "poly_mixed/2", "stage": "C"}
+{"id": "po-poly_mixed-t2-eval-000076", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(3) - p(-1) = -44, p'(0) = 0 and p(-2) - p(0) = 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(3) - p(-1) = -44, p'(0) = 0 and p(-2) - p(0) = 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 0, 5, -3", "ref": "-9, 0, 5, -3", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.000146, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "diff", "x": 3, "y": -1, "v": "-44"}, {"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "diff", "x": -2, "y": 0, "v": "44"}]}, "seed": 157317360102000146, "version": "po-v0", "tier": 2, "n_raw": 19, "small": 5, "valid_answers": ["-9, 0, 5, -3", "-8, 0, 5, -3", "-7, 0, 5, -3", "-6, 0, 5, -3", "-5, 0, 5, -3", "-4, 0, 5, -3", "-3, 0, 5, -3", "-2, 0, 5, -3", "-1, 0, 5, -3", "0, 0, 5, -3", "1, 0, 5, -3", "2, 0, 5, -3", "3, 0, 5, -3", "4, 0, 5, -3", "5, 0, 5, -3", "6, 0, 5, -3", "7, 0, 5, -3", "8, 0, 5, -3", "9, 0, 5, -3"]}, "cell": "poly_mixed/2", "stage": "C"}
+{"id": "po-poly_mixed-t2-eval-001844", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{1} p(x)\\,dx = 6, p''(1) = -18 and \\int_{0}^{2} p(x)\\,dx = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{1} p(x)\\,dx = 6, p''(1) = -18 and \\int_{0}^{2} p(x)\\,dx = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 7, -3, -2", "ref": "4, 7, -3, -2", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 2.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": 0, "hi": 1, "v": "6"}, {"t": "der", "x": 1, "m": 2, "v": "-18"}, {"t": "int", "lo": 0, "hi": 2, "v": "6"}]}, "seed": 157317360102003810, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 5, "valid_answers": ["4, 7, -3, -2", "5, 2, 3, -4", "6, -3, 9, -6"]}, "cell": "poly_mixed/2", "stage": "C"}
+{"id": "po-poly_mixed-t2-eval-000838", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(1) = -3, p(0) - p(-2) = -36 and p(0) - p(-1) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(1) = -3, p(0) - p(-2) = -36 and p(0) - p(-1) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 6, 4, -4", "ref": "-9, 6, 4, -4", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 3.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "val", "x": 1, "v": "-3"}, {"t": "diff", "x": 0, "y": -2, "v": "-36"}, {"t": "diff", "x": 0, "y": -1, "v": "-2"}]}, "seed": 157317360102001716, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 5, "valid_answers": ["-9, 6, 4, -4", "-3, 4, 1, -5", "3, 2, -2, -6", "9, 0, -5, -7"]}, "cell": "poly_mixed/2", "stage": "C"}
+{"id": "po-poly_mixed-t2-eval-001044", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = -\\frac{44}{3}, p'(0) = 5 and p''(2) = -80. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{-1}^{1} p(x)\\,dx = -\\frac{44}{3}, p'(0) = 5 and p''(2) = -80. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 5, 2, -7", "ref": "-8, 5, 2, -7", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "-44/3"}, {"t": "der", "x": 0, "m": 1, "v": "5"}, {"t": "der", "x": 2, "m": 2, "v": "-80"}]}, "seed": 157317360102002151, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 6, "valid_answers": ["-8, 5, 2, -7", "-6, 5, -4, -6"]}, "cell": "poly_mixed/2", "stage": "C"}
+{"id": "po-poly_mixed-t2-eval-001802", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(0) - p(-1) = -4, p(-2) - p(2) = 136 and p(2) = -96. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(0) - p(-1) = -4, p(-2) - p(2) = 136 and p(2) = -96. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -2, -6, -8", "ref": "-4, -2, -6, -8", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "diff", "x": 0, "y": -1, "v": "-4"}, {"t": "diff", "x": -2, "y": 2, "v": "136"}, {"t": "val", "x": 2, "v": "-96"}]}, "seed": 157317360102003730, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 8, "valid_answers": ["-4, -2, -6, -8", "8, -6, -9, -7"]}, "cell": "poly_mixed/2", "stage": "C"}
+{"id": "po-poly_mixed-t2-eval-002470", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(1) - p(3) = -90, p(-1) - p(1) = -10 and \\int_{-1}^{1} p(x)\\,dx = \\frac{44}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(1) - p(3) = -90, p(-1) - p(1) = -10 and \\int_{-1}^{1} p(x)\\,dx = \\frac{44}{3}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 4, 7, 1", "ref": "5, 4, 7, 1", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 3.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "diff", "x": 1, "y": 3, "v": "-90"}, {"t": "diff", "x": -1, "y": 1, "v": "-10"}, {"t": "int", "lo": -1, "hi": 1, "v": "44/3"}]}, "seed": 157317360102005118, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 6, "valid_answers": ["5, 4, 7, 1", "6, 3, 4, 2", "7, 2, 1, 3", "8, 1, -2, 4", "9, 0, -5, 5"]}, "cell": "poly_mixed/2", "stage": "C"}
+{"id": "po-poly_mixed-t2-eval-000093", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(1) = 34, p(2) - p(0) = 46 and p(0) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(1) = 34, p(2) - p(0) = 46 and p(0) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, -5, 8, 3", "ref": "5, -5, 8, 3", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 4.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 1, "m": 2, "v": "34"}, {"t": "diff", "x": 2, "y": 0, "v": "46"}, {"t": "val", "x": 0, "v": "5"}]}, "seed": 157317360102000172, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 5, "valid_answers": ["5, -5, 8, 3", "5, -3, 5, 4", "5, -1, 2, 5", "5, 1, -1, 6", "5, 3, -4, 7", "5, 5, -7, 8"]}, "cell": "poly_mixed/2", "stage": "C"}
+{"id": "po-poly_mixed-t2-eval-001283", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = \\frac{110}{3}, \\int_{-1}^{1} p(x)\\,dx = \\frac{44}{3} and p(0) - p(-1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies \\int_{0}^{2} p(x)\\,dx = \\frac{110}{3}, \\int_{-1}^{1} p(x)\\,dx = \\frac{44}{3} and p(0) - p(-1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 1, 4, 3", "ref": "6, 1, 4, 3", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "110/3"}, {"t": "int", "lo": -1, "hi": 1, "v": "44/3"}, {"t": "diff", "x": 0, "y": -1, "v": "0"}]}, "seed": 157317360102002636, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 6, "valid_answers": ["6, 1, 4, 3", "7, -8, 1, 9"]}, "cell": "poly_mixed/2", "stage": "C"}
+{"id": "po-poly_mixed-t3-eval-001197", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies \\int_{-1}^{1} p(x)\\,dx = -\\frac{44}{5}, p(0) = -3, \\int_{0}^{1} p(x)\\,dx = -\\frac{17}{5} and \\int_{-1}^{0} p(x)\\,dx = -\\frac{27}{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies \\int_{-1}^{1} p(x)\\,dx = -\\frac{44}{5}, p(0) = -3, \\int_{0}^{1} p(x)\\,dx = -\\frac{17}{5} and \\int_{-1}^{0} p(x)\\,dx = -\\frac{27}{5}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -1, -6, 6, 3", "ref": "-3, -1, -6, 6, 3", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 3.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "-44/5"}, {"t": "val", "x": 0, "v": "-3"}, {"t": "int", "lo": 0, "hi": 1, "v": "-17/5"}, {"t": "int", "lo": -1, "hi": 0, "v": "-27/5"}]}, "seed": 157317360103007036, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 3, "valid_answers": ["-3, -1, -6, 6, 3", "-3, -1, -3, 6, -2", "-3, 0, -6, 4, 3", "-3, 0, -3, 4, -2", "-3, 1, -6, 2, 3", "-3, 1, -3, 2, -2", "-3, 2, -6, 0, 3", "-3, 2, -3, 0, -2", "-3, 3, -6, -2, 3", "-3, 3, -3, -2, -2", "-3, 4, -6, -4, 3", "-3, 4, -3, -4, -2", "-3, 5, -6, -6, 3", "-3, 5, -3, -6, -2"]}, "cell": "poly_mixed/3", "stage": "B"}
+{"id": "po-poly_mixed-t3-eval-001359", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p''(1) = -90, p''(-2) = -216, p'(-2) = 108 and p(3) - p(2) = -455. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p''(1) = -90, p''(-2) = -216, p'(-2) = 108 and p(3) - p(2) = -455. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 0, 6, -5, -6", "ref": "-6, 0, 6, -5, -6", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 3.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": 1, "m": 2, "v": "-90"}, {"t": "der", "x": -2, "m": 2, "v": "-216"}, {"t": "der", "x": -2, "m": 1, "v": "108"}, {"t": "diff", "x": 3, "y": 2, "v": "-455"}]}, "seed": 157317360103007924, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 6, "valid_answers": ["-6, 0, 6, -5, -6", "-5, 0, 6, -5, -6", "-4, 0, 6, -5, -6", "-3, 0, 6, -5, -6", "-2, 0, 6, -5, -6", "-1, 0, 6, -5, -6", "0, 0, 6, -5, -6", "1, 0, 6, -5, -6", "2, 0, 6, -5, -6", "3, 0, 6, -5, -6", "4, 0, 6, -5, -6", "5, 0, 6, -5, -6", "6, 0, 6, -5, -6"]}, "cell": "poly_mixed/3", "stage": "B"}
+{"id": "po-poly_mixed-t3-eval-000951", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p''(0) = -12, p'(2) = -145, p''(2) = -228 and p''(1) = -48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p''(0) = -12, p'(2) = -145, p''(2) = -228 and p''(1) = -48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -1, -6, 6, -6", "ref": "-6, -1, -6, 6, -6", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 3.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": 0, "m": 2, "v": "-12"}, {"t": "der", "x": 2, "m": 1, "v": "-145"}, {"t": "der", "x": 2, "m": 2, "v": "-228"}, {"t": "der", "x": 1, "m": 2, "v": "-48"}]}, "seed": 157317360103005504, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 6, "valid_answers": ["-6, -1, -6, 6, -6", "-5, -1, -6, 6, -6", "-4, -1, -6, 6, -6", "-3, -1, -6, 6, -6", "-2, -1, -6, 6, -6", "-1, -1, -6, 6, -6", "0, -1, -6, 6, -6", "1, -1, -6, 6, -6", "2, -1, -6, 6, -6", "3, -1, -6, 6, -6", "4, -1, -6, 6, -6", "5, -1, -6, 6, -6", "6, -1, -6, 6, -6"]}, "cell": "poly_mixed/3", "stage": "B"}
+{"id": "po-poly_mixed-t3-eval-002048", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(3) - p(-1) = 224, p''(2) = 152, p''(-1) = 80 and p'(1) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(3) - p(-1) = 224, p''(2) = 152, p''(-1) = 80 and p'(1) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -4, 4, -4, 4", "ref": "-6, -4, 4, -4, 4", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 3.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "diff", "x": 3, "y": -1, "v": "224"}, {"t": "der", "x": 2, "m": 2, "v": "152"}, {"t": "der", "x": -1, "m": 2, "v": "80"}, {"t": "der", "x": 1, "m": 1, "v": "8"}]}, "seed": 157317360103011837, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 4, "valid_answers": ["-6, -4, 4, -4, 4", "-5, -4, 4, -4, 4", "-4, -4, 4, -4, 4", "-3, -4, 4, -4, 4", "-2, -4, 4, -4, 4", "-1, -4, 4, -4, 4", "0, -4, 4, -4, 4", "1, -4, 4, -4, 4", "2, -4, 4, -4, 4", "3, -4, 4, -4, 4", "4, -4, 4, -4, 4", "5, -4, 4, -4, 4", "6, -4, 4, -4, 4"]}, "cell": "poly_mixed/3", "stage": "B"}
+{"id": "po-poly_mixed-t3-eval-000332", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(2) - p(-1) = 15, p''(2) = 68, p(0) - p(3) = -105 and p(-1) - p(3) = -104. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(2) - p(-1) = 15, p''(2) = 68, p(0) - p(3) = -105 and p(-1) - p(3) = -104. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -4, -2, 2, 1", "ref": "-6, -4, -2, 2, 1", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 3.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "diff", "x": 2, "y": -1, "v": "15"}, {"t": "der", "x": 2, "m": 2, "v": "68"}, {"t": "diff", "x": 0, "y": 3, "v": "-105"}, {"t": "diff", "x": -1, "y": 3, "v": "-104"}]}, "seed": 157317360103001907, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 4, "valid_answers": ["-6, -4, -2, 2, 1", "-5, -4, -2, 2, 1", "-4, -4, -2, 2, 1", "-3, -4, -2, 2, 1", "-2, -4, -2, 2, 1", "-1, -4, -2, 2, 1", "0, -4, -2, 2, 1", "1, -4, -2, 2, 1", "2, -4, -2, 2, 1", "3, -4, -2, 2, 1", "4, -4, -2, 2, 1", "5, -4, -2, 2, 1", "6, -4, -2, 2, 1"]}, "cell": "poly_mixed/3", "stage": "B"}
+{"id": "po-poly_mixed-t3-eval-002185", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-1) - p(2) = -33, p(-2) = 100, p(1) - p(-1) = 0 and p(-2) - p(1) = 93. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-1) - p(2) = -33, p(-2) = 100, p(1) - p(-1) = 0 and p(-2) - p(1) = 93. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 5, 6, -5, 3", "ref": "-2, 5, 6, -5, 3", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 8e-06, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "diff", "x": -1, "y": 2, "v": "-33"}, {"t": "val", "x": -2, "v": "100"}, {"t": "diff", "x": 1, "y": -1, "v": "0"}, {"t": "diff", "x": -2, "y": 1, "v": "93"}]}, "seed": 157317360103012639, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 5, "valid_answers": ["-2, 5, 6, -5, 3", "2, 5, 1, -5, 4", "6, 5, -4, -5, 5"]}, "cell": "poly_mixed/3", "stage": "B"}
+{"id": "po-poly_mixed-t3-eval-001166", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-1) = -21, p''(2) = 94, p'(0) = 1 and p(3) - p(0) = 129. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-1) = -21, p''(2) = 94, p'(0) = 1 and p(3) - p(0) = 129. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 1, -1, -4, 3", "ref": "-6, 1, -1, -4, 3", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 3.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "-21"}, {"t": "der", "x": 2, "m": 2, "v": "94"}, {"t": "der", "x": 0, "m": 1, "v": "1"}, {"t": "diff", "x": 3, "y": 0, "v": "129"}]}, "seed": 157317360103006873, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 4, "valid_answers": ["-6, 1, -1, -4, 3", "-5, 1, -1, -4, 3", "-4, 1, -1, -4, 3", "-3, 1, -1, -4, 3", "-2, 1, -1, -4, 3", "-1, 1, -1, -4, 3", "0, 1, -1, -4, 3", "1, 1, -1, -4, 3", "2, 1, -1, -4, 3", "3, 1, -1, -4, 3", "4, 1, -1, -4, 3", "5, 1, -1, -4, 3", "6, 1, -1, -4, 3"]}, "cell": "poly_mixed/3", "stage": "B"}
+{"id": "po-poly_mixed-t3-eval-002200", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-2) = -221, p'(1) = 16, p''(-2) = 322 and p(-1) - p(3) = -400. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(-2) = -221, p'(1) = 16, p''(-2) = 322 and p(-1) - p(3) = -400. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 3, -1, -3, 6", "ref": "-6, 3, -1, -3, 6", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 3.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": -2, "m": 1, "v": "-221"}, {"t": "der", "x": 1, "m": 1, "v": "16"}, {"t": "der", "x": -2, "m": 2, "v": "322"}, {"t": "diff", "x": -1, "y": 3, "v": "-400"}]}, "seed": 157317360103012732, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 6, "valid_answers": ["-6, 3, -1, -3, 6", "-5, 3, -1, -3, 6", "-4, 3, -1, -3, 6", "-3, 3, -1, -3, 6", "-2, 3, -1, -3, 6", "-1, 3, -1, -3, 6", "0, 3, -1, -3, 6", "1, 3, -1, -3, 6", "2, 3, -1, -3, 6", "3, 3, -1, -3, 6", "4, 3, -1, -3, 6", "5, 3, -1, -3, 6", "6, 3, -1, -3, 6"]}, "cell": "poly_mixed/3", "stage": "B"}
+{"id": "po-poly_mixed-t3-eval-002494", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(1) = -15, p(2) - p(-1) = -42, p''(2) = -88 and p''(-2) = -16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p'(1) = -15, p(2) - p(-1) = -42, p''(2) = -88 and p''(-2) = -16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 2, -2, -3, -1", "ref": "-6, 2, -2, -3, -1", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 3.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "-15"}, {"t": "diff", "x": 2, "y": -1, "v": "-42"}, {"t": "der", "x": 2, "m": 2, "v": "-88"}, {"t": "der", "x": -2, "m": 2, "v": "-16"}]}, "seed": 157317360103014531, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 3, "valid_answers": ["-6, 2, -2, -3, -1", "-5, 2, -2, -3, -1", "-4, 2, -2, -3, -1", "-3, 2, -2, -3, -1", "-2, 2, -2, -3, -1", "-1, 2, -2, -3, -1", "0, 2, -2, -3, -1", "1, 2, -2, -3, -1", "2, 2, -2, -3, -1", "3, 2, -2, -3, -1", "4, 2, -2, -3, -1", "5, 2, -2, -3, -1", "6, 2, -2, -3, -1"]}, "cell": "poly_mixed/3", "stage": "B"}
+{"id": "po-poly_mixed-t3-eval-000458", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies \\int_{0}^{2} p(x)\\,dx = -\\frac{368}{15}, p'(0) = 3, p(1) = -4 and p''(1) = -46. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies \\int_{0}^{2} p(x)\\,dx = -\\frac{368}{15}, p'(0) = 3, p(1) = -4 and p''(1) = -46. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 3, 4, -3, -3", "ref": "-5, 3, 4, -3, -3", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "-368/15"}, {"t": "der", "x": 0, "m": 1, "v": "3"}, {"t": "val", "x": 1, "v": "-4"}, {"t": "der", "x": 1, "m": 2, "v": "-46"}]}, "seed": 157317360103002637, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 3, "valid_answers": ["-5, 3, 4, -3, -3", "-3, 3, 1, -2, -3", "-1, 3, -2, -1, -3", "1, 3, -5, 0, -3"]}, "cell": "poly_mixed/3", "stage": "B"}
+{"id": "po-poly_mixed-t3-eval-000960", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies \\int_{0}^{1} p(x)\\,dx = -\\frac{209}{60}, \\int_{-1}^{0} p(x)\\,dx = -\\frac{599}{60}, \\int_{1}^{2} p(x)\\,dx = -\\frac{479}{60} and \\int_{0}^{2} p(x)\\,dx = -\\frac{172}{15}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies \\int_{0}^{1} p(x)\\,dx = -\\frac{209}{60}, \\int_{-1}^{0} p(x)\\,dx = -\\frac{599}{60}, \\int_{1}^{2} p(x)\\,dx = -\\frac{479}{60} and \\int_{0}^{2} p(x)\\,dx = -\\frac{172}{15}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 6, -1, 1, -2", "ref": "-6, 6, -1, 1, -2", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "int", "lo": 0, "hi": 1, "v": "-209/60"}, {"t": "int", "lo": -1, "hi": 0, "v": "-599/60"}, {"t": "int", "lo": 1, "hi": 2, "v": "-479/60"}, {"t": "int", "lo": 0, "hi": 2, "v": "-172/15"}]}, "seed": 157317360103005547, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 5, "valid_answers": ["-6, 6, -1, 1, -2", "-5, 5, -4, 3, -2"]}, "cell": "poly_mixed/3", "stage": "B"}
+{"id": "po-poly_mixed-t3-eval-000818", "task": "cx_po_poly_mixed", "source": "cx_po_poly_mixed", "split": "eval", "problem": "Find integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies \\int_{-1}^{1} p(x)\\,dx = -\\frac{48}{5}, p(-2) = 100, p(0) = -6 and p''(0) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -6 <= a, b, c, d, e <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies \\int_{-1}^{1} p(x)\\,dx = -\\frac{48}{5}, p(-2) = 100, p(0) = -6 and p''(0) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -5, 0, 0, 6", "ref": "-6, -5, 0, 0, 6", "family": "po_poly_mixed", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 8e-06, "solvable": true, "holdout": false, "meta": {"family": "po_poly_mixed", "rung": "witness", "params": {"d": 4, "B": 6, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "-48/5"}, {"t": "val", "x": -2, "v": "100"}, {"t": "val", "x": 0, "v": "-6"}, {"t": "der", "x": 0, "m": 2, "v": "0"}]}, "seed": 157317360103004648, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 6, "valid_answers": ["-6, -5, 0, 0, 6", "-6, -1, 0, -1, 6", "-6, 3, 0, -2, 6"]}, "cell": "poly_mixed/3", "stage": "B"}
+{"id": "po-poly_roots-t1-eval-000053", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-3) = p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-3) = p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -5, 8, 3", "ref": "-6, -5, 8, 3", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 26, "bucket": null, "guess_rate": 0.000311, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 3, "B": 8, "cons": [{"t": "root", "x": -3, "v": "0"}, {"t": "root", "x": 1, "v": "0"}]}, "seed": 140069831801000179, "version": "po-v0", "tier": 1, "n_raw": 26, "small": 3, "valid_answers": ["-6, -5, 8, 3", "-6, -2, 6, 2", "-6, 1, 4, 1", "-6, 4, 2, 0", "-6, 7, 0, -1", "-3, -7, 7, 3", "-3, -4, 5, 2", "-3, -1, 3, 1", "-3, 2, 1, 0", "-3, 5, -1, -1", "-3, 8, -3, -2", "0, -6, 4, 2", "0, -3, 2, 1", "0, 3, -2, -1", "0, 6, -4, -2", "3, -8, 3, 2", "3, -5, 1, 1", "3, -2, -1, 0", "3, 1, -3, -1", "3, 4, -5, -2", "3, 7, -7, -3", "6, -7, 0, 1", "6, -4, -2, 0", "6, -1, -4, -1", "6, 2, -6, -2", "6, 5, -8, -3"]}, "cell": "poly_roots/1", "stage": "B"}
+{"id": "po-poly_roots-t1-eval-000052", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d with -4 <= a, b, c, d <= 4, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-3) = p(-1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -4 <= a, b, c, d <= 4, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-3) = p(-1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -4, -1, 0", "ref": "-3, -4, -1, 0", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000914, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 3, "B": 4, "cons": [{"t": "root", "x": -3, "v": "0"}, {"t": "root", "x": -1, "v": "0"}]}, "seed": 140069831801000177, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 3, "valid_answers": ["-3, -4, -1, 0", "-3, -1, 3, 1", "0, -3, -4, -1", "0, 3, 4, 1", "3, 1, -3, -1", "3, 4, 1, 0"]}, "cell": "poly_roots/1", "stage": "B"}
+{"id": "po-poly_roots-t1-eval-000061", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(2) = p(3) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(2) = p(3) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -1, 4, -1", "ref": "-6, -1, 4, -1", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 7.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 3, "B": 8, "cons": [{"t": "root", "x": 2, "v": "0"}, {"t": "root", "x": 3, "v": "0"}]}, "seed": 140069831801000208, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 6, "valid_answers": ["-6, -1, 4, -1", "-6, 5, -1, 0", "0, -6, 5, -1", "0, 6, -5, 1", "6, -5, 1, 0", "6, 1, -4, 1"]}, "cell": "poly_roots/1", "stage": "B"}
+{"id": "po-poly_roots-t2-eval-000093", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d with -11 <= a, b, c, d <= 11, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-4) = p(2) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -11 <= a, b, c, d <= 11, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-4) = p(2) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -6, 3, 1", "ref": "-8, -6, 3, 1", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 2.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 3, "B": 11, "cons": [{"t": "root", "x": -4, "v": "0"}, {"t": "root", "x": 2, "v": "0"}]}, "seed": 140069831802000279, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 8, "valid_answers": ["-8, -6, 3, 1", "-8, 2, 1, 0", "-8, 10, -1, -1", "0, -8, 2, 1", "0, 8, -2, -1", "8, -10, 1, 1", "8, -2, -1, 0", "8, 6, -3, -1"]}, "cell": "poly_roots/2", "stage": "B"}
+{"id": "po-poly_roots-t2-eval-000054", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-3) = p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-3) = p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -5, 8, 3", "ref": "-6, -5, 8, 3", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 26, "bucket": null, "guess_rate": 0.000311, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 3, "B": 8, "cons": [{"t": "root", "x": -3, "v": "0"}, {"t": "root", "x": 1, "v": "0"}]}, "seed": 140069831802000129, "version": "po-v0", "tier": 2, "n_raw": 26, "small": 3, "valid_answers": ["-6, -5, 8, 3", "-6, -2, 6, 2", "-6, 1, 4, 1", "-6, 4, 2, 0", "-6, 7, 0, -1", "-3, -7, 7, 3", "-3, -4, 5, 2", "-3, -1, 3, 1", "-3, 2, 1, 0", "-3, 5, -1, -1", "-3, 8, -3, -2", "0, -6, 4, 2", "0, -3, 2, 1", "0, 3, -2, -1", "0, 6, -4, -2", "3, -8, 3, 2", "3, -5, 1, 1", "3, -2, -1, 0", "3, 1, -3, -1", "3, 4, -5, -2", "3, 7, -7, -3", "6, -7, 0, 1", "6, -4, -2, 0", "6, -1, -4, -1", "6, 2, -6, -2", "6, 5, -8, -3"]}, "cell": "poly_roots/2", "stage": "B"}
+{"id": "po-poly_roots-t2-eval-000033", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-4) = p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-4) = p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -2, 8, 2", "ref": "-8, -2, 8, 2", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 0.000138, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "root", "x": -4, "v": "0"}, {"t": "root", "x": 1, "v": "0"}]}, "seed": 140069831802000059, "version": "po-v0", "tier": 2, "n_raw": 18, "small": 4, "valid_answers": ["-8, -2, 8, 2", "-8, 2, 5, 1", "-8, 6, 2, 0", "-4, -5, 7, 2", "-4, -1, 4, 1", "-4, 3, 1, 0", "-4, 7, -2, -1", "0, -8, 6, 2", "0, -4, 3, 1", "0, 4, -3, -1", "0, 8, -6, -2", "4, -7, 2, 1", "4, -3, -1, 0", "4, 1, -4, -1", "4, 5, -7, -2", "8, -6, -2, 0", "8, -2, -5, -1", "8, 2, -8, -2"]}, "cell": "poly_roots/2", "stage": "B"}
+{"id": "po-poly_roots-t2-eval-000104", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d with -12 <= a, b, c, d <= 12, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-2) = p(4) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -12 <= a, b, c, d <= 12, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-2) = p(4) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -10, -1, 1", "ref": "-8, -10, -1, 1", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 3, "B": 12, "cons": [{"t": "root", "x": -2, "v": "0"}, {"t": "root", "x": 4, "v": "0"}]}, "seed": 140069831802000319, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 8, "valid_answers": ["-8, -10, -1, 1", "-8, -2, 1, 0", "-8, 6, 3, -1", "0, -8, -2, 1", "0, 8, 2, -1", "8, -6, -3, 1", "8, 2, -1, 0", "8, 10, 1, -1"]}, "cell": "poly_roots/2", "stage": "B"}
+{"id": "po-poly_roots-t2-eval-000131", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d with -11 <= a, b, c, d <= 11, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-5) = p(2) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -11 <= a, b, c, d <= 11, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-5) = p(2) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -7, 4, 1", "ref": "-10, -7, 4, 1", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 2.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 3, "B": 11, "cons": [{"t": "root", "x": -5, "v": "0"}, {"t": "root", "x": 2, "v": "0"}]}, "seed": 140069831802000483, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 10, "valid_answers": ["-10, -7, 4, 1", "-10, 3, 1, 0", "0, -10, 3, 1", "0, 10, -3, -1", "10, -3, -1, 0", "10, 7, -4, -1"]}, "cell": "poly_roots/2", "stage": "B"}
+{"id": "po-poly_roots-t2-eval-000035", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-1) = p(3) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-1) = p(3) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -9, 1, 1", "ref": "-9, -9, 1, 1", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 38, "bucket": null, "guess_rate": 0.000292, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "root", "x": -1, "v": "0"}, {"t": "root", "x": 3, "v": "0"}]}, "seed": 140069831802000062, "version": "po-v0", "tier": 2, "n_raw": 38, "small": 3, "valid_answers": ["-9, -9, 1, 1", "-9, -6, 3, 0", "-9, -3, 5, -1", "-9, 0, 7, -2", "-9, 3, 9, -3", "-6, -7, 0, 1", "-6, -4, 2, 0", "-6, -1, 4, -1", "-6, 2, 6, -2", "-6, 5, 8, -3", "-3, -8, -3, 2", "-3, -5, -1, 1", "-3, -2, 1, 0", "-3, 1, 3, -1", "-3, 4, 5, -2", "-3, 7, 7, -3", "0, -9, -6, 3", "0, -6, -4, 2", "0, -3, -2, 1", "0, 3, 2, -1", "0, 6, 4, -2", "0, 9, 6, -3", "3, -7, -7, 3", "3, -4, -5, 2", "3, -1, -3, 1", "3, 2, -1, 0", "3, 5, 1, -1", "3, 8, 3, -2", "6, -5, -8, 3", "6, -2, -6, 2", "6, 1, -4, 1", "6, 4, -2, 0", "6, 7, 0, -1", "9, -3, -9, 3", "9, 0, -7, 2", "9, 3, -5, 1", "9, 6, -3, 0", "9, 9, -1, -1"]}, "cell": "poly_roots/2", "stage": "B"}
+{"id": "po-poly_roots-t2-eval-000062", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d with -7 <= a, b, c, d <= 7, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-5) = p(-1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -7 <= a, b, c, d <= 7, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-5) = p(-1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -6, -1, 0", "ref": "-5, -6, -1, 0", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000119, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 3, "B": 7, "cons": [{"t": "root", "x": -5, "v": "0"}, {"t": "root", "x": -1, "v": "0"}]}, "seed": 140069831802000149, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 5, "valid_answers": ["-5, -6, -1, 0", "-5, -1, 5, 1", "0, -5, -6, -1", "0, 5, 6, 1", "5, 1, -5, -1", "5, 6, 1, 0"]}, "cell": "poly_roots/2", "stage": "B"}
+{"id": "po-poly_roots-t2-eval-000013", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(1) = p(4) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(1) = p(4) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 2, 8, -2", "ref": "-8, 2, 8, -2", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 9.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 3, "B": 10, "cons": [{"t": "root", "x": 1, "v": "0"}, {"t": "root", "x": 4, "v": "0"}]}, "seed": 140069831802000021, "version": "po-v0", "tier": 2, "n_raw": 18, "small": 4, "valid_answers": ["-8, 2, 8, -2", "-8, 6, 3, -1", "-8, 10, -2, 0", "-4, -3, 9, -2", "-4, 1, 4, -1", "-4, 5, -1, 0", "-4, 9, -6, 1", "0, -8, 10, -2", "0, -4, 5, -1", "0, 4, -5, 1", "0, 8, -10, 2", "4, -9, 6, -1", "4, -5, 1, 0", "4, -1, -4, 1", "4, 3, -9, 2", "8, -10, 2, 0", "8, -6, -3, 1", "8, -2, -8, 2"]}, "cell": "poly_roots/2", "stage": "B"}
+{"id": "po-poly_roots-t2-eval-000021", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-3) = p(-2) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-3) = p(-2) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -5, -1, 0", "ref": "-6, -5, -1, 0", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 4.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 3, "B": 10, "cons": [{"t": "root", "x": -3, "v": "0"}, {"t": "root", "x": -2, "v": "0"}]}, "seed": 140069831802000034, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 6, "valid_answers": ["-6, -5, -1, 0", "-6, 1, 4, 1", "-6, 7, 9, 2", "0, -6, -5, -1", "0, 6, 5, 1", "6, -7, -9, -2", "6, -1, -4, -1", "6, 5, 1, 0"]}, "cell": "poly_roots/2", "stage": "B"}
+{"id": "po-poly_roots-t2-eval-000088", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(2) = p(5) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(2) = p(5) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -3, 6, -1", "ref": "-10, -3, 6, -1", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 3.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 3, "B": 10, "cons": [{"t": "root", "x": 2, "v": "0"}, {"t": "root", "x": 5, "v": "0"}]}, "seed": 140069831802000236, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 10, "valid_answers": ["-10, -3, 6, -1", "-10, 7, -1, 0", "0, -10, 7, -1", "0, 10, -7, 1", "10, -7, 1, 0", "10, 3, -6, 1"]}, "cell": "poly_roots/2", "stage": "B"}
+{"id": "po-poly_roots-t2-eval-000023", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d with -10 <= a, b, c, d <= 10, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-5) = p(-2) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -10 <= a, b, c, d <= 10, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-5) = p(-2) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -7, -1, 0", "ref": "-10, -7, -1, 0", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 3.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 3, "B": 10, "cons": [{"t": "root", "x": -5, "v": "0"}, {"t": "root", "x": -2, "v": "0"}]}, "seed": 140069831802000039, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 10, "valid_answers": ["-10, -7, -1, 0", "-10, 3, 6, 1", "0, -10, -7, -1", "0, 10, 7, 1", "10, -3, -6, -1", "10, 7, 1, 0"]}, "cell": "poly_roots/2", "stage": "B"}
+{"id": "po-poly_roots-t2-eval-000133", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d with -8 <= a, b, c, d <= 8, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(2) = p(3) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -8 <= a, b, c, d <= 8, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(2) = p(3) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -1, 4, -1", "ref": "-6, -1, 4, -1", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 7.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 3, "B": 8, "cons": [{"t": "root", "x": 2, "v": "0"}, {"t": "root", "x": 3, "v": "0"}]}, "seed": 140069831802000500, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 6, "valid_answers": ["-6, -1, 4, -1", "-6, 5, -1, 0", "0, -6, 5, -1", "0, 6, -5, 1", "6, -5, 1, 0", "6, 1, -4, 1"]}, "cell": "poly_roots/2", "stage": "B"}
+{"id": "po-poly_roots-t3-eval-000058", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d, e with -7 <= a, b, c, d, e <= 7, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-1) = p(1) = p(3) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -7 <= a, b, c, d, e <= 7, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-1) = p(1) = p(3) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -1, 7, 1, -1", "ref": "-6, -1, 7, 1, -1", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 2.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 4, "B": 7, "cons": [{"t": "root", "x": -1, "v": "0"}, {"t": "root", "x": 1, "v": "0"}, {"t": "root", "x": 3, "v": "0"}]}, "seed": 140069831803000490, "version": "po-v0", "tier": 3, "n_raw": 20, "small": 3, "valid_answers": ["-6, -1, 7, 1, -1", "-6, 2, 6, -2, 0", "-6, 5, 5, -5, 1", "-3, -5, 5, 5, -2", "-3, -2, 4, 2, -1", "-3, 1, 3, -1, 0", "-3, 4, 2, -4, 1", "-3, 7, 1, -7, 2", "0, -6, 2, 6, -2", "0, -3, 1, 3, -1", "0, 3, -1, -3, 1", "0, 6, -2, -6, 2", "3, -7, -1, 7, -2", "3, -4, -2, 4, -1", "3, -1, -3, 1, 0", "3, 2, -4, -2, 1", "3, 5, -5, -5, 2", "6, -5, -5, 5, -1", "6, -2, -6, 2, 0", "6, 1, -7, -1, 1"]}, "cell": "poly_roots/3", "stage": "B"}
+{"id": "po-poly_roots-t3-eval-000029", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d, e with -9 <= a, b, c, d, e <= 9, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-4) = p(-2) = p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -9 <= a, b, c, d, e <= 9, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-4) = p(-2) = p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -6, 7, 6, 1", "ref": "-8, -6, 7, 6, 1", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 2e-06, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 4, "B": 9, "cons": [{"t": "root", "x": -4, "v": "0"}, {"t": "root", "x": -2, "v": "0"}, {"t": "root", "x": 1, "v": "0"}]}, "seed": 140069831803000106, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 8, "valid_answers": ["-8, -6, 7, 6, 1", "-8, 2, 5, 1, 0", "0, -8, 2, 5, 1", "0, 8, -2, -5, -1", "8, -2, -5, -1, 0", "8, 6, -7, -6, -1"]}, "cell": "poly_roots/3", "stage": "B"}
+{"id": "po-poly_roots-t3-eval-000047", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d, e with -8 <= a, b, c, d, e <= 8, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(1) = p(2) = p(4) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -8 <= a, b, c, d, e <= 8, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(1) = p(2) = p(4) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 6, 7, -6, 1", "ref": "-8, 6, 7, -6, 1", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 4, "B": 8, "cons": [{"t": "root", "x": 1, "v": "0"}, {"t": "root", "x": 2, "v": "0"}, {"t": "root", "x": 4, "v": "0"}]}, "seed": 140069831803000248, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 8, "valid_answers": ["-8, 6, 7, -6, 1", "8, -6, -7, 6, -1"]}, "cell": "poly_roots/3", "stage": "B"}
+{"id": "po-poly_roots-t3-eval-000062", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c, d, e with -9 <= a, b, c, d, e <= 9, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-3) = p(-1) = p(3) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -9 <= a, b, c, d, e <= 9, not all zero, such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-3) = p(-1) = p(3) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -9, 1, 1, 0", "ref": "-9, -9, 1, 1, 0", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 2e-06, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 4, "B": 9, "cons": [{"t": "root", "x": -3, "v": "0"}, {"t": "root", "x": -1, "v": "0"}, {"t": "root", "x": 3, "v": "0"}]}, "seed": 140069831803000520, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 9, "valid_answers": ["-9, -9, 1, 1, 0", "0, -9, -9, 1, 1", "0, 9, 9, -1, -1", "9, 9, -1, -1, 0"]}, "cell": "poly_roots/3", "stage": "B"}
+{"id": "po-poly_slope-t0-eval-000580", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = 9 and p(0) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = 9 and p(0) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, -1, 5", "ref": "4, -1, 5", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.003005, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "der", "x": 1, "m": 1, "v": "9"}, {"t": "val", "x": 0, "v": "4"}]}, "seed": 396053805500001061, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["4, -1, 5", "4, 1, 4", "4, 3, 3", "4, 5, 2"]}, "cell": "poly_slope/0", "stage": "B"}
+{"id": "po-poly_slope-t0-eval-000854", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = -4 and p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = -4 and p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -4, 5", "ref": "-1, -4, 5", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.005259, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "der", "x": 0, "m": 1, "v": "-4"}, {"t": "val", "x": 1, "v": "0"}]}, "seed": 396053805500003099, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 4, "valid_answers": ["-1, -4, 5", "0, -4, 4", "1, -4, 3", "2, -4, 2", "3, -4, 1", "4, -4, 0", "5, -4, -1"]}, "cell": "poly_slope/0", "stage": "B"}
+{"id": "po-poly_slope-t0-eval-000841", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = -12 and p(1) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = -12 and p(1) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -2, -5", "ref": "3, -2, -5", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001503, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "der", "x": 1, "m": 1, "v": "-12"}, {"t": "val", "x": 1, "v": "-4"}]}, "seed": 396053805500002910, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["3, -2, -5", "4, -4, -4"]}, "cell": "poly_slope/0", "stage": "B"}
+{"id": "po-poly_slope-t0-eval-000907", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = -12 and p(0) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = -12 and p(0) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -4, -4", "ref": "3, -4, -4", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001503, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "der", "x": 1, "m": 1, "v": "-12"}, {"t": "val", "x": 0, "v": "3"}]}, "seed": 396053805500004512, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["3, -4, -4", "3, -2, -5"]}, "cell": "poly_slope/0", "stage": "B"}
+{"id": "po-poly_slope-t0-eval-000534", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = -5 and p(1) = -11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = -5 and p(1) = -11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -5, -1", "ref": "-5, -5, -1", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.003757, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "der", "x": 0, "m": 1, "v": "-5"}, {"t": "val", "x": 1, "v": "-11"}]}, "seed": 396053805500000897, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 5, "valid_answers": ["-5, -5, -1", "-4, -5, -2", "-3, -5, -3", "-2, -5, -4", "-1, -5, -5"]}, "cell": "poly_slope/0", "stage": "B"}
+{"id": "po-poly_slope-t0-eval-000941", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = 13 and p(0) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = 13 and p(0) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 3, 5", "ref": "4, 3, 5", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001503, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "der", "x": 1, "m": 1, "v": "13"}, {"t": "val", "x": 0, "v": "4"}]}, "seed": 396053805500007440, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["4, 3, 5", "4, 5, 4"]}, "cell": "poly_slope/0", "stage": "B"}
+{"id": "po-poly_slope-t0-eval-000061", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = -3 and p(1) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = -3 and p(1) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -3, 2", "ref": "-5, -3, 2", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.006011, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "der", "x": 0, "m": 1, "v": "-3"}, {"t": "val", "x": 1, "v": "-6"}]}, "seed": 396053805500000067, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 3, "valid_answers": ["-5, -3, 2", "-4, -3, 1", "-3, -3, 0", "-2, -3, -1", "-1, -3, -2", "0, -3, -3", "1, -3, -4", "2, -3, -5"]}, "cell": "poly_slope/0", "stage": "B"}
+{"id": "po-poly_slope-t0-eval-000795", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = -2 and p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = -2 and p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 4, -3", "ref": "-1, 4, -3", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.003757, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "der", "x": 1, "m": 1, "v": "-2"}, {"t": "val", "x": 1, "v": "0"}]}, "seed": 396053805500002385, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 1, "valid_answers": ["-1, 4, -3", "0, 2, -2", "1, 0, -1", "2, -2, 0", "3, -4, 1"]}, "cell": "poly_slope/0", "stage": "B"}
+{"id": "po-poly_slope-t0-eval-000611", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = 3 and p(0) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = 3 and p(0) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -5, 4", "ref": "-3, -5, 4", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.004508, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "der", "x": 1, "m": 1, "v": "3"}, {"t": "val", "x": 0, "v": "-3"}]}, "seed": 396053805500001185, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 3, "valid_answers": ["-3, -5, 4", "-3, -3, 3", "-3, -1, 2", "-3, 1, 1", "-3, 3, 0", "-3, 5, -1"]}, "cell": "poly_slope/0", "stage": "B"}
+{"id": "po-poly_slope-t0-eval-000468", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = -3 and p(0) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = -3 and p(0) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, -5, 1", "ref": "5, -5, 1", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.004508, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "der", "x": 1, "m": 1, "v": "-3"}, {"t": "val", "x": 0, "v": "5"}]}, "seed": 396053805500000718, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 5, "valid_answers": ["5, -5, 1", "5, -3, 0", "5, -1, -1", "5, 1, -2", "5, 3, -3", "5, 5, -4"]}, "cell": "poly_slope/0", "stage": "B"}
+{"id": "po-poly_slope-t0-eval-000769", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = 8 and p(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = 8 and p(1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 4, 2", "ref": "0, 4, 2", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.003005, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "der", "x": 1, "m": 1, "v": "8"}, {"t": "val", "x": 1, "v": "6"}]}, "seed": 396053805500002120, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["0, 4, 2", "1, 2, 3", "2, 0, 4", "3, -2, 5"]}, "cell": "poly_slope/0", "stage": "B"}
+{"id": "po-poly_slope-t0-eval-000166", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = 3 and p(0) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = 3 and p(0) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 3, -5", "ref": "-3, 3, -5", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.008264, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "der", "x": 0, "m": 1, "v": "3"}, {"t": "val", "x": 0, "v": "-3"}]}, "seed": 396053805500000198, "version": "po-v0", "tier": 0, "n_raw": 11, "small": 3, "valid_answers": ["-3, 3, -5", "-3, 3, -4", "-3, 3, -3", "-3, 3, -2", "-3, 3, -1", "-3, 3, 0", "-3, 3, 1", "-3, 3, 2", "-3, 3, 3", "-3, 3, 4", "-3, 3, 5"]}, "cell": "poly_slope/0", "stage": "B"}
+{"id": "po-poly_slope-t1-eval-001367", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 12 and p(-1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 12 and p(-1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, -6, -9", "ref": "6, -6, -9", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000583, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "der", "x": -1, "m": 1, "v": "12"}, {"t": "val", "x": -1, "v": "3"}]}, "seed": 396053805501001468, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 8, "valid_answers": ["6, -6, -9", "7, -4, -8", "8, -2, -7", "9, 0, -6"]}, "cell": "poly_slope/1", "stage": "B"}
+{"id": "po-poly_slope-t1-eval-001600", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = 7 and p(0) = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = 7 and p(0) = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 7, -9", "ref": "-7, 7, -9", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.00277, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "der", "x": 0, "m": 1, "v": "7"}, {"t": "val", "x": 0, "v": "-7"}]}, "seed": 396053805501001738, "version": "po-v0", "tier": 1, "n_raw": 19, "small": 7, "valid_answers": ["-7, 7, -9", "-7, 7, -8", "-7, 7, -7", "-7, 7, -6", "-7, 7, -5", "-7, 7, -4", "-7, 7, -3", "-7, 7, -2", "-7, 7, -1", "-7, 7, 0", "-7, 7, 1", "-7, 7, 2", "-7, 7, 3", "-7, 7, 4", "-7, 7, 5", "-7, 7, 6", "-7, 7, 7", "-7, 7, 8", "-7, 7, 9"]}, "cell": "poly_slope/1", "stage": "B"}
+{"id": "po-poly_slope-t1-eval-000111", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(2) = -3 and p'(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(2) = -3 and p'(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 1, -1", "ref": "-9, 1, -1", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.00277, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "der", "x": 2, "m": 1, "v": "-3"}, {"t": "der", "x": 0, "m": 1, "v": "1"}]}, "seed": 396053805501000116, "version": "po-v0", "tier": 1, "n_raw": 19, "small": 1, "valid_answers": ["-9, 1, -1", "-8, 1, -1", "-7, 1, -1", "-6, 1, -1", "-5, 1, -1", "-4, 1, -1", "-3, 1, -1", "-2, 1, -1", "-1, 1, -1", "0, 1, -1", "1, 1, -1", "2, 1, -1", "3, 1, -1", "4, 1, -1", "5, 1, -1", "6, 1, -1", "7, 1, -1", "8, 1, -1", "9, 1, -1"]}, "cell": "poly_slope/1", "stage": "B"}
+{"id": "po-poly_slope-t1-eval-001928", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 3 and p(0) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 3 and p(0) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -9, -6", "ref": "-2, -9, -6", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.001458, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "der", "x": -1, "m": 1, "v": "3"}, {"t": "val", "x": 0, "v": "-2"}]}, "seed": 396053805501002118, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 2, "valid_answers": ["-2, -9, -6", "-2, -7, -5", "-2, -5, -4", "-2, -3, -3", "-2, -1, -2", "-2, 1, -1", "-2, 3, 0", "-2, 5, 1", "-2, 7, 2", "-2, 9, 3"]}, "cell": "poly_slope/1", "stage": "B"}
+{"id": "po-poly_slope-t1-eval-002250", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = 10 and p(2) = 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = 10 and p(2) = 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -8, 9", "ref": "-5, -8, 9", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.001312, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "der", "x": 1, "m": 1, "v": "10"}, {"t": "val", "x": 2, "v": "15"}]}, "seed": 396053805501002499, "version": "po-v0", "tier": 1, "n_raw": 9, "small": 5, "valid_answers": ["-5, -8, 9", "-5, -6, 8", "-5, -4, 7", "-5, -2, 6", "-5, 0, 5", "-5, 2, 4", "-5, 4, 3", "-5, 6, 2", "-5, 8, 1"]}, "cell": "poly_slope/1", "stage": "B"}
+{"id": "po-poly_slope-t1-eval-000670", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = -15 and p(1) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = -15 and p(1) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 3, -9", "ref": "0, 3, -9", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.001021, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "der", "x": 1, "m": 1, "v": "-15"}, {"t": "val", "x": 1, "v": "-6"}]}, "seed": 396053805501000699, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 5, "valid_answers": ["0, 3, -9", "1, 1, -8", "2, -1, -7", "3, -3, -6", "4, -5, -5", "5, -7, -4", "6, -9, -3"]}, "cell": "poly_slope/1", "stage": "B"}
+{"id": "po-poly_slope-t1-eval-000194", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = -1 and p(-1) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = -1 and p(-1) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -9, -4", "ref": "3, -9, -4", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.001021, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "der", "x": -1, "m": 1, "v": "-1"}, {"t": "val", "x": -1, "v": "8"}]}, "seed": 396053805501000205, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 5, "valid_answers": ["3, -9, -4", "4, -7, -3", "5, -5, -2", "6, -3, -1", "7, -1, 0", "8, 1, 1", "9, 3, 2"]}, "cell": "poly_slope/1", "stage": "B"}
+{"id": "po-poly_slope-t1-eval-000415", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = -18 and p(-1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = -18 and p(-1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -8, 5", "ref": "-7, -8, 5", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000729, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "der", "x": -1, "m": 1, "v": "-18"}, {"t": "val", "x": -1, "v": "6"}]}, "seed": 396053805501000436, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 6, "valid_answers": ["-7, -8, 5", "-6, -6, 6", "-5, -4, 7", "-4, -2, 8", "-3, 0, 9"]}, "cell": "poly_slope/1", "stage": "B"}
+{"id": "po-poly_slope-t1-eval-001690", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = -6 and p(2) = -28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = -6 and p(2) = -28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -6, -2", "ref": "-8, -6, -2", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000729, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "der", "x": 0, "m": 1, "v": "-6"}, {"t": "val", "x": 2, "v": "-28"}]}, "seed": 396053805501001839, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 6, "valid_answers": ["-8, -6, -2", "-4, -6, -3", "0, -6, -4", "4, -6, -5", "8, -6, -6"]}, "cell": "poly_slope/1", "stage": "B"}
+{"id": "po-poly_slope-t1-eval-001736", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(2) = -27 and p(1) = -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(2) = -27 and p(1) = -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 9, -9", "ref": "-5, 9, -9", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000729, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "der", "x": 2, "m": 1, "v": "-27"}, {"t": "val", "x": 1, "v": "-5"}]}, "seed": 396053805501001891, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 6, "valid_answers": ["-5, 9, -9", "-2, 5, -8", "1, 1, -7", "4, -3, -6", "7, -7, -5"]}, "cell": "poly_slope/1", "stage": "B"}
+{"id": "po-poly_slope-t1-eval-000009", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 4 and p'(1) = -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(-1) = 4 and p'(1) = -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -8, -6", "ref": "-9, -8, -6", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.00277, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "der", "x": -1, "m": 1, "v": "4"}, {"t": "der", "x": 1, "m": 1, "v": "-20"}]}, "seed": 396053805501000009, "version": "po-v0", "tier": 1, "n_raw": 19, "small": 8, "valid_answers": ["-9, -8, -6", "-8, -8, -6", "-7, -8, -6", "-6, -8, -6", "-5, -8, -6", "-4, -8, -6", "-3, -8, -6", "-2, -8, -6", "-1, -8, -6", "0, -8, -6", "1, -8, -6", "2, -8, -6", "3, -8, -6", "4, -8, -6", "5, -8, -6", "6, -8, -6", "7, -8, -6", "8, -8, -6", "9, -8, -6"]}, "cell": "poly_slope/1", "stage": "B"}
+{"id": "po-poly_slope-t1-eval-000367", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = -7 and p(-1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(1) = -7 and p(-1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -9, 1", "ref": "-8, -9, 1", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000875, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "der", "x": 1, "m": 1, "v": "-7"}, {"t": "val", "x": -1, "v": "2"}]}, "seed": 396053805501000385, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 3, "valid_answers": ["-8, -9, 1", "-5, -7, 0", "-2, -5, -1", "1, -3, -2", "4, -1, -3", "7, 1, -4"]}, "cell": "poly_slope/1", "stage": "B"}
+{"id": "po-poly_slope-t2-eval-000509", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(2) = 57, p''(1) = 26 and p'(1) = 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(2) = 57, p''(1) = 26 and p'(1) = 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 5, -5, 6", "ref": "-6, 5, -5, 6", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 2, "m": 1, "v": "57"}, {"t": "der", "x": 1, "m": 2, "v": "26"}, {"t": "der", "x": 1, "m": 1, "v": "13"}]}, "seed": 396053805502000743, "version": "po-v0", "tier": 2, "n_raw": 13, "small": 6, "valid_answers": ["-6, 5, -5, 6", "-5, 5, -5, 6", "-4, 5, -5, 6", "-3, 5, -5, 6", "-2, 5, -5, 6", "-1, 5, -5, 6", "0, 5, -5, 6", "1, 5, -5, 6", "2, 5, -5, 6", "3, 5, -5, 6", "4, 5, -5, 6", "5, 5, -5, 6", "6, 5, -5, 6"]}, "cell": "poly_slope/2", "stage": "B"}
+{"id": "po-poly_slope-t2-eval-001865", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 7, p'(-1) = -1 and p(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 7, p'(-1) = -1 and p(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 6, 2, -1", "ref": "-5, 6, 2, -1", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000175, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "7"}, {"t": "der", "x": -1, "m": 1, "v": "-1"}, {"t": "val", "x": 1, "v": "2"}]}, "seed": 396053805502002709, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 2, "valid_answers": ["-5, 6, 2, -1", "-3, 3, 2, 0", "-1, 0, 2, 1", "1, -3, 2, 2", "3, -6, 2, 3"]}, "cell": "poly_slope/2", "stage": "B"}
+{"id": "po-poly_slope-t2-eval-000296", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = -26, p''(0) = -12 and p'(2) = -65. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = -26, p''(0) = -12 and p'(2) = -65. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -5, -6, -3", "ref": "-6, -5, -6, -3", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "-26"}, {"t": "der", "x": 0, "m": 2, "v": "-12"}, {"t": "der", "x": 2, "m": 1, "v": "-65"}]}, "seed": 396053805502000424, "version": "po-v0", "tier": 2, "n_raw": 13, "small": 6, "valid_answers": ["-6, -5, -6, -3", "-5, -5, -6, -3", "-4, -5, -6, -3", "-3, -5, -6, -3", "-2, -5, -6, -3", "-1, -5, -6, -3", "0, -5, -6, -3", "1, -5, -6, -3", "2, -5, -6, -3", "3, -5, -6, -3", "4, -5, -6, -3", "5, -5, -6, -3", "6, -5, -6, -3"]}, "cell": "poly_slope/2", "stage": "B"}
+{"id": "po-poly_slope-t2-eval-001761", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(2) = 18, p''(-1) = -36 and p'(0) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(2) = 18, p''(-1) = -36 and p'(0) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -6, -6, 4", "ref": "-6, -6, -6, 4", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 2, "m": 1, "v": "18"}, {"t": "der", "x": -1, "m": 2, "v": "-36"}, {"t": "der", "x": 0, "m": 1, "v": "-6"}]}, "seed": 396053805502002560, "version": "po-v0", "tier": 2, "n_raw": 13, "small": 6, "valid_answers": ["-6, -6, -6, 4", "-5, -6, -6, 4", "-4, -6, -6, 4", "-3, -6, -6, 4", "-2, -6, -6, 4", "-1, -6, -6, 4", "0, -6, -6, 4", "1, -6, -6, 4", "2, -6, -6, 4", "3, -6, -6, 4", "4, -6, -6, 4", "5, -6, -6, 4", "6, -6, -6, 4"]}, "cell": "poly_slope/2", "stage": "B"}
+{"id": "po-poly_slope-t2-eval-001432", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(2) = 10, p'(0) = 2 and p''(0) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(2) = 10, p'(0) = 2 and p''(0) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 2, -4, 2", "ref": "-6, 2, -4, 2", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 2, "m": 1, "v": "10"}, {"t": "der", "x": 0, "m": 1, "v": "2"}, {"t": "der", "x": 0, "m": 2, "v": "-8"}]}, "seed": 396053805502002082, "version": "po-v0", "tier": 2, "n_raw": 13, "small": 4, "valid_answers": ["-6, 2, -4, 2", "-5, 2, -4, 2", "-4, 2, -4, 2", "-3, 2, -4, 2", "-2, 2, -4, 2", "-1, 2, -4, 2", "0, 2, -4, 2", "1, 2, -4, 2", "2, 2, -4, 2", "3, 2, -4, 2", "4, 2, -4, 2", "5, 2, -4, 2", "6, 2, -4, 2"]}, "cell": "poly_slope/2", "stage": "B"}
+{"id": "po-poly_slope-t2-eval-001257", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(2) = 43, p(2) = 28 and p(0) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(2) = 43, p(2) = 28 and p(0) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, -5, 0, 4", "ref": "6, -5, 0, 4", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 2, "m": 1, "v": "43"}, {"t": "val", "x": 2, "v": "28"}, {"t": "val", "x": 0, "v": "6"}]}, "seed": 396053805502001817, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 6, "valid_answers": ["6, -5, 0, 4", "6, -1, -4, 5"]}, "cell": "poly_slope/2", "stage": "B"}
+{"id": "po-poly_slope-t2-eval-000098", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = -4, p(-1) = -1 and p'(-1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = -4, p(-1) = -1 and p'(-1) = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -6, -2, 2", "ref": "-3, -6, -2, 2", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000175, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "-4"}, {"t": "val", "x": -1, "v": "-1"}, {"t": "der", "x": -1, "m": 1, "v": "4"}]}, "seed": 396053805502000151, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 2, "valid_answers": ["-3, -6, -2, 2", "-1, -3, -2, 1", "1, 0, -2, 0", "3, 3, -2, -1", "5, 6, -2, -2"]}, "cell": "poly_slope/2", "stage": "B"}
+{"id": "po-poly_slope-t2-eval-000837", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(2) = -22, p''(-1) = -2 and p'(0) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(2) = -22, p''(-1) = -2 and p'(0) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 6, -4, -1", "ref": "-6, 6, -4, -1", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 2, "m": 1, "v": "-22"}, {"t": "der", "x": -1, "m": 2, "v": "-2"}, {"t": "der", "x": 0, "m": 1, "v": "6"}]}, "seed": 396053805502001227, "version": "po-v0", "tier": 2, "n_raw": 13, "small": 6, "valid_answers": ["-6, 6, -4, -1", "-5, 6, -4, -1", "-4, 6, -4, -1", "-3, 6, -4, -1", "-2, 6, -4, -1", "-1, 6, -4, -1", "0, 6, -4, -1", "1, 6, -4, -1", "2, 6, -4, -1", "3, 6, -4, -1", "4, 6, -4, -1", "5, 6, -4, -1", "6, 6, -4, -1"]}, "cell": "poly_slope/2", "stage": "B"}
+{"id": "po-poly_slope-t2-eval-002419", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = -22, p''(-1) = 16 and p'(2) = -66. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = -22, p''(-1) = 16 and p'(2) = -66. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -2, -4, -4", "ref": "-6, -2, -4, -4", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "-22"}, {"t": "der", "x": -1, "m": 2, "v": "16"}, {"t": "der", "x": 2, "m": 1, "v": "-66"}]}, "seed": 396053805502003483, "version": "po-v0", "tier": 2, "n_raw": 13, "small": 4, "valid_answers": ["-6, -2, -4, -4", "-5, -2, -4, -4", "-4, -2, -4, -4", "-3, -2, -4, -4", "-2, -2, -4, -4", "-1, -2, -4, -4", "0, -2, -4, -4", "1, -2, -4, -4", "2, -2, -4, -4", "3, -2, -4, -4", "4, -2, -4, -4", "5, -2, -4, -4", "6, -2, -4, -4"]}, "cell": "poly_slope/2", "stage": "B"}
+{"id": "po-poly_slope-t2-eval-002129", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = -2, p'(2) = -66 and p''(-1) = 28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(0) = -2, p'(2) = -66 and p''(-1) = 28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -2, -1, -5", "ref": "-6, -2, -1, -5", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 0, "m": 1, "v": "-2"}, {"t": "der", "x": 2, "m": 1, "v": "-66"}, {"t": "der", "x": -1, "m": 2, "v": "28"}]}, "seed": 396053805502003080, "version": "po-v0", "tier": 2, "n_raw": 13, "small": 5, "valid_answers": ["-6, -2, -1, -5", "-5, -2, -1, -5", "-4, -2, -1, -5", "-3, -2, -1, -5", "-2, -2, -1, -5", "-1, -2, -1, -5", "0, -2, -1, -5", "1, -2, -1, -5", "2, -2, -1, -5", "3, -2, -1, -5", "4, -2, -1, -5", "5, -2, -1, -5", "6, -2, -1, -5"]}, "cell": "poly_slope/2", "stage": "B"}
+{"id": "po-poly_slope-t2-eval-002142", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 9, p(-1) = -16 and p(0) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = 9, p(-1) = -16 and p(0) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 1, -5, 6", "ref": "-4, 1, -5, 6", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "9"}, {"t": "val", "x": -1, "v": "-16"}, {"t": "val", "x": 0, "v": "-4"}]}, "seed": 396053805502003099, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 6, "valid_answers": ["-4, 1, -5, 6", "-4, 6, -3, 3"]}, "cell": "poly_slope/2", "stage": "B"}
+{"id": "po-poly_slope-t2-eval-002467", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = -11, p''(1) = -24 and p(1) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p'(1) = -11, p''(1) = -24 and p(1) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 4, -3, -3", "ref": "-6, 4, -3, -3", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00014, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "der", "x": 1, "m": 1, "v": "-11"}, {"t": "der", "x": 1, "m": 2, "v": "-24"}, {"t": "val", "x": 1, "v": "-8"}]}, "seed": 396053805502003548, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 5, "valid_answers": ["-6, 4, -3, -3", "-5, 1, 0, -4", "-4, -2, 3, -5", "-3, -5, 6, -6"]}, "cell": "poly_slope/2", "stage": "B"}
+{"id": "po-poly_slope-t3-eval-002494", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(1) = -4, p''(0) = -10 and p(3) = -25. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(1) = -4, p''(0) = -10 and p(3) = -25. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 0, -5, 1", "ref": "-7, 0, -5, 1", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 4.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 1, "m": 2, "v": "-4"}, {"t": "der", "x": 0, "m": 2, "v": "-10"}, {"t": "val", "x": 3, "v": "-25"}]}, "seed": 396053805503003670, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 5, "valid_answers": ["-7, 0, -5, 1", "-4, -1, -5, 1", "-1, -2, -5, 1", "2, -3, -5, 1", "5, -4, -5, 1", "8, -5, -5, 1"]}, "cell": "poly_slope/3", "stage": "B"}
+{"id": "po-poly_slope-t3-eval-000053", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(3) = -106, p''(0) = -16 and p(3) = -216. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(3) = -106, p''(0) = -16 and p(3) = -216. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 0, -8, -5", "ref": "-9, 0, -8, -5", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 5.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 3, "m": 2, "v": "-106"}, {"t": "der", "x": 0, "m": 2, "v": "-16"}, {"t": "val", "x": 3, "v": "-216"}]}, "seed": 396053805503000077, "version": "po-v0", "tier": 3, "n_raw": 7, "small": 8, "valid_answers": ["-9, 0, -8, -5", "-6, -1, -8, -5", "-3, -2, -8, -5", "0, -3, -8, -5", "3, -4, -8, -5", "6, -5, -8, -5", "9, -6, -8, -5"]}, "cell": "poly_slope/3", "stage": "B"}
+{"id": "po-poly_slope-t3-eval-000772", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(2) = 52, p(2) = 60 and p''(3) = 70. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(2) = 52, p(2) = 60 and p''(3) = 70. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 6, 8, 3", "ref": "-8, 6, 8, 3", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 6.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 2, "m": 2, "v": "52"}, {"t": "val", "x": 2, "v": "60"}, {"t": "der", "x": 3, "m": 2, "v": "70"}]}, "seed": 396053805503001122, "version": "po-v0", "tier": 3, "n_raw": 9, "small": 8, "valid_answers": ["-8, 6, 8, 3", "-6, 5, 8, 3", "-4, 4, 8, 3", "-2, 3, 8, 3", "0, 2, 8, 3", "2, 1, 8, 3", "4, 0, 8, 3", "6, -1, 8, 3", "8, -2, 8, 3"]}, "cell": "poly_slope/3", "stage": "B"}
+{"id": "po-poly_slope-t3-eval-000981", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(2) = -2, p''(0) = 10 and p(0) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(2) = -2, p''(0) = 10 and p(0) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -9, 5, -1", "ref": "-4, -9, 5, -1", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.000146, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 2, "m": 2, "v": "-2"}, {"t": "der", "x": 0, "m": 2, "v": "10"}, {"t": "val", "x": 0, "v": "-4"}]}, "seed": 396053805503001419, "version": "po-v0", "tier": 3, "n_raw": 19, "small": 5, "valid_answers": ["-4, -9, 5, -1", "-4, -8, 5, -1", "-4, -7, 5, -1", "-4, -6, 5, -1", "-4, -5, 5, -1", "-4, -4, 5, -1", "-4, -3, 5, -1", "-4, -2, 5, -1", "-4, -1, 5, -1", "-4, 0, 5, -1", "-4, 1, 5, -1", "-4, 2, 5, -1", "-4, 3, 5, -1", "-4, 4, 5, -1", "-4, 5, 5, -1", "-4, 6, 5, -1", "-4, 7, 5, -1", "-4, 8, 5, -1", "-4, 9, 5, -1"]}, "cell": "poly_slope/3", "stage": "B"}
+{"id": "po-poly_slope-t3-eval-002387", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(-2) = 10, p(0) = 0 and p(2) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(-2) = 10, p(0) = 0 and p(2) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -7, 5, 0", "ref": "0, -7, 5, 0", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": -2, "m": 2, "v": "10"}, {"t": "val", "x": 0, "v": "0"}, {"t": "val", "x": 2, "v": "6"}]}, "seed": 396053805503003517, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 7, "valid_answers": ["0, -7, 5, 0", "0, 9, -1, -1"]}, "cell": "poly_slope/3", "stage": "B"}
+{"id": "po-poly_slope-t3-eval-000305", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(-2) = 50, p''(-1) = 20 and p'(2) = -85. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(-2) = 50, p''(-1) = 20 and p'(2) = -85. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -5, -5, -5", "ref": "-9, -5, -5, -5", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.000146, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": -2, "m": 2, "v": "50"}, {"t": "der", "x": -1, "m": 2, "v": "20"}, {"t": "der", "x": 2, "m": 1, "v": "-85"}]}, "seed": 396053805503000427, "version": "po-v0", "tier": 3, "n_raw": 19, "small": 5, "valid_answers": ["-9, -5, -5, -5", "-8, -5, -5, -5", "-7, -5, -5, -5", "-6, -5, -5, -5", "-5, -5, -5, -5", "-4, -5, -5, -5", "-3, -5, -5, -5", "-2, -5, -5, -5", "-1, -5, -5, -5", "0, -5, -5, -5", "1, -5, -5, -5", "2, -5, -5, -5", "3, -5, -5, -5", "4, -5, -5, -5", "5, -5, -5, -5", "6, -5, -5, -5", "7, -5, -5, -5", "8, -5, -5, -5", "9, -5, -5, -5"]}, "cell": "poly_slope/3", "stage": "B"}
+{"id": "po-poly_slope-t3-eval-001948", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(0) = 18, p(0) = -6 and p(1) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(0) = 18, p(0) = -6 and p(1) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -9, 9, 3", "ref": "-6, -9, 9, 3", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.0001, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 0, "m": 2, "v": "18"}, {"t": "val", "x": 0, "v": "-6"}, {"t": "val", "x": 1, "v": "-3"}]}, "seed": 396053805503002863, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 9, "valid_answers": ["-6, -9, 9, 3", "-6, -8, 9, 2", "-6, -7, 9, 1", "-6, -6, 9, 0", "-6, -5, 9, -1", "-6, -4, 9, -2", "-6, -3, 9, -3", "-6, -2, 9, -4", "-6, -1, 9, -5", "-6, 0, 9, -6", "-6, 1, 9, -7", "-6, 2, 9, -8", "-6, 3, 9, -9"]}, "cell": "poly_slope/3", "stage": "B"}
+{"id": "po-poly_slope-t3-eval-001667", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(-1) = 14, p(2) = -23 and p''(1) = -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(-1) = 14, p(2) = -23 and p''(1) = -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 9, -2, -3", "ref": "-9, 9, -2, -3", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 7.7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": -1, "m": 2, "v": "14"}, {"t": "val", "x": 2, "v": "-23"}, {"t": "der", "x": 1, "m": 2, "v": "-22"}]}, "seed": 396053805503002462, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 3, "valid_answers": ["-9, 9, -2, -3", "-7, 8, -2, -3", "-5, 7, -2, -3", "-3, 6, -2, -3", "-1, 5, -2, -3", "1, 4, -2, -3", "3, 3, -2, -3", "5, 2, -2, -3", "7, 1, -2, -3", "9, 0, -2, -3"]}, "cell": "poly_slope/3", "stage": "B"}
+{"id": "po-poly_slope-t3-eval-000680", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(3) = 22, p''(0) = 4 and p(1) = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(3) = 22, p''(0) = 4 and p(1) = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 9, 2, 1", "ref": "0, 9, 2, 1", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 7.7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 3, "m": 2, "v": "22"}, {"t": "der", "x": 0, "m": 2, "v": "4"}, {"t": "val", "x": 1, "v": "12"}]}, "seed": 396053805503000972, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 5, "valid_answers": ["0, 9, 2, 1", "1, 8, 2, 1", "2, 7, 2, 1", "3, 6, 2, 1", "4, 5, 2, 1", "5, 4, 2, 1", "6, 3, 2, 1", "7, 2, 2, 1", "8, 1, 2, 1", "9, 0, 2, 1"]}, "cell": "poly_slope/3", "stage": "B"}
+{"id": "po-poly_slope-t3-eval-001031", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(-2) = -102, p''(0) = 6 and p'(-1) = 23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(-2) = -102, p''(0) = 6 and p'(-1) = 23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 2, 3, 9", "ref": "-9, 2, 3, 9", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.000146, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": -2, "m": 2, "v": "-102"}, {"t": "der", "x": 0, "m": 2, "v": "6"}, {"t": "der", "x": -1, "m": 1, "v": "23"}]}, "seed": 396053805503001512, "version": "po-v0", "tier": 3, "n_raw": 19, "small": 9, "valid_answers": ["-9, 2, 3, 9", "-8, 2, 3, 9", "-7, 2, 3, 9", "-6, 2, 3, 9", "-5, 2, 3, 9", "-4, 2, 3, 9", "-3, 2, 3, 9", "-2, 2, 3, 9", "-1, 2, 3, 9", "0, 2, 3, 9", "1, 2, 3, 9", "2, 2, 3, 9", "3, 2, 3, 9", "4, 2, 3, 9", "5, 2, 3, 9", "6, 2, 3, 9", "7, 2, 3, 9", "8, 2, 3, 9", "9, 2, 3, 9"]}, "cell": "poly_slope/3", "stage": "B"}
+{"id": "po-poly_slope-t3-eval-001734", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(-2) = 20, p(0) = -6 and p''(3) = -40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(-2) = 20, p(0) = -6 and p''(3) = -40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -9, -2, -2", "ref": "-6, -9, -2, -2", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.000146, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": -2, "m": 2, "v": "20"}, {"t": "val", "x": 0, "v": "-6"}, {"t": "der", "x": 3, "m": 2, "v": "-40"}]}, "seed": 396053805503002560, "version": "po-v0", "tier": 3, "n_raw": 19, "small": 6, "valid_answers": ["-6, -9, -2, -2", "-6, -8, -2, -2", "-6, -7, -2, -2", "-6, -6, -2, -2", "-6, -5, -2, -2", "-6, -4, -2, -2", "-6, -3, -2, -2", "-6, -2, -2, -2", "-6, -1, -2, -2", "-6, 0, -2, -2", "-6, 1, -2, -2", "-6, 2, -2, -2", "-6, 3, -2, -2", "-6, 4, -2, -2", "-6, 5, -2, -2", "-6, 6, -2, -2", "-6, 7, -2, -2", "-6, 8, -2, -2", "-6, 9, -2, -2"]}, "cell": "poly_slope/3", "stage": "B"}
+{"id": "po-poly_slope-t3-eval-000336", "task": "cx_po_poly_slope", "source": "cx_po_poly_slope", "split": "eval", "problem": "Find integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(0) = 4, p''(-1) = -20 and p'(1) = 23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -9 <= a, b, c, d <= 9 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p''(0) = 4, p''(-1) = -20 and p'(1) = 23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 7, 2, 4", "ref": "-9, 7, 2, 4", "family": "po_poly_slope", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.000146, "solvable": true, "holdout": false, "meta": {"family": "po_poly_slope", "rung": "witness", "params": {"d": 3, "B": 9, "cons": [{"t": "der", "x": 0, "m": 2, "v": "4"}, {"t": "der", "x": -1, "m": 2, "v": "-20"}, {"t": "der", "x": 1, "m": 1, "v": "23"}]}, "seed": 396053805503000482, "version": "po-v0", "tier": 3, "n_raw": 19, "small": 7, "valid_answers": ["-9, 7, 2, 4", "-8, 7, 2, 4", "-7, 7, 2, 4", "-6, 7, 2, 4", "-5, 7, 2, 4", "-4, 7, 2, 4", "-3, 7, 2, 4", "-2, 7, 2, 4", "-1, 7, 2, 4", "0, 7, 2, 4", "1, 7, 2, 4", "2, 7, 2, 4", "3, 7, 2, 4", "4, 7, 2, 4", "5, 7, 2, 4", "6, 7, 2, 4", "7, 7, 2, 4", "8, 7, 2, 4", "9, 7, 2, 4"]}, "cell": "poly_slope/3", "stage": "B"}
+{"id": "po-poly_values-t0-eval-000867", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = -3 and p(-1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = -3 and p(-1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -3, 5", "ref": "-5, -3, 5", "family": "po_poly_values", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.008264, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "val", "x": 1, "v": "-3"}, {"t": "val", "x": -1, "v": "3"}]}, "seed": 358655629900001689, "version": "po-v0", "tier": 0, "n_raw": 11, "small": 3, "valid_answers": ["-5, -3, 5", "-4, -3, 4", "-3, -3, 3", "-2, -3, 2", "-1, -3, 1", "0, -3, 0", "1, -3, -1", "2, -3, -2", "3, -3, -3", "4, -3, -4", "5, -3, -5"]}, "cell": "poly_values/0", "stage": "A"}
+{"id": "po-poly_values-t0-eval-000212", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(-1) = 5 and p(1) = 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(-1) = 5 and p(1) = 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 4, 5", "ref": "4, 4, 5", "family": "po_poly_values", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001503, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "val", "x": -1, "v": "5"}, {"t": "val", "x": 1, "v": "13"}]}, "seed": 358655629900000235, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["4, 4, 5", "5, 4, 4"]}, "cell": "poly_values/0", "stage": "A"}
+{"id": "po-poly_values-t0-eval-000574", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = 4 and p(0) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = 4 and p(0) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -1, 5", "ref": "0, -1, 5", "family": "po_poly_values", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.005259, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "val", "x": 1, "v": "4"}, {"t": "val", "x": 0, "v": "0"}]}, "seed": 358655629900000815, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 2, "valid_answers": ["0, -1, 5", "0, 0, 4", "0, 1, 3", "0, 2, 2", "0, 3, 1", "0, 4, 0", "0, 5, -1"]}, "cell": "poly_values/0", "stage": "A"}
+{"id": "po-poly_values-t0-eval-001238", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -4 and p(-1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -4 and p(-1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -5, 4", "ref": "-4, -5, 4", "family": "po_poly_values", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001503, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "val", "x": 0, "v": "-4"}, {"t": "val", "x": -1, "v": "5"}]}, "seed": 358655629900009069, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-4, -5, 4", "-4, -4, 5"]}, "cell": "poly_values/0", "stage": "A"}
+{"id": "po-poly_values-t0-eval-000460", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = -8 and p(0) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = -8 and p(0) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -5, 0", "ref": "-3, -5, 0", "family": "po_poly_values", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.004508, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "val", "x": 1, "v": "-8"}, {"t": "val", "x": 0, "v": "-3"}]}, "seed": 358655629900000594, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 3, "valid_answers": ["-3, -5, 0", "-3, -4, -1", "-3, -3, -2", "-3, -2, -3", "-3, -1, -4", "-3, 0, -5"]}, "cell": "poly_values/0", "stage": "A"}
+{"id": "po-poly_values-t0-eval-000718", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = 2 and p(0) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = 2 and p(0) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 0, 5", "ref": "-3, 0, 5", "family": "po_poly_values", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.004508, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "val", "x": 1, "v": "2"}, {"t": "val", "x": 0, "v": "-3"}]}, "seed": 358655629900001142, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 3, "valid_answers": ["-3, 0, 5", "-3, 1, 4", "-3, 2, 3", "-3, 3, 2", "-3, 4, 1", "-3, 5, 0"]}, "cell": "poly_values/0", "stage": "A"}
+{"id": "po-poly_values-t0-eval-000107", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -1 and p(-1) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -1 and p(-1) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, -5", "ref": "-1, 0, -5", "family": "po_poly_values", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.004508, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "val", "x": 0, "v": "-1"}, {"t": "val", "x": -1, "v": "-6"}]}, "seed": 358655629900000109, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 3, "valid_answers": ["-1, 0, -5", "-1, 1, -4", "-1, 2, -3", "-1, 3, -2", "-1, 4, -1", "-1, 5, 0"]}, "cell": "poly_values/0", "stage": "A"}
+{"id": "po-poly_values-t0-eval-000230", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(-1) = 2 and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(-1) = 2 and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -5, -4", "ref": "1, -5, -4", "family": "po_poly_values", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.007513, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "val", "x": -1, "v": "2"}, {"t": "val", "x": 0, "v": "1"}]}, "seed": 358655629900000255, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 1, "valid_answers": ["1, -5, -4", "1, -4, -3", "1, -3, -2", "1, -2, -1", "1, -1, 0", "1, 0, 1", "1, 1, 2", "1, 2, 3", "1, 3, 4", "1, 4, 5"]}, "cell": "poly_values/0", "stage": "A"}
+{"id": "po-poly_values-t0-eval-000177", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -4 and p(1) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -4 and p(1) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -5, 3", "ref": "-4, -5, 3", "family": "po_poly_values", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.006762, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "val", "x": 0, "v": "-4"}, {"t": "val", "x": 1, "v": "-6"}]}, "seed": 358655629900000195, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 4, "valid_answers": ["-4, -5, 3", "-4, -4, 2", "-4, -3, 1", "-4, -2, 0", "-4, -1, -1", "-4, 0, -2", "-4, 1, -3", "-4, 2, -4", "-4, 3, -5"]}, "cell": "poly_values/0", "stage": "A"}
+{"id": "po-poly_values-t0-eval-000181", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = 7 and p(0) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = 7 and p(0) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -1, 5", "ref": "3, -1, 5", "family": "po_poly_values", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.005259, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "val", "x": 1, "v": "7"}, {"t": "val", "x": 0, "v": "3"}]}, "seed": 358655629900000199, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 3, "valid_answers": ["3, -1, 5", "3, 0, 4", "3, 1, 3", "3, 2, 2", "3, 3, 1", "3, 4, 0", "3, 5, -1"]}, "cell": "poly_values/0", "stage": "A"}
+{"id": "po-poly_values-t0-eval-000683", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -4 and p(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -4 and p(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 2, 5", "ref": "-4, 2, 5", "family": "po_poly_values", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.003005, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "val", "x": 0, "v": "-4"}, {"t": "val", "x": 1, "v": "3"}]}, "seed": 358655629900001044, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["-4, 2, 5", "-4, 3, 4", "-4, 4, 3", "-4, 5, 2"]}, "cell": "poly_values/0", "stage": "A"}
+{"id": "po-poly_values-t0-eval-000999", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -3 and p(1) = -11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -3 and p(1) = -11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -5, -3", "ref": "-3, -5, -3", "family": "po_poly_values", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002254, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "val", "x": 0, "v": "-3"}, {"t": "val", "x": 1, "v": "-11"}]}, "seed": 358655629900002463, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-3, -5, -3", "-3, -4, -4", "-3, -3, -5"]}, "cell": "poly_values/0", "stage": "A"}
+{"id": "po-poly_values-t1-eval-001658", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(2) = -15 and p(0) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(2) = -15 and p(0) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -9, 0", "ref": "3, -9, 0", "family": "po_poly_values", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.001458, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "val", "x": 2, "v": "-15"}, {"t": "val", "x": 0, "v": "3"}]}, "seed": 358655629901001796, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 3, "valid_answers": ["3, -9, 0", "3, -7, -1", "3, -5, -2", "3, -3, -3", "3, -1, -4", "3, 1, -5", "3, 3, -6", "3, 5, -7", "3, 7, -8", "3, 9, -9"]}, "cell": "poly_values/1", "stage": "B"}
+{"id": "po-poly_values-t1-eval-001915", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = 2 and p(-2) = -34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = 2 and p(-2) = -34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 9, -3", "ref": "-4, 9, -3", "family": "po_poly_values", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.001021, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "val", "x": 1, "v": "2"}, {"t": "val", "x": -2, "v": "-34"}]}, "seed": 358655629901002084, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 6, "valid_answers": ["-4, 9, -3", "-2, 8, -4", "0, 7, -5", "2, 6, -6", "4, 5, -7", "6, 4, -8", "8, 3, -9"]}, "cell": "poly_values/1", "stage": "B"}
+{"id": "po-poly_values-t1-eval-002018", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = -13 and p(2) = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = -13 and p(2) = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -5, 0", "ref": "-8, -5, 0", "family": "po_poly_values", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000292, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "val", "x": 1, "v": "-13"}, {"t": "val", "x": 2, "v": "-18"}]}, "seed": 358655629901002210, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 8, "valid_answers": ["-8, -5, 0", "-6, -8, 1"]}, "cell": "poly_values/1", "stage": "B"}
+{"id": "po-poly_values-t1-eval-001927", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(-2) = -6 and p(-1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(-2) = -6 and p(-1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, -7, -6", "ref": "4, -7, -6", "family": "po_poly_values", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000437, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "val", "x": -2, "v": "-6"}, {"t": "val", "x": -1, "v": "5"}]}, "seed": 358655629901002096, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 6, "valid_answers": ["4, -7, -6", "6, -4, -5", "8, -1, -4"]}, "cell": "poly_values/1", "stage": "B"}
+{"id": "po-poly_values-t1-eval-000112", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -9 and p(-1) = -13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -9 and p(-1) = -13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -5, -9", "ref": "-9, -5, -9", "family": "po_poly_values", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 0.002187, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "val", "x": 0, "v": "-9"}, {"t": "val", "x": -1, "v": "-13"}]}, "seed": 358655629901000113, "version": "po-v0", "tier": 1, "n_raw": 15, "small": 9, "valid_answers": ["-9, -5, -9", "-9, -4, -8", "-9, -3, -7", "-9, -2, -6", "-9, -1, -5", "-9, 0, -4", "-9, 1, -3", "-9, 2, -2", "-9, 3, -1", "-9, 4, 0", "-9, 5, 1", "-9, 6, 2", "-9, 7, 3", "-9, 8, 4", "-9, 9, 5"]}, "cell": "poly_values/1", "stage": "B"}
+{"id": "po-poly_values-t1-eval-001939", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(2) = 1 and p(1) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(2) = 1 and p(1) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 8, -2", "ref": "-7, 8, -2", "family": "po_poly_values", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000875, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "val", "x": 2, "v": "1"}, {"t": "val", "x": 1, "v": "-1"}]}, "seed": 358655629901002109, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 1, "valid_answers": ["-7, 8, -2", "-5, 5, -1", "-3, 2, 0", "-1, -1, 1", "1, -4, 2", "3, -7, 3"]}, "cell": "poly_values/1", "stage": "B"}
+{"id": "po-poly_values-t1-eval-002143", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(2) = 18 and p(-2) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(2) = 18 and p(-2) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 6, 3", "ref": "-6, 6, 3", "family": "po_poly_values", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000583, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "val", "x": 2, "v": "18"}, {"t": "val", "x": -2, "v": "-6"}]}, "seed": 358655629901002361, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 6, "valid_answers": ["-6, 6, 3", "-2, 6, 2", "2, 6, 1", "6, 6, 0"]}, "cell": "poly_values/1", "stage": "B"}
+{"id": "po-poly_values-t1-eval-001306", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = 7 and p(0) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(1) = 7 and p(0) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, -8, 9", "ref": "6, -8, 9", "family": "po_poly_values", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 0.002624, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "val", "x": 1, "v": "7"}, {"t": "val", "x": 0, "v": "6"}]}, "seed": 358655629901001409, "version": "po-v0", "tier": 1, "n_raw": 18, "small": 6, "valid_answers": ["6, -8, 9", "6, -7, 8", "6, -6, 7", "6, -5, 6", "6, -4, 5", "6, -3, 4", "6, -2, 3", "6, -1, 2", "6, 0, 1", "6, 1, 0", "6, 2, -1", "6, 3, -2", "6, 4, -3", "6, 5, -4", "6, 6, -5", "6, 7, -6", "6, 8, -7", "6, 9, -8"]}, "cell": "poly_values/1", "stage": "B"}
+{"id": "po-poly_values-t1-eval-000110", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -7 and p(-2) = -29. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -7 and p(-2) = -29. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -7, -9", "ref": "-7, -7, -9", "family": "po_poly_values", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.001312, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "val", "x": 0, "v": "-7"}, {"t": "val", "x": -2, "v": "-29"}]}, "seed": 358655629901000111, "version": "po-v0", "tier": 1, "n_raw": 9, "small": 7, "valid_answers": ["-7, -7, -9", "-7, -5, -8", "-7, -3, -7", "-7, -1, -6", "-7, 1, -5", "-7, 3, -4", "-7, 5, -3", "-7, 7, -2", "-7, 9, -1"]}, "cell": "poly_values/1", "stage": "B"}
+{"id": "po-poly_values-t1-eval-001400", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -9 and p(2) = -15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -9 and p(2) = -15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -9, 3", "ref": "-9, -9, 3", "family": "po_poly_values", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.001458, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "val", "x": 0, "v": "-9"}, {"t": "val", "x": 2, "v": "-15"}]}, "seed": 358655629901001515, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 9, "valid_answers": ["-9, -9, 3", "-9, -7, 2", "-9, -5, 1", "-9, -3, 0", "-9, -1, -1", "-9, 1, -2", "-9, 3, -3", "-9, 5, -4", "-9, 7, -5", "-9, 9, -6"]}, "cell": "poly_values/1", "stage": "B"}
+{"id": "po-poly_values-t1-eval-001838", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(-1) = 11 and p(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(-1) = 11 and p(1) = 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -4, 9", "ref": "-2, -4, 9", "family": "po_poly_values", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.00175, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "val", "x": -1, "v": "11"}, {"t": "val", "x": 1, "v": "3"}]}, "seed": 358655629901001999, "version": "po-v0", "tier": 1, "n_raw": 12, "small": 4, "valid_answers": ["-2, -4, 9", "-1, -4, 8", "0, -4, 7", "1, -4, 6", "2, -4, 5", "3, -4, 4", "4, -4, 3", "5, -4, 2", "6, -4, 1", "7, -4, 0", "8, -4, -1", "9, -4, -2"]}, "cell": "poly_values/1", "stage": "B"}
+{"id": "po-poly_values-t1-eval-001970", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(-1) = -5 and p(1) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -9 <= a, b, c <= 9 such that the polynomial p(x) = a + bx + cx^2 satisfies p(-1) = -5 and p(1) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 1, 5", "ref": "-9, 1, 5", "family": "po_poly_values", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 0.002187, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 2, "B": 9, "cons": [{"t": "val", "x": -1, "v": "-5"}, {"t": "val", "x": 1, "v": "-3"}]}, "seed": 358655629901002152, "version": "po-v0", "tier": 1, "n_raw": 15, "small": 2, "valid_answers": ["-9, 1, 5", "-8, 1, 4", "-7, 1, 3", "-6, 1, 2", "-5, 1, 1", "-4, 1, 0", "-3, 1, -1", "-2, 1, -2", "-1, 1, -3", "0, 1, -4", "1, 1, -5", "2, 1, -6", "3, 1, -7", "4, 1, -8", "5, 1, -9"]}, "cell": "poly_values/1", "stage": "B"}
+{"id": "po-poly_values-t2-eval-001828", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(2) = 34, p(1) = 4 and p(0) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(2) = 34, p(1) = 4 and p(0) = -4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 1, 5, 2", "ref": "-4, 1, 5, 2", "family": "po_poly_values", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "val", "x": 2, "v": "34"}, {"t": "val", "x": 1, "v": "4"}, {"t": "val", "x": 0, "v": "-4"}]}, "seed": 358655629902002105, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 4, "valid_answers": ["-4, 1, 5, 2", "-4, 3, 2, 3", "-4, 5, -1, 4"]}, "cell": "poly_values/2", "stage": "B"}
+{"id": "po-poly_values-t2-eval-001695", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-1) = 16, p(1) = 0 and p(-2) = 66. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-1) = 16, p(1) = 0 and p(-2) = 66. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -4, 6, -4", "ref": "2, -4, 6, -4", "family": "po_poly_values", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "val", "x": -1, "v": "16"}, {"t": "val", "x": 1, "v": "0"}, {"t": "val", "x": -2, "v": "66"}]}, "seed": 358655629902001945, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 5, "valid_answers": ["2, -4, 6, -4", "4, -3, 4, -5", "6, -2, 2, -6"]}, "cell": "poly_values/2", "stage": "B"}
+{"id": "po-poly_values-t2-eval-000976", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-1) = 5, p(0) = -4 and p(2) = -28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-1) = 5, p(0) = -4 and p(2) = -28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -6, 1, -2", "ref": "-4, -6, 1, -2", "family": "po_poly_values", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000175, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "val", "x": -1, "v": "5"}, {"t": "val", "x": 0, "v": "-4"}, {"t": "val", "x": 2, "v": "-28"}]}, "seed": 358655629902001110, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 4, "valid_answers": ["-4, -6, 1, -2", "-4, -4, 2, -3", "-4, -2, 3, -4", "-4, 0, 4, -5", "-4, 2, 5, -6"]}, "cell": "poly_values/2", "stage": "B"}
+{"id": "po-poly_values-t2-eval-002418", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-1) = 4, p(2) = 28 and p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-1) = 4, p(2) = 28 and p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -4, 6, 2", "ref": "-4, -4, 6, 2", "family": "po_poly_values", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "val", "x": -1, "v": "4"}, {"t": "val", "x": 2, "v": "28"}, {"t": "val", "x": 1, "v": "0"}]}, "seed": 358655629902002802, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 5, "valid_answers": ["-4, -4, 6, 2", "-2, -5, 4, 3", "0, -6, 2, 4"]}, "cell": "poly_values/2", "stage": "B"}
+{"id": "po-poly_values-t2-eval-000409", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(0) = 6, p(1) = 11 and p(-1) = 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(0) = 6, p(1) = 11 and p(-1) = 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, -6, 5, 6", "ref": "6, -6, 5, 6", "family": "po_poly_values", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000455, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "val", "x": 0, "v": "6"}, {"t": "val", "x": 1, "v": "11"}, {"t": "val", "x": -1, "v": "11"}]}, "seed": 358655629902000467, "version": "po-v0", "tier": 2, "n_raw": 13, "small": 6, "valid_answers": ["6, -6, 5, 6", "6, -5, 5, 5", "6, -4, 5, 4", "6, -3, 5, 3", "6, -2, 5, 2", "6, -1, 5, 1", "6, 0, 5, 0", "6, 1, 5, -1", "6, 2, 5, -2", "6, 3, 5, -3", "6, 4, 5, -4", "6, 5, 5, -5", "6, 6, 5, -6"]}, "cell": "poly_values/2", "stage": "B"}
+{"id": "po-poly_values-t2-eval-000855", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-2) = -43, p(2) = 41 and p(0) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-2) = -43, p(2) = 41 and p(0) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -3, 0, 6", "ref": "-1, -3, 0, 6", "family": "po_poly_values", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "val", "x": -2, "v": "-43"}, {"t": "val", "x": 2, "v": "41"}, {"t": "val", "x": 0, "v": "-1"}]}, "seed": 358655629902000970, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 5, "valid_answers": ["-1, -3, 0, 6", "-1, 1, 0, 5", "-1, 5, 0, 4"]}, "cell": "poly_values/2", "stage": "B"}
+{"id": "po-poly_values-t2-eval-002059", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-1) = 5, p(1) = -11 and p(2) = -34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-1) = 5, p(1) = -11 and p(2) = -34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -4, 3, -4", "ref": "-6, -4, 3, -4", "family": "po_poly_values", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "val", "x": -1, "v": "5"}, {"t": "val", "x": 1, "v": "-11"}, {"t": "val", "x": 2, "v": "-34"}]}, "seed": 358655629902002382, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 5, "valid_answers": ["-6, -4, 3, -4", "-4, -5, 1, -3", "-2, -6, -1, -2"]}, "cell": "poly_values/2", "stage": "B"}
+{"id": "po-poly_values-t2-eval-001524", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(1) = 9, p(0) = 6 and p(-2) = -66. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(1) = 9, p(0) = 6 and p(-2) = -66. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 2, -5, 6", "ref": "6, 2, -5, 6", "family": "po_poly_values", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "val", "x": 1, "v": "9"}, {"t": "val", "x": 0, "v": "6"}, {"t": "val", "x": -2, "v": "-66"}]}, "seed": 358655629902001743, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 6, "valid_answers": ["6, 2, -5, 6", "6, 4, -6, 5"]}, "cell": "poly_values/2", "stage": "B"}
+{"id": "po-poly_values-t2-eval-002091", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(2) = -50, p(0) = -4 and p(1) = -10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(2) = -50, p(0) = -4 and p(1) = -10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -1, 1, -6", "ref": "-4, -1, 1, -6", "family": "po_poly_values", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "val", "x": 2, "v": "-50"}, {"t": "val", "x": 0, "v": "-4"}, {"t": "val", "x": 1, "v": "-10"}]}, "seed": 358655629902002417, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 5, "valid_answers": ["-4, -1, 1, -6", "-4, 1, -2, -5", "-4, 3, -5, -4"]}, "cell": "poly_values/2", "stage": "B"}
+{"id": "po-poly_values-t2-eval-000199", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-2) = -10, p(1) = 2 and p(2) = 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(-2) = -10, p(1) = 2 and p(2) = 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 5, 1, 0", "ref": "-4, 5, 1, 0", "family": "po_poly_values", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000105, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "val", "x": -2, "v": "-10"}, {"t": "val", "x": 1, "v": "2"}, {"t": "val", "x": 2, "v": "10"}]}, "seed": 358655629902000227, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 1, "valid_answers": ["-4, 5, 1, 0", "0, 1, 0, 1", "4, -3, -1, 2"]}, "cell": "poly_values/2", "stage": "B"}
+{"id": "po-poly_values-t2-eval-000189", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(0) = 0, p(-1) = 12 and p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(0) = 0, p(-1) = 12 and p(1) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -6, 6, 0", "ref": "0, -6, 6, 0", "family": "po_poly_values", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000245, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "val", "x": 0, "v": "0"}, {"t": "val", "x": -1, "v": "12"}, {"t": "val", "x": 1, "v": "0"}]}, "seed": 358655629902000216, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 6, "valid_answers": ["0, -6, 6, 0", "0, -5, 6, -1", "0, -4, 6, -2", "0, -3, 6, -3", "0, -2, 6, -4", "0, -1, 6, -5", "0, 0, 6, -6"]}, "cell": "poly_values/2", "stage": "B"}
+{"id": "po-poly_values-t2-eval-001472", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(1) = 15, p(0) = 6 and p(-2) = -48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d with -6 <= a, b, c, d <= 6 such that the polynomial p(x) = a + bx + cx^2 + dx^3 satisfies p(1) = 15, p(0) = 6 and p(-2) = -48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 3, 0, 6", "ref": "6, 3, 0, 6", "family": "po_poly_values", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 3, "B": 6, "cons": [{"t": "val", "x": 1, "v": "15"}, {"t": "val", "x": 0, "v": "6"}, {"t": "val", "x": -2, "v": "-48"}]}, "seed": 358655629902001678, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 6, "valid_answers": ["6, 3, 0, 6", "6, 5, -1, 5"]}, "cell": "poly_values/2", "stage": "B"}
+{"id": "po-poly_values-t3-eval-000450", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-2) = 4, p(0) = 0 and p(1) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-2) = 4, p(0) = 0 and p(1) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -4, -3, 3, 2", "ref": "0, -4, -3, 3, 2", "family": "po_poly_values", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.000322, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 4, "B": 4, "cons": [{"t": "val", "x": -2, "v": "4"}, {"t": "val", "x": 0, "v": "0"}, {"t": "val", "x": 1, "v": "-2"}]}, "seed": 358655629903000602, "version": "po-v0", "tier": 3, "n_raw": 19, "small": 1, "valid_answers": ["0, -4, -3, 3, 2", "0, -4, -1, 2, 1", "0, -4, 1, 1, 0", "0, -4, 3, 0, -1", "0, -2, -4, 2, 2", "0, -2, -2, 1, 1", "0, -2, 0, 0, 0", "0, -2, 2, -1, -1", "0, -2, 4, -2, -2", "0, 0, -3, 0, 1", "0, 0, -1, -1, 0", "0, 0, 1, -2, -1", "0, 0, 3, -3, -2", "0, 2, -4, -1, 1", "0, 2, -2, -2, 0", "0, 2, 0, -3, -1", "0, 2, 2, -4, -2", "0, 4, -3, -3, 0", "0, 4, -1, -4, -1"]}, "cell": "poly_values/3", "stage": "B"}
+{"id": "po-poly_values-t3-eval-002046", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-1) = 5, p(-2) = 58 and p(0) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-1) = 5, p(-2) = 58 and p(0) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -3, -3, -2, 3", "ref": "0, -3, -3, -2, 3", "family": "po_poly_values", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000135, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 4, "B": 4, "cons": [{"t": "val", "x": -1, "v": "5"}, {"t": "val", "x": -2, "v": "58"}, {"t": "val", "x": 0, "v": "0"}]}, "seed": 358655629903002811, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 3, "valid_answers": ["0, -3, -3, -2, 3", "0, -3, -1, 1, 4", "0, -1, -2, -4, 2", "0, -1, 0, -1, 3", "0, -1, 2, 2, 4", "0, 1, 1, -3, 2", "0, 1, 3, 0, 3", "0, 3, 4, -2, 2"]}, "cell": "poly_values/3", "stage": "B"}
+{"id": "po-poly_values-t3-eval-000653", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(2) = 26, p(0) = -4 and p(1) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(2) = 26, p(0) = -4 and p(1) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -3, 3, 3, 0", "ref": "-4, -3, 3, 3, 0", "family": "po_poly_values", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.000169, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 4, "B": 4, "cons": [{"t": "val", "x": 2, "v": "26"}, {"t": "val", "x": 0, "v": "-4"}, {"t": "val", "x": 1, "v": "-1"}]}, "seed": 358655629903000888, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 4, "valid_answers": ["-4, -3, 3, 3, 0", "-4, -1, 0, 4, 0", "-4, -1, 2, 1, 1", "-4, -1, 4, -2, 2", "-4, 1, -1, 2, 1", "-4, 1, 1, -1, 2", "-4, 1, 3, -4, 3", "-4, 3, -4, 3, 1", "-4, 3, -2, 0, 2", "-4, 3, 0, -3, 3"]}, "cell": "poly_values/3", "stage": "B"}
+{"id": "po-poly_values-t3-eval-000511", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(0) = -1, p(1) = 8 and p(3) = 350. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(0) = -1, p(1) = 8 and p(3) = 350. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 0, 3, 3, 3", "ref": "-1, 0, 3, 3, 3", "family": "po_poly_values", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 5.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 4, "B": 4, "cons": [{"t": "val", "x": 0, "v": "-1"}, {"t": "val", "x": 1, "v": "8"}, {"t": "val", "x": 3, "v": "350"}]}, "seed": 358655629903000690, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 3, "valid_answers": ["-1, 0, 3, 3, 3", "-1, 3, -1, 4, 3", "-1, 3, 2, 0, 4"]}, "cell": "poly_values/3", "stage": "B"}
+{"id": "po-poly_values-t3-eval-002025", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(3) = -103, p(0) = 2 and p(-2) = -78. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(3) = -103, p(0) = 2 and p(-2) = -78. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 4, -4, 3, -2", "ref": "2, 4, -4, 3, -2", "family": "po_poly_values", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 3.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 4, "B": 4, "cons": [{"t": "val", "x": 3, "v": "-103"}, {"t": "val", "x": 0, "v": "2"}, {"t": "val", "x": -2, "v": "-78"}]}, "seed": 358655629903002785, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 4, "valid_answers": ["2, 4, -4, 3, -2", "2, 4, 2, 4, -3"]}, "cell": "poly_values/3", "stage": "B"}
+{"id": "po-poly_values-t3-eval-000637", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-2) = 45, p(-1) = 6 and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-2) = 45, p(-1) = 6 and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -4, -3, -2, 2", "ref": "1, -4, -3, -2, 2", "family": "po_poly_values", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000152, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 4, "B": 4, "cons": [{"t": "val", "x": -2, "v": "45"}, {"t": "val", "x": -1, "v": "6"}, {"t": "val", "x": 0, "v": "1"}]}, "seed": 358655629903000869, "version": "po-v0", "tier": 3, "n_raw": 9, "small": 2, "valid_answers": ["1, -4, -3, -2, 2", "1, -4, -1, 1, 3", "1, -4, 1, 4, 4", "1, -2, -2, -4, 1", "1, -2, 0, -1, 2", "1, -2, 2, 2, 3", "1, 0, 1, -3, 1", "1, 0, 3, 0, 2", "1, 2, 4, -2, 1"]}, "cell": "poly_values/3", "stage": "B"}
+{"id": "po-poly_values-t3-eval-000809", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(1) = 1, p(-1) = -1 and p(2) = 32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(1) = 1, p(-1) = -1 and p(2) = 32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -2, 4, 3, 0", "ref": "-4, -2, 4, 3, 0", "family": "po_poly_values", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.000237, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 4, "B": 4, "cons": [{"t": "val", "x": 1, "v": "1"}, {"t": "val", "x": -1, "v": "-1"}, {"t": "val", "x": 2, "v": "32"}]}, "seed": 358655629903001102, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 2, "valid_answers": ["-4, -2, 4, 3, 0", "-4, 0, 3, 1, 1", "-4, 2, 2, -1, 2", "-4, 4, 1, -3, 3", "-2, -3, 2, 4, 0", "-2, -1, 1, 2, 1", "-2, 1, 0, 0, 2", "-2, 3, -1, -2, 3", "0, -2, -1, 3, 1", "0, 0, -2, 1, 2", "0, 2, -3, -1, 3", "0, 4, -4, -3, 4", "2, -3, -3, 4, 1", "2, -1, -4, 2, 2"]}, "cell": "poly_values/3", "stage": "B"}
+{"id": "po-poly_values-t3-eval-001740", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-1) = 6, p(2) = -12 and p(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(-1) = 6, p(2) = -12 and p(1) = 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 2, 4, -4, 0", "ref": "0, 2, 4, -4, 0", "family": "po_poly_values", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000135, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 4, "B": 4, "cons": [{"t": "val", "x": -1, "v": "6"}, {"t": "val", "x": 2, "v": "-12"}, {"t": "val", "x": 1, "v": "2"}]}, "seed": 358655629903002393, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 3, "valid_answers": ["0, 2, 4, -4, 0", "2, -3, 4, 1, -2", "2, -1, 3, -1, -1", "2, 1, 2, -3, 0", "4, -4, 2, 2, -2", "4, -2, 1, 0, -1", "4, 0, 0, -2, 0", "4, 2, -1, -4, 1"]}, "cell": "poly_values/3", "stage": "B"}
+{"id": "po-poly_values-t3-eval-001785", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(2) = -63, p(-1) = 6 and p(-2) = 17. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(2) = -63, p(-1) = 6 and p(-2) = 17. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -4, 3, -4, -2", "ref": "-3, -4, 3, -4, -2", "family": "po_poly_values", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 3.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 4, "B": 4, "cons": [{"t": "val", "x": 2, "v": "-63"}, {"t": "val", "x": -1, "v": "6"}, {"t": "val", "x": -2, "v": "17"}]}, "seed": 358655629903002458, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 4, "valid_answers": ["-3, -4, 3, -4, -2", "1, -4, -2, -4, -1"]}, "cell": "poly_values/3", "stage": "B"}
+{"id": "po-poly_values-t3-eval-000617", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(0) = 1, p(2) = -67 and p(-1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(0) = 1, p(2) = -67 and p(-1) = 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -4, -3, -4, -1", "ref": "1, -4, -3, -4, -1", "family": "po_poly_values", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000135, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 4, "B": 4, "cons": [{"t": "val", "x": 0, "v": "1"}, {"t": "val", "x": 2, "v": "-67"}, {"t": "val", "x": -1, "v": "5"}]}, "seed": 358655629903000841, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 3, "valid_answers": ["1, -4, -3, -4, -1", "1, -4, -1, -3, -2", "1, -4, 1, -2, -3", "1, -4, 3, -1, -4", "1, -2, 0, -4, -2", "1, -2, 2, -3, -3", "1, -2, 4, -2, -4", "1, 0, 3, -4, -3"]}, "cell": "poly_values/3", "stage": "B"}
+{"id": "po-poly_values-t3-eval-001992", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(1) = -4, p(-1) = -2 and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(1) = -4, p(-1) = -2 and p(0) = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -4, -4, 3, 0", "ref": "1, -4, -4, 3, 0", "family": "po_poly_values", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 0.000677, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 4, "B": 4, "cons": [{"t": "val", "x": 1, "v": "-4"}, {"t": "val", "x": -1, "v": "-2"}, {"t": "val", "x": 0, "v": "1"}]}, "seed": 358655629903002737, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 2, "valid_answers": ["1, -4, -4, 3, 0", "1, -4, -3, 3, -1", "1, -4, -2, 3, -2", "1, -4, -1, 3, -3", "1, -4, 0, 3, -4", "1, -3, -4, 2, 0", "1, -3, -3, 2, -1", "1, -3, -2, 2, -2", "1, -3, -1, 2, -3", "1, -3, 0, 2, -4", "1, -2, -4, 1, 0", "1, -2, -3, 1, -1", "1, -2, -2, 1, -2", "1, -2, -1, 1, -3", "1, -2, 0, 1, -4", "1, -1, -4, 0, 0", "1, -1, -3, 0, -1", "1, -1, -2, 0, -2", "1, -1, -1, 0, -3", "1, -1, 0, 0, -4", "1, 0, -4, -1, 0", "1, 0, -3, -1, -1", "1, 0, -2, -1, -2", "1, 0, -1, -1, -3", "1, 0, 0, -1, -4", "1, 1, -4, -2, 0", "1, 1, -3, -2, -1", "1, 1, -2, -2, -2", "1, 1, -1, -2, -3", "1, 1, 0, -2, -4", "1, 2, -4, -3, 0", "1, 2, -3, -3, -1", "1, 2, -2, -3, -2", "1, 2, -1, -3, -3", "1, 2, 0, -3, -4", "1, 3, -4, -4, 0", "1, 3, -3, -4, -1", "1, 3, -2, -4, -2", "1, 3, -1, -4, -3", "1, 3, 0, -4, -4"]}, "cell": "poly_values/3", "stage": "B"}
+{"id": "po-poly_values-t3-eval-000631", "task": "cx_po_poly_values", "source": "cx_po_poly_values", "split": "eval", "problem": "Find integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(1) = -10, p(0) = -1 and p(-2) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c, d, e with -4 <= a, b, c, d, e <= 4 such that the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 satisfies p(1) = -10, p(0) = -1 and p(-2) = -1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -4, -4, -1, 0", "ref": "-1, -4, -4, -1, 0", "family": "po_poly_values", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000119, "solvable": true, "holdout": false, "meta": {"family": "po_poly_values", "rung": "witness", "params": {"d": 4, "B": 4, "cons": [{"t": "val", "x": 1, "v": "-10"}, {"t": "val", "x": 0, "v": "-1"}, {"t": "val", "x": -2, "v": "-1"}]}, "seed": 358655629903000861, "version": "po-v0", "tier": 3, "n_raw": 7, "small": 3, "valid_answers": ["-1, -4, -4, -1, 0", "-1, -4, -2, -2, -1", "-1, -4, 0, -3, -2", "-1, -4, 2, -4, -3", "-1, -2, -3, -3, -1", "-1, -2, -1, -4, -2", "-1, 0, -4, -4, -1"]}, "cell": "poly_values/3", "stage": "B"}
+{"id": "po-pyth_leg-t1-eval-000094", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 165^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 165^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "52, 173", "ref": "52, 173", "family": "po_pyth_leg", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 165}, "seed": 397698761601000499, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 173, "valid_answers": ["52, 173", "88, 187", "144, 219", "220, 275", "280, 325", "396, 429", "532, 557", "900, 915", "1232, 1243", "1508, 1517", "2720, 2725", "4536, 4539", "13612, 13613"]}, "cell": "pyth_leg/1", "stage": "B"}
+{"id": "po-pyth_leg-t1-eval-000085", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 110^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 110^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "96, 146", "ref": "96, 146", "family": "po_pyth_leg", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 110}, "seed": 397698761601000246, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 146, "valid_answers": ["96, 146", "264, 286", "600, 610", "3024, 3026"]}, "cell": "pyth_leg/1", "stage": "B"}
+{"id": "po-pyth_leg-t1-eval-000006", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 147^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 147^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "140, 203", "ref": "140, 203", "family": "po_pyth_leg", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 147}, "seed": 397698761601000006, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 203, "valid_answers": ["140, 203", "196, 245", "504, 525", "1196, 1205", "1540, 1547", "3600, 3603", "10804, 10805"]}, "cell": "pyth_leg/1", "stage": "B"}
+{"id": "po-pyth_leg-t1-eval-000013", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 98^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 98^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "336, 350", "ref": "336, 350", "family": "po_pyth_leg", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 98}, "seed": 397698761601000014, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 350, "valid_answers": ["336, 350", "2400, 2402"]}, "cell": "pyth_leg/1", "stage": "B"}
+{"id": "po-pyth_leg-t1-eval-000055", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 143^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 143^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "24, 145", "ref": "24, 145", "family": "po_pyth_leg", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 143}, "seed": 397698761601000103, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 145, "valid_answers": ["24, 145", "780, 793", "924, 935", "10224, 10225"]}, "cell": "pyth_leg/1", "stage": "B"}
+{"id": "po-pyth_leg-t2-eval-000024", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 546^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 546^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "272, 610", "ref": "272, 610", "family": "po_pyth_leg", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 546}, "seed": 397698761602000029, "version": "po-v0", "tier": 2, "n_raw": 13, "small": 610, "valid_answers": ["272, 610", "360, 654", "520, 754", "728, 910", "1120, 1246", "1472, 1570", "1872, 1950", "3528, 3570", "5720, 5746", "8272, 8290", "10640, 10654", "24840, 24846", "74528, 74530"]}, "cell": "pyth_leg/2", "stage": "C"}
+{"id": "po-pyth_leg-t2-eval-000188", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 588^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 588^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "91, 595", "ref": "91, 595", "family": "po_pyth_leg", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 588}, "seed": 397698761602000428, "version": "po-v0", "tier": 2, "n_raw": 22, "small": 595, "valid_answers": ["91, 595", "245, 637", "441, 735", "560, 812", "784, 980", "945, 1113", "1309, 1435", "1715, 1813", "2016, 2100", "2365, 2437", "3059, 3115", "4095, 4137", "4784, 4820", "6160, 6188", "7191, 7215", "9595, 9613", "12341, 12355", "14400, 14412", "21605, 21613", "28809, 28815", "43216, 43220", "86435, 86437"]}, "cell": "pyth_leg/2", "stage": "C"}
+{"id": "po-pyth_leg-t2-eval-000173", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 438^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 438^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "584, 730", "ref": "584, 730", "family": "po_pyth_leg", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 438}, "seed": 397698761602000373, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 730, "valid_answers": ["584, 730", "5320, 5338", "15984, 15990", "47960, 47962"]}, "cell": "pyth_leg/2", "stage": "C"}
+{"id": "po-pyth_leg-t2-eval-000055", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 384^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 384^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "112, 400", "ref": "112, 400", "family": "po_pyth_leg", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 384}, "seed": 397698761602000080, "version": "po-v0", "tier": 2, "n_raw": 19, "small": 400, "valid_answers": ["112, 400", "160, 416", "288, 480", "440, 584", "512, 640", "720, 816", "988, 1060", "1120, 1184", "1512, 1560", "2030, 2066", "2288, 2320", "3060, 3084", "4087, 4105", "4600, 4616", "6138, 6150", "9212, 9220", "12285, 12291", "18430, 18434", "36863, 36865"]}, "cell": "pyth_leg/2", "stage": "C"}
+{"id": "po-pyth_leg-t2-eval-000191", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 201^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 201^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "268, 335", "ref": "268, 335", "family": "po_pyth_leg", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 201}, "seed": 397698761602000443, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 335, "valid_answers": ["268, 335", "2240, 2249", "6732, 6735", "20200, 20201"]}, "cell": "pyth_leg/2", "stage": "C"}
+{"id": "po-pyth_leg-t2-eval-000140", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 565^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 565^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1356, 1469", "ref": "1356, 1469", "family": "po_pyth_leg", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 565}, "seed": 397698761602000270, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 1469, "valid_answers": ["1356, 1469", "6372, 6397", "31920, 31925", "159612, 159613"]}, "cell": "pyth_leg/2", "stage": "C"}
+{"id": "po-pyth_leg-t2-eval-000296", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 225^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 225^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "120, 255", "ref": "120, 255", "family": "po_pyth_leg", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 225}, "seed": 397698761602001674, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 255, "valid_answers": ["120, 255", "140, 265", "272, 353", "300, 375", "540, 585", "924, 951", "1000, 1025", "1680, 1695", "2808, 2817", "5060, 5065", "8436, 8439", "25312, 25313"]}, "cell": "pyth_leg/2", "stage": "C"}
+{"id": "po-pyth_leg-t2-eval-000236", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 318^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 318^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "424, 530", "ref": "424, 530", "family": "po_pyth_leg", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 318}, "seed": 397698761602000654, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 530, "valid_answers": ["424, 530", "2800, 2818", "8424, 8430", "25280, 25282"]}, "cell": "pyth_leg/2", "stage": "C"}
+{"id": "po-pyth_leg-t2-eval-000096", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 355^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 355^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "852, 923", "ref": "852, 923", "family": "po_pyth_leg", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 355}, "seed": 397698761602000156, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 923, "valid_answers": ["852, 923", "2508, 2533", "12600, 12605", "63012, 63013"]}, "cell": "pyth_leg/2", "stage": "C"}
+{"id": "po-pyth_leg-t2-eval-000035", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 494^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 494^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "192, 530", "ref": "192, 530", "family": "po_pyth_leg", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 494}, "seed": 397698761602000050, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 530, "valid_answers": ["192, 530", "3192, 3230", "4680, 4706", "61008, 61010"]}, "cell": "pyth_leg/2", "stage": "C"}
+{"id": "po-pyth_leg-t2-eval-000059", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 243^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 243^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "324, 405", "ref": "324, 405", "family": "po_pyth_leg", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 243}, "seed": 397698761602000091, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 405, "valid_answers": ["324, 405", "1080, 1107", "3276, 3285", "9840, 9843", "29524, 29525"]}, "cell": "pyth_leg/2", "stage": "C"}
+{"id": "po-pyth_leg-t2-eval-000289", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 301^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 301^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "900, 949", "ref": "900, 949", "family": "po_pyth_leg", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 301}, "seed": 397698761602001471, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 949, "valid_answers": ["900, 949", "1032, 1075", "6468, 6475", "45300, 45301"]}, "cell": "pyth_leg/2", "stage": "C"}
+{"id": "po-pyth_leg-t3-eval-001023", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 1176^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 1176^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "182, 1190", "ref": "182, 1190", "family": "po_pyth_leg", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 37, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 1176}, "seed": 397698761603003909, "version": "po-v0", "tier": 3, "n_raw": 37, "small": 1190, "valid_answers": ["182, 1190", "343, 1225", "490, 1274", "665, 1351", "693, 1365", "882, 1470", "1120, 1624", "1568, 1960", "1890, 2226", "2205, 2499", "2257, 2545", "2618, 2870", "2975, 3199", "3430, 3626", "4032, 4200", "4730, 4874", "5425, 5551", "6118, 6230", "7007, 7105", "7155, 7251", "8190, 8274", "9568, 9640", "12320, 12376", "14382, 14430", "16443, 16485", "19190, 19226", "21593, 21625", "24682, 24710", "28800, 28824", "38407, 38425", "43210, 43226", "49385, 49399", "57618, 57630", "86432, 86440", "115245, 115251", "172870, 172874", "345743, 345745"]}, "cell": "pyth_leg/3", "stage": "C"}
+{"id": "po-pyth_leg-t3-eval-000580", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 1245^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 1245^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "664, 1411", "ref": "664, 1411", "family": "po_pyth_leg", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 1245}, "seed": 397698761603001050, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 1411, "valid_answers": ["664, 1411", "1660, 2075", "2988, 3237", "3332, 3557", "9296, 9379", "10296, 10371", "17200, 17245", "30988, 31013", "51660, 51675", "86108, 86117", "155000, 155005", "258336, 258339", "775012, 775013"]}, "cell": "pyth_leg/3", "stage": "C"}
+{"id": "po-pyth_leg-t3-eval-000780", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 1518^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 1518^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "560, 1618", "ref": "560, 1618", "family": "po_pyth_leg", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 1518}, "seed": 397698761603001855, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 1618, "valid_answers": ["560, 1618", "1224, 1950", "2024, 2530", "2576, 2990", "4640, 4882", "5720, 5918", "8280, 8418", "17424, 17490", "25024, 25070", "52360, 52382", "64000, 64018", "192024, 192030", "576080, 576082"]}, "cell": "pyth_leg/3", "stage": "C"}
+{"id": "po-pyth_leg-t3-eval-000302", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 1845^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 1845^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "172, 1853", "ref": "172, 1853", "family": "po_pyth_leg", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 1845}, "seed": 397698761603000448, "version": "po-v0", "tier": 3, "n_raw": 22, "small": 1853, "valid_answers": ["172, 1853", "984, 2091", "1148, 2173", "2184, 2859", "2460, 3075", "4000, 4405", "4428, 4797", "7452, 7677", "8200, 8405", "12540, 12675", "13776, 13899", "20972, 21053", "22656, 22731", "37800, 37845", "41492, 41533", "63024, 63051", "68068, 68093", "113460, 113475", "189108, 189117", "340400, 340405", "567336, 567339", "1702012, 1702013"]}, "cell": "pyth_leg/3", "stage": "C"}
+{"id": "po-pyth_leg-t3-eval-001034", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 1970^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 1970^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4728, 5122", "ref": "4728, 5122", "family": "po_pyth_leg", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 1970}, "seed": 397698761603004028, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 5122, "valid_answers": ["4728, 5122", "38784, 38834", "194040, 194050", "970224, 970226"]}, "cell": "pyth_leg/3", "stage": "C"}
+{"id": "po-pyth_leg-t3-eval-000585", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 1922^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 1922^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "29760, 29822", "ref": "29760, 29822", "family": "po_pyth_leg", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 1922}, "seed": 397698761603001066, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 29822, "valid_answers": ["29760, 29822", "923520, 923522"]}, "cell": "pyth_leg/3", "stage": "C"}
+{"id": "po-pyth_leg-t3-eval-000662", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 672^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 672^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "90, 678", "ref": "90, 678", "family": "po_pyth_leg", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 672}, "seed": 397698761603001321, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 678, "valid_answers": ["90, 678", "104, 680", "185, 697", "196, 700", "280, 728", "380, 772", "396, 780", "504, 840", "621, 915", "640, 928", "754, 1010", "770, 1022", "896, 1120", "1054, 1250", "1080, 1272", "1260, 1428", "1496, 1640", "1700, 1828", "1729, 1855", "1960, 2072", "2255, 2353", "2304, 2400", "2646, 2730", "3100, 3172", "3496, 3560", "4004, 4060", "4680, 4728", "5355, 5397", "6254, 6290", "7040, 7072", "8050, 8078", "9396, 9420", "12535, 12553", "14104, 14120", "16121, 16135", "18810, 18822", "28220, 28228", "37629, 37635", "56446, 56450", "112895, 112897"]}, "cell": "pyth_leg/3", "stage": "C"}
+{"id": "po-pyth_leg-t3-eval-000199", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 1740^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 1740^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "59, 1741", "ref": "59, 1741", "family": "po_pyth_leg", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 40, "bucket": null, "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 1740}, "seed": 397698761603000275, "version": "po-v0", "tier": 3, "n_raw": 40, "small": 1741, "valid_answers": ["59, 1741", "319, 1769", "725, 1885", "928, 1972", "1232, 2132", "1305, 2175", "1827, 2523", "2223, 2823", "2320, 2900", "2639, 3161", "3139, 3589", "4025, 4385", "4176, 4524", "4896, 5196", "5075, 5365", "6409, 6641", "7469, 7669", "8320, 8500", "8613, 8787", "10017, 10167", "12555, 12675", "12992, 13108", "15088, 15188", "16775, 16865", "20989, 21061", "25200, 25260", "26071, 26129", "30251, 30301", "37825, 37865", "42032, 42068", "50445, 50475", "63063, 63087", "75680, 75700", "84091, 84109", "126144, 126156", "151375, 151385", "189221, 189229", "252297, 252303", "378448, 378452", "756899, 756901"]}, "cell": "pyth_leg/3", "stage": "C"}
+{"id": "po-pyth_leg-t3-eval-000908", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 675^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 675^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "52, 677", "ref": "52, 677", "family": "po_pyth_leg", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 17, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 675}, "seed": 397698761603002619, "version": "po-v0", "tier": 3, "n_raw": 17, "small": 677, "valid_answers": ["52, 677", "360, 765", "420, 795", "816, 1059", "900, 1125", "1620, 1755", "1760, 1885", "2772, 2853", "3000, 3075", "5040, 5085", "8424, 8451", "9100, 9125", "15180, 15195", "25308, 25317", "45560, 45565", "75936, 75939", "227812, 227813"]}, "cell": "pyth_leg/3", "stage": "C"}
+{"id": "po-pyth_leg-t3-eval-000391", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 1698^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 1698^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2264, 2830", "ref": "2264, 2830", "family": "po_pyth_leg", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 1698}, "seed": 397698761603000604, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 2830, "valid_answers": ["2264, 2830", "80080, 80098", "240264, 240270", "720800, 720802"]}, "cell": "pyth_leg/3", "stage": "C"}
+{"id": "po-pyth_leg-t3-eval-000841", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 1864^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 1864^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1398, 2330", "ref": "1398, 2330", "family": "po_pyth_leg", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 1864}, "seed": 397698761603002147, "version": "po-v0", "tier": 3, "n_raw": 7, "small": 2330, "valid_answers": ["1398, 2330", "3495, 3961", "54273, 54305", "108570, 108586", "217152, 217160", "434310, 434314", "868623, 868625"]}, "cell": "pyth_leg/3", "stage": "C"}
+{"id": "po-pyth_leg-t3-eval-000151", "task": "cx_po_pyth_leg", "source": "cx_po_pyth_leg", "split": "eval", "problem": "Find positive integers b, c such that 1956^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers b, c such that 1956^2 + b^2 = c^2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "815, 2119", "ref": "815, 2119", "family": "po_pyth_leg", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_leg", "rung": "witness", "params": {"a": 1956}, "seed": 397698761603000203, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 2119, "valid_answers": ["815, 2119", "1467, 2445", "2608, 3260", "5705, 6031", "26533, 26605", "53120, 53156", "79695, 79719", "106267, 106285", "159408, 159420", "239117, 239125", "318825, 318831", "478240, 478244", "956483, 956485"]}, "cell": "pyth_leg/3", "stage": "C"}
+{"id": "po-pyth_perim-t0-eval-000021", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 360. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 360. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "36, 160, 164", "ref": "36, 160, 164", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00037, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 360}, "seed": 387040090700000176, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 150, "valid_answers": ["36, 160, 164", "60, 144, 156", "72, 135, 153", "90, 120, 150"]}, "cell": "pyth_perim/0", "stage": "C"}
+{"id": "po-pyth_perim-t0-eval-000012", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 420. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 420. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "28, 195, 197", "ref": "28, 195, 197", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.00034, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 420}, "seed": 387040090700000050, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 174, "valid_answers": ["28, 195, 197", "60, 175, 185", "70, 168, 182", "105, 140, 175", "120, 126, 174"]}, "cell": "pyth_perim/0", "stage": "C"}
+{"id": "po-pyth_perim-t0-eval-000019", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 528. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 528. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "44, 240, 244", "ref": "44, 240, 244", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000129, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 528}, "seed": 387040090700000152, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 219, "valid_answers": ["44, 240, 244", "132, 176, 220", "144, 165, 219"]}, "cell": "pyth_perim/0", "stage": "C"}
+{"id": "po-pyth_perim-t0-eval-000032", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 312. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 312. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "24, 143, 145", "ref": "24, 143, 145", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000247, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 312}, "seed": 387040090700000787, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 130, "valid_answers": ["24, 143, 145", "78, 104, 130"]}, "cell": "pyth_perim/0", "stage": "C"}
+{"id": "po-pyth_perim-t0-eval-000018", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 510. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 510. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "60, 221, 229", "ref": "60, 221, 229", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 9.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 510}, "seed": 387040090700000146, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 221, "valid_answers": ["60, 221, 229", "85, 204, 221"]}, "cell": "pyth_perim/0", "stage": "C"}
+{"id": "po-pyth_perim-t1-eval-000005", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1008. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1008. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "112, 441, 455", "ref": "112, 441, 455", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 5.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 1008}, "seed": 387040090701000030, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 420, "valid_answers": ["112, 441, 455", "126, 432, 450", "144, 420, 444", "224, 360, 424", "252, 336, 420"]}, "cell": "pyth_perim/1", "stage": "C"}
+{"id": "po-pyth_perim-t1-eval-000078", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1044. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1044. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "203, 396, 445", "ref": "203, 396, 445", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 1044}, "seed": 387040090701000952, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 435, "valid_answers": ["203, 396, 445", "261, 348, 435"]}, "cell": "pyth_perim/1", "stage": "C"}
+{"id": "po-pyth_perim-t1-eval-000071", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1160. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1160. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "232, 435, 493", "ref": "232, 435, 493", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 1160}, "seed": 387040090701000772, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 481, "valid_answers": ["232, 435, 493", "319, 360, 481"]}, "cell": "pyth_perim/1", "stage": "C"}
+{"id": "po-pyth_perim-t1-eval-000006", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1188. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1188. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "99, 540, 549", "ref": "99, 540, 549", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 3.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 1188}, "seed": 387040090701000037, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 495, "valid_answers": ["99, 540, 549", "216, 462, 510", "220, 459, 509", "297, 396, 495"]}, "cell": "pyth_perim/1", "stage": "C"}
+{"id": "po-pyth_perim-t1-eval-000081", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 840. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 840. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "40, 399, 401", "ref": "40, 399, 401", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000136, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 840}, "seed": 387040090701001437, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 348, "valid_answers": ["40, 399, 401", "56, 390, 394", "105, 360, 375", "120, 350, 370", "140, 336, 364", "168, 315, 357", "210, 280, 350", "240, 252, 348"]}, "cell": "pyth_perim/1", "stage": "C"}
+{"id": "po-pyth_perim-t1-eval-000049", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1480. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1480. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "111, 680, 689", "ref": "111, 680, 689", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 1480}, "seed": 387040090701000381, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 629, "valid_answers": ["111, 680, 689", "296, 555, 629"]}, "cell": "pyth_perim/1", "stage": "C"}
+{"id": "po-pyth_perim-t1-eval-000066", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1488. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1488. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "336, 527, 625", "ref": "336, 527, 625", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 1488}, "seed": 387040090701000632, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 620, "valid_answers": ["336, 527, 625", "372, 496, 620"]}, "cell": "pyth_perim/1", "stage": "C"}
+{"id": "po-pyth_perim-t1-eval-000032", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1428. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1428. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "204, 595, 629", "ref": "204, 595, 629", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 1428}, "seed": 387040090701000227, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 595, "valid_answers": ["204, 595, 629", "252, 561, 615", "272, 546, 610", "357, 476, 595"]}, "cell": "pyth_perim/1", "stage": "C"}
+{"id": "po-pyth_perim-t1-eval-000053", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 918. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 918. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "51, 432, 435", "ref": "51, 432, 435", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 918}, "seed": 387040090701000406, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 389, "valid_answers": ["51, 432, 435", "189, 340, 389"]}, "cell": "pyth_perim/1", "stage": "C"}
+{"id": "po-pyth_perim-t1-eval-000015", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1248. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1248. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "96, 572, 580", "ref": "96, 572, 580", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 2.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 1248}, "seed": 387040090701000095, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 520, "valid_answers": ["96, 572, 580", "234, 480, 534", "312, 416, 520"]}, "cell": "pyth_perim/1", "stage": "C"}
+{"id": "po-pyth_perim-t1-eval-000020", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 690. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 690. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "115, 276, 299", "ref": "115, 276, 299", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 690}, "seed": 387040090701000130, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 289, "valid_answers": ["115, 276, 299", "161, 240, 289"]}, "cell": "pyth_perim/1", "stage": "C"}
+{"id": "po-pyth_perim-t1-eval-000025", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1380. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1380. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "230, 552, 598", "ref": "230, 552, 598", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 1.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 1380}, "seed": 387040090701000197, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 575, "valid_answers": ["230, 552, 598", "322, 480, 578", "345, 460, 575"]}, "cell": "pyth_perim/1", "stage": "C"}
+{"id": "po-pyth_perim-t2-eval-000091", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 2688. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 2688. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "336, 1152, 1200", "ref": "336, 1152, 1200", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 7e-06, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 2688}, "seed": 387040090702000621, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 1120, "valid_answers": ["336, 1152, 1200", "384, 1120, 1184", "640, 924, 1124", "672, 896, 1120"]}, "cell": "pyth_perim/2", "stage": "C"}
+{"id": "po-pyth_perim-t2-eval-000166", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1530. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1530. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "85, 720, 725", "ref": "85, 720, 725", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 2.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 1530}, "seed": 387040090702001505, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 663, "valid_answers": ["85, 720, 725", "153, 680, 697", "180, 663, 687", "255, 612, 663"]}, "cell": "pyth_perim/2", "stage": "C"}
+{"id": "po-pyth_perim-t2-eval-000044", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 3640. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 3640. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "260, 1680, 1700", "ref": "260, 1680, 1700", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 5e-06, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 3640}, "seed": 387040090702000323, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 1508, "valid_answers": ["260, 1680, 1700", "455, 1560, 1625", "728, 1365, 1547", "840, 1274, 1526", "936, 1190, 1514", "1040, 1092, 1508"]}, "cell": "pyth_perim/2", "stage": "C"}
+{"id": "po-pyth_perim-t2-eval-000049", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 3000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 3000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "500, 1200, 1300", "ref": "500, 1200, 1300", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 3000}, "seed": 387040090702000334, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 1250, "valid_answers": ["500, 1200, 1300", "600, 1125, 1275", "750, 1000, 1250"]}, "cell": "pyth_perim/2", "stage": "C"}
+{"id": "po-pyth_perim-t2-eval-000085", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 3906. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 3906. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "868, 1395, 1643", "ref": "868, 1395, 1643", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2e-06, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 3906}, "seed": 387040090702000558, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 1623, "valid_answers": ["868, 1395, 1643", "1023, 1260, 1623"]}, "cell": "pyth_perim/2", "stage": "C"}
+{"id": "po-pyth_perim-t2-eval-000004", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 3440. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 3440. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "240, 1591, 1609", "ref": "240, 1591, 1609", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2e-06, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 3440}, "seed": 387040090702000026, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 1462, "valid_answers": ["240, 1591, 1609", "688, 1290, 1462"]}, "cell": "pyth_perim/2", "stage": "C"}
+{"id": "po-pyth_perim-t2-eval-000058", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1800. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1800. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "180, 800, 820", "ref": "180, 800, 820", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 1.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 1800}, "seed": 387040090702000376, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 746, "valid_answers": ["180, 800, 820", "300, 720, 780", "360, 675, 765", "450, 600, 750", "504, 550, 746"]}, "cell": "pyth_perim/2", "stage": "C"}
+{"id": "po-pyth_perim-t2-eval-000213", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1890. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1890. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "189, 840, 861", "ref": "189, 840, 861", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 1.7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 1890}, "seed": 387040090702002811, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 783, "valid_answers": ["189, 840, 861", "315, 756, 819", "420, 675, 795", "432, 665, 793", "540, 567, 783"]}, "cell": "pyth_perim/2", "stage": "C"}
+{"id": "po-pyth_perim-t2-eval-000191", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 2464. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 2464. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "308, 1056, 1100", "ref": "308, 1056, 1100", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 6e-06, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 2464}, "seed": 387040090702002262, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 1022, "valid_answers": ["308, 1056, 1100", "528, 896, 1040", "672, 770, 1022"]}, "cell": "pyth_perim/2", "stage": "C"}
+{"id": "po-pyth_perim-t2-eval-000088", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1512. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 1512. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "54, 728, 730", "ref": "54, 728, 730", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 2.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 1512}, "seed": 387040090702000596, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 630, "valid_answers": ["54, 728, 730", "189, 648, 675", "216, 630, 666", "336, 540, 636", "378, 504, 630"]}, "cell": "pyth_perim/2", "stage": "C"}
+{"id": "po-pyth_perim-t2-eval-000162", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 2590. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 2590. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "140, 1221, 1229", "ref": "140, 1221, 1229", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 2590}, "seed": 387040090702001405, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 1073, "valid_answers": ["140, 1221, 1229", "740, 777, 1073"]}, "cell": "pyth_perim/2", "stage": "C"}
+{"id": "po-pyth_perim-t2-eval-000059", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 2408. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 2408. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "301, 1032, 1075", "ref": "301, 1032, 1075", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 2408}, "seed": 387040090702000382, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 1009, "valid_answers": ["301, 1032, 1075", "559, 840, 1009"]}, "cell": "pyth_perim/2", "stage": "C"}
+{"id": "po-pyth_perim-t3-eval-000048", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 7904. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 7904. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1248, 3211, 3445", "ref": "1248, 3211, 3445", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 7904}, "seed": 387040090703000287, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 3280, "valid_answers": ["1248, 3211, 3445", "1482, 3040, 3382", "2128, 2496, 3280"]}, "cell": "pyth_perim/3", "stage": "C"}
+{"id": "po-pyth_perim-t3-eval-000525", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 8874. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 8874. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "493, 4176, 4205", "ref": "493, 4176, 4205", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 8874}, "seed": 387040090703005258, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 3897, "valid_answers": ["493, 4176, 4205", "1071, 3828, 3975", "1305, 3672, 3897"]}, "cell": "pyth_perim/3", "stage": "C"}
+{"id": "po-pyth_perim-t3-eval-000060", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 7380. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 7380. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "656, 3330, 3394", "ref": "656, 3330, 3394", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 7380}, "seed": 387040090703000368, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 3063, "valid_answers": ["656, 3330, 3394", "738, 3280, 3362", "1230, 2952, 3198", "1845, 2460, 3075", "1980, 2337, 3063"]}, "cell": "pyth_perim/3", "stage": "C"}
+{"id": "po-pyth_perim-t3-eval-000401", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 8624. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 8624. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "880, 3822, 3922", "ref": "880, 3822, 3922", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 8624}, "seed": 387040090703003121, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 3577, "valid_answers": ["880, 3822, 3922", "1078, 3696, 3850", "1848, 3136, 3640", "2352, 2695, 3577"]}, "cell": "pyth_perim/3", "stage": "C"}
+{"id": "po-pyth_perim-t3-eval-000487", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 9184. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 9184. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1148, 3936, 4100", "ref": "1148, 3936, 4100", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 9184}, "seed": 387040090703004484, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 3812, "valid_answers": ["1148, 3936, 4100", "2460, 2912, 3812"]}, "cell": "pyth_perim/3", "stage": "C"}
+{"id": "po-pyth_perim-t3-eval-000056", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 6768. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 6768. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "141, 3312, 3315", "ref": "141, 3312, 3315", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 6768}, "seed": 387040090703000341, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 2820, "valid_answers": ["141, 3312, 3315", "752, 2961, 3055", "1584, 2350, 2834", "1692, 2256, 2820"]}, "cell": "pyth_perim/3", "stage": "C"}
+{"id": "po-pyth_perim-t3-eval-000185", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 8742. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 8742. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "93, 4324, 4325", "ref": "93, 4324, 4325", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 8742}, "seed": 387040090703001089, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 3651, "valid_answers": ["93, 4324, 4325", "2115, 2976, 3651"]}, "cell": "pyth_perim/3", "stage": "C"}
+{"id": "po-pyth_perim-t3-eval-000447", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 5124. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 5124. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "732, 2135, 2257", "ref": "732, 2135, 2257", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 5124}, "seed": 387040090703003685, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 2125, "valid_answers": ["732, 2135, 2257", "1281, 1708, 2135", "1403, 1596, 2125"]}, "cell": "pyth_perim/3", "stage": "C"}
+{"id": "po-pyth_perim-t3-eval-000079", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 9810. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 9810. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "981, 4360, 4469", "ref": "981, 4360, 4469", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 9810}, "seed": 387040090703000463, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 4251, "valid_answers": ["981, 4360, 4469", "1635, 3924, 4251"]}, "cell": "pyth_perim/3", "stage": "C"}
+{"id": "po-pyth_perim-t3-eval-000515", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 7590. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 7590. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1265, 3036, 3289", "ref": "1265, 3036, 3289", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 7590}, "seed": 387040090703005053, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 3145, "valid_answers": ["1265, 3036, 3289", "1540, 2829, 3221", "1771, 2640, 3179", "2024, 2415, 3151", "2145, 2300, 3145"]}, "cell": "pyth_perim/3", "stage": "C"}
+{"id": "po-pyth_perim-t3-eval-000556", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 9960. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 9960. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1660, 3984, 4316", "ref": "1660, 3984, 4316", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 9960}, "seed": 387040090703006205, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 4129, "valid_answers": ["1660, 3984, 4316", "1992, 3735, 4233", "2490, 3320, 4150", "2760, 3071, 4129"]}, "cell": "pyth_perim/3", "stage": "C"}
+{"id": "po-pyth_perim-t3-eval-000444", "task": "cx_po_pyth_perim", "source": "cx_po_pyth_perim", "split": "eval", "problem": "Find positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 5820. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers a <= b < c with a^2 + b^2 = c^2 (the sides of a right triangle) and a + b + c = 5820. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "970, 2328, 2522", "ref": "970, 2328, 2522", "family": "po_pyth_perim", "domain": "geometry", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_pyth_perim", "rung": "witness", "params": {"P": 5820}, "seed": 387040090703003610, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 2425, "valid_answers": ["970, 2328, 2522", "1455, 1940, 2425"]}, "cell": "pyth_perim/3", "stage": "C"}
+{"id": "po-quad_rootdiff-t0-eval-000056", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -19 <= b, c <= 19 such that x^2 + bx + c has two integer roots that differ by 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -19 <= b, c <= 19 such that x^2 + bx + c has two integer roots that differ by 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 15", "ref": "-8, 15", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.005917, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 19, "D": 2, "cond": null}, "seed": 177091180800000100, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 1, "valid_answers": ["-8, 15", "-6, 8", "-4, 3", "-2, 0", "0, -1", "2, 0", "4, 3", "6, 8", "8, 15"]}, "cell": "quad_rootdiff/0", "stage": "B"}
+{"id": "po-quad_rootdiff-t1-eval-000195", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -39 <= b, c <= 39 such that x^2 + bx + c has two integer roots that differ by 7. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -39 <= b, c <= 39 such that x^2 + bx + c has two integer roots that differ by 7. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-13, 30", "ref": "-13, 30", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000481, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 39, "D": 7, "cond": "pos"}, "seed": 177091180801000272, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 9, "valid_answers": ["-13, 30", "-11, 18", "-9, 8"]}, "cell": "quad_rootdiff/1", "stage": "B"}
+{"id": "po-quad_rootdiff-t1-eval-000302", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -31 <= b, c <= 31 such that x^2 + bx + c has two integer roots that differ by 2. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -31 <= b, c <= 31 such that x^2 + bx + c has two integer roots that differ by 2. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 3", "ref": "4, 3", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001008, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 31, "D": 2, "cond": "neg"}, "seed": 177091180801000626, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 4, "valid_answers": ["4, 3", "6, 8", "8, 15", "10, 24"]}, "cell": "quad_rootdiff/1", "stage": "B"}
+{"id": "po-quad_rootdiff-t1-eval-000262", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -33 <= b, c <= 33 such that x^2 + bx + c has two integer roots that differ by 4. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -33 <= b, c <= 33 such that x^2 + bx + c has two integer roots that differ by 4. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, 32", "ref": "-12, 32", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000891, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 33, "D": 4, "cond": "pos"}, "seed": 177091180801000446, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 6, "valid_answers": ["-12, 32", "-10, 21", "-8, 12", "-6, 5"]}, "cell": "quad_rootdiff/1", "stage": "B"}
+{"id": "po-quad_rootdiff-t1-eval-000112", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -23 <= b, c <= 23 such that x^2 + bx + c has two integer roots that differ by 2. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -23 <= b, c <= 23 such that x^2 + bx + c has two integer roots that differ by 2. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 3", "ref": "4, 3", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001358, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 23, "D": 2, "cond": "neg"}, "seed": 177091180801000138, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 4, "valid_answers": ["4, 3", "6, 8", "8, 15"]}, "cell": "quad_rootdiff/1", "stage": "B"}
+{"id": "po-quad_rootdiff-t1-eval-000014", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -35 <= b, c <= 35 such that x^2 + bx + c has two integer roots that differ by 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -35 <= b, c <= 35 such that x^2 + bx + c has two integer roots that differ by 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, 32", "ref": "-12, 32", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.002579, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 35, "D": 4, "cond": null}, "seed": 177091180801000015, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 3, "valid_answers": ["-12, 32", "-10, 21", "-8, 12", "-6, 5", "-4, 0", "-2, -3", "0, -4", "2, -3", "4, 0", "6, 5", "8, 12", "10, 21", "12, 32"]}, "cell": "quad_rootdiff/1", "stage": "B"}
+{"id": "po-quad_rootdiff-t1-eval-000148", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -20 <= b, c <= 20 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -20 <= b, c <= 20 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8, 7", "ref": "8, 7", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00119, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 20, "D": 6, "cond": "neg"}, "seed": 177091180801000189, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 8, "valid_answers": ["8, 7", "10, 16"]}, "cell": "quad_rootdiff/1", "stage": "B"}
+{"id": "po-quad_rootdiff-t1-eval-000250", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -21 <= b, c <= 21 such that x^2 + bx + c has two integer roots that differ by 4. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -21 <= b, c <= 21 such that x^2 + bx + c has two integer roots that differ by 4. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 21", "ref": "-10, 21", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001622, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 21, "D": 4, "cond": "pos"}, "seed": 177091180801000409, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 6, "valid_answers": ["-10, 21", "-8, 12", "-6, 5"]}, "cell": "quad_rootdiff/1", "stage": "B"}
+{"id": "po-quad_rootdiff-t1-eval-000074", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -35 <= b, c <= 35 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -35 <= b, c <= 35 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, 27", "ref": "-12, 27", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000595, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 35, "D": 6, "cond": "pos"}, "seed": 177091180801000088, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 8, "valid_answers": ["-12, 27", "-10, 16", "-8, 7"]}, "cell": "quad_rootdiff/1", "stage": "B"}
+{"id": "po-quad_rootdiff-t1-eval-000044", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -38 <= b, c <= 38 such that x^2 + bx + c has two integer roots that differ by 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -38 <= b, c <= 38 such that x^2 + bx + c has two integer roots that differ by 3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-11, 28", "ref": "-11, 28", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.002024, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 38, "D": 3, "cond": null}, "seed": 177091180801000046, "version": "po-v0", "tier": 1, "n_raw": 12, "small": 2, "valid_answers": ["-11, 28", "-9, 18", "-7, 10", "-5, 4", "-3, 0", "-1, -2", "1, -2", "3, 0", "5, 4", "7, 10", "9, 18", "11, 28"]}, "cell": "quad_rootdiff/1", "stage": "B"}
+{"id": "po-quad_rootdiff-t1-eval-000113", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -28 <= b, c <= 28 such that x^2 + bx + c has two integer roots that differ by 4. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -28 <= b, c <= 28 such that x^2 + bx + c has two integer roots that differ by 4. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 5", "ref": "6, 5", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000923, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 28, "D": 4, "cond": "neg"}, "seed": 177091180801000139, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 6, "valid_answers": ["6, 5", "8, 12", "10, 21"]}, "cell": "quad_rootdiff/1", "stage": "B"}
+{"id": "po-quad_rootdiff-t1-eval-000015", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -24 <= b, c <= 24 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -24 <= b, c <= 24 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8, 7", "ref": "8, 7", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000833, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 24, "D": 6, "cond": "neg"}, "seed": 177091180801000016, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 8, "valid_answers": ["8, 7", "10, 16"]}, "cell": "quad_rootdiff/1", "stage": "B"}
+{"id": "po-quad_rootdiff-t1-eval-000111", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -30 <= b, c <= 30 such that x^2 + bx + c has two integer roots that differ by 5. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -30 <= b, c <= 30 such that x^2 + bx + c has two integer roots that differ by 5. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-11, 24", "ref": "-11, 24", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000806, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 30, "D": 5, "cond": "pos"}, "seed": 177091180801000137, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 7, "valid_answers": ["-11, 24", "-9, 14", "-7, 6"]}, "cell": "quad_rootdiff/1", "stage": "B"}
+{"id": "po-quad_rootdiff-t2-eval-000084", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -31 <= b, c <= 31 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -31 <= b, c <= 31 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, 27", "ref": "-12, 27", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000756, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 31, "D": 6, "cond": "pos"}, "seed": 177091180802000087, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 8, "valid_answers": ["-12, 27", "-10, 16", "-8, 7"]}, "cell": "quad_rootdiff/2", "stage": "B"}
+{"id": "po-quad_rootdiff-t2-eval-000114", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -40 <= b, c <= 40 such that x^2 + bx + c has two integer roots that differ by 5. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -40 <= b, c <= 40 such that x^2 + bx + c has two integer roots that differ by 5. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-13, 36", "ref": "-13, 36", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00061, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 40, "D": 5, "cond": "pos"}, "seed": 177091180802000124, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 7, "valid_answers": ["-13, 36", "-11, 24", "-9, 14", "-7, 6"]}, "cell": "quad_rootdiff/2", "stage": "B"}
+{"id": "po-quad_rootdiff-t2-eval-000036", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -60 <= b, c <= 60 such that x^2 + bx + c has two integer roots that differ by 7. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -60 <= b, c <= 60 such that x^2 + bx + c has two integer roots that differ by 7. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-17, 60", "ref": "-17, 60", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000342, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 60, "D": 7, "cond": "pos"}, "seed": 177091180802000036, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 9, "valid_answers": ["-17, 60", "-15, 44", "-13, 30", "-11, 18", "-9, 8"]}, "cell": "quad_rootdiff/2", "stage": "B"}
+{"id": "po-quad_rootdiff-t2-eval-000237", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -46 <= b, c <= 46 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -46 <= b, c <= 46 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-14, 40", "ref": "-14, 40", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000462, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 46, "D": 6, "cond": "pos"}, "seed": 177091180802000309, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 8, "valid_answers": ["-14, 40", "-12, 27", "-10, 16", "-8, 7"]}, "cell": "quad_rootdiff/2", "stage": "B"}
+{"id": "po-quad_rootdiff-t2-eval-000012", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -51 <= b, c <= 51 such that x^2 + bx + c has two integer roots that differ by 4. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -51 <= b, c <= 51 such that x^2 + bx + c has two integer roots that differ by 4. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 5", "ref": "6, 5", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000471, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 51, "D": 4, "cond": "neg"}, "seed": 177091180802000012, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 6, "valid_answers": ["6, 5", "8, 12", "10, 21", "12, 32", "14, 45"]}, "cell": "quad_rootdiff/2", "stage": "B"}
+{"id": "po-quad_rootdiff-t2-eval-000139", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -60 <= b, c <= 60 such that x^2 + bx + c has two integer roots that differ by 8. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -60 <= b, c <= 60 such that x^2 + bx + c has two integer roots that differ by 8. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-16, 48", "ref": "-16, 48", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000273, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 60, "D": 8, "cond": "pos"}, "seed": 177091180802000159, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 10, "valid_answers": ["-16, 48", "-14, 33", "-12, 20", "-10, 9"]}, "cell": "quad_rootdiff/2", "stage": "B"}
+{"id": "po-quad_rootdiff-t2-eval-000252", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -38 <= b, c <= 38 such that x^2 + bx + c has two integer roots that differ by 7. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -38 <= b, c <= 38 such that x^2 + bx + c has two integer roots that differ by 7. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9, 8", "ref": "9, 8", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000506, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 38, "D": 7, "cond": "neg"}, "seed": 177091180802000332, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 9, "valid_answers": ["9, 8", "11, 18", "13, 30"]}, "cell": "quad_rootdiff/2", "stage": "B"}
+{"id": "po-quad_rootdiff-t2-eval-000425", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -47 <= b, c <= 47 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -47 <= b, c <= 47 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-14, 40", "ref": "-14, 40", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000443, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 47, "D": 6, "cond": "pos"}, "seed": 177091180802000696, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 8, "valid_answers": ["-14, 40", "-12, 27", "-10, 16", "-8, 7"]}, "cell": "quad_rootdiff/2", "stage": "B"}
+{"id": "po-quad_rootdiff-t2-eval-000485", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -45 <= b, c <= 45 such that x^2 + bx + c has two integer roots that differ by 3. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -45 <= b, c <= 45 such that x^2 + bx + c has two integer roots that differ by 3. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-13, 40", "ref": "-13, 40", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000604, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 45, "D": 3, "cond": "pos"}, "seed": 177091180802000963, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 5, "valid_answers": ["-13, 40", "-11, 28", "-9, 18", "-7, 10", "-5, 4"]}, "cell": "quad_rootdiff/2", "stage": "B"}
+{"id": "po-quad_rootdiff-t2-eval-000454", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -30 <= b, c <= 30 such that x^2 + bx + c has two integer roots that differ by 5. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -30 <= b, c <= 30 such that x^2 + bx + c has two integer roots that differ by 5. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-11, 24", "ref": "-11, 24", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000806, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 30, "D": 5, "cond": "pos"}, "seed": 177091180802000825, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 7, "valid_answers": ["-11, 24", "-9, 14", "-7, 6"]}, "cell": "quad_rootdiff/2", "stage": "B"}
+{"id": "po-quad_rootdiff-t2-eval-000634", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -30 <= b, c <= 30 such that x^2 + bx + c has two integer roots that differ by 9. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -30 <= b, c <= 30 such that x^2 + bx + c has two integer roots that differ by 9. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 10", "ref": "11, 10", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000537, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 30, "D": 9, "cond": "neg"}, "seed": 177091180802002626, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 11, "valid_answers": ["11, 10", "13, 22"]}, "cell": "quad_rootdiff/2", "stage": "B"}
+{"id": "po-quad_rootdiff-t2-eval-000011", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -39 <= b, c <= 39 such that x^2 + bx + c has two integer roots that differ by 3. One root must be negative and the other positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -39 <= b, c <= 39 such that x^2 + bx + c has two integer roots that differ by 3. One root must be negative and the other positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -2", "ref": "-1, -2", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00032, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 39, "D": 3, "cond": "opp"}, "seed": 177091180802000011, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 2, "valid_answers": ["-1, -2", "1, -2"]}, "cell": "quad_rootdiff/2", "stage": "B"}
+{"id": "po-quad_rootdiff-t3-eval-000408", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -83 <= b, c <= 83 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -83 <= b, c <= 83 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8, 7", "ref": "8, 7", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000215, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 83, "D": 6, "cond": "neg"}, "seed": 177091180803000508, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 8, "valid_answers": ["8, 7", "10, 16", "12, 27", "14, 40", "16, 55", "18, 72"]}, "cell": "quad_rootdiff/3", "stage": "B"}
+{"id": "po-quad_rootdiff-t3-eval-000612", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -89 <= b, c <= 89 such that x^2 + bx + c has two integer roots that differ by 8. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -89 <= b, c <= 89 such that x^2 + bx + c has two integer roots that differ by 8. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 9", "ref": "10, 9", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000187, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 89, "D": 8, "cond": "neg"}, "seed": 177091180803000896, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 10, "valid_answers": ["10, 9", "12, 20", "14, 33", "16, 48", "18, 65", "20, 84"]}, "cell": "quad_rootdiff/3", "stage": "B"}
+{"id": "po-quad_rootdiff-t3-eval-001074", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -61 <= b, c <= 61 such that x^2 + bx + c has two integer roots that differ by 9. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -61 <= b, c <= 61 such that x^2 + bx + c has two integer roots that differ by 9. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-17, 52", "ref": "-17, 52", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000264, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 61, "D": 9, "cond": "pos"}, "seed": 177091180803003706, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 11, "valid_answers": ["-17, 52", "-15, 36", "-13, 22", "-11, 10"]}, "cell": "quad_rootdiff/3", "stage": "B"}
+{"id": "po-quad_rootdiff-t3-eval-000786", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -85 <= b, c <= 85 such that x^2 + bx + c has two integer roots that differ by 8. One root must be negative and the other positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -85 <= b, c <= 85 such that x^2 + bx + c has two integer roots that differ by 8. One root must be negative and the other positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -7", "ref": "-6, -7", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000239, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 85, "D": 8, "cond": "opp"}, "seed": 177091180803001366, "version": "po-v0", "tier": 3, "n_raw": 7, "small": 7, "valid_answers": ["-6, -7", "-4, -12", "-2, -15", "0, -16", "2, -15", "4, -12", "6, -7"]}, "cell": "quad_rootdiff/3", "stage": "B"}
+{"id": "po-quad_rootdiff-t3-eval-001022", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -77 <= b, c <= 77 such that x^2 + bx + c has two integer roots that differ by 12. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -77 <= b, c <= 77 such that x^2 + bx + c has two integer roots that differ by 12. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-20, 64", "ref": "-20, 64", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000166, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 77, "D": 12, "cond": "pos"}, "seed": 177091180803002904, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 14, "valid_answers": ["-20, 64", "-18, 45", "-16, 28", "-14, 13"]}, "cell": "quad_rootdiff/3", "stage": "B"}
+{"id": "po-quad_rootdiff-t3-eval-000952", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -62 <= b, c <= 62 such that x^2 + bx + c has two integer roots that differ by 6. One root must be negative and the other positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -62 <= b, c <= 62 such that x^2 + bx + c has two integer roots that differ by 6. One root must be negative and the other positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -5", "ref": "-4, -5", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.00032, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 62, "D": 6, "cond": "opp"}, "seed": 177091180803002263, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 5, "valid_answers": ["-4, -5", "-2, -8", "0, -9", "2, -8", "4, -5"]}, "cell": "quad_rootdiff/3", "stage": "B"}
+{"id": "po-quad_rootdiff-t3-eval-000648", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -79 <= b, c <= 79 such that x^2 + bx + c has two integer roots that differ by 4. One root must be negative and the other positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -79 <= b, c <= 79 such that x^2 + bx + c has two integer roots that differ by 4. One root must be negative and the other positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -3", "ref": "-2, -3", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000119, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 79, "D": 4, "cond": "opp"}, "seed": 177091180803000968, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 3, "valid_answers": ["-2, -3", "0, -4", "2, -3"]}, "cell": "quad_rootdiff/3", "stage": "B"}
+{"id": "po-quad_rootdiff-t3-eval-000630", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -77 <= b, c <= 77 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -77 <= b, c <= 77 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-18, 72", "ref": "-18, 72", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.00025, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 77, "D": 6, "cond": "pos"}, "seed": 177091180803000930, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 8, "valid_answers": ["-18, 72", "-16, 55", "-14, 40", "-12, 27", "-10, 16", "-8, 7"]}, "cell": "quad_rootdiff/3", "stage": "B"}
+{"id": "po-quad_rootdiff-t3-eval-000094", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -73 <= b, c <= 73 such that x^2 + bx + c has two integer roots that differ by 8. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -73 <= b, c <= 73 such that x^2 + bx + c has two integer roots that differ by 8. Both roots must be positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-18, 65", "ref": "-18, 65", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000231, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 73, "D": 8, "cond": "pos"}, "seed": 177091180803000098, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 10, "valid_answers": ["-18, 65", "-16, 48", "-14, 33", "-12, 20", "-10, 9"]}, "cell": "quad_rootdiff/3", "stage": "B"}
+{"id": "po-quad_rootdiff-t3-eval-000063", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -53 <= b, c <= 53 such that x^2 + bx + c has two integer roots that differ by 10. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -53 <= b, c <= 53 such that x^2 + bx + c has two integer roots that differ by 10. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "12, 11", "ref": "12, 11", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000262, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 53, "D": 10, "cond": "neg"}, "seed": 177091180803000066, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 12, "valid_answers": ["12, 11", "14, 24", "16, 39"]}, "cell": "quad_rootdiff/3", "stage": "B"}
+{"id": "po-quad_rootdiff-t3-eval-000755", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -69 <= b, c <= 69 such that x^2 + bx + c has two integer roots that differ by 12. One root must be negative and the other positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -69 <= b, c <= 69 such that x^2 + bx + c has two integer roots that differ by 12. One root must be negative and the other positive. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -11", "ref": "-10, -11", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.000569, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 69, "D": 12, "cond": "opp"}, "seed": 177091180803001274, "version": "po-v0", "tier": 3, "n_raw": 11, "small": 11, "valid_answers": ["-10, -11", "-8, -20", "-6, -27", "-4, -32", "-2, -35", "0, -36", "2, -35", "4, -32", "6, -27", "8, -20", "10, -11"]}, "cell": "quad_rootdiff/3", "stage": "B"}
+{"id": "po-quad_rootdiff-t3-eval-000495", "task": "cx_po_quad_rootdiff", "source": "cx_po_quad_rootdiff", "split": "eval", "problem": "Find integers b, c with -85 <= b, c <= 85 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers b, c with -85 <= b, c <= 85 such that x^2 + bx + c has two integer roots that differ by 6. Both roots must be negative. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8, 7", "ref": "8, 7", "family": "po_quad_rootdiff", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000205, "solvable": true, "holdout": false, "meta": {"family": "po_quad_rootdiff", "rung": "witness", "params": {"B": 85, "D": 6, "cond": "neg"}, "seed": 177091180803000652, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 8, "valid_answers": ["8, 7", "10, 16", "12, 27", "14, 40", "16, 55", "18, 72"]}, "cell": "quad_rootdiff/3", "stage": "B"}
+{"id": "po-recur_coef-t0-eval-000372", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 0, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 60. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 0, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 60. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -1", "ref": "-4, -1", "family": "po_recur_coef", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 0, "u1": 4, "m": 3, "B": 5, "T": 60}, "seed": 71449463300003575, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-4, -1", "4, -1"]}, "cell": "recur_coef/0", "stage": "A"}
+{"id": "po-recur_coef-t0-eval-000420", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -1, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -1, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 4", "ref": "-2, 4", "family": "po_recur_coef", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -1, "u1": 2, "m": 3, "B": 5, "T": 24}, "seed": 71449463300007698, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-2, 4", "4, 4"]}, "cell": "recur_coef/0", "stage": "A"}
+{"id": "po-recur_coef-t0-eval-000391", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 2, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 63. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 2, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 63. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -3", "ref": "-4, -3", "family": "po_recur_coef", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 2, "u1": 3, "m": 3, "B": 5, "T": 63}, "seed": 71449463300004162, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-4, -3", "3, 4"]}, "cell": "recur_coef/0", "stage": "A"}
+{"id": "po-recur_coef-t0-eval-000141", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -3, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -3, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 1", "ref": "-1, 1", "family": "po_recur_coef", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -3, "u1": 4, "m": 3, "B": 5, "T": 11}, "seed": 71449463300000629, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 1, "valid_answers": ["-1, 1", "3, 5"]}, "cell": "recur_coef/0", "stage": "A"}
+{"id": "po-recur_coef-t0-eval-000004", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 1, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 1, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -2", "ref": "-1, -2", "family": "po_recur_coef", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 1, "u1": 4, "m": 3, "B": 5, "T": -2}, "seed": 71449463300000013, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["-1, -2", "2, -3"]}, "cell": "recur_coef/0", "stage": "A"}
+{"id": "po-recur_coef-t0-eval-000298", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 2, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 2, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 4", "ref": "-1, 4", "family": "po_recur_coef", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 2, "u1": 4, "m": 3, "B": 5, "T": 12}, "seed": 71449463300001991, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["-1, 4", "0, 3"]}, "cell": "recur_coef/0", "stage": "A"}
+{"id": "po-recur_coef-t0-eval-000246", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 2, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 2, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 1", "ref": "-3, 1", "family": "po_recur_coef", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 2, "u1": 4, "m": 3, "B": 5, "T": 34}, "seed": 71449463300001345, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["-3, 1", "1, 5"]}, "cell": "recur_coef/0", "stage": "A"}
+{"id": "po-recur_coef-t0-eval-000076", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -1, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -1, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -5", "ref": "-3, -5", "family": "po_recur_coef", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -1, "u1": 4, "m": 3, "B": 5, "T": 1}, "seed": 71449463300000335, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 1, "valid_answers": ["-3, -5", "1, -1"]}, "cell": "recur_coef/0", "stage": "A"}
+{"id": "po-recur_coef-t0-eval-000061", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -2, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_3 = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -2, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_3 = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -3", "ref": "0, -3", "family": "po_recur_coef", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -2, "u1": 1, "m": 3, "B": 5, "T": -3}, "seed": 71449463300000281, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["0, -3", "1, 4"]}, "cell": "recur_coef/0", "stage": "A"}
+{"id": "po-recur_coef-t0-eval-000174", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 0, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 81. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 0, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 81. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 2", "ref": "-5, 2", "family": "po_recur_coef", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 0, "u1": 3, "m": 3, "B": 5, "T": 81}, "seed": 71449463300000795, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-5, 2", "5, 2"]}, "cell": "recur_coef/0", "stage": "A"}
+{"id": "po-recur_coef-t0-eval-000175", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 0, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 45. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 0, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 45. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -1", "ref": "-4, -1", "family": "po_recur_coef", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 0, "u1": 3, "m": 3, "B": 5, "T": 45}, "seed": 71449463300000800, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-4, -1", "4, -1"]}, "cell": "recur_coef/0", "stage": "A"}
+{"id": "po-recur_coef-t0-eval-000419", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 3, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 50. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 3, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 50. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 0", "ref": "-5, 0", "family": "po_recur_coef", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 3, "u1": 2, "m": 3, "B": 5, "T": 50}, "seed": 71449463300007356, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-5, 0", "5, 0"]}, "cell": "recur_coef/0", "stage": "A"}
+{"id": "po-recur_coef-t1-eval-001131", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = -1, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = -1, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -5", "ref": "-4, -5", "family": "po_recur_coef", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.00831, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -1, "u1": 3, "m": 3, "B": 9, "T": 13}, "seed": 71449463301002237, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 2, "valid_answers": ["-4, -5", "1, 5", "2, 1"]}, "cell": "recur_coef/1", "stage": "B"}
+{"id": "po-recur_coef-t1-eval-000079", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = -1, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 91. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = -1, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 91. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -1", "ref": "-5, -1", "family": "po_recur_coef", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00554, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -1, "u1": 4, "m": 3, "B": 9, "T": 91}, "seed": 71449463301000119, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 5, "valid_answers": ["-5, -1", "5, 9"]}, "cell": "recur_coef/1", "stage": "B"}
+{"id": "po-recur_coef-t1-eval-000363", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = -3, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 93. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = -3, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 93. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -5", "ref": "-9, -5", "family": "po_recur_coef", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.019391, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -3, "u1": 3, "m": 3, "B": 9, "T": 93}, "seed": 71449463301000563, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 4, "valid_answers": ["-9, -5", "-5, 1", "-4, 3", "-2, 9", "4, -5", "6, 1", "7, 3"]}, "cell": "recur_coef/1", "stage": "B"}
+{"id": "po-recur_coef-t1-eval-002248", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = 3, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 121. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = 3, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 121. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -5", "ref": "-6, -5", "family": "po_recur_coef", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00554, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 3, "u1": 1, "m": 3, "B": 9, "T": 121}, "seed": 71449463301013992, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 6, "valid_answers": ["-6, -5", "5, 6"]}, "cell": "recur_coef/1", "stage": "B"}
+{"id": "po-recur_coef-t1-eval-001232", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = 2, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 101. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = 2, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 101. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 7", "ref": "-8, 7", "family": "po_recur_coef", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00554, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 2, "u1": 3, "m": 3, "B": 9, "T": 101}, "seed": 71449463301002572, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 5, "valid_answers": ["-8, 7", "5, 2"]}, "cell": "recur_coef/1", "stage": "B"}
+{"id": "po-recur_coef-t1-eval-001519", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = 1, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = 1, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 8", "ref": "-3, 8", "family": "po_recur_coef", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00554, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 1, "u1": 4, "m": 3, "B": 9, "T": 44}, "seed": 71449463301003740, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 8, "valid_answers": ["-3, 8", "1, 8"]}, "cell": "recur_coef/1", "stage": "B"}
+{"id": "po-recur_coef-t1-eval-001956", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = 0, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 118. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = 0, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 118. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -5", "ref": "-8, -5", "family": "po_recur_coef", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00554, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 0, "u1": 2, "m": 3, "B": 9, "T": 118}, "seed": 71449463301006934, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 8, "valid_answers": ["-8, -5", "8, -5"]}, "cell": "recur_coef/1", "stage": "B"}
+{"id": "po-recur_coef-t1-eval-000660", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = -2, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = -2, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -2", "ref": "-6, -2", "family": "po_recur_coef", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.01108, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -2, "u1": 2, "m": 3, "B": 9, "T": 44}, "seed": 71449463301001079, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 4, "valid_answers": ["-6, -2", "-2, 6", "4, -2", "8, 6"]}, "cell": "recur_coef/1", "stage": "B"}
+{"id": "po-recur_coef-t1-eval-000088", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = -3, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_3 = -26. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = -3, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_3 = -26. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -4", "ref": "-3, -4", "family": "po_recur_coef", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00554, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -3, "u1": 2, "m": 3, "B": 9, "T": -26}, "seed": 71449463301000135, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["-3, -4", "8, 7"]}, "cell": "recur_coef/1", "stage": "B"}
+{"id": "po-recur_coef-t1-eval-000010", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = 0, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 54. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = 0, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 54. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -9", "ref": "-6, -9", "family": "po_recur_coef", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.01108, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 0, "u1": 2, "m": 3, "B": 9, "T": 54}, "seed": 71449463301000013, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 5, "valid_answers": ["-6, -9", "-5, 2", "5, 2", "6, -9"]}, "cell": "recur_coef/1", "stage": "B"}
+{"id": "po-recur_coef-t1-eval-000895", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = 0, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_3 = -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = 0, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_3 = -5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -9", "ref": "-2, -9", "family": "po_recur_coef", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.01385, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 0, "u1": 1, "m": 3, "B": 9, "T": -5}, "seed": 71449463301001604, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 5, "valid_answers": ["-2, -9", "-1, -6", "0, -5", "1, -6", "2, -9"]}, "cell": "recur_coef/1", "stage": "B"}
+{"id": "po-recur_coef-t1-eval-000770", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = -3, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = -69. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -9 <= p, q <= 9 such that the sequence with u_0 = -3, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = -69. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -9", "ref": "-7, -9", "family": "po_recur_coef", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.01108, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -3, "u1": 3, "m": 3, "B": 9, "T": -69}, "seed": 71449463301001304, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 8, "valid_answers": ["-7, -9", "-5, -8", "-3, -8", "-2, -9"]}, "cell": "recur_coef/1", "stage": "B"}
+{"id": "po-recur_coef-t2-eval-000205", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 3, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 3, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -4", "ref": "-2, -4", "family": "po_recur_coef", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 3, "u1": 1, "m": 4, "B": 6, "T": 8}, "seed": 71449463302000632, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 2, "valid_answers": ["-2, -4", "2, 0"]}, "cell": "recur_coef/2", "stage": "C"}
+{"id": "po-recur_coef-t2-eval-000733", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 1, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 1, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -4", "ref": "-2, -4", "family": "po_recur_coef", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 1, "u1": 4, "m": 4, "B": 6, "T": 32}, "seed": 71449463302004302, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 2, "valid_answers": ["-2, -4", "2, 0"]}, "cell": "recur_coef/2", "stage": "C"}
+{"id": "po-recur_coef-t2-eval-000191", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = -3, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_4 = -32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = -3, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_4 = -32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 0", "ref": "-2, 0", "family": "po_recur_coef", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -3, "u1": 4, "m": 4, "B": 6, "T": -32}, "seed": 71449463302000589, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 2, "valid_answers": ["-2, 0", "2, -4"]}, "cell": "recur_coef/2", "stage": "C"}
+{"id": "po-recur_coef-t2-eval-000207", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = -1, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = -1, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -3", "ref": "-3, -3", "family": "po_recur_coef", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017751, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -1, "u1": 1, "m": 4, "B": 6, "T": 9}, "seed": 71449463302000642, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 3, "valid_answers": ["-3, -3", "3, -6", "3, 3"]}, "cell": "recur_coef/2", "stage": "C"}
+{"id": "po-recur_coef-t2-eval-000219", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = -2, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 100. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = -2, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 100. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -1", "ref": "3, -1", "family": "po_recur_coef", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -2, "u1": 4, "m": 4, "B": 6, "T": 100}, "seed": 71449463302000680, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 3, "valid_answers": ["3, -1", "3, 4"]}, "cell": "recur_coef/2", "stage": "C"}
+{"id": "po-recur_coef-t2-eval-000858", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 3, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 3, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -4", "ref": "0, -4", "family": "po_recur_coef", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 3, "u1": 4, "m": 4, "B": 6, "T": 48}, "seed": 71449463302011801, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 4, "valid_answers": ["0, -4", "0, 4"]}, "cell": "recur_coef/2", "stage": "C"}
+{"id": "po-recur_coef-t2-eval-000317", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 0, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_4 = -24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 0, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_4 = -24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 1", "ref": "-2, 1", "family": "po_recur_coef", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 0, "u1": 2, "m": 4, "B": 6, "T": -24}, "seed": 71449463302001040, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 2, "valid_answers": ["-2, 1", "2, -5"]}, "cell": "recur_coef/2", "stage": "C"}
+{"id": "po-recur_coef-t2-eval-000842", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 0, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 0, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -6", "ref": "-2, -6", "family": "po_recur_coef", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017751, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 0, "u1": 3, "m": 4, "B": 6, "T": 48}, "seed": 71449463302009150, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 2, "valid_answers": ["-2, -6", "2, 2", "4, -6"]}, "cell": "recur_coef/2", "stage": "C"}
+{"id": "po-recur_coef-t2-eval-000470", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 3, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 38. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 3, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 38. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -3", "ref": "-1, -3", "family": "po_recur_coef", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 3, "u1": 4, "m": 4, "B": 6, "T": 38}, "seed": 71449463302001748, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 2, "valid_answers": ["-1, -3", "1, 2"]}, "cell": "recur_coef/2", "stage": "C"}
+{"id": "po-recur_coef-t2-eval-000040", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 1, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 556. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 1, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 556. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 6", "ref": "5, 6", "family": "po_recur_coef", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 1, "u1": 2, "m": 4, "B": 6, "T": 556}, "seed": 71449463302000135, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 6, "valid_answers": ["5, 6", "6, 2"]}, "cell": "recur_coef/2", "stage": "C"}
+{"id": "po-recur_coef-t2-eval-000653", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 1, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = 1, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_4 = 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -2", "ref": "-1, -2", "family": "po_recur_coef", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 1, "u1": 4, "m": 4, "B": 6, "T": 14}, "seed": 71449463302003167, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 1, "valid_answers": ["-1, -2", "1, 1"]}, "cell": "recur_coef/2", "stage": "C"}
+{"id": "po-recur_coef-t2-eval-000845", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = -2, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_4 = -46. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -6 <= p, q <= 6 such that the sequence with u_0 = -2, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_4 = -46. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 1", "ref": "-2, 1", "family": "po_recur_coef", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -2, "u1": 3, "m": 4, "B": 6, "T": -46}, "seed": 71449463302009363, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 2, "valid_answers": ["-2, 1", "2, -5"]}, "cell": "recur_coef/2", "stage": "C"}
+{"id": "po-recur_coef-t3-eval-000404", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -2, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -2, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -2", "ref": "-1, -2", "family": "po_recur_coef", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -2, "u1": 1, "m": 5, "B": 5, "T": 11}, "seed": 71449463303002816, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 1, "valid_answers": ["-1, -2", "-1, 1"]}, "cell": "recur_coef/3", "stage": "C"}
+{"id": "po-recur_coef-t3-eval-000236", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 0, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 1364. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 0, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 1364. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -4", "ref": "-5, -4", "family": "po_recur_coef", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 0, "u1": 4, "m": 5, "B": 5, "T": 1364}, "seed": 71449463303001062, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 5, "valid_answers": ["-5, -4", "5, -4"]}, "cell": "recur_coef/3", "stage": "C"}
+{"id": "po-recur_coef-t3-eval-000018", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 1, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 512. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 1, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 512. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 0", "ref": "-4, 0", "family": "po_recur_coef", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 1, "u1": 2, "m": 5, "B": 5, "T": 512}, "seed": 71449463303000048, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 4, "valid_answers": ["-4, 0", "4, 0"]}, "cell": "recur_coef/3", "stage": "C"}
+{"id": "po-recur_coef-t3-eval-000119", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -1, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 81. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -1, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 81. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 0", "ref": "-3, 0", "family": "po_recur_coef", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -1, "u1": 1, "m": 5, "B": 5, "T": 81}, "seed": 71449463303000484, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 3, "valid_answers": ["-3, 0", "3, 0"]}, "cell": "recur_coef/3", "stage": "C"}
+{"id": "po-recur_coef-t3-eval-000126", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 2, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 1875. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 2, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 1875. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 0", "ref": "-5, 0", "family": "po_recur_coef", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 2, "u1": 3, "m": 5, "B": 5, "T": 1875}, "seed": 71449463303000510, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 5, "valid_answers": ["-5, 0", "5, 0"]}, "cell": "recur_coef/3", "stage": "C"}
+{"id": "po-recur_coef-t3-eval-000394", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 0, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_5 = -19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 0, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_5 = -19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -5", "ref": "-2, -5", "family": "po_recur_coef", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 0, "u1": 1, "m": 5, "B": 5, "T": -19}, "seed": 71449463303002744, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 5, "valid_answers": ["-2, -5", "2, -5"]}, "cell": "recur_coef/3", "stage": "C"}
+{"id": "po-recur_coef-t3-eval-000159", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -3, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_5 = -218. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -3, u_1 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_5 = -218. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -5", "ref": "2, -5", "family": "po_recur_coef", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -3, "u1": 2, "m": 5, "B": 5, "T": -218}, "seed": 71449463303000660, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 5, "valid_answers": ["2, -5", "2, 5"]}, "cell": "recur_coef/3", "stage": "C"}
+{"id": "po-recur_coef-t3-eval-000208", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 0, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 363. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 0, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 363. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -3", "ref": "-4, -3", "family": "po_recur_coef", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 0, "u1": 3, "m": 5, "B": 5, "T": 363}, "seed": 71449463303000890, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 4, "valid_answers": ["-4, -3", "4, -3"]}, "cell": "recur_coef/3", "stage": "C"}
+{"id": "po-recur_coef-t3-eval-000096", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 3, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 81. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 3, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 81. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 0", "ref": "-3, 0", "family": "po_recur_coef", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 3, "u1": 1, "m": 5, "B": 5, "T": 81}, "seed": 71449463303000368, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 3, "valid_answers": ["-3, 0", "3, 0"]}, "cell": "recur_coef/3", "stage": "C"}
+{"id": "po-recur_coef-t3-eval-000215", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 2, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_5 = -27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 2, u_1 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_5 = -27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -3", "ref": "-1, -3", "family": "po_recur_coef", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 2, "u1": 3, "m": 5, "B": 5, "T": -27}, "seed": 71449463303000925, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 3, "valid_answers": ["-1, -3", "3, -3"]}, "cell": "recur_coef/3", "stage": "C"}
+{"id": "po-recur_coef-t3-eval-000094", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 0, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 164. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = 0, u_1 = 1 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 164. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -2", "ref": "-4, -2", "family": "po_recur_coef", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": 0, "u1": 1, "m": 5, "B": 5, "T": 164}, "seed": 71449463303000354, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 4, "valid_answers": ["-4, -2", "4, -2"]}, "cell": "recur_coef/3", "stage": "C"}
+{"id": "po-recur_coef-t3-eval-000251", "task": "cx_po_recur_coef", "source": "cx_po_recur_coef", "split": "eval", "problem": "Find integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -3, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 1024. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers p, q with -5 <= p, q <= 5 such that the sequence with u_0 = -3, u_1 = 4 and u_{n+2} = p u_{n+1} + q u_n has u_5 = 1024. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 0", "ref": "-4, 0", "family": "po_recur_coef", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016529, "solvable": true, "holdout": false, "meta": {"family": "po_recur_coef", "rung": "witness", "params": {"u0": -3, "u1": 4, "m": 5, "B": 5, "T": 1024}, "seed": 71449463303001201, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 4, "valid_answers": ["-4, 0", "4, 0"]}, "cell": "recur_coef/3", "stage": "C"}
+{"id": "po-recur_init-t0-eval-001168", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + u_n has u_3 = 60. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + u_n has u_3 = 60. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 10", "ref": "5, 10", "family": "po_recur_init", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004535, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 2, "q": 1, "m": 3, "B": 10, "T": 60}, "seed": 46805164800005743, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 10, "valid_answers": ["5, 10", "10, 8"]}, "cell": "recur_init/0", "stage": "A"}
+{"id": "po-recur_init-t0-eval-000821", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 3u_{n+1} - u_n has u_3 = 39. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 3u_{n+1} - u_n has u_3 = 39. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 3", "ref": "-5, 3", "family": "po_recur_init", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004535, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 3, "q": -1, "m": 3, "B": 10, "T": 39}, "seed": 46805164800002106, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-5, 3", "3, 6"]}, "cell": "recur_init/0", "stage": "A"}
+{"id": "po-recur_init-t0-eval-000281", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + u_n has u_3 = 27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + u_n has u_3 = 27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 9", "ref": "-9, 9", "family": "po_recur_init", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00907, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 2, "q": 1, "m": 3, "B": 10, "T": 27}, "seed": 46805164800000436, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 5, "valid_answers": ["-9, 9", "-4, 7", "1, 5", "6, 3"]}, "cell": "recur_init/0", "stage": "A"}
+{"id": "po-recur_init-t0-eval-000904", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} - u_n has u_3 = 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} - u_n has u_3 = 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 8", "ref": "-10, 8", "family": "po_recur_init", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004535, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 2, "q": -1, "m": 3, "B": 10, "T": 44}, "seed": 46805164800002541, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 10, "valid_answers": ["-10, 8", "-7, 10"]}, "cell": "recur_init/0", "stage": "A"}
+{"id": "po-recur_init-t0-eval-000026", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} - u_n has u_3 = -19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} - u_n has u_3 = -19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -1", "ref": "-8, -1", "family": "po_recur_init", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.011338, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": -1, "m": 3, "B": 10, "T": -19}, "seed": 46805164800000034, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 5, "valid_answers": ["-8, -1", "-5, -3", "-2, -5", "1, -7", "4, -9"]}, "cell": "recur_init/0", "stage": "A"}
+{"id": "po-recur_init-t0-eval-000346", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + u_n has u_3 = -50. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + u_n has u_3 = -50. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -6", "ref": "-10, -6", "family": "po_recur_init", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006803, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 2, "q": 1, "m": 3, "B": 10, "T": -50}, "seed": 46805164800000573, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 8, "valid_answers": ["-10, -6", "-5, -8", "0, -10"]}, "cell": "recur_init/0", "stage": "A"}
+{"id": "po-recur_init-t0-eval-000013", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -u_{n+1} + u_n has u_3 = -17. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -u_{n+1} + u_n has u_3 = -17. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -10", "ref": "-3, -10", "family": "po_recur_init", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.015873, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -1, "q": 1, "m": 3, "B": 10, "T": -17}, "seed": 46805164800000019, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 6, "valid_answers": ["-3, -10", "-1, -9", "1, -8", "3, -7", "5, -6", "7, -5", "9, -4"]}, "cell": "recur_init/0", "stage": "A"}
+{"id": "po-recur_init-t0-eval-000914", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 3u_{n+1} - u_n has u_3 = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 3u_{n+1} - u_n has u_3 = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -3", "ref": "-6, -3", "family": "po_recur_init", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006803, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 3, "q": -1, "m": 3, "B": 10, "T": -6}, "seed": 46805164800002609, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["-6, -3", "2, 0", "10, 3"]}, "cell": "recur_init/0", "stage": "A"}
+{"id": "po-recur_init-t0-eval-000373", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} - u_n has u_3 = -30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} - u_n has u_3 = -30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -10", "ref": "0, -10", "family": "po_recur_init", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00907, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 2, "q": -1, "m": 3, "B": 10, "T": -30}, "seed": 46805164800000638, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 6, "valid_answers": ["0, -10", "3, -8", "6, -6", "9, -4"]}, "cell": "recur_init/0", "stage": "A"}
+{"id": "po-recur_init-t0-eval-000313", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 3u_{n+1} - u_n has u_3 = 34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 3u_{n+1} - u_n has u_3 = 34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 2", "ref": "-6, 2", "family": "po_recur_init", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006803, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 3, "q": -1, "m": 3, "B": 10, "T": 34}, "seed": 46805164800000502, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-6, 2", "2, 5", "10, 8"]}, "cell": "recur_init/0", "stage": "A"}
+{"id": "po-recur_init-t0-eval-000042", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -u_{n+1} + 2u_n has u_3 = 23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -u_{n+1} + 2u_n has u_3 = 23. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 1", "ref": "-10, 1", "family": "po_recur_init", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.011338, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -1, "q": 2, "m": 3, "B": 10, "T": 23}, "seed": 46805164800000059, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 5, "valid_answers": ["-10, 1", "-7, 3", "-4, 5", "-1, 7", "2, 9"]}, "cell": "recur_init/0", "stage": "A"}
+{"id": "po-recur_init-t0-eval-000055", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_3 = 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -10 <= x, y <= 10 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_3 = 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -3", "ref": "-9, -3", "family": "po_recur_init", "domain": "sequence", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.015873, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": 2, "m": 3, "B": 10, "T": 18}, "seed": 46805164800000078, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 3, "valid_answers": ["-9, -3", "-6, -1", "-3, 1", "0, 3", "3, 5", "6, 7", "9, 9"]}, "cell": "recur_init/0", "stage": "A"}
+{"id": "po-recur_init-t1-eval-002137", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + u_n has u_4 = -171. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + u_n has u_4 = -171. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-15, 8", "ref": "-15, 8", "family": "po_recur_init", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002081, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": 1, "m": 4, "B": 15, "T": -171}, "seed": 46805164801010377, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 13, "valid_answers": ["-15, 8", "-3, 13"]}, "cell": "recur_init/1", "stage": "B"}
+{"id": "po-recur_init-t1-eval-000835", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -u_{n+1} + u_n has u_4 = -49. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -u_{n+1} + u_n has u_4 = -49. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-14, 7", "ref": "-14, 7", "family": "po_recur_init", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.005203, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -1, "q": 1, "m": 4, "B": 15, "T": -49}, "seed": 46805164801001579, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 11, "valid_answers": ["-14, 7", "-11, 9", "-8, 11", "-5, 13", "-2, 15"]}, "cell": "recur_init/1", "stage": "B"}
+{"id": "po-recur_init-t1-eval-001317", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -u_{n+1} + 2u_n has u_4 = 17. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -u_{n+1} + 2u_n has u_4 = 17. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -13", "ref": "-8, -13", "family": "po_recur_init", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.005203, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -1, "q": 2, "m": 4, "B": 15, "T": 17}, "seed": 46805164801003158, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 2, "valid_answers": ["-8, -13", "-3, -7", "2, -1", "7, 5", "12, 11"]}, "cell": "recur_init/1", "stage": "B"}
+{"id": "po-recur_init-t1-eval-000005", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} - u_n has u_4 = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} - u_n has u_4 = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-15, 9", "ref": "-15, 9", "family": "po_recur_init", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.008325, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": -1, "m": 4, "B": 15, "T": 9}, "seed": 46805164801000008, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 3, "valid_answers": ["-15, 9", "-11, 6", "-7, 3", "-3, 0", "1, -3", "5, -6", "9, -9", "13, -12"]}, "cell": "recur_init/1", "stage": "B"}
+{"id": "po-recur_init-t1-eval-000099", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_4 = 128. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_4 = 128. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -14", "ref": "-8, -14", "family": "po_recur_init", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.006243, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": 2, "m": 4, "B": 15, "T": 128}, "seed": 46805164801000148, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 5, "valid_answers": ["-8, -14", "-4, -11", "0, -8", "4, -5", "8, -2", "12, 1"]}, "cell": "recur_init/1", "stage": "B"}
+{"id": "po-recur_init-t1-eval-000120", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} - u_n has u_4 = -54. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} - u_n has u_4 = -54. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 15", "ref": "-2, 15", "family": "po_recur_init", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.005203, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": -1, "m": 4, "B": 15, "T": -54}, "seed": 46805164801000181, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 9, "valid_answers": ["-2, 15", "2, 12", "6, 9", "10, 6", "14, 3"]}, "cell": "recur_init/1", "stage": "B"}
+{"id": "po-recur_init-t1-eval-002015", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} - u_n has u_4 = -32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} - u_n has u_4 = -32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -14", "ref": "-8, -14", "family": "po_recur_init", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.006243, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 2, "q": -1, "m": 4, "B": 15, "T": -32}, "seed": 46805164801008392, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 5, "valid_answers": ["-8, -14", "-4, -11", "0, -8", "4, -5", "8, -2", "12, 1"]}, "cell": "recur_init/1", "stage": "B"}
+{"id": "po-recur_init-t1-eval-002386", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + u_n has u_4 = 139. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + u_n has u_4 = 139. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 12", "ref": "-1, 12", "family": "po_recur_init", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002081, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 2, "q": 1, "m": 4, "B": 15, "T": 139}, "seed": 46805164801024834, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 11, "valid_answers": ["-1, 12", "11, 7"]}, "cell": "recur_init/1", "stage": "B"}
+{"id": "po-recur_init-t1-eval-000333", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_4 = 276. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_4 = 276. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, -15", "ref": "3, -15", "family": "po_recur_init", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.004162, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": 2, "m": 4, "B": 15, "T": 276}, "seed": 46805164801000527, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 11, "valid_answers": ["3, -15", "7, -12", "11, -9", "15, -6"]}, "cell": "recur_init/1", "stage": "B"}
+{"id": "po-recur_init-t1-eval-001482", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + u_n has u_4 = 142. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + u_n has u_4 = 142. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -11", "ref": "2, -11", "family": "po_recur_init", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002081, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": 1, "m": 4, "B": 15, "T": 142}, "seed": 46805164801003915, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 11, "valid_answers": ["2, -11", "14, -6"]}, "cell": "recur_init/1", "stage": "B"}
+{"id": "po-recur_init-t1-eval-001582", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = u_{n+1} + u_n has u_4 = 62. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = u_{n+1} + u_n has u_4 = 62. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 14", "ref": "10, 14", "family": "po_recur_init", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002081, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 1, "q": 1, "m": 4, "B": 15, "T": 62}, "seed": 46805164801004511, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 13, "valid_answers": ["10, 14", "13, 12"]}, "cell": "recur_init/1", "stage": "B"}
+{"id": "po-recur_init-t1-eval-000411", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + u_n has u_4 = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -15 <= x, y <= 15 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + u_n has u_4 = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-13, 6", "ref": "-13, 6", "family": "po_recur_init", "domain": "sequence", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.003122, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 2, "q": 1, "m": 4, "B": 15, "T": 7}, "seed": 46805164801000660, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 1, "valid_answers": ["-13, 6", "-1, 1", "11, -4"]}, "cell": "recur_init/1", "stage": "B"}
+{"id": "po-recur_init-t2-eval-002140", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} - u_n has u_5 = 66. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} - u_n has u_5 = 66. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 18", "ref": "-6, 18", "family": "po_recur_init", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.003569, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": -1, "m": 5, "B": 20, "T": 66}, "seed": 46805164802006196, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 9, "valid_answers": ["-6, 18", "-1, 14", "4, 10", "9, 6", "14, 2", "19, -2"]}, "cell": "recur_init/2", "stage": "B"}
+{"id": "po-recur_init-t2-eval-000403", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = u_{n+1} + 2u_n has u_5 = -36. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = u_{n+1} + 2u_n has u_5 = -36. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-19, 14", "ref": "-19, 14", "family": "po_recur_init", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00238, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 1, "q": 2, "m": 5, "B": 20, "T": -36}, "seed": 46805164802000734, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 6, "valid_answers": ["-19, 14", "-8, 4", "3, -6", "14, -16"]}, "cell": "recur_init/2", "stage": "B"}
+{"id": "po-recur_init-t2-eval-001812", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + u_n has u_5 = 349. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + u_n has u_5 = 349. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-17, 5", "ref": "-17, 5", "family": "po_recur_init", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00119, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": 1, "m": 5, "B": 20, "T": 349}, "seed": 46805164802004607, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 17, "valid_answers": ["-17, 5", "12, 17"]}, "cell": "recur_init/2", "stage": "B"}
+{"id": "po-recur_init-t2-eval-002145", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = u_{n+1} + u_n has u_5 = 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = u_{n+1} + u_n has u_5 = 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-20, 15", "ref": "-20, 15", "family": "po_recur_init", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.005354, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 1, "q": 1, "m": 5, "B": 20, "T": 15}, "seed": 46805164802006220, "version": "po-v0", "tier": 2, "n_raw": 9, "small": 3, "valid_answers": ["-20, 15", "-15, 12", "-10, 9", "-5, 6", "0, 3", "5, 0", "10, -3", "15, -6", "20, -9"]}, "cell": "recur_init/2", "stage": "B"}
+{"id": "po-recur_init-t2-eval-001030", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + 2u_n has u_5 = -748. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + 2u_n has u_5 = -748. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-11, -9", "ref": "-11, -9", "family": "po_recur_init", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00119, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 2, "q": 2, "m": 5, "B": 20, "T": -748}, "seed": 46805164802002096, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 11, "valid_answers": ["-11, -9", "0, -17"]}, "cell": "recur_init/2", "stage": "B"}
+{"id": "po-recur_init-t2-eval-002148", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -u_{n+1} + 2u_n has u_5 = 199. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -u_{n+1} + 2u_n has u_5 = 199. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 9", "ref": "-10, 9", "family": "po_recur_init", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00119, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -1, "q": 2, "m": 5, "B": 20, "T": 199}, "seed": 46805164802006234, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 10, "valid_answers": ["-10, 9", "1, 19"]}, "cell": "recur_init/2", "stage": "B"}
+{"id": "po-recur_init-t2-eval-000069", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_5 = 736. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_5 = 736. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, 8", "ref": "-12, 8", "family": "po_recur_init", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00119, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": 2, "m": 5, "B": 20, "T": 736}, "seed": 46805164802000119, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 12, "valid_answers": ["-12, 8", "-1, 16"]}, "cell": "recur_init/2", "stage": "B"}
+{"id": "po-recur_init-t2-eval-001117", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + u_n has u_5 = 265. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + u_n has u_5 = 265. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-19, 17", "ref": "-19, 17", "family": "po_recur_init", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00119, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 2, "q": 1, "m": 5, "B": 20, "T": 265}, "seed": 46805164802002314, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 10, "valid_answers": ["-19, 17", "10, 5"]}, "cell": "recur_init/2", "stage": "B"}
+{"id": "po-recur_init-t2-eval-001166", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} - u_n has u_5 = 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} - u_n has u_5 = 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-14, 20", "ref": "-14, 20", "family": "po_recur_init", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.004164, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": -1, "m": 5, "B": 20, "T": 44}, "seed": 46805164802002433, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 6, "valid_answers": ["-14, 20", "-9, 16", "-4, 12", "1, 8", "6, 4", "11, 0", "16, -4"]}, "cell": "recur_init/2", "stage": "B"}
+{"id": "po-recur_init-t2-eval-001550", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} - u_n has u_5 = 74. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} - u_n has u_5 = 74. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 18", "ref": "-4, 18", "family": "po_recur_init", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002974, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": -1, "m": 5, "B": 20, "T": 74}, "seed": 46805164802003631, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 10, "valid_answers": ["-4, 18", "1, 14", "6, 10", "11, 6", "16, 2"]}, "cell": "recur_init/2", "stage": "B"}
+{"id": "po-recur_init-t2-eval-002489", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = u_{n+1} + 2u_n has u_5 = -40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = u_{n+1} + 2u_n has u_5 = -40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-15, 10", "ref": "-15, 10", "family": "po_recur_init", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00238, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 1, "q": 2, "m": 5, "B": 20, "T": -40}, "seed": 46805164802008184, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 4, "valid_answers": ["-15, 10", "-4, 0", "7, -10", "18, -20"]}, "cell": "recur_init/2", "stage": "B"}
+{"id": "po-recur_init-t2-eval-001847", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_5 = 756. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -20 <= x, y <= 20 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_5 = 756. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-14, 7", "ref": "-14, 7", "family": "po_recur_init", "domain": "sequence", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00119, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": 2, "m": 5, "B": 20, "T": 756}, "seed": 46805164802004796, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 14, "valid_answers": ["-14, 7", "-3, 15"]}, "cell": "recur_init/2", "stage": "B"}
+{"id": "po-recur_init-t3-eval-001598", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_6 = -1264. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_6 = -1264. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-28, -10", "ref": "-28, -10", "family": "po_recur_init", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001075, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": 2, "m": 6, "B": 30, "T": -1264}, "seed": 46805164803003948, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 12, "valid_answers": ["-28, -10", "-13, 1", "2, 12", "17, 23"]}, "cell": "recur_init/3", "stage": "B"}
+{"id": "po-recur_init-t3-eval-001198", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} - u_n has u_6 = -163. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} - u_n has u_6 = -163. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 28", "ref": "-1, 28", "family": "po_recur_init", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.001612, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": -1, "m": 6, "B": 30, "T": -163}, "seed": 46805164803002729, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 17, "valid_answers": ["-1, 28", "5, 23", "11, 18", "17, 13", "23, 8", "29, 3"]}, "cell": "recur_init/3", "stage": "B"}
+{"id": "po-recur_init-t3-eval-000657", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_6 = 2536. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_6 = 2536. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, -27", "ref": "-8, -27", "family": "po_recur_init", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000806, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": 2, "m": 6, "B": 30, "T": 2536}, "seed": 46805164803001343, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 16, "valid_answers": ["-8, -27", "7, -16", "22, -5"]}, "cell": "recur_init/3", "stage": "B"}
+{"id": "po-recur_init-t3-eval-001729", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_6 = 1824. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_6 = 1824. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, -24", "ref": "-12, -24", "family": "po_recur_init", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000806, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": 2, "m": 6, "B": 30, "T": 1824}, "seed": 46805164803004375, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 13, "valid_answers": ["-12, -24", "3, -13", "18, -2"]}, "cell": "recur_init/3", "stage": "B"}
+{"id": "po-recur_init-t3-eval-001084", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = u_{n+1} + u_n has u_6 = -350. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = u_{n+1} + u_n has u_6 = -350. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-30, -25", "ref": "-30, -25", "family": "po_recur_init", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000537, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 1, "q": 1, "m": 6, "B": 30, "T": -350}, "seed": 46805164803002409, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 30, "valid_answers": ["-30, -25", "-22, -30"]}, "cell": "recur_init/3", "stage": "B"}
+{"id": "po-recur_init-t3-eval-001871", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + 2u_n has u_6 = 1480. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + 2u_n has u_6 = 1480. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-20, 27", "ref": "-20, 27", "family": "po_recur_init", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001075, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 2, "q": 2, "m": 6, "B": 30, "T": 1480}, "seed": 46805164803004878, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 10, "valid_answers": ["-20, 27", "-5, 16", "10, 5", "25, -6"]}, "cell": "recur_init/3", "stage": "B"}
+{"id": "po-recur_init-t3-eval-000354", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -u_{n+1} + 2u_n has u_6 = -148. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -u_{n+1} + 2u_n has u_6 = -148. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-22, -16", "ref": "-22, -16", "family": "po_recur_init", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000806, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -1, "q": 2, "m": 6, "B": 30, "T": -148}, "seed": 46805164803000673, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 6, "valid_answers": ["-22, -16", "-1, 6", "20, 28"]}, "cell": "recur_init/3", "stage": "B"}
+{"id": "po-recur_init-t3-eval-000630", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -u_{n+1} + u_n has u_6 = 61. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -u_{n+1} + u_n has u_6 = 61. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-23, -22", "ref": "-23, -22", "family": "po_recur_init", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.001881, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -1, "q": 1, "m": 6, "B": 30, "T": 61}, "seed": 46805164803001291, "version": "po-v0", "tier": 3, "n_raw": 7, "small": 7, "valid_answers": ["-23, -22", "-15, -17", "-7, -12", "1, -7", "9, -2", "17, 3", "25, 8"]}, "cell": "recur_init/3", "stage": "B"}
+{"id": "po-recur_init-t3-eval-001928", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + 2u_n has u_6 = -4128. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + 2u_n has u_6 = -4128. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-21, -19", "ref": "-21, -19", "family": "po_recur_init", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000537, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 2, "q": 2, "m": 6, "B": 30, "T": -4128}, "seed": 46805164803005086, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 21, "valid_answers": ["-21, -19", "-6, -30"]}, "cell": "recur_init/3", "stage": "B"}
+{"id": "po-recur_init-t3-eval-001674", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_6 = -3144. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = -2u_{n+1} + 2u_n has u_6 = -3144. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-18, 13", "ref": "-18, 13", "family": "po_recur_init", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000537, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": -2, "q": 2, "m": 6, "B": 30, "T": -3144}, "seed": 46805164803004187, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 18, "valid_answers": ["-18, 13", "-3, 24"]}, "cell": "recur_init/3", "stage": "B"}
+{"id": "po-recur_init-t3-eval-000459", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = u_{n+1} + u_n has u_6 = -136. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = u_{n+1} + u_n has u_6 = -136. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-24, -2", "ref": "-24, -2", "family": "po_recur_init", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.001612, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 1, "q": 1, "m": 6, "B": 30, "T": -136}, "seed": 46805164803000901, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 12, "valid_answers": ["-24, -2", "-16, -7", "-8, -12", "0, -17", "8, -22", "16, -27"]}, "cell": "recur_init/3", "stage": "B"}
+{"id": "po-recur_init-t3-eval-002108", "task": "cx_po_recur_init", "source": "cx_po_recur_init", "split": "eval", "problem": "Find integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + 2u_n has u_6 = 2256. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -30 <= x, y <= 30 such that the sequence with u_0 = x, u_1 = y and u_{n+2} = 2u_{n+1} + 2u_n has u_6 = 2256. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 21", "ref": "-3, 21", "family": "po_recur_init", "domain": "sequence", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000806, "solvable": true, "holdout": false, "meta": {"family": "po_recur_init", "rung": "witness", "params": {"p": 2, "q": 2, "m": 6, "B": 30, "T": 2256}, "seed": 46805164803005649, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 12, "valid_answers": ["-3, 21", "12, 10", "27, -1"]}, "cell": "recur_init/3", "stage": "B"}
+{"id": "po-sfft-t0-eval-001104", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 3x - 6y = -13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 3x - 6y = -13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 2", "ref": "1, 2", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001875, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -3, "b": -6, "c": -13}, "seed": 167733942900003239, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["1, 2", "7, 8", "11, 4"]}, "cell": "sfft/0", "stage": "A"}
+{"id": "po-sfft-t0-eval-000572", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + x + 3y = 21. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + x + 3y = 21. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 5", "ref": "1, 5", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005102, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 1, "b": 3, "c": 21}, "seed": 167733942900001243, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["1, 5", "3, 3", "5, 2", "9, 1"]}, "cell": "sfft/0", "stage": "A"}
+{"id": "po-sfft-t0-eval-001208", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - x + 4y = 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - x + 4y = 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 3", "ref": "3, 3", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002743, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -1, "b": 4, "c": 18}, "seed": 167733942900003804, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["3, 3", "10, 2"]}, "cell": "sfft/0", "stage": "A"}
+{"id": "po-sfft-t0-eval-000927", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 5x - 2y = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 5x - 2y = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1", "ref": "1, 1", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.007561, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -5, "b": -2, "c": -6}, "seed": 167733942900002399, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 1, "valid_answers": ["1, 1", "3, 9", "4, 7", "6, 6"]}, "cell": "sfft/0", "stage": "A"}
+{"id": "po-sfft-t0-eval-000503", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 4x + 4y = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 4x + 4y = 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 2", "ref": "2, 2", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002551, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -4, "b": 4, "c": 4}, "seed": 167733942900001060, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["2, 2", "8, 3"]}, "cell": "sfft/0", "stage": "A"}
+{"id": "po-sfft-t0-eval-000963", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy = 16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy = 16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 16", "ref": "1, 16", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.019531, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 0, "b": 0, "c": 16}, "seed": 167733942900002561, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 4, "valid_answers": ["1, 16", "2, 8", "4, 4", "8, 2", "16, 1"]}, "cell": "sfft/0", "stage": "A"}
+{"id": "po-sfft-t0-eval-000537", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 3x - 4y = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 3x - 4y = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1", "ref": "1, 1", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.008, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -3, "b": -4, "c": -6}, "seed": 167733942900001165, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 1, "valid_answers": ["1, 1", "5, 9", "6, 6", "7, 5", "10, 4"]}, "cell": "sfft/0", "stage": "A"}
+{"id": "po-sfft-t0-eval-001667", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 6x - 5y = -15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 6x - 5y = -15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 1", "ref": "2, 1", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.001594, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -6, "b": -5, "c": -15}, "seed": 167733942900008883, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 2, "valid_answers": ["2, 1", "6, 21", "8, 11", "10, 9", "20, 7"]}, "cell": "sfft/0", "stage": "A"}
+{"id": "po-sfft-t0-eval-001239", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - x + 5y = 32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - x + 5y = 32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 4", "ref": "4, 4", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001082, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -1, "b": 5, "c": 32}, "seed": 167733942900003998, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["4, 4", "22, 2"]}, "cell": "sfft/0", "stage": "A"}
+{"id": "po-sfft-t0-eval-000967", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - x + 3y = 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - x + 3y = 13. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 3", "ref": "2, 3", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.005, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -1, "b": 3, "c": 13}, "seed": 167733942900002590, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["2, 3", "7, 2"]}, "cell": "sfft/0", "stage": "A"}
+{"id": "po-sfft-t0-eval-001505", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 6x + 3y = 38. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 6x + 3y = 38. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 8", "ref": "1, 8", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.00071, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 6, "b": 3, "c": 38}, "seed": 167733942900006257, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["1, 8", "4, 2", "5, 1"]}, "cell": "sfft/0", "stage": "A"}
+{"id": "po-sfft-t0-eval-000172", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 6x - 2y = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 6x - 2y = -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 12", "ref": "1, 12", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00277, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -6, "b": -2, "c": -18}, "seed": 167733942900000352, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["1, 12", "4, 3", "5, 4", "8, 5"]}, "cell": "sfft/0", "stage": "A"}
+{"id": "po-sfft-t1-eval-002121", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 4x - 9y = -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 4x - 9y = -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 2", "ref": "2, 2", "family": "po_sfft", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000992, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -4, "b": -9, "c": -22}, "seed": 167733942901002940, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 2, "valid_answers": ["2, 2", "10, 18", "11, 11", "16, 6", "23, 5"]}, "cell": "sfft/1", "stage": "B"}
+{"id": "po-sfft-t1-eval-000621", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 7x + 3y = 183. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 7x + 3y = 183. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 44", "ref": "1, 44", "family": "po_sfft", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 8.7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 7, "b": 3, "c": 183}, "seed": 167733942901000810, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 10, "valid_answers": ["1, 44", "3, 27", "9, 10", "14, 5"]}, "cell": "sfft/1", "stage": "B"}
+{"id": "po-sfft-t1-eval-000679", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - x = 64. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - x = 64. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 65", "ref": "1, 65", "family": "po_sfft", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.001657, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -1, "b": 0, "c": 64}, "seed": 167733942901000890, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 9, "valid_answers": ["1, 65", "2, 33", "4, 17", "8, 9", "16, 5", "32, 3", "64, 2"]}, "cell": "sfft/1", "stage": "B"}
+{"id": "po-sfft-t1-eval-002240", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 2x - 8y = -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 2x - 8y = -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 3", "ref": "4, 3", "family": "po_sfft", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00189, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -2, "b": -8, "c": -20}, "seed": 167733942901003132, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 4, "valid_answers": ["4, 3", "6, 4", "7, 6", "12, 1"]}, "cell": "sfft/1", "stage": "B"}
+{"id": "po-sfft-t1-eval-001319", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 5x - 9y = -46. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 5x - 9y = -46. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8, 6", "ref": "8, 6", "family": "po_sfft", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000181, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -5, "b": -9, "c": -46}, "seed": 167733942901001774, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 8, "valid_answers": ["8, 6", "10, 4"]}, "cell": "sfft/1", "stage": "B"}
+{"id": "po-sfft-t1-eval-000943", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 9x - 4y = 87. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 9x - 4y = 87. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 42", "ref": "5, 42", "family": "po_sfft", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000108, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 9, "b": -4, "c": 87}, "seed": 167733942901001249, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 8, "valid_answers": ["5, 42", "7, 8"]}, "cell": "sfft/1", "stage": "B"}
+{"id": "po-sfft-t1-eval-001356", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 8x + 6y = 26. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 8x + 6y = 26. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 6", "ref": "5, 6", "family": "po_sfft", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000258, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -8, "b": 6, "c": 26}, "seed": 167733942901001827, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 6, "valid_answers": ["5, 6", "16, 7"]}, "cell": "sfft/1", "stage": "B"}
+{"id": "po-sfft-t1-eval-000059", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 2x + 5y = 32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 2x + 5y = 32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 5", "ref": "1, 5", "family": "po_sfft", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001249, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 2, "b": 5, "c": 32}, "seed": 167733942901000074, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 4, "valid_answers": ["1, 5", "2, 4", "9, 1"]}, "cell": "sfft/1", "stage": "B"}
+{"id": "po-sfft-t1-eval-001524", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 3x - 6y = 36. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 3x - 6y = 36. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, 15", "ref": "7, 15", "family": "po_sfft", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000756, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 3, "b": -6, "c": 36}, "seed": 167733942901002060, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 8, "valid_answers": ["7, 15", "8, 6", "9, 3"]}, "cell": "sfft/1", "stage": "B"}
+{"id": "po-sfft-t1-eval-001549", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 2x - 2y = 58. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 2x - 2y = 58. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 52", "ref": "3, 52", "family": "po_sfft", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.001377, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 2, "b": -2, "c": 58}, "seed": 167733942901002095, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 8, "valid_answers": ["3, 52", "4, 25", "5, 16", "8, 7", "11, 4", "20, 1"]}, "cell": "sfft/1", "stage": "B"}
+{"id": "po-sfft-t1-eval-001966", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + x + 6y = 58. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + x + 6y = 58. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 7", "ref": "2, 7", "family": "po_sfft", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000595, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 1, "b": 6, "c": 58}, "seed": 167733942901002713, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 7, "valid_answers": ["2, 7", "10, 3", "26, 1"]}, "cell": "sfft/1", "stage": "B"}
+{"id": "po-sfft-t1-eval-001165", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 6x - 5y = -30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 6x - 5y = -30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 9", "ref": "1, 9", "family": "po_sfft", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000793, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 6, "b": -5, "c": -30}, "seed": 167733942901001559, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 9, "valid_answers": ["1, 9", "2, 14", "3, 24", "4, 54"]}, "cell": "sfft/1", "stage": "B"}
+{"id": "po-sfft-t2-eval-002164", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 8x + y = 76. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 8x + y = 76. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 42", "ref": "1, 42", "family": "po_sfft", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000578, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -8, "b": 1, "c": 76}, "seed": 167733942902002426, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 16, "valid_answers": ["1, 42", "3, 25", "16, 12", "33, 10", "67, 9"]}, "cell": "sfft/2", "stage": "B"}
+{"id": "po-sfft-t2-eval-000596", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 11x + 4y = 256. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 11x + 4y = 256. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 49", "ref": "1, 49", "family": "po_sfft", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 7.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 11, "b": 4, "c": 256}, "seed": 167733942902000673, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 11, "valid_answers": ["1, 49", "2, 39", "6, 19", "8, 14", "11, 9", "16, 4", "21, 1"]}, "cell": "sfft/2", "stage": "B"}
+{"id": "po-sfft-t2-eval-000704", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 11x - 7y = 441. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 11x - 7y = 441. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8, 353", "ref": "8, 353", "family": "po_sfft", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 2.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 11, "b": -7, "c": 441}, "seed": 167733942902000797, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 20, "valid_answers": ["8, 353", "9, 171", "11, 80", "14, 41", "20, 17", "21, 15", "33, 3", "35, 2"]}, "cell": "sfft/2", "stage": "B"}
+{"id": "po-sfft-t2-eval-000093", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 2x - 8y = 54. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 2x - 8y = 54. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9, 72", "ref": "9, 72", "family": "po_sfft", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.00125, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -2, "b": -8, "c": 54}, "seed": 167733942902000109, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 15, "valid_answers": ["9, 72", "10, 37", "13, 16", "15, 12", "18, 9", "22, 7", "43, 4", "78, 3"]}, "cell": "sfft/2", "stage": "B"}
+{"id": "po-sfft-t2-eval-001536", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 12x - 12y = 110. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 12x - 12y = 110. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 5", "ref": "10, 5", "family": "po_sfft", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 12, "b": -12, "c": 110}, "seed": 167733942902001721, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 10, "valid_answers": ["10, 5", "11, 22"]}, "cell": "sfft/2", "stage": "B"}
+{"id": "po-sfft-t2-eval-000831", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 4x + 12y = 384. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 4x + 12y = 384. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 23", "ref": "4, 23", "family": "po_sfft", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 4, "b": 12, "c": 384}, "seed": 167733942902000940, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 14, "valid_answers": ["4, 23", "6, 20", "12, 14", "15, 12", "24, 8", "36, 5", "42, 4", "60, 2"]}, "cell": "sfft/2", "stage": "B"}
+{"id": "po-sfft-t2-eval-002137", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 11x + 6y = 122. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 11x + 6y = 122. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 19", "ref": "1, 19", "family": "po_sfft", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000119, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -11, "b": 6, "c": 122}, "seed": 167733942902002398, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 15, "valid_answers": ["1, 19", "2, 18", "8, 15", "22, 13", "50, 12"]}, "cell": "sfft/2", "stage": "B"}
+{"id": "po-sfft-t2-eval-001987", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - y = 96. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - y = 96. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 96", "ref": "2, 96", "family": "po_sfft", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.001275, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 0, "b": -1, "c": 96}, "seed": 167733942902002227, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 12, "valid_answers": ["2, 96", "3, 48", "4, 32", "5, 24", "7, 16", "9, 12", "13, 8", "17, 6", "25, 4", "33, 3", "49, 2", "97, 1"]}, "cell": "sfft/2", "stage": "B"}
+{"id": "po-sfft-t2-eval-001018", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 9x + 10y = 180. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 9x + 10y = 180. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 9", "ref": "5, 9", "family": "po_sfft", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 3.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 9, "b": 10, "c": 180}, "seed": 167733942902001150, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 8, "valid_answers": ["5, 9", "8, 6", "17, 1"]}, "cell": "sfft/2", "stage": "B"}
+{"id": "po-sfft-t2-eval-002424", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 11x + 9y = 73. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 11x + 9y = 73. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 9", "ref": "4, 9", "family": "po_sfft", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -11, "b": 9, "c": 73}, "seed": 167733942902002724, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 9, "valid_answers": ["4, 9", "17, 10"]}, "cell": "sfft/2", "stage": "B"}
+{"id": "po-sfft-t2-eval-001571", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 4x + 8y = 227. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 4x + 8y = 227. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 19", "ref": "5, 19", "family": "po_sfft", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 6.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -4, "b": 8, "c": 227}, "seed": 167733942902001760, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 17, "valid_answers": ["5, 19", "7, 17", "31, 9", "57, 7", "187, 5"]}, "cell": "sfft/2", "stage": "B"}
+{"id": "po-sfft-t2-eval-000262", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 7x - 8y = -86. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 7x - 8y = -86. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 12", "ref": "2, 12", "family": "po_sfft", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.000406, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -7, "b": -8, "c": -86}, "seed": 167733942902000294, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 12, "valid_answers": ["2, 12", "3, 13", "5, 17", "6, 22", "7, 37", "13, 1", "14, 2", "18, 4", "23, 5", "38, 6"]}, "cell": "sfft/2", "stage": "B"}
+{"id": "po-sfft-t3-eval-001303", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - x + 12y = 887. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - x + 12y = 887. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "13, 36", "ref": "13, 36", "family": "po_sfft", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 6e-06, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -1, "b": 12, "c": 887}, "seed": 167733942903001376, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 26, "valid_answers": ["13, 36", "23, 26", "113, 8", "163, 6", "863, 2"]}, "cell": "sfft/3", "stage": "B"}
+{"id": "po-sfft-t3-eval-002164", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 15x + 4y = 126. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 15x + 4y = 126. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 26", "ref": "2, 26", "family": "po_sfft", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000119, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -15, "b": 4, "c": 126}, "seed": 167733942903002276, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 18, "valid_answers": ["2, 26", "7, 21", "18, 18", "29, 17", "62, 16"]}, "cell": "sfft/3", "stage": "B"}
+{"id": "po-sfft-t3-eval-002103", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 6x = 312. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 6x = 312. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 318", "ref": "1, 318", "family": "po_sfft", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.000158, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -6, "b": 0, "c": 312}, "seed": 167733942903002213, "version": "po-v0", "tier": 3, "n_raw": 16, "small": 24, "valid_answers": ["1, 318", "2, 162", "3, 110", "4, 84", "6, 58", "8, 45", "12, 32", "13, 30", "24, 19", "26, 18", "39, 14", "52, 12", "78, 10", "104, 9", "156, 8", "312, 7"]}, "cell": "sfft/3", "stage": "B"}
+{"id": "po-sfft-t3-eval-000103", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + x - 12y = -27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + x - 12y = -27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9, 12", "ref": "9, 12", "family": "po_sfft", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00074, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 1, "b": -12, "c": -27}, "seed": 167733942903000112, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 12, "valid_answers": ["9, 12", "11, 38"]}, "cell": "sfft/3", "stage": "B"}
+{"id": "po-sfft-t3-eval-000326", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 15x - 10y = 1223. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 15x - 10y = 1223. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 1058", "ref": "11, 1058", "family": "po_sfft", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 2e-06, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 15, "b": -10, "c": 1223}, "seed": 167733942903000351, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 39, "valid_answers": ["11, 1058", "39, 22", "47, 14"]}, "cell": "sfft/3", "stage": "B"}
+{"id": "po-sfft-t3-eval-001763", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - y = 120. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - y = 120. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 120", "ref": "2, 120", "family": "po_sfft", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.001093, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 0, "b": -1, "c": 120}, "seed": 167733942903001856, "version": "po-v0", "tier": 3, "n_raw": 16, "small": 12, "valid_answers": ["2, 120", "3, 60", "4, 40", "5, 30", "6, 24", "7, 20", "9, 15", "11, 12", "13, 10", "16, 8", "21, 6", "25, 5", "31, 4", "41, 3", "61, 2", "121, 1"]}, "cell": "sfft/3", "stage": "B"}
+{"id": "po-sfft-t3-eval-000943", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 2x + 5y = 920. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 2x + 5y = 920. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 153", "ref": "1, 153", "family": "po_sfft", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 1.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 2, "b": 5, "c": 920}, "seed": 167733942903001004, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 28, "valid_answers": ["1, 153", "5, 91", "10, 60", "25, 29", "26, 28", "57, 13", "88, 8", "150, 4", "181, 3", "305, 1"]}, "cell": "sfft/3", "stage": "B"}
+{"id": "po-sfft-t3-eval-000037", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 12x + 11y = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 12x + 11y = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1", "ref": "1, 1", "family": "po_sfft", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.00025, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -12, "b": 11, "c": 0}, "seed": 167733942903000041, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 1, "valid_answers": ["1, 1", "11, 6", "22, 8", "33, 9", "55, 10", "121, 11"]}, "cell": "sfft/3", "stage": "B"}
+{"id": "po-sfft-t3-eval-001895", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 12x - 12y = 624. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 12x - 12y = 624. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "13, 468", "ref": "13, 468", "family": "po_sfft", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 2.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 12, "b": -12, "c": 624}, "seed": 167733942903001995, "version": "po-v0", "tier": 3, "n_raw": 15, "small": 27, "valid_answers": ["13, 468", "14, 228", "15, 148", "16, 108", "17, 84", "18, 68", "20, 48", "22, 36", "24, 28", "27, 20", "28, 18", "32, 12", "36, 8", "42, 4", "44, 3"]}, "cell": "sfft/3", "stage": "B"}
+{"id": "po-sfft-t3-eval-001013", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 10x + 10y = 1376. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy - 10x + 10y = 1376. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 126", "ref": "1, 126", "family": "po_sfft", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -10, "b": 10, "c": 1376}, "seed": 167733942903001077, "version": "po-v0", "tier": 3, "n_raw": 9, "small": 39, "valid_answers": ["1, 126", "12, 68", "19, 54", "34, 39", "48, 32", "106, 21", "309, 14", "628, 12", "1266, 11"]}, "cell": "sfft/3", "stage": "B"}
+{"id": "po-sfft-t3-eval-001792", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 2x + 4y = 40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 2x + 4y = 40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 6", "ref": "2, 6", "family": "po_sfft", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001372, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 2, "b": 4, "c": 40}, "seed": 167733942903001885, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 4, "valid_answers": ["2, 6", "4, 4", "8, 2", "12, 1"]}, "cell": "sfft/3", "stage": "B"}
+{"id": "po-sfft-t3-eval-002151", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 15x - 9y = 1124. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind positive integers x, y such that xy + 15x - 9y = 1124. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10, 974", "ref": "10, 974", "family": "po_sfft", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 2e-06, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 15, "b": -9, "c": 1124}, "seed": 167733942903002263, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 32, "valid_answers": ["10, 974", "32, 28", "52, 8"]}, "cell": "sfft/3", "stage": "B"}
+{"id": "po-sigma_inv-t1-eval-000059", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 1170. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 1170. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "360", "ref": "360", "family": "po_sigma_inv", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.003419, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 1170}, "seed": 44459245801000136, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 360, "valid_answers": ["360", "522", "778", "801"]}, "cell": "sigma_inv/1", "stage": "B"}
+{"id": "po-sigma_inv-t1-eval-000142", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 440. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 440. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "327", "ref": "327", "family": "po_sigma_inv", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004545, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 440}, "seed": 44459245801000785, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 327, "valid_answers": ["327", "439"]}, "cell": "sigma_inv/1", "stage": "B"}
+{"id": "po-sigma_inv-t1-eval-000010", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 1200. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 1200. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "456", "ref": "456", "family": "po_sigma_inv", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.005833, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 1200}, "seed": 44459245801000018, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 456, "valid_answers": ["456", "632", "686", "783", "995", "1043", "1121"]}, "cell": "sigma_inv/1", "stage": "B"}
+{"id": "po-sigma_inv-t1-eval-000115", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 1216. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 1216. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "777", "ref": "777", "family": "po_sigma_inv", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002467, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 1216}, "seed": 44459245801000384, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 777, "valid_answers": ["777", "1057", "1147"]}, "cell": "sigma_inv/1", "stage": "B"}
+{"id": "po-sigma_inv-t1-eval-000061", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 884. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 884. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "603", "ref": "603", "family": "po_sigma_inv", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002262, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 884}, "seed": 44459245801000141, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 603, "valid_answers": ["603", "883"]}, "cell": "sigma_inv/1", "stage": "B"}
+{"id": "po-sigma_inv-t1-eval-000101", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 1240. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 1240. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "432", "ref": "432", "family": "po_sigma_inv", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002419, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 1240}, "seed": 44459245801000317, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 432, "valid_answers": ["432", "675", "1159"]}, "cell": "sigma_inv/1", "stage": "B"}
+{"id": "po-sigma_inv-t1-eval-000102", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 810. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 810. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "424", "ref": "424", "family": "po_sigma_inv", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.003704, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 810}, "seed": 44459245801000320, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 424, "valid_answers": ["424", "538", "809"]}, "cell": "sigma_inv/1", "stage": "B"}
+{"id": "po-sigma_inv-t1-eval-000012", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 588. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 588. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "260", "ref": "260", "family": "po_sigma_inv", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.008503, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 588}, "seed": 44459245801000020, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 260, "valid_answers": ["260", "332", "485", "533", "587"]}, "cell": "sigma_inv/1", "stage": "B"}
+{"id": "po-sigma_inv-t1-eval-000125", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 762. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 762. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "320", "ref": "320", "family": "po_sigma_inv", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002625, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 762}, "seed": 44459245801000497, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 320, "valid_answers": ["320", "761"]}, "cell": "sigma_inv/1", "stage": "B"}
+{"id": "po-sigma_inv-t1-eval-000089", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 880. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 880. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "763", "ref": "763", "family": "po_sigma_inv", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002273, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 880}, "seed": 44459245801000259, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 763, "valid_answers": ["763", "817"]}, "cell": "sigma_inv/1", "stage": "B"}
+{"id": "po-sigma_inv-t1-eval-000019", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 432. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 432. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "230", "ref": "230", "family": "po_sigma_inv", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.018519, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 432}, "seed": 44459245801000030, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 230, "valid_answers": ["230", "238", "255", "321", "355", "371", "391", "431"]}, "cell": "sigma_inv/1", "stage": "B"}
+{"id": "po-sigma_inv-t1-eval-000022", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 768. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 768. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "434", "ref": "434", "family": "po_sigma_inv", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.007812, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 768}, "seed": 44459245801000038, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 434, "valid_answers": ["434", "465", "483", "573", "635", "713"]}, "cell": "sigma_inv/1", "stage": "B"}
+{"id": "po-sigma_inv-t2-eval-000149", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 3312. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 3312. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1918", "ref": "1918", "family": "po_sigma_inv", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.00151, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 3312}, "seed": 44459245802000294, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 1918, "valid_answers": ["1918", "2055", "2206", "2481", "3151"]}, "cell": "sigma_inv/2", "stage": "B"}
+{"id": "po-sigma_inv-t2-eval-000259", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 3864. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 3864. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1644", "ref": "1644", "family": "po_sigma_inv", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000776, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 3864}, "seed": 44459245802000624, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 1644, "valid_answers": ["1644", "3215", "3863"]}, "cell": "sigma_inv/2", "stage": "B"}
+{"id": "po-sigma_inv-t2-eval-000414", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 2648. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 2648. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1983", "ref": "1983", "family": "po_sigma_inv", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000755, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 2648}, "seed": 44459245802002881, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 1983, "valid_answers": ["1983", "2647"]}, "cell": "sigma_inv/2", "stage": "B"}
+{"id": "po-sigma_inv-t2-eval-000089", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 3510. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 3510. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1224", "ref": "1224", "family": "po_sigma_inv", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00114, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 3510}, "seed": 44459245802000144, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 1224, "valid_answers": ["1224", "1602", "1864", "2421"]}, "cell": "sigma_inv/2", "stage": "B"}
+{"id": "po-sigma_inv-t2-eval-000307", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 1188. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 1188. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "985", "ref": "985", "family": "po_sigma_inv", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001684, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 1188}, "seed": 44459245802001007, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 985, "valid_answers": ["985", "1187"]}, "cell": "sigma_inv/2", "stage": "B"}
+{"id": "po-sigma_inv-t2-eval-000302", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 4494. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 4494. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2564", "ref": "2564", "family": "po_sigma_inv", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000445, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 4494}, "seed": 44459245802000937, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 2564, "valid_answers": ["2564", "4493"]}, "cell": "sigma_inv/2", "stage": "B"}
+{"id": "po-sigma_inv-t2-eval-000126", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 4284. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 4284. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2020", "ref": "2020", "family": "po_sigma_inv", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.001401, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 4284}, "seed": 44459245802000226, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 2020, "valid_answers": ["2020", "2144", "2626", "2854", "4141", "4283"]}, "cell": "sigma_inv/2", "stage": "B"}
+{"id": "po-sigma_inv-t2-eval-000304", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 1968. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 1968. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "978", "ref": "978", "family": "po_sigma_inv", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001524, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 1968}, "seed": 44459245802001000, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 978, "valid_answers": ["978", "1473", "1793"]}, "cell": "sigma_inv/2", "stage": "B"}
+{"id": "po-sigma_inv-t2-eval-000354", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 4520. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 4520. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3387", "ref": "3387", "family": "po_sigma_inv", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000442, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 4520}, "seed": 44459245802001469, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 3387, "valid_answers": ["3387", "4519"]}, "cell": "sigma_inv/2", "stage": "B"}
+{"id": "po-sigma_inv-t2-eval-000030", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 3024. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 3024. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1020", "ref": "1020", "family": "po_sigma_inv", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 26, "bucket": null, "guess_rate": 0.008598, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 3024}, "seed": 44459245802000046, "version": "po-v0", "tier": 2, "n_raw": 26, "small": 1020, "valid_answers": ["1020", "1056", "1120", "1230", "1284", "1326", "1420", "1430", "1484", "1504", "1506", "1564", "1670", "1724", "1826", "1846", "1886", "2067", "2091", "2255", "2431", "2515", "2761", "2839", "2911", "3023"]}, "cell": "sigma_inv/2", "stage": "B"}
+{"id": "po-sigma_inv-t2-eval-000189", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 1200. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 1200. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "456", "ref": "456", "family": "po_sigma_inv", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.005833, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 1200}, "seed": 44459245802000400, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 456, "valid_answers": ["456", "632", "686", "783", "995", "1043", "1121"]}, "cell": "sigma_inv/2", "stage": "B"}
+{"id": "po-sigma_inv-t2-eval-000338", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 2196. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 2196. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1014", "ref": "1014", "family": "po_sigma_inv", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000911, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 2196}, "seed": 44459245802001294, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 1014, "valid_answers": ["1014", "1859"]}, "cell": "sigma_inv/2", "stage": "B"}
+{"id": "po-sigma_inv-t3-eval-000698", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 7152. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 7152. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4766", "ref": "4766", "family": "po_sigma_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000419, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 7152}, "seed": 44459245803001609, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 4766, "valid_answers": ["4766", "5361", "7151"]}, "cell": "sigma_inv/3", "stage": "B"}
+{"id": "po-sigma_inv-t3-eval-000433", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 16000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 16000. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9261", "ref": "9261", "family": "po_sigma_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000375, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 16000}, "seed": 44459245803000780, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 9261, "valid_answers": ["9261", "10479", "11343", "13993", "15469", "15721"]}, "cell": "sigma_inv/3", "stage": "B"}
+{"id": "po-sigma_inv-t3-eval-000869", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 6968. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 6968. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5223", "ref": "5223", "family": "po_sigma_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000431, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 6968}, "seed": 44459245803002315, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 5223, "valid_answers": ["5223", "5887", "6967"]}, "cell": "sigma_inv/3", "stage": "B"}
+{"id": "po-sigma_inv-t3-eval-000799", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 15810. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 15810. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7808", "ref": "7808", "family": "po_sigma_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000253, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 15810}, "seed": 44459245803001985, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 7808, "valid_answers": ["7808", "8144", "12725", "15809"]}, "cell": "sigma_inv/3", "stage": "B"}
+{"id": "po-sigma_inv-t3-eval-001028", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 7824. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 7824. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6515", "ref": "6515", "family": "po_sigma_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000383, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 7824}, "seed": 44459245803003265, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 6515, "valid_answers": ["6515", "6839", "7823"]}, "cell": "sigma_inv/3", "stage": "B"}
+{"id": "po-sigma_inv-t3-eval-000074", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 13152. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 13152. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7658", "ref": "7658", "family": "po_sigma_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000304, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 13152}, "seed": 44459245803000097, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 7658, "valid_answers": ["7658", "8205", "12581", "13151"]}, "cell": "sigma_inv/3", "stage": "B"}
+{"id": "po-sigma_inv-t3-eval-000792", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 9204. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 9204. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6134", "ref": "6134", "family": "po_sigma_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000217, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 9204}, "seed": 44459245803001963, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 6134, "valid_answers": ["6134", "9203"]}, "cell": "sigma_inv/3", "stage": "B"}
+{"id": "po-sigma_inv-t3-eval-001520", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 7868. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 7868. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4492", "ref": "4492", "family": "po_sigma_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000254, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 7868}, "seed": 44459245803015867, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 4492, "valid_answers": ["4492", "7867"]}, "cell": "sigma_inv/3", "stage": "B"}
+{"id": "po-sigma_inv-t3-eval-001172", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 8670. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 8670. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4616", "ref": "4616", "family": "po_sigma_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000231, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 8670}, "seed": 44459245803004461, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 4616, "valid_answers": ["4616", "8669"]}, "cell": "sigma_inv/3", "stage": "B"}
+{"id": "po-sigma_inv-t3-eval-000054", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 5040. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 5040. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1560", "ref": "1560", "family": "po_sigma_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 33, "bucket": null, "guess_rate": 0.006548, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 5040}, "seed": 44459245803000065, "version": "po-v0", "tier": 3, "n_raw": 33, "small": 1560, "valid_answers": ["1560", "1740", "1824", "1836", "1992", "2148", "2214", "2262", "2296", "2392", "2470", "2492", "2514", "2528", "2596", "2668", "2876", "3058", "3154", "3471", "3567", "3777", "3835", "3895", "4147", "4195", "4199", "4609", "4667", "4769", "4843", "4897", "5039"]}, "cell": "sigma_inv/3", "stage": "B"}
+{"id": "po-sigma_inv-t3-eval-001120", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 10008. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 10008. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6094", "ref": "6094", "family": "po_sigma_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0003, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 10008}, "seed": 44459245803003956, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 6094, "valid_answers": ["6094", "8335", "10007"]}, "cell": "sigma_inv/3", "stage": "B"}
+{"id": "po-sigma_inv-t3-eval-000721", "task": "cx_po_sigma_inv", "source": "cx_po_sigma_inv", "split": "eval", "problem": "Find a positive integer n whose divisors (including 1 and n) add up to 9456. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n whose divisors (including 1 and n) add up to 9456. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4722", "ref": "4722", "family": "po_sigma_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000317, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 9456}, "seed": 44459245803001683, "version": "po-v0", "tier": 3, "n_raw": 3, "small": 4722, "valid_answers": ["4722", "8267", "8657"]}, "cell": "sigma_inv/3", "stage": "B"}
+{"id": "po-sqrt_mod-t0-eval-000215", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 390 such that x^2 \\equiv 94 \\pmod{391}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 390 such that x^2 \\equiv 94 \\pmod{391}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "133", "ref": "133", "family": "po_sqrt_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.01023, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 391, "a": 94}, "seed": 109058477100000361, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 133, "valid_answers": ["133", "156", "235", "258"]}, "cell": "sqrt_mod/0", "stage": "A"}
+{"id": "po-sqrt_mod-t0-eval-000505", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 246 such that x^2 \\equiv 168 \\pmod{247}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 246 such that x^2 \\equiv 168 \\pmod{247}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "34", "ref": "34", "family": "po_sqrt_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.016194, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 247, "a": 168}, "seed": 109058477100002148, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 34, "valid_answers": ["34", "99", "148", "213"]}, "cell": "sqrt_mod/0", "stage": "A"}
+{"id": "po-sqrt_mod-t0-eval-000095", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 246 such that x^2 \\equiv 87 \\pmod{247}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 246 such that x^2 \\equiv 87 \\pmod{247}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "88", "ref": "88", "family": "po_sqrt_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.016194, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 247, "a": 87}, "seed": 109058477100000134, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 88, "valid_answers": ["88", "107", "140", "159"]}, "cell": "sqrt_mod/0", "stage": "A"}
+{"id": "po-sqrt_mod-t0-eval-000186", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 436 such that x^2 \\equiv 372 \\pmod{437}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 436 such that x^2 \\equiv 372 \\pmod{437}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "140", "ref": "140", "family": "po_sqrt_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.009153, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 437, "a": 372}, "seed": 109058477100000298, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 140, "valid_answers": ["140", "159", "278", "297"]}, "cell": "sqrt_mod/0", "stage": "A"}
+{"id": "po-sqrt_mod-t0-eval-000052", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 298 such that x^2 \\equiv 82 \\pmod{299}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 298 such that x^2 \\equiv 82 \\pmod{299}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "63", "ref": "63", "family": "po_sqrt_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.013378, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 299, "a": 82}, "seed": 109058477100000072, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 63, "valid_answers": ["63", "132", "167", "236"]}, "cell": "sqrt_mod/0", "stage": "A"}
+{"id": "po-sqrt_mod-t0-eval-000049", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 252 such that x^2 \\equiv 27 \\pmod{253}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 252 such that x^2 \\equiv 27 \\pmod{253}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "48", "ref": "48", "family": "po_sqrt_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.01581, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 253, "a": 27}, "seed": 109058477100000065, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 48, "valid_answers": ["48", "117", "136", "205"]}, "cell": "sqrt_mod/0", "stage": "A"}
+{"id": "po-sqrt_mod-t0-eval-000041", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 436 such that x^2 \\equiv 197 \\pmod{437}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 436 such that x^2 \\equiv 197 \\pmod{437}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "144", "ref": "144", "family": "po_sqrt_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.009153, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 437, "a": 197}, "seed": 109058477100000053, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 144, "valid_answers": ["144", "201", "236", "293"]}, "cell": "sqrt_mod/0", "stage": "A"}
+{"id": "po-sqrt_mod-t0-eval-000420", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 220 such that x^2 \\equiv 101 \\pmod{221}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 220 such that x^2 \\equiv 101 \\pmod{221}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "72", "ref": "72", "family": "po_sqrt_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0181, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 221, "a": 101}, "seed": 109058477100001253, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 72, "valid_answers": ["72", "98", "123", "149"]}, "cell": "sqrt_mod/0", "stage": "A"}
+{"id": "po-sqrt_mod-t0-eval-000188", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 252 such that x^2 \\equiv 100 \\pmod{253}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 252 such that x^2 \\equiv 100 \\pmod{253}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10", "ref": "10", "family": "po_sqrt_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.01581, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 253, "a": 100}, "seed": 109058477100000305, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 10, "valid_answers": ["10", "56", "197", "243"]}, "cell": "sqrt_mod/0", "stage": "A"}
+{"id": "po-sqrt_mod-t0-eval-000082", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 252 such that x^2 \\equiv 31 \\pmod{253}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 252 such that x^2 \\equiv 31 \\pmod{253}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "36", "ref": "36", "family": "po_sqrt_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.01581, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 253, "a": 31}, "seed": 109058477100000116, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 36, "valid_answers": ["36", "102", "151", "217"]}, "cell": "sqrt_mod/0", "stage": "A"}
+{"id": "po-sqrt_mod-t0-eval-000012", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 436 such that x^2 \\equiv 377 \\pmod{437}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 436 such that x^2 \\equiv 377 \\pmod{437}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "72", "ref": "72", "family": "po_sqrt_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.009153, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 437, "a": 377}, "seed": 109058477100000021, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 72, "valid_answers": ["72", "118", "319", "365"]}, "cell": "sqrt_mod/0", "stage": "A"}
+{"id": "po-sqrt_mod-t0-eval-000246", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 220 such that x^2 \\equiv 25 \\pmod{221}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 220 such that x^2 \\equiv 25 \\pmod{221}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5", "ref": "5", "family": "po_sqrt_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0181, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 221, "a": 25}, "seed": 109058477100000432, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 5, "valid_answers": ["5", "73", "148", "216"]}, "cell": "sqrt_mod/0", "stage": "A"}
+{"id": "po-sqrt_mod-t1-eval-001710", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 588 such that x^2 \\equiv 529 \\pmod{589}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 588 such that x^2 \\equiv 529 \\pmod{589}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "23", "ref": "23", "family": "po_sqrt_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.006791, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 589, "a": 529}, "seed": 109058477101002401, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 23, "valid_answers": ["23", "194", "395", "566"]}, "cell": "sqrt_mod/1", "stage": "B"}
+{"id": "po-sqrt_mod-t1-eval-001156", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 702 such that x^2 \\equiv 514 \\pmod{703}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 702 such that x^2 \\equiv 514 \\pmod{703}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "210", "ref": "210", "family": "po_sqrt_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00569, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 703, "a": 514}, "seed": 109058477101001409, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 210, "valid_answers": ["210", "284", "419", "493"]}, "cell": "sqrt_mod/1", "stage": "B"}
+{"id": "po-sqrt_mod-t1-eval-001345", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 628 such that x^2 \\equiv 270 \\pmod{629}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 628 such that x^2 \\equiv 270 \\pmod{629}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "245", "ref": "245", "family": "po_sqrt_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.006359, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 629, "a": 270}, "seed": 109058477101001710, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 245, "valid_answers": ["245", "282", "347", "384"]}, "cell": "sqrt_mod/1", "stage": "B"}
+{"id": "po-sqrt_mod-t1-eval-000269", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 1516 such that x^2 \\equiv 657 \\pmod{1517}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 1516 such that x^2 \\equiv 657 \\pmod{1517}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "204", "ref": "204", "family": "po_sqrt_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.002637, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 1517, "a": 657}, "seed": 109058477101000276, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 204, "valid_answers": ["204", "573", "944", "1313"]}, "cell": "sqrt_mod/1", "stage": "B"}
+{"id": "po-sqrt_mod-t1-eval-001863", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 1146 such that x^2 \\equiv 506 \\pmod{1147}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 1146 such that x^2 \\equiv 506 \\pmod{1147}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "79", "ref": "79", "family": "po_sqrt_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.003487, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 1147, "a": 506}, "seed": 109058477101002719, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 79, "valid_answers": ["79", "513", "634", "1068"]}, "cell": "sqrt_mod/1", "stage": "B"}
+{"id": "po-sqrt_mod-t1-eval-002045", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 778 such that x^2 \\equiv 739 \\pmod{779}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 778 such that x^2 \\equiv 739 \\pmod{779}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "165", "ref": "165", "family": "po_sqrt_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005135, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 779, "a": 739}, "seed": 109058477101003147, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 165, "valid_answers": ["165", "329", "450", "614"]}, "cell": "sqrt_mod/1", "stage": "B"}
+{"id": "po-sqrt_mod-t1-eval-002040", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 492 such that x^2 \\equiv 339 \\pmod{493}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 492 such that x^2 \\equiv 339 \\pmod{493}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "123", "ref": "123", "family": "po_sqrt_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.008114, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 493, "a": 339}, "seed": 109058477101003134, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 123, "valid_answers": ["123", "225", "268", "370"]}, "cell": "sqrt_mod/1", "stage": "B"}
+{"id": "po-sqrt_mod-t1-eval-001972", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 1270 such that x^2 \\equiv 882 \\pmod{1271}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 1270 such that x^2 \\equiv 882 \\pmod{1271}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "111", "ref": "111", "family": "po_sqrt_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.003147, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 1271, "a": 882}, "seed": 109058477101002993, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 111, "valid_answers": ["111", "545", "726", "1160"]}, "cell": "sqrt_mod/1", "stage": "B"}
+{"id": "po-sqrt_mod-t1-eval-000279", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 898 such that x^2 \\equiv 59 \\pmod{899}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 898 such that x^2 \\equiv 59 \\pmod{899}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "144", "ref": "144", "family": "po_sqrt_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.004449, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 899, "a": 59}, "seed": 109058477101000287, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 144, "valid_answers": ["144", "175", "724", "755"]}, "cell": "sqrt_mod/1", "stage": "B"}
+{"id": "po-sqrt_mod-t1-eval-002220", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 942 such that x^2 \\equiv 811 \\pmod{943}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 942 such that x^2 \\equiv 811 \\pmod{943}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "150", "ref": "150", "family": "po_sqrt_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.004242, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 943, "a": 811}, "seed": 109058477101003601, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 150, "valid_answers": ["150", "219", "724", "793"]}, "cell": "sqrt_mod/1", "stage": "B"}
+{"id": "po-sqrt_mod-t1-eval-001265", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 526 such that x^2 \\equiv 222 \\pmod{527}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 526 such that x^2 \\equiv 222 \\pmod{527}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "118", "ref": "118", "family": "po_sqrt_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00759, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 527, "a": 222}, "seed": 109058477101001570, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 118, "valid_answers": ["118", "254", "273", "409"]}, "cell": "sqrt_mod/1", "stage": "B"}
+{"id": "po-sqrt_mod-t1-eval-001575", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 1188 such that x^2 \\equiv 91 \\pmod{1189}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 1188 such that x^2 \\equiv 91 \\pmod{1189}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "85", "ref": "85", "family": "po_sqrt_mod", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.003364, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 1189, "a": 91}, "seed": 109058477101002143, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 85, "valid_answers": ["85", "495", "694", "1104"]}, "cell": "sqrt_mod/1", "stage": "B"}
+{"id": "po-sqrt_mod-t2-eval-001949", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 1234 such that x^2 \\equiv 339 \\pmod{1235}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 1234 such that x^2 \\equiv 339 \\pmod{1235}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "53", "ref": "53", "family": "po_sqrt_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.006478, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 1235, "a": 339}, "seed": 109058477102004369, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 53, "valid_answers": ["53", "118", "547", "612", "623", "688", "1117", "1182"]}, "cell": "sqrt_mod/2", "stage": "B"}
+{"id": "po-sqrt_mod-t2-eval-001465", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 2430 such that x^2 \\equiv 1600 \\pmod{2431}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 2430 such that x^2 \\equiv 1600 \\pmod{2431}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "40", "ref": "40", "family": "po_sqrt_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.003291, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 2431, "a": 1600}, "seed": 109058477102002510, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 40, "valid_answers": ["40", "521", "623", "1184", "1247", "1808", "1910", "2391"]}, "cell": "sqrt_mod/2", "stage": "B"}
+{"id": "po-sqrt_mod-t2-eval-000419", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 934 such that x^2 \\equiv 174 \\pmod{935}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 934 such that x^2 \\equiv 174 \\pmod{935}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "168", "ref": "168", "family": "po_sqrt_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.008556, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 935, "a": 174}, "seed": 109058477102000508, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 168, "valid_answers": ["168", "223", "338", "393", "542", "597", "712", "767"]}, "cell": "sqrt_mod/2", "stage": "B"}
+{"id": "po-sqrt_mod-t2-eval-000486", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 2260 such that x^2 \\equiv 501 \\pmod{2261}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 2260 such that x^2 \\equiv 501 \\pmod{2261}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "369", "ref": "369", "family": "po_sqrt_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.003538, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 2261, "a": 501}, "seed": 109058477102000600, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 369, "valid_answers": ["369", "464", "600", "828", "1433", "1661", "1797", "1892"]}, "cell": "sqrt_mod/2", "stage": "B"}
+{"id": "po-sqrt_mod-t2-eval-000459", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 1234 such that x^2 \\equiv 64 \\pmod{1235}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 1234 such that x^2 \\equiv 64 \\pmod{1235}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8", "ref": "8", "family": "po_sqrt_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.006478, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 1235, "a": 64}, "seed": 109058477102000561, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 8, "valid_answers": ["8", "122", "372", "502", "733", "863", "1113", "1227"]}, "cell": "sqrt_mod/2", "stage": "B"}
+{"id": "po-sqrt_mod-t2-eval-000578", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 1000 such that x^2 \\equiv 753 \\pmod{1001}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 1000 such that x^2 \\equiv 753 \\pmod{1001}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "268", "ref": "268", "family": "po_sqrt_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.007992, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 1001, "a": 753}, "seed": 109058477102000741, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 268, "valid_answers": ["268", "359", "411", "499", "502", "590", "642", "733"]}, "cell": "sqrt_mod/2", "stage": "B"}
+{"id": "po-sqrt_mod-t2-eval-000478", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 2430 such that x^2 \\equiv 276 \\pmod{2431}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 2430 such that x^2 \\equiv 276 \\pmod{2431}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "87", "ref": "87", "family": "po_sqrt_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.003291, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 2431, "a": 276}, "seed": 109058477102000586, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 87, "valid_answers": ["87", "100", "342", "529", "1902", "2089", "2331", "2344"]}, "cell": "sqrt_mod/2", "stage": "B"}
+{"id": "po-sqrt_mod-t2-eval-000599", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 4198 such that x^2 \\equiv 841 \\pmod{4199}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 4198 such that x^2 \\equiv 841 \\pmod{4199}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "29", "ref": "29", "family": "po_sqrt_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.001905, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 4199, "a": 841}, "seed": 109058477102000766, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 29, "valid_answers": ["29", "770", "998", "1739", "2460", "3201", "3429", "4170"]}, "cell": "sqrt_mod/2", "stage": "B"}
+{"id": "po-sqrt_mod-t2-eval-000118", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 1044 such that x^2 \\equiv 339 \\pmod{1045}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 1044 such that x^2 \\equiv 339 \\pmod{1045}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "118", "ref": "118", "family": "po_sqrt_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.007656, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 1045, "a": 339}, "seed": 109058477102000130, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 118, "valid_answers": ["118", "327", "338", "498", "547", "707", "718", "927"]}, "cell": "sqrt_mod/2", "stage": "B"}
+{"id": "po-sqrt_mod-t2-eval-001403", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 2260 such that x^2 \\equiv 548 \\pmod{2261}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 2260 such that x^2 \\equiv 548 \\pmod{2261}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "53", "ref": "53", "family": "po_sqrt_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.003538, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 2261, "a": 548}, "seed": 109058477102002377, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 53, "valid_answers": ["53", "270", "661", "984", "1277", "1600", "1991", "2208"]}, "cell": "sqrt_mod/2", "stage": "B"}
+{"id": "po-sqrt_mod-t2-eval-000418", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 2430 such that x^2 \\equiv 1192 \\pmod{2431}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 2430 such that x^2 \\equiv 1192 \\pmod{2431}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "244", "ref": "244", "family": "po_sqrt_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.003291, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 2431, "a": 1192}, "seed": 109058477102000507, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 244, "valid_answers": ["244", "504", "640", "1043", "1388", "1791", "1927", "2187"]}, "cell": "sqrt_mod/2", "stage": "B"}
+{"id": "po-sqrt_mod-t2-eval-002223", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 4198 such that x^2 \\equiv 172 \\pmod{4199}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 4198 such that x^2 \\equiv 172 \\pmod{4199}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "113", "ref": "113", "family": "po_sqrt_mod", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.001905, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 4199, "a": 172}, "seed": 109058477102006103, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 113, "valid_answers": ["113", "550", "1179", "1842", "2357", "3020", "3649", "4086"]}, "cell": "sqrt_mod/2", "stage": "B"}
+{"id": "po-sqrt_mod-t3-eval-001243", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 3450 such that x^2 \\equiv 2055 \\pmod{3451}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 3450 such that x^2 \\equiv 2055 \\pmod{3451}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "401", "ref": "401", "family": "po_sqrt_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.002318, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 3451, "a": 2055}, "seed": 109058477103001335, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 401, "valid_answers": ["401", "894", "1010", "1503", "1948", "2441", "2557", "3050"]}, "cell": "sqrt_mod/3", "stage": "B"}
+{"id": "po-sqrt_mod-t3-eval-000928", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 7162 such that x^2 \\equiv 3418 \\pmod{7163}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 7162 such that x^2 \\equiv 3418 \\pmod{7163}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1165", "ref": "1165", "family": "po_sqrt_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.001117, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 7163, "a": 3418}, "seed": 109058477103000972, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 1165, "valid_answers": ["1165", "2267", "2673", "3388", "3775", "4490", "4896", "5998"]}, "cell": "sqrt_mod/3", "stage": "B"}
+{"id": "po-sqrt_mod-t3-eval-002258", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 3058 such that x^2 \\equiv 1600 \\pmod{3059}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 3058 such that x^2 \\equiv 1600 \\pmod{3059}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "40", "ref": "40", "family": "po_sqrt_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.002615, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 3059, "a": 1600}, "seed": 109058477103002552, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 40, "valid_answers": ["40", "397", "891", "1328", "1731", "2168", "2662", "3019"]}, "cell": "sqrt_mod/3", "stage": "B"}
+{"id": "po-sqrt_mod-t3-eval-002205", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 5422 such that x^2 \\equiv 2014 \\pmod{5423}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 5422 such that x^2 \\equiv 2014 \\pmod{5423}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "947", "ref": "947", "family": "po_sqrt_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.001475, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 5423, "a": 2014}, "seed": 109058477103002486, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 947, "valid_answers": ["947", "1440", "2069", "2562", "2861", "3354", "3983", "4476"]}, "cell": "sqrt_mod/3", "stage": "B"}
+{"id": "po-sqrt_mod-t3-eval-001566", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 7162 such that x^2 \\equiv 4702 \\pmod{7163}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 7162 such that x^2 \\equiv 4702 \\pmod{7163}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "263", "ref": "263", "family": "po_sqrt_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.001117, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 7163, "a": 4702}, "seed": 109058477103001713, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 263, "valid_answers": ["263", "491", "1713", "2467", "4696", "5450", "6672", "6900"]}, "cell": "sqrt_mod/3", "stage": "B"}
+{"id": "po-sqrt_mod-t3-eval-001842", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 3552 such that x^2 \\equiv 2874 \\pmod{3553}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 3552 such that x^2 \\equiv 2874 \\pmod{3553}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "237", "ref": "237", "family": "po_sqrt_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.002252, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 3553, "a": 2874}, "seed": 109058477103002040, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 237, "valid_answers": ["237", "390", "732", "1359", "2194", "2821", "3163", "3316"]}, "cell": "sqrt_mod/3", "stage": "B"}
+{"id": "po-sqrt_mod-t3-eval-000755", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 1770 such that x^2 \\equiv 1457 \\pmod{1771}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 1770 such that x^2 \\equiv 1457 \\pmod{1771}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "125", "ref": "125", "family": "po_sqrt_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.004517, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 1771, "a": 1457}, "seed": 109058477103000783, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 125, "valid_answers": ["125", "447", "631", "818", "953", "1140", "1324", "1646"]}, "cell": "sqrt_mod/3", "stage": "B"}
+{"id": "po-sqrt_mod-t3-eval-001789", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 6060 such that x^2 \\equiv 4618 \\pmod{6061}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 6060 such that x^2 \\equiv 4618 \\pmod{6061}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "151", "ref": "151", "family": "po_sqrt_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.00132, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 6061, "a": 4618}, "seed": 109058477103001975, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 151, "valid_answers": ["151", "267", "2604", "3022", "3039", "3457", "5794", "5910"]}, "cell": "sqrt_mod/3", "stage": "B"}
+{"id": "po-sqrt_mod-t3-eval-002005", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 1770 such that x^2 \\equiv 1016 \\pmod{1771}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 1770 such that x^2 \\equiv 1016 \\pmod{1771}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "90", "ref": "90", "family": "po_sqrt_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.004517, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 1771, "a": 1016}, "seed": 109058477103002239, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 90, "valid_answers": ["90", "251", "596", "757", "1014", "1175", "1520", "1681"]}, "cell": "sqrt_mod/3", "stage": "B"}
+{"id": "po-sqrt_mod-t3-eval-000872", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 2638 such that x^2 \\equiv 932 \\pmod{2639}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 2638 such that x^2 \\equiv 932 \\pmod{2639}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "491", "ref": "491", "family": "po_sqrt_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.003031, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 2639, "a": 932}, "seed": 109058477103000910, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 491, "valid_answers": ["491", "582", "1245", "1303", "1336", "1394", "2057", "2148"]}, "cell": "sqrt_mod/3", "stage": "B"}
+{"id": "po-sqrt_mod-t3-eval-001696", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 2260 such that x^2 \\equiv 100 \\pmod{2261}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 2260 such that x^2 \\equiv 100 \\pmod{2261}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "10", "ref": "10", "family": "po_sqrt_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.003538, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 2261, "a": 100}, "seed": 109058477103001862, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 10, "valid_answers": ["10", "143", "333", "466", "1795", "1928", "2118", "2251"]}, "cell": "sqrt_mod/3", "stage": "B"}
+{"id": "po-sqrt_mod-t3-eval-000281", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 2232 such that x^2 \\equiv 268 \\pmod{2233}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer x with 0 <= x <= 2232 such that x^2 \\equiv 268 \\pmod{2233}. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "255", "ref": "255", "family": "po_sqrt_mod", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.003583, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 2233, "a": 268}, "seed": 109058477103000285, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 255, "valid_answers": ["255", "284", "354", "893", "1340", "1879", "1949", "1978"]}, "cell": "sqrt_mod/3", "stage": "B"}
+{"id": "po-stats_multiset-t1-eval-000086", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 5, median is 3 and unique mode is 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 5, median is 3 and unique mode is 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 2, 3, 6, 12", "ref": "2, 2, 3, 6, 12", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000485, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 5, "hi": 12, "c": {"sum": 25, "median": 3, "mode": 2}}, "seed": 309711794201000213, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 10, "valid_answers": ["2, 2, 3, 6, 12", "2, 2, 3, 7, 11", "2, 2, 3, 8, 10"]}, "cell": "stats_multiset/1", "stage": "B"}
+{"id": "po-stats_multiset-t1-eval-000058", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 8, median is 7 and unique mode is 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 8, median is 7 and unique mode is 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 7, 7, 11, 12", "ref": "3, 7, 7, 11, 12", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.001454, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 5, "hi": 12, "c": {"sum": 40, "median": 7, "mode": 7}}, "seed": 309711794201000102, "version": "po-v0", "tier": 1, "n_raw": 9, "small": 10, "valid_answers": ["3, 7, 7, 11, 12", "4, 7, 7, 10, 12", "5, 7, 7, 9, 12", "5, 7, 7, 10, 11", "6, 7, 7, 8, 12", "6, 7, 7, 9, 11", "7, 7, 7, 7, 12", "7, 7, 7, 8, 11", "7, 7, 7, 9, 10"]}, "cell": "stats_multiset/1", "stage": "B"}
+{"id": "po-stats_multiset-t1-eval-000051", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 7, median is 11 and unique mode is 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 7, median is 11 and unique mode is 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, 11, 11, 12", "ref": "0, 1, 11, 11, 12", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000485, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 5, "hi": 12, "c": {"sum": 35, "median": 11, "mode": 11}}, "seed": 309711794201000086, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 11, "valid_answers": ["0, 1, 11, 11, 12", "0, 2, 11, 11, 11", "1, 1, 11, 11, 11"]}, "cell": "stats_multiset/1", "stage": "B"}
+{"id": "po-stats_multiset-t1-eval-000092", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 6, median is 8 and unique mode is 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 6, median is 8 and unique mode is 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 4, 8, 9, 9", "ref": "0, 4, 8, 9, 9", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000323, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 5, "hi": 12, "c": {"sum": 30, "median": 8, "mode": 9}}, "seed": 309711794201000259, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 9, "valid_answers": ["0, 4, 8, 9, 9", "1, 3, 8, 9, 9"]}, "cell": "stats_multiset/1", "stage": "B"}
+{"id": "po-stats_multiset-t1-eval-000069", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 7, median is 6 and unique mode is 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 7, median is 6 and unique mode is 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 5, 6, 11, 11", "ref": "2, 5, 6, 11, 11", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000323, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 5, "hi": 12, "c": {"sum": 35, "median": 6, "mode": 11}}, "seed": 309711794201000135, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 11, "valid_answers": ["2, 5, 6, 11, 11", "3, 4, 6, 11, 11"]}, "cell": "stats_multiset/1", "stage": "B"}
+{"id": "po-stats_multiset-t1-eval-000120", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 1, median is 0 and unique mode is 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 1, median is 0 and unique mode is 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, 0, 0, 5", "ref": "0, 0, 0, 0, 5", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000485, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 5, "hi": 12, "c": {"sum": 5, "median": 0, "mode": 0}}, "seed": 309711794201000791, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 3, "valid_answers": ["0, 0, 0, 0, 5", "0, 0, 0, 1, 4", "0, 0, 0, 2, 3"]}, "cell": "stats_multiset/1", "stage": "B"}
+{"id": "po-stats_multiset-t1-eval-000088", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 8, median is 8 and unique mode is 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 8, median is 8 and unique mode is 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 7, 8, 11, 11", "ref": "3, 7, 8, 11, 11", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000323, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 5, "hi": 12, "c": {"sum": 40, "median": 8, "mode": 11}}, "seed": 309711794201000220, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 11, "valid_answers": ["3, 7, 8, 11, 11", "4, 6, 8, 11, 11"]}, "cell": "stats_multiset/1", "stage": "B"}
+{"id": "po-stats_multiset-t1-eval-000080", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 6, median is 7 and unique mode is 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 6, median is 7 and unique mode is 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 6, 7, 8, 8", "ref": "1, 6, 7, 8, 8", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000485, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 5, "hi": 12, "c": {"sum": 30, "median": 7, "mode": 8}}, "seed": 309711794201000187, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 8, "valid_answers": ["1, 6, 7, 8, 8", "2, 5, 7, 8, 8", "3, 4, 7, 8, 8"]}, "cell": "stats_multiset/1", "stage": "B"}
+{"id": "po-stats_multiset-t1-eval-000033", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 6, median is 9 and unique mode is 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 6, median is 9 and unique mode is 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, 9, 9, 11", "ref": "0, 1, 9, 9, 11", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000646, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 5, "hi": 12, "c": {"sum": 30, "median": 9, "mode": 9}}, "seed": 309711794201000049, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 9, "valid_answers": ["0, 1, 9, 9, 11", "0, 2, 9, 9, 10", "0, 3, 9, 9, 9", "1, 2, 9, 9, 9"]}, "cell": "stats_multiset/1", "stage": "B"}
+{"id": "po-stats_multiset-t1-eval-000053", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 4, median is 4 and unique mode is 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 4, median is 4 and unique mode is 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, 4, 5, 11", "ref": "0, 0, 4, 5, 11", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000485, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 5, "hi": 12, "c": {"sum": 20, "median": 4, "mode": 0}}, "seed": 309711794201000088, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 9, "valid_answers": ["0, 0, 4, 5, 11", "0, 0, 4, 6, 10", "0, 0, 4, 7, 9"]}, "cell": "stats_multiset/1", "stage": "B"}
+{"id": "po-stats_multiset-t1-eval-000044", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 5, median is 4 and unique mode is 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 5, median is 4 and unique mode is 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 3, 4, 9, 9", "ref": "0, 3, 4, 9, 9", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000323, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 5, "hi": 12, "c": {"sum": 25, "median": 4, "mode": 9}}, "seed": 309711794201000072, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 9, "valid_answers": ["0, 3, 4, 9, 9", "1, 2, 4, 9, 9"]}, "cell": "stats_multiset/1", "stage": "B"}
+{"id": "po-stats_multiset-t1-eval-000017", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 5, median is 4 and unique mode is 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e, each between 0 and 12, whose mean is 5, median is 4 and unique mode is 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 4, 7, 12", "ref": "1, 1, 4, 7, 12", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000485, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 5, "hi": 12, "c": {"sum": 25, "median": 4, "mode": 1}}, "seed": 309711794201000022, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 10, "valid_answers": ["1, 1, 4, 7, 12", "1, 1, 4, 8, 11", "1, 1, 4, 9, 10"]}, "cell": "stats_multiset/1", "stage": "B"}
+{"id": "po-stats_multiset-t2-eval-000102", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 5, median is 5 and unique mode is 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 5, median is 5 and unique mode is 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 4, 4, 5, 6, 7, 9", "ref": "0, 4, 4, 5, 6, 7, 9", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 12, "c": {"sum": 35, "median": 5, "mode": 4}}, "seed": 309711794202000305, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 7, "valid_answers": ["0, 4, 4, 5, 6, 7, 9", "1, 4, 4, 5, 6, 7, 8", "4, 4, 4, 5, 5, 6, 7"]}, "cell": "stats_multiset/2", "stage": "B"}
+{"id": "po-stats_multiset-t2-eval-000097", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 7, median is 9 and unique mode is 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 7, median is 9 and unique mode is 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, 7, 9, 11, 11, 11", "ref": "0, 0, 7, 9, 11, 11, 11", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.000318, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 12, "c": {"sum": 49, "median": 9, "mode": 11}}, "seed": 309711794202000280, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 11, "valid_answers": ["0, 0, 7, 9, 11, 11, 11", "0, 1, 5, 9, 11, 11, 12", "0, 1, 6, 9, 11, 11, 11", "0, 1, 7, 9, 10, 11, 11", "0, 2, 4, 9, 11, 11, 12", "0, 2, 5, 9, 11, 11, 11", "0, 2, 6, 9, 10, 11, 11", "0, 3, 4, 9, 11, 11, 11", "0, 3, 5, 9, 10, 11, 11", "1, 1, 5, 9, 11, 11, 11", "1, 2, 3, 9, 11, 11, 12", "1, 2, 4, 9, 11, 11, 11", "1, 2, 5, 9, 10, 11, 11", "1, 3, 3, 9, 11, 11, 11", "1, 3, 4, 9, 10, 11, 11", "2, 2, 3, 9, 11, 11, 11"]}, "cell": "stats_multiset/2", "stage": "B"}
+{"id": "po-stats_multiset-t2-eval-000210", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 8, median is 6 and unique mode is 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 8, median is 6 and unique mode is 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 6, 6, 6, 11, 12, 12", "ref": "3, 6, 6, 6, 11, 12, 12", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000179, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 12, "c": {"sum": 56, "median": 6, "mode": 6}}, "seed": 309711794202004220, "version": "po-v0", "tier": 2, "n_raw": 9, "small": 11, "valid_answers": ["3, 6, 6, 6, 11, 12, 12", "4, 6, 6, 6, 10, 12, 12", "4, 6, 6, 6, 11, 11, 12", "5, 6, 6, 6, 9, 12, 12", "5, 6, 6, 6, 10, 11, 12", "6, 6, 6, 6, 8, 12, 12", "6, 6, 6, 6, 9, 11, 12", "6, 6, 6, 6, 10, 10, 12", "6, 6, 6, 6, 10, 11, 11"]}, "cell": "stats_multiset/2", "stage": "B"}
+{"id": "po-stats_multiset-t2-eval-000127", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 2, median is 1 and unique mode is 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 2, median is 1 and unique mode is 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, 1, 1, 1, 1, 10", "ref": "0, 0, 1, 1, 1, 1, 10", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 23, "bucket": null, "guess_rate": 0.000456, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 12, "c": {"sum": 14, "median": 1, "mode": 1}}, "seed": 309711794202000413, "version": "po-v0", "tier": 2, "n_raw": 23, "small": 4, "valid_answers": ["0, 0, 1, 1, 1, 1, 10", "0, 0, 1, 1, 1, 2, 9", "0, 0, 1, 1, 1, 3, 8", "0, 0, 1, 1, 1, 4, 7", "0, 0, 1, 1, 1, 5, 6", "0, 1, 1, 1, 1, 1, 9", "0, 1, 1, 1, 1, 2, 8", "0, 1, 1, 1, 1, 3, 7", "0, 1, 1, 1, 1, 4, 6", "0, 1, 1, 1, 1, 5, 5", "0, 1, 1, 1, 2, 2, 7", "0, 1, 1, 1, 2, 3, 6", "0, 1, 1, 1, 2, 4, 5", "0, 1, 1, 1, 3, 3, 5", "0, 1, 1, 1, 3, 4, 4", "1, 1, 1, 1, 1, 1, 8", "1, 1, 1, 1, 1, 2, 7", "1, 1, 1, 1, 1, 3, 6", "1, 1, 1, 1, 1, 4, 5", "1, 1, 1, 1, 2, 2, 6", "1, 1, 1, 1, 2, 3, 5", "1, 1, 1, 1, 2, 4, 4", "1, 1, 1, 1, 3, 3, 4"]}, "cell": "stats_multiset/2", "stage": "B"}
+{"id": "po-stats_multiset-t2-eval-000132", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 6, median is 4 and unique mode is 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 6, median is 4 and unique mode is 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 2, 3, 4, 10, 10, 12", "ref": "1, 2, 3, 4, 10, 10, 12", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 7.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 12, "c": {"sum": 42, "median": 4, "mode": 10}}, "seed": 309711794202000468, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 10, "valid_answers": ["1, 2, 3, 4, 10, 10, 12", "1, 3, 4, 4, 10, 10, 10", "2, 2, 4, 4, 10, 10, 10", "2, 3, 3, 4, 10, 10, 10"]}, "cell": "stats_multiset/2", "stage": "B"}
+{"id": "po-stats_multiset-t2-eval-000089", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 6, median is 5 and unique mode is 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 6, median is 5 and unique mode is 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, 4, 5, 11, 11, 11", "ref": "0, 0, 4, 5, 11, 11, 11", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000258, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 12, "c": {"sum": 42, "median": 5, "mode": 11}}, "seed": 309711794202000241, "version": "po-v0", "tier": 2, "n_raw": 13, "small": 11, "valid_answers": ["0, 0, 4, 5, 11, 11, 11", "0, 1, 2, 5, 11, 11, 12", "0, 1, 3, 5, 11, 11, 11", "0, 1, 4, 5, 10, 11, 11", "0, 2, 2, 5, 11, 11, 11", "0, 2, 3, 5, 10, 11, 11", "0, 2, 4, 5, 9, 11, 11", "0, 3, 4, 5, 8, 11, 11", "1, 1, 2, 5, 11, 11, 11", "1, 2, 3, 5, 9, 11, 11", "1, 2, 4, 5, 8, 11, 11", "1, 3, 4, 5, 7, 11, 11", "2, 3, 4, 5, 6, 11, 11"]}, "cell": "stats_multiset/2", "stage": "B"}
+{"id": "po-stats_multiset-t2-eval-000120", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 8, median is 8 and unique mode is 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 8, median is 8 and unique mode is 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 6, 7, 8, 9, 9, 12", "ref": "5, 6, 7, 8, 9, 9, 12", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 12, "c": {"sum": 56, "median": 8, "mode": 9}}, "seed": 309711794202000391, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 9, "valid_answers": ["5, 6, 7, 8, 9, 9, 12", "6, 7, 8, 8, 9, 9, 9"]}, "cell": "stats_multiset/2", "stage": "B"}
+{"id": "po-stats_multiset-t2-eval-000069", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 5, median is 2 and unique mode is 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 5, median is 2 and unique mode is 1. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, 1, 2, 8, 11, 12", "ref": "0, 1, 1, 2, 8, 11, 12", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000159, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 12, "c": {"sum": 35, "median": 2, "mode": 1}}, "seed": 309711794202000163, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 11, "valid_answers": ["0, 1, 1, 2, 8, 11, 12", "0, 1, 1, 2, 9, 10, 12", "1, 1, 1, 2, 6, 12, 12", "1, 1, 1, 2, 7, 11, 12", "1, 1, 1, 2, 8, 10, 12", "1, 1, 1, 2, 8, 11, 11", "1, 1, 1, 2, 9, 9, 12", "1, 1, 1, 2, 9, 10, 11"]}, "cell": "stats_multiset/2", "stage": "B"}
+{"id": "po-stats_multiset-t2-eval-000144", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 3, median is 3 and unique mode is 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 3, median is 3 and unique mode is 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, 3, 3, 5, 5, 5", "ref": "0, 0, 3, 3, 5, 5, 5", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 12, "c": {"sum": 21, "median": 3, "mode": 5}}, "seed": 309711794202000633, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 5, "valid_answers": ["0, 0, 3, 3, 5, 5, 5", "0, 1, 2, 3, 5, 5, 5"]}, "cell": "stats_multiset/2", "stage": "B"}
+{"id": "po-stats_multiset-t2-eval-000028", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 7, median is 4 and unique mode is 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 7, median is 4 and unique mode is 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 4, 4, 4, 11, 12, 12", "ref": "2, 4, 4, 4, 11, 12, 12", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000119, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 12, "c": {"sum": 49, "median": 4, "mode": 4}}, "seed": 309711794202000051, "version": "po-v0", "tier": 2, "n_raw": 6, "small": 11, "valid_answers": ["2, 4, 4, 4, 11, 12, 12", "3, 4, 4, 4, 10, 12, 12", "3, 4, 4, 4, 11, 11, 12", "4, 4, 4, 4, 9, 12, 12", "4, 4, 4, 4, 10, 11, 12", "4, 4, 4, 4, 11, 11, 11"]}, "cell": "stats_multiset/2", "stage": "B"}
+{"id": "po-stats_multiset-t2-eval-000095", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 8, median is 10 and unique mode is 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 8, median is 10 and unique mode is 4. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 4, 4, 10, 10, 12, 12", "ref": "4, 4, 4, 10, 10, 12, 12", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 12, "c": {"sum": 56, "median": 10, "mode": 4}}, "seed": 309711794202000263, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 12, "valid_answers": ["4, 4, 4, 10, 10, 12, 12", "4, 4, 4, 10, 11, 11, 12"]}, "cell": "stats_multiset/2", "stage": "B"}
+{"id": "po-stats_multiset-t2-eval-000042", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 9, median is 10 and unique mode is 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 12, whose mean is 9, median is 10 and unique mode is 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 9, 10, 10, 11, 11, 11", "ref": "1, 9, 10, 10, 11, 11, 11", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 0.000357, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 12, "c": {"sum": 63, "median": 10, "mode": 11}}, "seed": 309711794202000081, "version": "po-v0", "tier": 2, "n_raw": 18, "small": 11, "valid_answers": ["1, 9, 10, 10, 11, 11, 11", "2, 8, 9, 10, 11, 11, 12", "2, 8, 10, 10, 11, 11, 11", "2, 9, 9, 10, 11, 11, 11", "3, 7, 9, 10, 11, 11, 12", "3, 7, 10, 10, 11, 11, 11", "3, 8, 9, 10, 11, 11, 11", "4, 6, 9, 10, 11, 11, 12", "4, 6, 10, 10, 11, 11, 11", "4, 7, 8, 10, 11, 11, 12", "4, 7, 9, 10, 11, 11, 11", "4, 8, 8, 10, 11, 11, 11", "5, 5, 10, 10, 11, 11, 11", "5, 6, 8, 10, 11, 11, 12", "5, 6, 9, 10, 11, 11, 11", "5, 7, 8, 10, 11, 11, 11", "6, 6, 8, 10, 11, 11, 11", "6, 7, 7, 10, 11, 11, 11"]}, "cell": "stats_multiset/2", "stage": "B"}
+{"id": "po-stats_multiset-t3-eval-000425", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 9, median is 10, unique mode is 14 and range (largest minus smallest) is 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 9, median is 10, unique mode is 14 and range (largest minus smallest) is 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 1, 10, 10, 14, 14, 14", "ref": "0, 1, 10, 10, 14, 14, 14", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 9.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 15, "c": {"sum": 63, "median": 10, "mode": 14, "range": 14}}, "seed": 309711794203000579, "version": "po-v0", "tier": 3, "n_raw": 16, "small": 14, "valid_answers": ["0, 1, 10, 10, 14, 14, 14", "0, 2, 9, 10, 14, 14, 14", "0, 3, 8, 10, 14, 14, 14", "0, 3, 9, 10, 13, 14, 14", "0, 4, 7, 10, 14, 14, 14", "0, 4, 8, 10, 13, 14, 14", "0, 4, 9, 10, 12, 14, 14", "0, 5, 6, 10, 14, 14, 14", "0, 5, 7, 10, 13, 14, 14", "0, 5, 8, 10, 12, 14, 14", "0, 5, 9, 10, 11, 14, 14", "0, 6, 7, 10, 12, 14, 14", "0, 6, 8, 10, 11, 14, 14", "1, 2, 7, 10, 14, 14, 15", "1, 3, 6, 10, 14, 14, 15", "1, 4, 5, 10, 14, 14, 15"]}, "cell": "stats_multiset/3", "stage": "B"}
+{"id": "po-stats_multiset-t3-eval-000005", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 8, median is 10, unique mode is 10 and range (largest minus smallest) is 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 8, median is 10, unique mode is 10 and range (largest minus smallest) is 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 2, 10, 10, 10, 12, 12", "ref": "0, 2, 10, 10, 10, 12, 12", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 32, "bucket": null, "guess_rate": 0.000188, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 15, "c": {"sum": 56, "median": 10, "mode": 10, "range": 12}}, "seed": 309711794203000005, "version": "po-v0", "tier": 3, "n_raw": 32, "small": 12, "valid_answers": ["0, 2, 10, 10, 10, 12, 12", "0, 3, 10, 10, 10, 11, 12", "0, 4, 9, 10, 10, 11, 12", "0, 4, 10, 10, 10, 10, 12", "0, 5, 8, 10, 10, 11, 12", "0, 5, 9, 10, 10, 10, 12", "0, 6, 7, 10, 10, 11, 12", "0, 6, 8, 10, 10, 10, 12", "0, 7, 7, 10, 10, 10, 12", "1, 1, 10, 10, 10, 11, 13", "1, 2, 8, 10, 10, 12, 13", "1, 2, 9, 10, 10, 11, 13", "1, 2, 10, 10, 10, 10, 13", "1, 3, 7, 10, 10, 12, 13", "1, 3, 8, 10, 10, 11, 13", "1, 3, 9, 10, 10, 10, 13", "1, 4, 6, 10, 10, 12, 13", "1, 4, 7, 10, 10, 11, 13", "1, 4, 8, 10, 10, 10, 13", "1, 5, 6, 10, 10, 11, 13", "1, 5, 7, 10, 10, 10, 13", "1, 6, 6, 10, 10, 10, 13", "2, 2, 8, 10, 10, 10, 14", "2, 3, 4, 10, 10, 13, 14", "2, 3, 5, 10, 10, 12, 14", "2, 3, 6, 10, 10, 11, 14", "2, 3, 7, 10, 10, 10, 14", "2, 4, 5, 10, 10, 11, 14", "2, 4, 6, 10, 10, 10, 14", "2, 5, 5, 10, 10, 10, 14", "3, 3, 5, 10, 10, 10, 15", "3, 4, 4, 10, 10, 10, 15"]}, "cell": "stats_multiset/3", "stage": "B"}
+{"id": "po-stats_multiset-t3-eval-001569", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 9, median is 9, unique mode is 7 and range (largest minus smallest) is 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 9, median is 9, unique mode is 7 and range (largest minus smallest) is 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 7, 7, 9, 11, 14, 15", "ref": "0, 7, 7, 9, 11, 14, 15", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 15, "c": {"sum": 63, "median": 9, "mode": 7, "range": 15}}, "seed": 309711794203006694, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 15, "valid_answers": ["0, 7, 7, 9, 11, 14, 15", "0, 7, 7, 9, 12, 13, 15"]}, "cell": "stats_multiset/3", "stage": "B"}
+{"id": "po-stats_multiset-t3-eval-001607", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 5, median is 4, unique mode is 9 and range (largest minus smallest) is 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 5, median is 4, unique mode is 9 and range (largest minus smallest) is 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 2, 4, 9, 9, 9", "ref": "1, 1, 2, 4, 9, 9, 9", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 15, "c": {"sum": 35, "median": 4, "mode": 9, "range": 8}}, "seed": 309711794203007229, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 9, "valid_answers": ["1, 1, 2, 4, 9, 9, 9", "1, 2, 3, 4, 7, 9, 9"]}, "cell": "stats_multiset/3", "stage": "B"}
+{"id": "po-stats_multiset-t3-eval-001685", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 4, median is 3, unique mode is 2 and range (largest minus smallest) is 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 4, median is 3, unique mode is 2 and range (largest minus smallest) is 5. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 2, 2, 3, 5, 7, 7", "ref": "2, 2, 2, 3, 5, 7, 7", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 15, "c": {"sum": 28, "median": 3, "mode": 2, "range": 5}}, "seed": 309711794203009192, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 7, "valid_answers": ["2, 2, 2, 3, 5, 7, 7", "2, 2, 2, 3, 6, 6, 7"]}, "cell": "stats_multiset/3", "stage": "B"}
+{"id": "po-stats_multiset-t3-eval-001839", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 5, median is 2, unique mode is 2 and range (largest minus smallest) is 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 5, median is 2, unique mode is 2 and range (largest minus smallest) is 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 2, 2, 2, 8, 10, 10", "ref": "1, 2, 2, 2, 8, 10, 10", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 3.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 15, "c": {"sum": 35, "median": 2, "mode": 2, "range": 9}}, "seed": 309711794203013716, "version": "po-v0", "tier": 3, "n_raw": 6, "small": 10, "valid_answers": ["1, 2, 2, 2, 8, 10, 10", "1, 2, 2, 2, 9, 9, 10", "2, 2, 2, 2, 5, 11, 11", "2, 2, 2, 2, 6, 10, 11", "2, 2, 2, 2, 7, 9, 11", "2, 2, 2, 2, 8, 8, 11"]}, "cell": "stats_multiset/3", "stage": "B"}
+{"id": "po-stats_multiset-t3-eval-001384", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 11, median is 13, unique mode is 14 and range (largest minus smallest) is 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 11, median is 13, unique mode is 14 and range (largest minus smallest) is 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 9, 13, 13, 14, 14, 14", "ref": "0, 9, 13, 13, 14, 14, 14", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 2.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 15, "c": {"sum": 77, "median": 13, "mode": 14, "range": 14}}, "seed": 309711794203004326, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 14, "valid_answers": ["0, 9, 13, 13, 14, 14, 14", "0, 10, 12, 13, 14, 14, 14", "0, 11, 11, 13, 14, 14, 14", "1, 8, 12, 13, 14, 14, 15", "1, 9, 11, 13, 14, 14, 15"]}, "cell": "stats_multiset/3", "stage": "B"}
+{"id": "po-stats_multiset-t3-eval-000310", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 5, median is 4, unique mode is 0 and range (largest minus smallest) is 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 5, median is 4, unique mode is 0 and range (largest minus smallest) is 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, 0, 4, 7, 12, 12", "ref": "0, 0, 0, 4, 7, 12, 12", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 6.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 15, "c": {"sum": 35, "median": 4, "mode": 0, "range": 12}}, "seed": 309711794203000393, "version": "po-v0", "tier": 3, "n_raw": 11, "small": 12, "valid_answers": ["0, 0, 0, 4, 7, 12, 12", "0, 0, 0, 4, 8, 11, 12", "0, 0, 0, 4, 9, 10, 12", "0, 0, 1, 4, 7, 11, 12", "0, 0, 1, 4, 8, 10, 12", "0, 0, 2, 4, 6, 11, 12", "0, 0, 2, 4, 7, 10, 12", "0, 0, 2, 4, 8, 9, 12", "0, 0, 3, 4, 5, 11, 12", "0, 0, 3, 4, 6, 10, 12", "0, 0, 3, 4, 7, 9, 12"]}, "cell": "stats_multiset/3", "stage": "B"}
+{"id": "po-stats_multiset-t3-eval-001827", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 7, median is 3, unique mode is 2 and range (largest minus smallest) is 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 7, median is 3, unique mode is 2 and range (largest minus smallest) is 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 2, 2, 3, 12, 14, 14", "ref": "2, 2, 2, 3, 12, 14, 14", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 15, "c": {"sum": 49, "median": 3, "mode": 2, "range": 12}}, "seed": 309711794203013312, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 14, "valid_answers": ["2, 2, 2, 3, 12, 14, 14", "2, 2, 2, 3, 13, 13, 14"]}, "cell": "stats_multiset/3", "stage": "B"}
+{"id": "po-stats_multiset-t3-eval-001730", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 8, median is 11, unique mode is 12 and range (largest minus smallest) is 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 8, median is 11, unique mode is 12 and range (largest minus smallest) is 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 1, 7, 11, 12, 12, 12", "ref": "1, 1, 7, 11, 12, 12, 12", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 2.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 15, "c": {"sum": 56, "median": 11, "mode": 12, "range": 11}}, "seed": 309711794203010397, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 12, "valid_answers": ["1, 1, 7, 11, 12, 12, 12", "1, 2, 6, 11, 12, 12, 12", "1, 3, 5, 11, 12, 12, 12", "1, 4, 4, 11, 12, 12, 12"]}, "cell": "stats_multiset/3", "stage": "B"}
+{"id": "po-stats_multiset-t3-eval-001831", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 8, median is 8, unique mode is 3 and range (largest minus smallest) is 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 8, median is 8, unique mode is 3 and range (largest minus smallest) is 10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3, 3, 6, 8, 11, 12, 13", "ref": "3, 3, 6, 8, 11, 12, 13", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 15, "c": {"sum": 56, "median": 8, "mode": 3, "range": 10}}, "seed": 309711794203013453, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 13, "valid_answers": ["3, 3, 6, 8, 11, 12, 13", "3, 3, 7, 8, 10, 12, 13"]}, "cell": "stats_multiset/3", "stage": "B"}
+{"id": "po-stats_multiset-t3-eval-000574", "task": "cx_po_stats_multiset", "source": "cx_po_stats_multiset", "split": "eval", "problem": "Find integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 10, median is 11, unique mode is 14 and range (largest minus smallest) is 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a <= b <= c <= d <= e <= f <= g, each between 0 and 15, whose mean is 10, median is 11, unique mode is 14 and range (largest minus smallest) is 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ..., d = ..., e = ..., f = ..., g = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 4, 11, 11, 14, 14, 14", "ref": "2, 4, 11, 11, 14, 14, 14", "family": "po_stats_multiset", "domain": "combinatorics", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 6.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_stats_multiset", "rung": "witness", "params": {"n": 7, "hi": 15, "c": {"sum": 70, "median": 11, "mode": 14, "range": 12}}, "seed": 309711794203000845, "version": "po-v0", "tier": 3, "n_raw": 11, "small": 14, "valid_answers": ["2, 4, 11, 11, 14, 14, 14", "2, 5, 10, 11, 14, 14, 14", "2, 6, 9, 11, 14, 14, 14", "2, 6, 10, 11, 13, 14, 14", "2, 7, 8, 11, 14, 14, 14", "2, 7, 9, 11, 13, 14, 14", "2, 7, 10, 11, 12, 14, 14", "2, 8, 9, 11, 12, 14, 14", "3, 4, 9, 11, 14, 14, 15", "3, 5, 8, 11, 14, 14, 15", "3, 6, 7, 11, 14, 14, 15"]}, "cell": "stats_multiset/3", "stage": "B"}
+{"id": "po-tau_range-t0-eval-001535", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 18540 <= n <= 18863 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 18540 <= n <= 18863 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "18549", "ref": "18549", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.009259, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 18540, "U": 18863, "k": 10}, "seed": 193876035900004658, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 18549, "valid_answers": ["18549", "18608", "18736"]}, "cell": "tau_range/0", "stage": "A"}
+{"id": "po-tau_range-t0-eval-001637", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 14146 <= n <= 14545 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 14146 <= n <= 14545 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "14192", "ref": "14192", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.01, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 14146, "U": 14545, "k": 10}, "seed": 193876035900004998, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 14192, "valid_answers": ["14192", "14375", "14499", "14512"]}, "cell": "tau_range/0", "stage": "A"}
+{"id": "po-tau_range-t0-eval-001550", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 18066 <= n <= 18443 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 18066 <= n <= 18443 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "18125", "ref": "18125", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.007937, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 18066, "U": 18443, "k": 10}, "seed": 193876035900004707, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 18125, "valid_answers": ["18125", "18387", "18416"]}, "cell": "tau_range/0", "stage": "A"}
+{"id": "po-tau_range-t0-eval-001269", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 11627 <= n <= 12192 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 11627 <= n <= 12192 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11632", "ref": "11632", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.017668, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 11627, "U": 12192, "k": 10}, "seed": 193876035900003807, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 11632, "valid_answers": ["11632", "11728", "11824", "11875", "11888", "12005", "12016", "12069", "12112", "12176"]}, "cell": "tau_range/0", "stage": "A"}
+{"id": "po-tau_range-t0-eval-000288", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 5786 <= n <= 6103 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 5786 <= n <= 6103 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5872", "ref": "5872", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.012579, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 5786, "U": 6103, "k": 10}, "seed": 193876035900000807, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 5872, "valid_answers": ["5872", "5913", "5968", "6064"]}, "cell": "tau_range/0", "stage": "A"}
+{"id": "po-tau_range-t0-eval-000671", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 6369 <= n <= 6969 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 6369 <= n <= 6969 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6399", "ref": "6399", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.014975, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 6369, "U": 6969, "k": 10}, "seed": 193876035900001997, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 6399, "valid_answers": ["6399", "6416", "6544", "6704", "6723", "6736", "6875", "6896", "6928"]}, "cell": "tau_range/0", "stage": "A"}
+{"id": "po-tau_range-t0-eval-001786", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 4406 <= n <= 4747 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 4406 <= n <= 4747 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4432", "ref": "4432", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.011696, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 4406, "U": 4747, "k": 10}, "seed": 193876035900005449, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4432, "valid_answers": ["4432", "4496", "4528", "4688"]}, "cell": "tau_range/0", "stage": "A"}
+{"id": "po-tau_range-t0-eval-001930", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 15176 <= n <= 15652 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 15176 <= n <= 15652 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "15248", "ref": "15248", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.012579, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 15176, "U": 15652, "k": 10}, "seed": 193876035900005901, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 15248, "valid_answers": ["15248", "15471", "15472", "15536", "15632", "15633"]}, "cell": "tau_range/0", "stage": "A"}
+{"id": "po-tau_range-t0-eval-001573", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 17538 <= n <= 17945 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 17538 <= n <= 17945 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "17552", "ref": "17552", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.009804, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 17538, "U": 17945, "k": 10}, "seed": 193876035900004776, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 17552, "valid_answers": ["17552", "17648", "17744", "17872"]}, "cell": "tau_range/0", "stage": "A"}
+{"id": "po-tau_range-t0-eval-000435", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 2081 <= n <= 2583 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 2081 <= n <= 2583 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2096", "ref": "2096", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.015905, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 2081, "U": 2583, "k": 10}, "seed": 193876035900001265, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 2096, "valid_answers": ["2096", "2192", "2224", "2349", "2384", "2416", "2511", "2512"]}, "cell": "tau_range/0", "stage": "A"}
+{"id": "po-tau_range-t0-eval-000942", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 12726 <= n <= 13061 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 12726 <= n <= 13061 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "12752", "ref": "12752", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.008929, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 12726, "U": 13061, "k": 10}, "seed": 193876035900002825, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 12752, "valid_answers": ["12752", "12944", "12976"]}, "cell": "tau_range/0", "stage": "A"}
+{"id": "po-tau_range-t0-eval-000201", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 3322 <= n <= 3899 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 3322 <= n <= 3899 that has exactly 10 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "3376", "ref": "3376", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.015571, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 3322, "U": 3899, "k": 10}, "seed": 193876035900000554, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 3376, "valid_answers": ["3376", "3483", "3568", "3632", "3664", "3728", "3807", "3824", "3856"]}, "cell": "tau_range/0", "stage": "A"}
+{"id": "po-tau_range-t1-eval-000175", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 47546 <= n <= 48327 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 47546 <= n <= 48327 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "47552", "ref": "47552", "family": "po_tau_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002558, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 47546, "U": 48327, "k": 14}, "seed": 193876035901000570, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 47552, "valid_answers": ["47552", "48064"]}, "cell": "tau_range/1", "stage": "C"}
+{"id": "po-tau_range-t1-eval-000044", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 6743 <= n <= 7803 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 6743 <= n <= 7803 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6848", "ref": "6848", "family": "po_tau_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002828, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 6743, "U": 7803, "k": 14}, "seed": 193876035901000125, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 6848, "valid_answers": ["6848", "6976", "7232"]}, "cell": "tau_range/1", "stage": "C"}
+{"id": "po-tau_range-t1-eval-000597", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 43891 <= n <= 45245 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 43891 <= n <= 45245 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "44224", "ref": "44224", "family": "po_tau_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002214, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 43891, "U": 45245, "k": 14}, "seed": 193876035901001891, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 44224, "valid_answers": ["44224", "44469", "44864"]}, "cell": "tau_range/1", "stage": "C"}
+{"id": "po-tau_range-t1-eval-001685", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 8292 <= n <= 9627 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 8292 <= n <= 9627 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8384", "ref": "8384", "family": "po_tau_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.003743, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 8292, "U": 9627, "k": 14}, "seed": 193876035901005187, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 8384, "valid_answers": ["8384", "8768", "8896", "9477", "9536"]}, "cell": "tau_range/1", "stage": "C"}
+{"id": "po-tau_range-t1-eval-002316", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 42458 <= n <= 43429 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 42458 <= n <= 43429 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "43011", "ref": "43011", "family": "po_tau_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.003086, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 42458, "U": 43429, "k": 14}, "seed": 193876035901007119, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 43011, "valid_answers": ["43011", "43072", "43328"]}, "cell": "tau_range/1", "stage": "C"}
+{"id": "po-tau_range-t1-eval-001497", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 35273 <= n <= 36350 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 35273 <= n <= 36350 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "35648", "ref": "35648", "family": "po_tau_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001855, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 35273, "U": 36350, "k": 14}, "seed": 193876035901004625, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 35648, "valid_answers": ["35648", "36032"]}, "cell": "tau_range/1", "stage": "C"}
+{"id": "po-tau_range-t1-eval-000556", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 25951 <= n <= 27372 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 25951 <= n <= 27372 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "26176", "ref": "26176", "family": "po_tau_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.002813, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 25951, "U": 27372, "k": 14}, "seed": 193876035901001771, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 26176, "valid_answers": ["26176", "26816", "26944", "26973"]}, "cell": "tau_range/1", "stage": "C"}
+{"id": "po-tau_range-t1-eval-002180", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 34222 <= n <= 34991 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 34222 <= n <= 34991 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "34263", "ref": "34263", "family": "po_tau_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002597, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 34222, "U": 34991, "k": 14}, "seed": 193876035901006701, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 34263, "valid_answers": ["34263", "34624"]}, "cell": "tau_range/1", "stage": "C"}
+{"id": "po-tau_range-t1-eval-001718", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 5129 <= n <= 6281 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 5129 <= n <= 6281 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5312", "ref": "5312", "family": "po_tau_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002602, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 5129, "U": 6281, "k": 14}, "seed": 193876035901005292, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 5312, "valid_answers": ["5312", "5696", "6208"]}, "cell": "tau_range/1", "stage": "C"}
+{"id": "po-tau_range-t1-eval-001181", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 41912 <= n <= 42995 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 41912 <= n <= 42995 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "42176", "ref": "42176", "family": "po_tau_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001845, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 41912, "U": 42995, "k": 14}, "seed": 193876035901003660, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 42176, "valid_answers": ["42176", "42304"]}, "cell": "tau_range/1", "stage": "C"}
+{"id": "po-tau_range-t1-eval-002335", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 46016 <= n <= 47244 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 46016 <= n <= 47244 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "46016", "ref": "46016", "family": "po_tau_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.003255, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 46016, "U": 47244, "k": 14}, "seed": 193876035901007191, "version": "po-v0", "tier": 1, "n_raw": 4, "small": 46016, "valid_answers": ["46016", "46528", "46875", "46912"]}, "cell": "tau_range/1", "stage": "C"}
+{"id": "po-tau_range-t1-eval-001085", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 25904 <= n <= 26944 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 25904 <= n <= 26944 that has exactly 14 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "26176", "ref": "26176", "family": "po_tau_range", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002882, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 25904, "U": 26944, "k": 14}, "seed": 193876035901003353, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 26176, "valid_answers": ["26176", "26816", "26944"]}, "cell": "tau_range/1", "stage": "C"}
+{"id": "po-tau_range-t2-eval-000396", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 44187 <= n <= 46327 that has exactly 18 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 44187 <= n <= 46327 that has exactly 18 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "44244", "ref": "44244", "family": "po_tau_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 28, "bucket": null, "guess_rate": 0.013078, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 44187, "U": 46327, "k": 18}, "seed": 193876035902000969, "version": "po-v0", "tier": 2, "n_raw": 28, "small": 44244, "valid_answers": ["44244", "44288", "44300", "44316", "44325", "44492", "44532", "44541", "44649", "44652", "44764", "44775", "44884", "44900", "44964", "45292", "45324", "45325", "45423", "45668", "45700", "45824", "45927", "45972", "46044", "46100", "46188", "46300"]}, "cell": "tau_range/2", "stage": "C"}
+{"id": "po-tau_range-t2-eval-001574", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 16367 <= n <= 18446 that has exactly 18 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 16367 <= n <= 18446 that has exactly 18 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "16425", "ref": "16425", "family": "po_tau_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 30, "bucket": null, "guess_rate": 0.014423, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 16367, "U": 18446, "k": 18}, "seed": 193876035902003919, "version": "po-v0", "tier": 2, "n_raw": 30, "small": 16425, "valid_answers": ["16425", "16428", "16452", "16562", "16596", "16668", "16700", "16731", "16812", "16820", "16928", "17152", "17244", "17298", "17300", "17444", "17532", "17676", "17775", "17787", "17900", "17908", "17964", "18050", "18081", "18100", "18108", "18176", "18207", "18324"]}, "cell": "tau_range/2", "stage": "C"}
+{"id": "po-tau_range-t2-eval-001537", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 46898 <= n <= 48546 that has exactly 18 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 46898 <= n <= 48546 that has exactly 18 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "46908", "ref": "46908", "family": "po_tau_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.009703, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 46898, "U": 48546, "k": 18}, "seed": 193876035902003830, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 46908, "valid_answers": ["46908", "46948", "47052", "47068", "47151", "47187", "47236", "47396", "47475", "47484", "47556", "47772", "47900", "47996", "48050", "48069"]}, "cell": "tau_range/2", "stage": "C"}
+{"id": "po-tau_range-t2-eval-002382", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 29718 <= n <= 31827 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 29718 <= n <= 31827 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "29840", "ref": "29840", "family": "po_tau_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 36, "bucket": null, "guess_rate": 0.017062, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 29718, "U": 31827, "k": 20}, "seed": 193876035902005845, "version": "po-v0", "tier": 2, "n_raw": 36, "small": 29840, "valid_answers": ["29840", "30051", "30128", "30208", "30256", "30288", "30320", "30352", "30448", "30537", "30544", "30640", "30704", "30736", "30768", "30832", "30864", "30942", "30992", "31024", "31056", "31088", "31120", "31232", "31266", "31312", "31344", "31376", "31408", "31472", "31504", "31632", "31671", "31696", "31728", "31760"]}, "cell": "tau_range/2", "stage": "C"}
+{"id": "po-tau_range-t2-eval-000212", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 13432 <= n <= 15457 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 13432 <= n <= 15457 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "13446", "ref": "13446", "family": "po_tau_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 32, "bucket": null, "guess_rate": 0.015795, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 13432, "U": 15457, "k": 20}, "seed": 193876035902000499, "version": "po-v0", "tier": 2, "n_raw": 32, "small": 13446, "valid_answers": ["13446", "13488", "13584", "13616", "13750", "13840", "13904", "13936", "14064", "14224", "14288", "14320", "14384", "14406", "14416", "14418", "14480", "14608", "14672", "14736", "14768", "14848", "14928", "14985", "15024", "15088", "15147", "15184", "15216", "15280", "15344", "15440"]}, "cell": "tau_range/2", "stage": "C"}
+{"id": "po-tau_range-t2-eval-002236", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 61724 <= n <= 64258 that has exactly 18 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 61724 <= n <= 64258 that has exactly 18 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "61852", "ref": "61852", "family": "po_tau_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 28, "bucket": null, "guess_rate": 0.011045, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 61724, "U": 64258, "k": 18}, "seed": 193876035902005495, "version": "po-v0", "tier": 2, "n_raw": 28, "small": 61852, "valid_answers": ["61852", "61893", "61900", "61956", "62028", "62092", "62132", "62325", "62361", "62388", "62658", "62676", "62892", "63075", "63100", "63108", "63175", "63225", "63324", "63404", "63675", "63916", "63948", "63972", "64100", "64188", "64251", "64256"]}, "cell": "tau_range/2", "stage": "C"}
+{"id": "po-tau_range-t2-eval-002439", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 9257 <= n <= 10767 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 9257 <= n <= 10767 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9264", "ref": "9264", "family": "po_tau_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 22, "bucket": null, "guess_rate": 0.01456, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 9257, "U": 10767, "k": 20}, "seed": 193876035902005985, "version": "po-v0", "tier": 2, "n_raw": 22, "small": 9264, "valid_answers": ["9264", "9296", "9315", "9328", "9424", "9456", "9552", "9558", "9639", "9728", "9776", "9882", "9968", "10064", "10125", "10128", "10160", "10384", "10480", "10672", "10704", "10736"]}, "cell": "tau_range/2", "stage": "C"}
+{"id": "po-tau_range-t2-eval-001422", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 44035 <= n <= 46531 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 44035 <= n <= 46531 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "44112", "ref": "44112", "family": "po_tau_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 38, "bucket": null, "guess_rate": 0.015218, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 44035, "U": 46531, "k": 20}, "seed": 193876035902003518, "version": "po-v0", "tier": 2, "n_raw": 38, "small": 44112, "valid_answers": ["44112", "44144", "44145", "44176", "44336", "44368", "44464", "44560", "44592", "44631", "44793", "44874", "44912", "44976", "45008", "45040", "45168", "45232", "45279", "45296", "45424", "45456", "45520", "45522", "45568", "45680", "45744", "45765", "45808", "45846", "45872", "45904", "46096", "46160", "46250", "46288", "46384", "46416"]}, "cell": "tau_range/2", "stage": "C"}
+{"id": "po-tau_range-t2-eval-000414", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 55436 <= n <= 57776 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 55436 <= n <= 57776 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "55485", "ref": "55485", "family": "po_tau_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 36, "bucket": null, "guess_rate": 0.015378, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 55436, "U": 57776, "k": 20}, "seed": 193876035902001016, "version": "po-v0", "tier": 2, "n_raw": 36, "small": 55485, "valid_answers": ["55485", "55568", "55792", "55808", "55809", "55824", "55888", "55952", "56048", "56080", "56208", "56214", "56295", "56336", "56368", "56457", "56538", "56688", "56720", "56816", "56875", "56943", "56976", "57008", "57104", "57186", "57264", "57267", "57392", "57424", "57520", "57584", "57616", "57648", "57753", "57776"]}, "cell": "tau_range/2", "stage": "C"}
+{"id": "po-tau_range-t2-eval-000052", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 54998 <= n <= 57639 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 54998 <= n <= 57639 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "54999", "ref": "54999", "family": "po_tau_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 39, "bucket": null, "guess_rate": 0.014762, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 54998, "U": 57639, "k": 20}, "seed": 193876035902000118, "version": "po-v0", "tier": 2, "n_raw": 39, "small": 54999, "valid_answers": ["54999", "55024", "55088", "55248", "55280", "55344", "55485", "55568", "55792", "55808", "55809", "55824", "55888", "55952", "56048", "56080", "56208", "56214", "56295", "56336", "56368", "56457", "56538", "56688", "56720", "56816", "56875", "56943", "56976", "57008", "57104", "57186", "57264", "57267", "57392", "57424", "57520", "57584", "57616"]}, "cell": "tau_range/2", "stage": "C"}
+{"id": "po-tau_range-t2-eval-001828", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 37542 <= n <= 40356 that has exactly 18 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 37542 <= n <= 40356 that has exactly 18 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "37575", "ref": "37575", "family": "po_tau_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 38, "bucket": null, "guess_rate": 0.013499, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 37542, "U": 40356, "k": 18}, "seed": 193876035902004482, "version": "po-v0", "tier": 2, "n_raw": 38, "small": 37575, "valid_answers": ["37575", "37764", "37828", "37836", "37845", "37900", "37975", "38144", "38196", "38236", "38268", "38300", "38332", "38484", "38612", "38656", "38900", "38925", "39004", "39132", "39249", "39276", "39325", "39348", "39492", "39675", "39700", "39708", "39762", "39884", "39924", "40100", "40172", "40192", "40204", "40212", "40275", "40293"]}, "cell": "tau_range/2", "stage": "C"}
+{"id": "po-tau_range-t2-eval-002414", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 64012 <= n <= 65873 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 64012 <= n <= 65873 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "64071", "ref": "64071", "family": "po_tau_range", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 28, "bucket": null, "guess_rate": 0.015038, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 64012, "U": 65873, "k": 20}, "seed": 193876035902005924, "version": "po-v0", "tier": 2, "n_raw": 28, "small": 64071, "valid_answers": ["64071", "64144", "64233", "64314", "64496", "64528", "64592", "64624", "64688", "64719", "64720", "64827", "64880", "64962", "64976", "65008", "65024", "65043", "65104", "65328", "65392", "65552", "65616", "65648", "65680", "65744", "65840", "65872"]}, "cell": "tau_range/2", "stage": "C"}
+{"id": "po-tau_range-t3-eval-000824", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 87034 <= n <= 90171 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 87034 <= n <= 90171 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "87152", "ref": "87152", "family": "po_tau_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 51, "bucket": null, "guess_rate": 0.016252, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 87034, "U": 90171, "k": 20}, "seed": 193876035903002355, "version": "po-v0", "tier": 3, "n_raw": 51, "small": 87152, "valid_answers": ["87152", "87280", "87344", "87376", "87399", "87440", "87504", "87561", "87568", "87642", "87760", "87824", "87846", "87888", "87952", "88125", "88144", "88208", "88240", "88528", "88576", "88614", "88624", "88656", "88688", "88720", "88750", "88784", "88976", "89019", "89072", "89264", "89328", "89360", "89375", "89392", "89552", "89584", "89616", "89648", "89744", "89776", "89808", "89840", "89872", "89904", "89991", "90032", "90064", "90096", "90128"]}, "cell": "tau_range/3", "stage": "C"}
+{"id": "po-tau_range-t3-eval-002016", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 16659 <= n <= 20712 that has exactly 30 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 16659 <= n <= 20712 that has exactly 30 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "17172", "ref": "17172", "family": "po_tau_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.003947, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 16659, "U": 20712, "k": 30}, "seed": 193876035903005605, "version": "po-v0", "tier": 3, "n_raw": 16, "small": 17172, "valid_answers": ["17172", "17200", "17328", "17500", "18032", "18288", "18800", "18864", "18928", "19116", "19602", "19728", "19764", "19845", "20000", "20016"]}, "cell": "tau_range/3", "stage": "C"}
+{"id": "po-tau_range-t3-eval-001046", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 12765 <= n <= 16118 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 12765 <= n <= 16118 that has exactly 20 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "12784", "ref": "12784", "family": "po_tau_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 53, "bucket": null, "guess_rate": 0.015802, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 12765, "U": 16118, "k": 20}, "seed": 193876035903002903, "version": "po-v0", "tier": 3, "n_raw": 53, "small": 12784, "valid_answers": ["12784", "12798", "12848", "12912", "13008", "13040", "13041", "13072", "13125", "13296", "13360", "13446", "13488", "13584", "13616", "13750", "13840", "13904", "13936", "14064", "14224", "14288", "14320", "14384", "14406", "14416", "14418", "14480", "14608", "14672", "14736", "14768", "14848", "14928", "14985", "15024", "15088", "15147", "15184", "15216", "15280", "15344", "15440", "15568", "15664", "15714", "15760", "15824", "15872", "15888", "15920", "16048", "16112"]}, "cell": "tau_range/3", "stage": "C"}
+{"id": "po-tau_range-t3-eval-002148", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 53042 <= n <= 58501 that has exactly 30 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 53042 <= n <= 58501 that has exactly 30 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "53712", "ref": "53712", "family": "po_tau_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.00293, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 53042, "U": 58501, "k": 30}, "seed": 193876035903006002, "version": "po-v0", "tier": 3, "n_raw": 16, "small": 53712, "valid_answers": ["53712", "54108", "54576", "54800", "55152", "55600", "55664", "56016", "56052", "56144", "57168", "57232", "57500", "57744", "57996", "58482"]}, "cell": "tau_range/3", "stage": "C"}
+{"id": "po-tau_range-t3-eval-001998", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 91307 <= n <= 95224 that has exactly 30 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 91307 <= n <= 95224 that has exactly 30 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "91600", "ref": "91600", "family": "po_tau_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 0.003828, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 91307, "U": 95224, "k": 30}, "seed": 193876035903005561, "version": "po-v0", "tier": 3, "n_raw": 15, "small": 91600, "valid_answers": ["91600", "91692", "91875", "92304", "92500", "92592", "93104", "93168", "93200", "94032", "94192", "94896", "94932", "95175", "95184"]}, "cell": "tau_range/3", "stage": "C"}
+{"id": "po-tau_range-t3-eval-001292", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 72739 <= n <= 75948 that has exactly 30 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 72739 <= n <= 75948 that has exactly 30 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "73125", "ref": "73125", "family": "po_tau_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.003115, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 72739, "U": 75948, "k": 30}, "seed": 193876035903003633, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 73125, "valid_answers": ["73125", "73296", "73548", "74196", "74925", "75024", "75088", "75312", "75411", "75492"]}, "cell": "tau_range/3", "stage": "C"}
+{"id": "po-tau_range-t3-eval-000104", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 58826 <= n <= 62042 that has exactly 30 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 58826 <= n <= 62042 that has exactly 30 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "58896", "ref": "58896", "family": "po_tau_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.004041, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 58826, "U": 62042, "k": 30}, "seed": 193876035903000304, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 58896, "valid_answers": ["58896", "59248", "59600", "60016", "60112", "60336", "60400", "60624", "61250", "61875", "61884", "61936", "61952"]}, "cell": "tau_range/3", "stage": "C"}
+{"id": "po-tau_range-t3-eval-000663", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 36540 <= n <= 41897 that has exactly 28 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 36540 <= n <= 41897 that has exactly 28 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "36672", "ref": "36672", "family": "po_tau_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 0.0028, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 36540, "U": 41897, "k": 28}, "seed": 193876035903001889, "version": "po-v0", "tier": 3, "n_raw": 15, "small": 36672, "valid_answers": ["36672", "37056", "37184", "37312", "37696", "37824", "38208", "39104", "39872", "40095", "40256", "40512", "40640", "40960", "41536"]}, "cell": "tau_range/3", "stage": "C"}
+{"id": "po-tau_range-t3-eval-001933", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 70050 <= n <= 75586 that has exactly 22 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 70050 <= n <= 75586 that has exactly 22 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "72704", "ref": "72704", "family": "po_tau_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000361, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 70050, "U": 75586, "k": 22}, "seed": 193876035903005391, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 72704, "valid_answers": ["72704", "74752"]}, "cell": "tau_range/3", "stage": "C"}
+{"id": "po-tau_range-t3-eval-000887", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 26936 <= n <= 32025 that has exactly 22 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 26936 <= n <= 32025 that has exactly 22 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "29696", "ref": "29696", "family": "po_tau_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000393, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 26936, "U": 32025, "k": 22}, "seed": 193876035903002522, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 29696, "valid_answers": ["29696", "31744"]}, "cell": "tau_range/3", "stage": "C"}
+{"id": "po-tau_range-t3-eval-002166", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 48569 <= n <= 54288 that has exactly 28 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 48569 <= n <= 54288 that has exactly 28 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "48832", "ref": "48832", "family": "po_tau_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.003322, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 48569, "U": 54288, "k": 28}, "seed": 193876035903006052, "version": "po-v0", "tier": 3, "n_raw": 19, "small": 48832, "valid_answers": ["48832", "49088", "49344", "49856", "49984", "50240", "50496", "50624", "50752", "51136", "51392", "51648", "52032", "52160", "52288", "53184", "53440", "53946", "53952"]}, "cell": "tau_range/3", "stage": "C"}
+{"id": "po-tau_range-t3-eval-000278", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 59145 <= n <= 64859 that has exactly 28 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind an integer n with 59145 <= n <= 64859 that has exactly 28 positive divisors. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "59712", "ref": "59712", "family": "po_tau_range", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 20, "bucket": "9-20", "guess_rate": 0.0035, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 59145, "U": 64859, "k": 28}, "seed": 193876035903000770, "version": "po-v0", "tier": 3, "n_raw": 20, "small": 59712, "valid_answers": ["59712", "59778", "60096", "60352", "60736", "60864", "61120", "61376", "61760", "61965", "62272", "62656", "62694", "63040", "63296", "63552", "63680", "64192", "64448", "64704"]}, "cell": "tau_range/3", "stage": "C"}
+{"id": "po-totient_inv-t3-eval-000101", "task": "cx_po_totient_inv", "source": "cx_po_totient_inv", "split": "eval", "problem": "Find a positive integer n with \\varphi(n) = 292, where \\varphi is Euler's totient function. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n with \\varphi(n) = 292, where \\varphi is Euler's totient function. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "293", "ref": "293", "family": "po_totient_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_totient_inv", "rung": "witness", "params": {"m": 292}, "seed": 55511483003000622, "version": "po-v0", "tier": 3, "n_raw": 2, "small": 293, "valid_answers": ["293", "586"]}, "cell": "totient_inv/3", "stage": "B"}
+{"id": "po-totient_inv-t3-eval-000002", "task": "cx_po_totient_inv", "source": "cx_po_totient_inv", "split": "eval", "problem": "Find a positive integer n with \\varphi(n) = 420, where \\varphi is Euler's totient function. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n with \\varphi(n) = 420, where \\varphi is Euler's totient function. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "421", "ref": "421", "family": "po_totient_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 3.7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_totient_inv", "rung": "witness", "params": {"m": 420}, "seed": 55511483003000002, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 421, "valid_answers": ["421", "473", "497", "539", "633", "639", "842", "844", "946", "994", "1078", "1266", "1278"]}, "cell": "totient_inv/3", "stage": "B"}
+{"id": "po-totient_inv-t3-eval-000024", "task": "cx_po_totient_inv", "source": "cx_po_totient_inv", "split": "eval", "problem": "Find a positive integer n with \\varphi(n) = 400, where \\varphi is Euler's totient function. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n with \\varphi(n) = 400, where \\varphi is Euler's totient function. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "401", "ref": "401", "family": "po_totient_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 4.1e-05, "solvable": true, "holdout": false, "meta": {"family": "po_totient_inv", "rung": "witness", "params": {"m": 400}, "seed": 55511483003000034, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 401, "valid_answers": ["401", "451", "505", "802", "808", "825", "902", "1000", "1010", "1100", "1212", "1500", "1650"]}, "cell": "totient_inv/3", "stage": "B"}
+{"id": "po-totient_inv-t3-eval-000095", "task": "cx_po_totient_inv", "source": "cx_po_totient_inv", "split": "eval", "problem": "Find a positive integer n with \\varphi(n) = 468, where \\varphi is Euler's totient function. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind a positive integer n with \\varphi(n) = 468, where \\varphi is Euler's totient function. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{n = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "553", "ref": "553", "family": "po_totient_inv", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 9e-06, "solvable": true, "holdout": false, "meta": {"family": "po_totient_inv", "rung": "witness", "params": {"m": 468}, "seed": 55511483003000462, "version": "po-v0", "tier": 3, "n_raw": 4, "small": 553, "valid_answers": ["553", "711", "1106", "1422"]}, "cell": "totient_inv/3", "stage": "B"}
+{"id": "po-two_sq_comp-t0-eval-000016", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (2^2 + 5^2)(3^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (2^2 + 5^2)(3^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "25, 19", "ref": "25, 19", "family": "po_two_sq_comp", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.003788, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[2, 5], [3, 5]]}, "seed": 115722981400000034, "version": "po-v0", "tier": 0, "n_raw": 16, "small": 25, "valid_answers": ["25, 19", "31, 5"]}, "cell": "two_sq_comp/0", "stage": "B"}
+{"id": "po-two_sq_comp-t0-eval-000080", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 4^2)(3^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 4^2)(3^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "16, 13", "ref": "16, 13", "family": "po_two_sq_comp", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.012987, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 4], [3, 4]]}, "seed": 115722981400000269, "version": "po-v0", "tier": 0, "n_raw": 24, "small": 16, "valid_answers": ["16, 13", "19, 8", "20, 5"]}, "cell": "two_sq_comp/0", "stage": "B"}
+{"id": "po-two_sq_comp-t0-eval-000051", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (5^2 + 5^2)(1^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (5^2 + 5^2)(1^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "30, 20", "ref": "30, 20", "family": "po_two_sq_comp", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.004267, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[5, 5], [1, 5]]}, "seed": 115722981400000129, "version": "po-v0", "tier": 0, "n_raw": 24, "small": 30, "valid_answers": ["30, 20", "34, 12", "36, 2"]}, "cell": "two_sq_comp/0", "stage": "B"}
+{"id": "po-two_sq_comp-t0-eval-000000", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (2^2 + 3^2)(2^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (2^2 + 3^2)(2^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "12, 5", "ref": "12, 5", "family": "po_two_sq_comp", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.019048, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[2, 3], [2, 3]]}, "seed": 115722981400000000, "version": "po-v0", "tier": 0, "n_raw": 12, "small": 12, "valid_answers": ["12, 5", "13, 0"]}, "cell": "two_sq_comp/0", "stage": "B"}
+{"id": "po-two_sq_comp-t0-eval-000098", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (5^2 + 5^2)(4^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (5^2 + 5^2)(4^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "32, 24", "ref": "32, 24", "family": "po_two_sq_comp", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002323, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[5, 5], [4, 4]]}, "seed": 115722981400000477, "version": "po-v0", "tier": 0, "n_raw": 12, "small": 32, "valid_answers": ["32, 24", "40, 0"]}, "cell": "two_sq_comp/0", "stage": "B"}
+{"id": "po-two_sq_comp-t0-eval-000007", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 4^2)(2^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 4^2)(2^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 10", "ref": "11, 10", "family": "po_two_sq_comp", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.016667, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 4], [2, 3]]}, "seed": 115722981400000012, "version": "po-v0", "tier": 0, "n_raw": 16, "small": 11, "valid_answers": ["11, 10", "14, 5"]}, "cell": "two_sq_comp/0", "stage": "B"}
+{"id": "po-two_sq_comp-t0-eval-000022", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (5^2 + 5^2)(2^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (5^2 + 5^2)(2^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "33, 19", "ref": "33, 19", "family": "po_two_sq_comp", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.003846, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[5, 5], [2, 5]]}, "seed": 115722981400000048, "version": "po-v0", "tier": 0, "n_raw": 24, "small": 33, "valid_answers": ["33, 19", "35, 15", "37, 9"]}, "cell": "two_sq_comp/0", "stage": "B"}
+{"id": "po-two_sq_comp-t0-eval-000034", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (3^2 + 4^2)(4^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (3^2 + 4^2)(4^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "20, 20", "ref": "20, 20", "family": "po_two_sq_comp", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004598, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[3, 4], [4, 4]]}, "seed": 115722981400000072, "version": "po-v0", "tier": 0, "n_raw": 12, "small": 20, "valid_answers": ["20, 20", "28, 4"]}, "cell": "two_sq_comp/0", "stage": "B"}
+{"id": "po-two_sq_comp-t0-eval-000083", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (2^2 + 3^2)(3^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (2^2 + 3^2)(3^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "19, 9", "ref": "19, 9", "family": "po_two_sq_comp", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.007905, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[2, 3], [3, 5]]}, "seed": 115722981400000292, "version": "po-v0", "tier": 0, "n_raw": 16, "small": 19, "valid_answers": ["19, 9", "21, 1"]}, "cell": "two_sq_comp/0", "stage": "B"}
+{"id": "po-two_sq_comp-t0-eval-000114", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (3^2 + 3^2)(5^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (3^2 + 3^2)(5^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "24, 18", "ref": "24, 18", "family": "po_two_sq_comp", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004032, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[3, 3], [5, 5]]}, "seed": 115722981400001352, "version": "po-v0", "tier": 0, "n_raw": 12, "small": 24, "valid_answers": ["24, 18", "30, 0"]}, "cell": "two_sq_comp/0", "stage": "B"}
+{"id": "po-two_sq_comp-t0-eval-000024", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (4^2 + 5^2)(3^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (4^2 + 5^2)(3^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "35, 13", "ref": "35, 13", "family": "po_two_sq_comp", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002699, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[4, 5], [3, 5]]}, "seed": 115722981400000053, "version": "po-v0", "tier": 0, "n_raw": 16, "small": 35, "valid_answers": ["35, 13", "37, 5"]}, "cell": "two_sq_comp/0", "stage": "B"}
+{"id": "po-two_sq_comp-t0-eval-000111", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (2^2 + 2^2)(5^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (2^2 + 2^2)(5^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "16, 12", "ref": "16, 12", "family": "po_two_sq_comp", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.008658, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[2, 2], [5, 5]]}, "seed": 115722981400000873, "version": "po-v0", "tier": 0, "n_raw": 12, "small": 16, "valid_answers": ["16, 12", "20, 0"]}, "cell": "two_sq_comp/0", "stage": "B"}
+{"id": "po-two_sq_comp-t1-eval-000369", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 7^2)(1^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 7^2)(1^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "25, 15", "ref": "25, 15", "family": "po_two_sq_comp", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006452, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 7], [1, 4]]}, "seed": 115722981401000576, "version": "po-v0", "tier": 1, "n_raw": 24, "small": 25, "valid_answers": ["25, 15", "27, 11", "29, 3"]}, "cell": "two_sq_comp/1", "stage": "B"}
+{"id": "po-two_sq_comp-t1-eval-000236", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (6^2 + 7^2)(1^2 + 7^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (6^2 + 7^2)(1^2 + 7^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "49, 43", "ref": "49, 43", "family": "po_two_sq_comp", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001809, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[6, 7], [1, 7]]}, "seed": 115722981401000342, "version": "po-v0", "tier": 1, "n_raw": 32, "small": 49, "valid_answers": ["49, 43", "55, 35", "61, 23", "65, 5"]}, "cell": "two_sq_comp/1", "stage": "B"}
+{"id": "po-two_sq_comp-t1-eval-000323", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 3^2)(5^2 + 7^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 3^2)(5^2 + 7^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "22, 16", "ref": "22, 16", "family": "po_two_sq_comp", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004926, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 3], [5, 7]]}, "seed": 115722981401000497, "version": "po-v0", "tier": 1, "n_raw": 16, "small": 22, "valid_answers": ["22, 16", "26, 8"]}, "cell": "two_sq_comp/1", "stage": "B"}
+{"id": "po-two_sq_comp-t1-eval-000801", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (3^2 + 4^2)(6^2 + 8^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (3^2 + 4^2)(6^2 + 8^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "40, 30", "ref": "40, 30", "family": "po_two_sq_comp", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002262, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[3, 4], [6, 8]]}, "seed": 115722981401002524, "version": "po-v0", "tier": 1, "n_raw": 20, "small": 40, "valid_answers": ["40, 30", "48, 14", "50, 0"]}, "cell": "two_sq_comp/1", "stage": "B"}
+{"id": "po-two_sq_comp-t1-eval-000497", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (3^2 + 4^2)(4^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (3^2 + 4^2)(4^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "20, 20", "ref": "20, 20", "family": "po_two_sq_comp", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004598, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[3, 4], [4, 4]]}, "seed": 115722981401000919, "version": "po-v0", "tier": 1, "n_raw": 12, "small": 20, "valid_answers": ["20, 20", "28, 4"]}, "cell": "two_sq_comp/1", "stage": "B"}
+{"id": "po-two_sq_comp-t1-eval-000212", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 6^2)(4^2 + 8^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 6^2)(4^2 + 8^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "44, 32", "ref": "44, 32", "family": "po_two_sq_comp", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001299, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 6], [4, 8]]}, "seed": 115722981401000295, "version": "po-v0", "tier": 1, "n_raw": 16, "small": 44, "valid_answers": ["44, 32", "52, 16"]}, "cell": "two_sq_comp/1", "stage": "B"}
+{"id": "po-two_sq_comp-t1-eval-000426", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (7^2 + 7^2)(5^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (7^2 + 7^2)(5^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "56, 42", "ref": "56, 42", "family": "po_two_sq_comp", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000782, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[7, 7], [5, 5]]}, "seed": 115722981401000723, "version": "po-v0", "tier": 1, "n_raw": 12, "small": 56, "valid_answers": ["56, 42", "70, 0"]}, "cell": "two_sq_comp/1", "stage": "B"}
+{"id": "po-two_sq_comp-t1-eval-000888", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (3^2 + 3^2)(6^2 + 8^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (3^2 + 3^2)(6^2 + 8^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "30, 30", "ref": "30, 30", "family": "po_two_sq_comp", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002114, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[3, 3], [6, 8]]}, "seed": 115722981401004772, "version": "po-v0", "tier": 1, "n_raw": 12, "small": 30, "valid_answers": ["30, 30", "42, 6"]}, "cell": "two_sq_comp/1", "stage": "B"}
+{"id": "po-two_sq_comp-t1-eval-000203", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 8^2)(7^2 + 7^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 8^2)(7^2 + 7^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "63, 49", "ref": "63, 49", "family": "po_two_sq_comp", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000617, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 8], [7, 7]]}, "seed": 115722981401000278, "version": "po-v0", "tier": 1, "n_raw": 16, "small": 63, "valid_answers": ["63, 49", "77, 21"]}, "cell": "two_sq_comp/1", "stage": "B"}
+{"id": "po-two_sq_comp-t1-eval-000658", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (5^2 + 7^2)(1^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (5^2 + 7^2)(1^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "27, 23", "ref": "27, 23", "family": "po_two_sq_comp", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.003003, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[5, 7], [1, 4]]}, "seed": 115722981401001441, "version": "po-v0", "tier": 1, "n_raw": 16, "small": 27, "valid_answers": ["27, 23", "33, 13"]}, "cell": "two_sq_comp/1", "stage": "B"}
+{"id": "po-two_sq_comp-t1-eval-000071", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (3^2 + 7^2)(5^2 + 7^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (3^2 + 7^2)(5^2 + 7^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "56, 34", "ref": "56, 34", "family": "po_two_sq_comp", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000905, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[3, 7], [5, 7]]}, "seed": 115722981401000087, "version": "po-v0", "tier": 1, "n_raw": 16, "small": 56, "valid_answers": ["56, 34", "64, 14"]}, "cell": "two_sq_comp/1", "stage": "B"}
+{"id": "po-two_sq_comp-t1-eval-000694", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (2^2 + 3^2)(5^2 + 6^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (2^2 + 3^2)(5^2 + 6^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "27, 8", "ref": "27, 8", "family": "po_two_sq_comp", "domain": "number", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004598, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[2, 3], [5, 6]]}, "seed": 115722981401001658, "version": "po-v0", "tier": 1, "n_raw": 16, "small": 27, "valid_answers": ["27, 8", "28, 3"]}, "cell": "two_sq_comp/1", "stage": "B"}
+{"id": "po-two_sq_comp-t2-eval-001049", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 3^2)(1^2 + 1^2)(2^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 3^2)(1^2 + 1^2)(2^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "16, 12", "ref": "16, 12", "family": "po_two_sq_comp", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.008658, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 3], [1, 1], [2, 4]]}, "seed": 115722981402001465, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 16, "valid_answers": ["16, 12", "20, 0"]}, "cell": "two_sq_comp/2", "stage": "C"}
+{"id": "po-two_sq_comp-t2-eval-001447", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (3^2 + 4^2)(2^2 + 4^2)(1^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (3^2 + 4^2)(2^2 + 4^2)(1^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "50, 50", "ref": "50, 50", "family": "po_two_sq_comp", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001174, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[3, 4], [2, 4], [1, 3]]}, "seed": 115722981402002262, "version": "po-v0", "tier": 2, "n_raw": 20, "small": 50, "valid_answers": ["50, 50", "62, 34", "70, 10"]}, "cell": "two_sq_comp/2", "stage": "C"}
+{"id": "po-two_sq_comp-t2-eval-002441", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (2^2 + 3^2)(2^2 + 5^2)(1^2 + 1^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (2^2 + 3^2)(2^2 + 5^2)(1^2 + 1^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "23, 15", "ref": "23, 15", "family": "po_two_sq_comp", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.004926, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[2, 3], [2, 5], [1, 1]]}, "seed": 115722981402007202, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 23, "valid_answers": ["23, 15", "27, 5"]}, "cell": "two_sq_comp/2", "stage": "C"}
+{"id": "po-two_sq_comp-t2-eval-001093", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 4^2)(4^2 + 5^2)(1^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 4^2)(4^2 + 5^2)(1^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "101, 89", "ref": "101, 89", "family": "po_two_sq_comp", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000436, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 4], [4, 5], [1, 5]]}, "seed": 115722981402001541, "version": "po-v0", "tier": 2, "n_raw": 32, "small": 101, "valid_answers": ["101, 89", "109, 79", "121, 59", "131, 31"]}, "cell": "two_sq_comp/2", "stage": "C"}
+{"id": "po-two_sq_comp-t2-eval-001423", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (4^2 + 4^2)(3^2 + 4^2)(4^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (4^2 + 4^2)(3^2 + 4^2)(4^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "132, 124", "ref": "132, 124", "family": "po_two_sq_comp", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.00018, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[4, 4], [3, 4], [4, 5]]}, "seed": 115722981402002195, "version": "po-v0", "tier": 2, "n_raw": 24, "small": 132, "valid_answers": ["132, 124", "156, 92", "180, 20"]}, "cell": "two_sq_comp/2", "stage": "C"}
+{"id": "po-two_sq_comp-t2-eval-000246", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (4^2 + 4^2)(1^2 + 4^2)(5^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (4^2 + 4^2)(1^2 + 4^2)(5^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "128, 104", "ref": "128, 104", "family": "po_two_sq_comp", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000219, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[4, 4], [1, 4], [5, 5]]}, "seed": 115722981402000276, "version": "po-v0", "tier": 2, "n_raw": 24, "small": 128, "valid_answers": ["128, 104", "152, 64", "160, 40"]}, "cell": "two_sq_comp/2", "stage": "C"}
+{"id": "po-two_sq_comp-t2-eval-000689", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (2^2 + 3^2)(1^2 + 2^2)(2^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (2^2 + 3^2)(1^2 + 2^2)(2^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "34, 27", "ref": "34, 27", "family": "po_two_sq_comp", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00404, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[2, 3], [1, 2], [2, 5]]}, "seed": 115722981402000854, "version": "po-v0", "tier": 2, "n_raw": 32, "small": 34, "valid_answers": ["34, 27", "38, 21", "42, 11", "43, 6"]}, "cell": "two_sq_comp/2", "stage": "C"}
+{"id": "po-two_sq_comp-t2-eval-001252", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 2^2)(1^2 + 5^2)(3^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 2^2)(1^2 + 5^2)(3^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "42, 24", "ref": "42, 24", "family": "po_two_sq_comp", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001633, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 2], [1, 5], [3, 3]]}, "seed": 115722981402001815, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 42, "valid_answers": ["42, 24", "48, 6"]}, "cell": "two_sq_comp/2", "stage": "C"}
+{"id": "po-two_sq_comp-t2-eval-000583", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 4^2)(4^2 + 5^2)(1^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 4^2)(4^2 + 5^2)(1^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "61, 57", "ref": "61, 57", "family": "po_two_sq_comp", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.00112, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 4], [4, 5], [1, 3]]}, "seed": 115722981402000699, "version": "po-v0", "tier": 2, "n_raw": 32, "small": 61, "valid_answers": ["61, 57", "69, 47", "79, 27", "83, 9"]}, "cell": "two_sq_comp/2", "stage": "C"}
+{"id": "po-two_sq_comp-t2-eval-002087", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 3^2)(2^2 + 5^2)(2^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 3^2)(2^2 + 5^2)(2^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "49, 37", "ref": "49, 37", "family": "po_two_sq_comp", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.002048, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 3], [2, 5], [2, 3]]}, "seed": 115722981402004449, "version": "po-v0", "tier": 2, "n_raw": 32, "small": 49, "valid_answers": ["49, 37", "53, 31", "59, 17", "61, 7"]}, "cell": "two_sq_comp/2", "stage": "C"}
+{"id": "po-two_sq_comp-t2-eval-000298", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (4^2 + 5^2)(5^2 + 5^2)(2^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (4^2 + 5^2)(5^2 + 5^2)(2^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "170, 110", "ref": "170, 110", "family": "po_two_sq_comp", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000193, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[4, 5], [5, 5], [2, 4]]}, "seed": 115722981402000336, "version": "po-v0", "tier": 2, "n_raw": 32, "small": 170, "valid_answers": ["170, 110", "190, 70", "194, 58", "202, 14"]}, "cell": "two_sq_comp/2", "stage": "C"}
+{"id": "po-two_sq_comp-t2-eval-000797", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 4^2)(1^2 + 2^2)(1^2 + 1^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 4^2)(1^2 + 2^2)(1^2 + 1^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "11, 7", "ref": "11, 7", "family": "po_two_sq_comp", "domain": "number", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.019048, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 4], [1, 2], [1, 1]]}, "seed": 115722981402001014, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 11, "valid_answers": ["11, 7", "13, 1"]}, "cell": "two_sq_comp/2", "stage": "C"}
+{"id": "po-two_sq_comp-t3-eval-000387", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (4^2 + 6^2)(5^2 + 6^2)(2^2 + 7^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (4^2 + 6^2)(5^2 + 6^2)(2^2 + 7^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "346, 220", "ref": "346, 220", "family": "po_two_sq_comp", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 4.7e-05, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[4, 6], [5, 6], [2, 7]]}, "seed": 115722981403000395, "version": "po-v0", "tier": 3, "n_raw": 32, "small": 346, "valid_answers": ["346, 220", "380, 154", "404, 70", "410, 4"]}, "cell": "two_sq_comp/3", "stage": "C"}
+{"id": "po-two_sq_comp-t3-eval-001020", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 3^2)(1^2 + 4^2)(2^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 3^2)(1^2 + 4^2)(2^2 + 5^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "57, 41", "ref": "57, 41", "family": "po_two_sq_comp", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001565, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 3], [1, 4], [2, 5]]}, "seed": 115722981403001065, "version": "po-v0", "tier": 3, "n_raw": 32, "small": 57, "valid_answers": ["57, 41", "63, 31", "67, 21", "69, 13"]}, "cell": "two_sq_comp/3", "stage": "C"}
+{"id": "po-two_sq_comp-t3-eval-002255", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 7^2)(1^2 + 6^2)(6^2 + 6^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 7^2)(1^2 + 6^2)(6^2 + 6^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "264, 252", "ref": "264, 252", "family": "po_two_sq_comp", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 4.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 7], [1, 6], [6, 6]]}, "seed": 115722981403002392, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 264, "valid_answers": ["264, 252", "324, 168", "360, 60"]}, "cell": "two_sq_comp/3", "stage": "C"}
+{"id": "po-two_sq_comp-t3-eval-000279", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (4^2 + 6^2)(3^2 + 8^2)(1^2 + 7^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (4^2 + 6^2)(3^2 + 8^2)(1^2 + 7^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "314, 302", "ref": "314, 302", "family": "po_two_sq_comp", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 6.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[4, 6], [3, 8], [1, 7]]}, "seed": 115722981403000286, "version": "po-v0", "tier": 3, "n_raw": 48, "small": 314, "valid_answers": ["314, 302", "370, 230", "386, 202", "406, 158", "430, 70", "434, 38"]}, "cell": "two_sq_comp/3", "stage": "C"}
+{"id": "po-two_sq_comp-t3-eval-000381", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 7^2)(4^2 + 5^2)(4^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 7^2)(4^2 + 5^2)(4^2 + 4^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "200, 160", "ref": "200, 160", "family": "po_two_sq_comp", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 7], [4, 5], [4, 4]]}, "seed": 115722981403000389, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 200, "valid_answers": ["200, 160", "248, 64", "256, 8"]}, "cell": "two_sq_comp/3", "stage": "C"}
+{"id": "po-two_sq_comp-t3-eval-001849", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (6^2 + 8^2)(3^2 + 6^2)(2^2 + 2^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (6^2 + 8^2)(3^2 + 6^2)(2^2 + 2^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "156, 108", "ref": "156, 108", "family": "po_two_sq_comp", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00011, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[6, 8], [3, 6], [2, 2]]}, "seed": 115722981403001956, "version": "po-v0", "tier": 3, "n_raw": 16, "small": 156, "valid_answers": ["156, 108", "180, 60"]}, "cell": "two_sq_comp/3", "stage": "C"}
+{"id": "po-two_sq_comp-t3-eval-002385", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (2^2 + 3^2)(2^2 + 6^2)(3^2 + 6^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (2^2 + 3^2)(2^2 + 6^2)(3^2 + 6^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "114, 102", "ref": "114, 102", "family": "po_two_sq_comp", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000255, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[2, 3], [2, 6], [3, 6]]}, "seed": 115722981403002533, "version": "po-v0", "tier": 3, "n_raw": 24, "small": 114, "valid_answers": ["114, 102", "138, 66", "150, 30"]}, "cell": "two_sq_comp/3", "stage": "C"}
+{"id": "po-two_sq_comp-t3-eval-000161", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (1^2 + 1^2)(2^2 + 7^2)(1^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (1^2 + 1^2)(2^2 + 7^2)(1^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "24, 22", "ref": "24, 22", "family": "po_two_sq_comp", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.003565, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[1, 1], [2, 7], [1, 3]]}, "seed": 115722981403000164, "version": "po-v0", "tier": 3, "n_raw": 16, "small": 24, "valid_answers": ["24, 22", "32, 6"]}, "cell": "two_sq_comp/3", "stage": "C"}
+{"id": "po-two_sq_comp-t3-eval-002313", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (2^2 + 6^2)(2^2 + 7^2)(3^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (2^2 + 6^2)(2^2 + 7^2)(3^2 + 3^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "144, 132", "ref": "144, 132", "family": "po_two_sq_comp", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000104, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[2, 6], [2, 7], [3, 3]]}, "seed": 115722981403002455, "version": "po-v0", "tier": 3, "n_raw": 16, "small": 144, "valid_answers": ["144, 132", "192, 36"]}, "cell": "two_sq_comp/3", "stage": "C"}
+{"id": "po-two_sq_comp-t3-eval-001869", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (2^2 + 8^2)(6^2 + 8^2)(4^2 + 8^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (2^2 + 8^2)(6^2 + 8^2)(4^2 + 8^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "560, 480", "ref": "560, 480", "family": "po_two_sq_comp", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[2, 8], [6, 8], [4, 8]]}, "seed": 115722981403001976, "version": "po-v0", "tier": 3, "n_raw": 32, "small": 560, "valid_answers": ["560, 480", "672, 304", "720, 160", "736, 48"]}, "cell": "two_sq_comp/3", "stage": "C"}
+{"id": "po-two_sq_comp-t3-eval-001624", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (4^2 + 6^2)(2^2 + 7^2)(6^2 + 8^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (4^2 + 6^2)(2^2 + 7^2)(6^2 + 8^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "400, 340", "ref": "400, 340", "family": "po_two_sq_comp", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 4.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[4, 6], [2, 7], [6, 8]]}, "seed": 115722981403001714, "version": "po-v0", "tier": 3, "n_raw": 48, "small": 400, "valid_answers": ["400, 340", "428, 304", "496, 172", "500, 160", "512, 116", "524, 32"]}, "cell": "two_sq_comp/3", "stage": "C"}
+{"id": "po-two_sq_comp-t3-eval-001560", "task": "cx_po_two_sq_comp", "source": "cx_po_two_sq_comp", "split": "eval", "problem": "Find integers x, y such that x^2 + y^2 = (7^2 + 8^2)(1^2 + 2^2)(6^2 + 7^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y such that x^2 + y^2 = (7^2 + 8^2)(1^2 + 2^2)(6^2 + 7^2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "180, 125", "ref": "180, 125", "family": "po_two_sq_comp", "domain": "number", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.000247, "solvable": true, "holdout": false, "meta": {"family": "po_two_sq_comp", "rung": "witness", "params": {"fac": [[7, 8], [1, 2], [6, 7]]}, "seed": 115722981403001648, "version": "po-v0", "tier": 3, "n_raw": 48, "small": 180, "valid_answers": ["180, 125", "195, 100", "197, 96", "208, 69", "216, 37", "219, 8"]}, "cell": "two_sq_comp/3", "stage": "C"}
+{"id": "po-vec_cross-t0-eval-000484", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that (-1, -2, 1) \\times (x, y, z) = (-12, 7, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that (-1, -2, 1) \\times (x, y, z) = (-12, 7, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 0, 6", "ref": "1, 0, 6", "family": "po_vec_cross", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001821, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-1, -2, 1], "B": 6, "w": [-12, 7, 2]}, "seed": 43056419100000507, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["1, 0, 6", "2, 2, 5", "3, 4, 4", "4, 6, 3"]}, "cell": "vec_cross/0", "stage": "C"}
+{"id": "po-vec_cross-t0-eval-000011", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that (-1, 2, 1) \\times (x, y, z) = (-7, -8, 9). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that (-1, 2, 1) \\times (x, y, z) = (-7, -8, 9). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 3, -2", "ref": "-6, 3, -2", "family": "po_vec_cross", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002276, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-1, 2, 1], "B": 6, "w": [-7, -8, 9]}, "seed": 43056419100000013, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 4, "valid_answers": ["-6, 3, -2", "-5, 1, -3", "-4, -1, -4", "-3, -3, -5", "-2, -5, -6"]}, "cell": "vec_cross/0", "stage": "C"}
+{"id": "po-vec_cross-t0-eval-000088", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that (0, -1, 0) \\times (x, y, z) = (-4, 0, -2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that (0, -1, 0) \\times (x, y, z) = (-4, 0, -2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -6, 4", "ref": "-2, -6, 4", "family": "po_vec_cross", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.005917, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [0, -1, 0], "B": 6, "w": [-4, 0, -2]}, "seed": 43056419100000094, "version": "po-v0", "tier": 0, "n_raw": 13, "small": 4, "valid_answers": ["-2, -6, 4", "-2, -5, 4", "-2, -4, 4", "-2, -3, 4", "-2, -2, 4", "-2, -1, 4", "-2, 0, 4", "-2, 1, 4", "-2, 2, 4", "-2, 3, 4", "-2, 4, 4", "-2, 5, 4", "-2, 6, 4"]}, "cell": "vec_cross/0", "stage": "C"}
+{"id": "po-vec_cross-t0-eval-002251", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that (-2, -1, 0) \\times (x, y, z) = (4, -8, 16). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that (-2, -1, 0) \\times (x, y, z) = (4, -8, 16). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, -6, -4", "ref": "4, -6, -4", "family": "po_vec_cross", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00091, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-2, -1, 0], "B": 6, "w": [4, -8, 16]}, "seed": 43056419100002435, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 6, "valid_answers": ["4, -6, -4", "6, -5, -4"]}, "cell": "vec_cross/0", "stage": "C"}
+{"id": "po-vec_cross-t0-eval-001119", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that (0, -1, -2) \\times (x, y, z) = (6, 12, -6). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that (0, -1, -2) \\times (x, y, z) = (6, 12, -6). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 0, -6", "ref": "-6, 0, -6", "family": "po_vec_cross", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.003186, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [0, -1, -2], "B": 6, "w": [6, 12, -6]}, "seed": 43056419100001189, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 6, "valid_answers": ["-6, 0, -6", "-6, 1, -4", "-6, 2, -2", "-6, 3, 0", "-6, 4, 2", "-6, 5, 4", "-6, 6, 6"]}, "cell": "vec_cross/0", "stage": "C"}
+{"id": "po-vec_cross-t0-eval-001566", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that (1, 0, -2) \\times (x, y, z) = (-10, -7, -5). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that (1, 0, -2) \\times (x, y, z) = (-10, -7, -5). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -5, 5", "ref": "1, -5, 5", "family": "po_vec_cross", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.002731, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [1, 0, -2], "B": 6, "w": [-10, -7, -5]}, "seed": 43056419100001689, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 5, "valid_answers": ["1, -5, 5", "2, -5, 3", "3, -5, 1", "4, -5, -1", "5, -5, -3", "6, -5, -5"]}, "cell": "vec_cross/0", "stage": "C"}
+{"id": "po-vec_cross-t0-eval-001908", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that (2, 0, 2) \\times (x, y, z) = (-2, -10, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that (2, 0, 2) \\times (x, y, z) = (-2, -10, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 1, -1", "ref": "-6, 1, -1", "family": "po_vec_cross", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.003641, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [2, 0, 2], "B": 6, "w": [-2, -10, 2]}, "seed": 43056419100002066, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 3, "valid_answers": ["-6, 1, -1", "-5, 1, 0", "-4, 1, 1", "-3, 1, 2", "-2, 1, 3", "-1, 1, 4", "0, 1, 5", "1, 1, 6"]}, "cell": "vec_cross/0", "stage": "C"}
+{"id": "po-vec_cross-t0-eval-002007", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that (-1, 2, 2) \\times (x, y, z) = (-2, 8, -9). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that (-1, 2, 2) \\times (x, y, z) = (-2, 8, -9). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 5, 4", "ref": "2, 5, 4", "family": "po_vec_cross", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002276, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-1, 2, 2], "B": 6, "w": [-2, 8, -9]}, "seed": 43056419100002170, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 3, "valid_answers": ["2, 5, 4", "3, 3, 2", "4, 1, 0", "5, -1, -2", "6, -3, -4"]}, "cell": "vec_cross/0", "stage": "C"}
+{"id": "po-vec_cross-t0-eval-001682", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that (1, -2, 1) \\times (x, y, z) = (-3, 3, 9). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that (1, -2, 1) \\times (x, y, z) = (-3, 3, 9). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 5, -1", "ref": "2, 5, -1", "family": "po_vec_cross", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002276, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [1, -2, 1], "B": 6, "w": [-3, 3, 9]}, "seed": 43056419100001820, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 3, "valid_answers": ["2, 5, -1", "3, 3, 0", "4, 1, 1", "5, -1, 2", "6, -3, 3"]}, "cell": "vec_cross/0", "stage": "C"}
+{"id": "po-vec_cross-t0-eval-001346", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that (-1, -1, -2) \\times (x, y, z) = (6, -2, -2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that (-1, -1, -2) \\times (x, y, z) = (6, -2, -2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 0, -6", "ref": "-2, 0, -6", "family": "po_vec_cross", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.003186, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-1, -1, -2], "B": 6, "w": [6, -2, -2]}, "seed": 43056419100001439, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 2, "valid_answers": ["-2, 0, -6", "-1, 1, -4", "0, 2, -2", "1, 3, 0", "2, 4, 2", "3, 5, 4", "4, 6, 6"]}, "cell": "vec_cross/0", "stage": "C"}
+{"id": "po-vec_cross-t0-eval-001932", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that (2, -1, 2) \\times (x, y, z) = (-5, -4, 3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that (2, -1, 2) \\times (x, y, z) = (-5, -4, 3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 4, -3", "ref": "-5, 4, -3", "family": "po_vec_cross", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002276, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [2, -1, 2], "B": 6, "w": [-5, -4, 3]}, "seed": 43056419100002092, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 2, "valid_answers": ["-5, 4, -3", "-3, 3, -1", "-1, 2, 1", "1, 1, 3", "3, 0, 5"]}, "cell": "vec_cross/0", "stage": "C"}
+{"id": "po-vec_cross-t0-eval-001850", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that (-2, 2, 0) \\times (x, y, z) = (-6, -6, -10). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that (-2, 2, 0) \\times (x, y, z) = (-6, -6, -10). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 6, -3", "ref": "-1, 6, -3", "family": "po_vec_cross", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.003641, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-2, 2, 0], "B": 6, "w": [-6, -6, -10]}, "seed": 43056419100002005, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 3, "valid_answers": ["-1, 6, -3", "0, 5, -3", "1, 4, -3", "2, 3, -3", "3, 2, -3", "4, 1, -3", "5, 0, -3", "6, -1, -3"]}, "cell": "vec_cross/0", "stage": "C"}
+{"id": "po-vec_cross-t1-eval-000515", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that (-2, -2, -1) \\times (x, y, z) = (-10, 12, -4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that (-2, -2, -1) \\times (x, y, z) = (-10, 12, -4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, -8, 1", "ref": "-10, -8, 1", "family": "po_vec_cross", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.00108, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-2, -2, -1], "B": 10, "w": [-10, 12, -4]}, "seed": 43056419101000530, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 4, "valid_answers": ["-10, -8, 1", "-8, -6, 2", "-6, -4, 3", "-4, -2, 4", "-2, 0, 5", "0, 2, 6", "2, 4, 7", "4, 6, 8", "6, 8, 9", "8, 10, 10"]}, "cell": "vec_cross/1", "stage": "B"}
+{"id": "po-vec_cross-t1-eval-002333", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that (2, -1, 1) \\times (x, y, z) = (12, 13, -11). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that (2, -1, 1) \\times (x, y, z) = (12, 13, -11). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -2, -10", "ref": "-7, -2, -10", "family": "po_vec_cross", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000972, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [2, -1, 1], "B": 10, "w": [12, 13, -11]}, "seed": 43056419101002410, "version": "po-v0", "tier": 1, "n_raw": 9, "small": 6, "valid_answers": ["-7, -2, -10", "-5, -3, -9", "-3, -4, -8", "-1, -5, -7", "1, -6, -6", "3, -7, -5", "5, -8, -4", "7, -9, -3", "9, -10, -2"]}, "cell": "vec_cross/1", "stage": "B"}
+{"id": "po-vec_cross-t1-eval-001698", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that (1, 2, 1) \\times (x, y, z) = (-13, 5, 3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that (1, 2, 1) \\times (x, y, z) = (-13, 5, 3). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -7, -10", "ref": "-5, -7, -10", "family": "po_vec_cross", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000972, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [1, 2, 1], "B": 10, "w": [-13, 5, 3]}, "seed": 43056419101001756, "version": "po-v0", "tier": 1, "n_raw": 9, "small": 5, "valid_answers": ["-5, -7, -10", "-4, -5, -9", "-3, -3, -8", "-2, -1, -7", "-1, 1, -6", "0, 3, -5", "1, 5, -4", "2, 7, -3", "3, 9, -2"]}, "cell": "vec_cross/1", "stage": "B"}
+{"id": "po-vec_cross-t1-eval-001666", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that (2, -2, -2) \\times (x, y, z) = (18, 16, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that (2, -2, -2) \\times (x, y, z) = (18, 16, 2). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 10, 1", "ref": "-9, 10, 1", "family": "po_vec_cross", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.001296, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [2, -2, -2], "B": 10, "w": [18, 16, 2]}, "seed": 43056419101001723, "version": "po-v0", "tier": 1, "n_raw": 12, "small": 5, "valid_answers": ["-9, 10, 1", "-8, 9, 0", "-7, 8, -1", "-6, 7, -2", "-5, 6, -3", "-4, 5, -4", "-3, 4, -5", "-2, 3, -6", "-1, 2, -7", "0, 1, -8", "1, 0, -9", "2, -1, -10"]}, "cell": "vec_cross/1", "stage": "B"}
+{"id": "po-vec_cross-t1-eval-000445", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that (1, 0, 1) \\times (x, y, z) = (-10, 0, 10). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that (1, 0, 1) \\times (x, y, z) = (-10, 0, 10). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 10, -10", "ref": "-10, 10, -10", "family": "po_vec_cross", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.002268, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [1, 0, 1], "B": 10, "w": [-10, 0, 10]}, "seed": 43056419101000458, "version": "po-v0", "tier": 1, "n_raw": 21, "small": 10, "valid_answers": ["-10, 10, -10", "-9, 10, -9", "-8, 10, -8", "-7, 10, -7", "-6, 10, -6", "-5, 10, -5", "-4, 10, -4", "-3, 10, -3", "-2, 10, -2", "-1, 10, -1", "0, 10, 0", "1, 10, 1", "2, 10, 2", "3, 10, 3", "4, 10, 4", "5, 10, 5", "6, 10, 6", "7, 10, 7", "8, 10, 8", "9, 10, 9", "10, 10, 10"]}, "cell": "vec_cross/1", "stage": "B"}
+{"id": "po-vec_cross-t1-eval-000031", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that (0, 2, 0) \\times (x, y, z) = (8, 0, -16). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that (0, 2, 0) \\times (x, y, z) = (8, 0, -16). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8, -10, 4", "ref": "8, -10, 4", "family": "po_vec_cross", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.002268, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [0, 2, 0], "B": 10, "w": [8, 0, -16]}, "seed": 43056419101000032, "version": "po-v0", "tier": 1, "n_raw": 21, "small": 8, "valid_answers": ["8, -10, 4", "8, -9, 4", "8, -8, 4", "8, -7, 4", "8, -6, 4", "8, -5, 4", "8, -4, 4", "8, -3, 4", "8, -2, 4", "8, -1, 4", "8, 0, 4", "8, 1, 4", "8, 2, 4", "8, 3, 4", "8, 4, 4", "8, 5, 4", "8, 6, 4", "8, 7, 4", "8, 8, 4", "8, 9, 4", "8, 10, 4"]}, "cell": "vec_cross/1", "stage": "B"}
+{"id": "po-vec_cross-t1-eval-001152", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that (-2, 2, -1) \\times (x, y, z) = (-14, -8, 12). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that (-2, 2, -1) \\times (x, y, z) = (-14, -8, 12). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 4, -9", "ref": "-10, 4, -9", "family": "po_vec_cross", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000864, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-2, 2, -1], "B": 10, "w": [-14, -8, 12]}, "seed": 43056419101001188, "version": "po-v0", "tier": 1, "n_raw": 8, "small": 5, "valid_answers": ["-10, 4, -9", "-8, 2, -8", "-6, 0, -7", "-4, -2, -6", "-2, -4, -5", "0, -6, -4", "2, -8, -3", "4, -10, -2"]}, "cell": "vec_cross/1", "stage": "B"}
+{"id": "po-vec_cross-t1-eval-001072", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that (0, 0, -2) \\times (x, y, z) = (10, -10, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that (0, 0, -2) \\times (x, y, z) = (10, -10, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, 5, -10", "ref": "5, 5, -10", "family": "po_vec_cross", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.002268, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [0, 0, -2], "B": 10, "w": [10, -10, 0]}, "seed": 43056419101001105, "version": "po-v0", "tier": 1, "n_raw": 21, "small": 5, "valid_answers": ["5, 5, -10", "5, 5, -9", "5, 5, -8", "5, 5, -7", "5, 5, -6", "5, 5, -5", "5, 5, -4", "5, 5, -3", "5, 5, -2", "5, 5, -1", "5, 5, 0", "5, 5, 1", "5, 5, 2", "5, 5, 3", "5, 5, 4", "5, 5, 5", "5, 5, 6", "5, 5, 7", "5, 5, 8", "5, 5, 9", "5, 5, 10"]}, "cell": "vec_cross/1", "stage": "B"}
+{"id": "po-vec_cross-t1-eval-002444", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that (1, -2, 0) \\times (x, y, z) = (-18, -9, 4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that (1, -2, 0) \\times (x, y, z) = (-18, -9, 4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 10, 9", "ref": "-3, 10, 9", "family": "po_vec_cross", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.001188, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [1, -2, 0], "B": 10, "w": [-18, -9, 4]}, "seed": 43056419101002528, "version": "po-v0", "tier": 1, "n_raw": 11, "small": 9, "valid_answers": ["-3, 10, 9", "-2, 8, 9", "-1, 6, 9", "0, 4, 9", "1, 2, 9", "2, 0, 9", "3, -2, 9", "4, -4, 9", "5, -6, 9", "6, -8, 9", "7, -10, 9"]}, "cell": "vec_cross/1", "stage": "B"}
+{"id": "po-vec_cross-t1-eval-001512", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that (2, 1, 0) \\times (x, y, z) = (8, -16, 7). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that (2, 1, 0) \\times (x, y, z) = (8, -16, 7). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -1, 8", "ref": "-9, -1, 8", "family": "po_vec_cross", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.00108, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [2, 1, 0], "B": 10, "w": [8, -16, 7]}, "seed": 43056419101001562, "version": "po-v0", "tier": 1, "n_raw": 10, "small": 8, "valid_answers": ["-9, -1, 8", "-7, 0, 8", "-5, 1, 8", "-3, 2, 8", "-1, 3, 8", "1, 4, 8", "3, 5, 8", "5, 6, 8", "7, 7, 8", "9, 8, 8"]}, "cell": "vec_cross/1", "stage": "B"}
+{"id": "po-vec_cross-t1-eval-001321", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that (2, -1, 0) \\times (x, y, z) = (7, 14, 13). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that (2, -1, 0) \\times (x, y, z) = (7, 14, 13). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, 10, -7", "ref": "-7, 10, -7", "family": "po_vec_cross", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000972, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [2, -1, 0], "B": 10, "w": [7, 14, 13]}, "seed": 43056419101001365, "version": "po-v0", "tier": 1, "n_raw": 9, "small": 7, "valid_answers": ["-7, 10, -7", "-5, 9, -7", "-3, 8, -7", "-1, 7, -7", "1, 6, -7", "3, 5, -7", "5, 4, -7", "7, 3, -7", "9, 2, -7"]}, "cell": "vec_cross/1", "stage": "B"}
+{"id": "po-vec_cross-t1-eval-001052", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that (0, 2, -1) \\times (x, y, z) = (-13, 6, 12). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that (0, 2, -1) \\times (x, y, z) = (-13, 6, 12). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -9, -2", "ref": "-6, -9, -2", "family": "po_vec_cross", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000972, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [0, 2, -1], "B": 10, "w": [-13, 6, 12]}, "seed": 43056419101001085, "version": "po-v0", "tier": 1, "n_raw": 9, "small": 6, "valid_answers": ["-6, -9, -2", "-6, -7, -3", "-6, -5, -4", "-6, -3, -5", "-6, -1, -6", "-6, 1, -7", "-6, 3, -8", "-6, 5, -9", "-6, 7, -10"]}, "cell": "vec_cross/1", "stage": "B"}
+{"id": "po-vec_cross-t2-eval-000014", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -12 <= x, y, z <= 12 such that (1, -3, -3) \\times (x, y, z) = (36, 13, -1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -12 <= x, y, z <= 12 such that (1, -3, -3) \\times (x, y, z) = (36, 13, -1). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 11, -1", "ref": "-4, 11, -1", "family": "po_vec_cross", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000256, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [1, -3, -3], "B": 12, "w": [36, 13, -1]}, "seed": 43056419102000014, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 7, "valid_answers": ["-4, 11, -1", "-3, 8, -4", "-2, 5, -7", "-1, 2, -10"]}, "cell": "vec_cross/2", "stage": "B"}
+{"id": "po-vec_cross-t2-eval-001280", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -12 <= x, y, z <= 12 such that (3, -1, 3) \\times (x, y, z) = (-38, -21, 31). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -12 <= x, y, z <= 12 such that (3, -1, 3) \\times (x, y, z) = (-38, -21, 31). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 12, 2", "ref": "-5, 12, 2", "family": "po_vec_cross", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000256, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [3, -1, 3], "B": 12, "w": [-38, -21, 31]}, "seed": 43056419102001315, "version": "po-v0", "tier": 2, "n_raw": 4, "small": 10, "valid_answers": ["-5, 12, 2", "-2, 11, 5", "1, 10, 8", "4, 9, 11"]}, "cell": "vec_cross/2", "stage": "B"}
+{"id": "po-vec_cross-t2-eval-000939", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -12 <= x, y, z <= 12 such that (2, -3, -1) \\times (x, y, z) = (5, 5, -5). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -12 <= x, y, z <= 12 such that (2, -3, -1) \\times (x, y, z) = (5, 5, -5). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 11, 2", "ref": "-9, 11, 2", "family": "po_vec_cross", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000512, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [2, -3, -1], "B": 12, "w": [5, 5, -5]}, "seed": 43056419102000961, "version": "po-v0", "tier": 2, "n_raw": 8, "small": 2, "valid_answers": ["-9, 11, 2", "-7, 8, 1", "-5, 5, 0", "-3, 2, -1", "-1, -1, -2", "1, -4, -3", "3, -7, -4", "5, -10, -5"]}, "cell": "vec_cross/2", "stage": "B"}
+{"id": "po-vec_cross-t2-eval-002450", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -12 <= x, y, z <= 12 such that (-1, -3, -3) \\times (x, y, z) = (30, 4, -14). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -12 <= x, y, z <= 12 such that (-1, -3, -3) \\times (x, y, z) = (30, 4, -14). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -1, -11", "ref": "-5, -1, -11", "family": "po_vec_cross", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.00032, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-1, -3, -3], "B": 12, "w": [30, 4, -14]}, "seed": 43056419102002517, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 5, "valid_answers": ["-5, -1, -11", "-4, 2, -8", "-3, 5, -5", "-2, 8, -2", "-1, 11, 1"]}, "cell": "vec_cross/2", "stage": "B"}
+{"id": "po-vec_cross-t2-eval-002468", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -12 <= x, y, z <= 12 such that (-1, -2, -3) \\times (x, y, z) = (10, -29, 16). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -12 <= x, y, z <= 12 such that (-1, -2, -3) \\times (x, y, z) = (10, -29, 16). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, -4, -11", "ref": "6, -4, -11", "family": "po_vec_cross", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000448, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-1, -2, -3], "B": 12, "w": [10, -29, 16]}, "seed": 43056419102002535, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 8, "valid_answers": ["6, -4, -11", "7, -2, -8", "8, 0, -5", "9, 2, -2", "10, 4, 1", "11, 6, 4", "12, 8, 7"]}, "cell": "vec_cross/2", "stage": "B"}
+{"id": "po-vec_cross-t2-eval-001319", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -12 <= x, y, z <= 12 such that (1, 1, -1) \\times (x, y, z) = (-2, 15, 13). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -12 <= x, y, z <= 12 such that (1, 1, -1) \\times (x, y, z) = (-2, 15, 13). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, 1, -3", "ref": "-12, 1, -3", "family": "po_vec_cross", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.00064, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [1, 1, -1], "B": 12, "w": [-2, 15, 13]}, "seed": 43056419102001355, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 8, "valid_answers": ["-12, 1, -3", "-11, 2, -4", "-10, 3, -5", "-9, 4, -6", "-8, 5, -7", "-7, 6, -8", "-6, 7, -9", "-5, 8, -10", "-4, 9, -11", "-3, 10, -12"]}, "cell": "vec_cross/2", "stage": "B"}
+{"id": "po-vec_cross-t2-eval-002401", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -12 <= x, y, z <= 12 such that (-3, -1, -3) \\times (x, y, z) = (-21, 36, 9). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -12 <= x, y, z <= 12 such that (-3, -1, -3) \\times (x, y, z) = (-21, 36, 9). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, -7, 0", "ref": "-12, -7, 0", "family": "po_vec_cross", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.00032, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-3, -1, -3], "B": 12, "w": [-21, 36, 9]}, "seed": 43056419102002464, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 6, "valid_answers": ["-12, -7, 0", "-9, -6, 3", "-6, -5, 6", "-3, -4, 9", "0, -3, 12"]}, "cell": "vec_cross/2", "stage": "B"}
+{"id": "po-vec_cross-t2-eval-000908", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -12 <= x, y, z <= 12 such that (3, 0, 3) \\times (x, y, z) = (24, 27, -24). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -12 <= x, y, z <= 12 such that (3, 0, 3) \\times (x, y, z) = (24, 27, -24). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -8, -12", "ref": "-3, -8, -12", "family": "po_vec_cross", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 16, "bucket": "9-20", "guess_rate": 0.001024, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [3, 0, 3], "B": 12, "w": [24, 27, -24]}, "seed": 43056419102000929, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 8, "valid_answers": ["-3, -8, -12", "-2, -8, -11", "-1, -8, -10", "0, -8, -9", "1, -8, -8", "2, -8, -7", "3, -8, -6", "4, -8, -5", "5, -8, -4", "6, -8, -3", "7, -8, -2", "8, -8, -1", "9, -8, 0", "10, -8, 1", "11, -8, 2", "12, -8, 3"]}, "cell": "vec_cross/2", "stage": "B"}
+{"id": "po-vec_cross-t2-eval-000202", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -12 <= x, y, z <= 12 such that (-3, -2, -1) \\times (x, y, z) = (-4, -5, 22). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -12 <= x, y, z <= 12 such that (-3, -2, -1) \\times (x, y, z) = (-4, -5, 22). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -12, -4", "ref": "-7, -12, -4", "family": "po_vec_cross", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000448, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-3, -2, -1], "B": 12, "w": [-4, -5, 22]}, "seed": 43056419102000205, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 5, "valid_answers": ["-7, -12, -4", "-4, -10, -3", "-1, -8, -2", "2, -6, -1", "5, -4, 0", "8, -2, 1", "11, 0, 2"]}, "cell": "vec_cross/2", "stage": "B"}
+{"id": "po-vec_cross-t2-eval-000181", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -12 <= x, y, z <= 12 such that (-3, 3, 0) \\times (x, y, z) = (-9, -9, -42). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -12 <= x, y, z <= 12 such that (-3, 3, 0) \\times (x, y, z) = (-9, -9, -42). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 12, -3", "ref": "2, 12, -3", "family": "po_vec_cross", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.000704, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-3, 3, 0], "B": 12, "w": [-9, -9, -42]}, "seed": 43056419102000183, "version": "po-v0", "tier": 2, "n_raw": 11, "small": 7, "valid_answers": ["2, 12, -3", "3, 11, -3", "4, 10, -3", "5, 9, -3", "6, 8, -3", "7, 7, -3", "8, 6, -3", "9, 5, -3", "10, 4, -3", "11, 3, -3", "12, 2, -3"]}, "cell": "vec_cross/2", "stage": "B"}
+{"id": "po-vec_cross-t2-eval-000819", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -12 <= x, y, z <= 12 such that (2, -1, 1) \\times (x, y, z) = (-9, 10, 28). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -12 <= x, y, z <= 12 such that (2, -1, 1) \\times (x, y, z) = (-9, 10, 28). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 12, -3", "ref": "4, 12, -3", "family": "po_vec_cross", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.00032, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [2, -1, 1], "B": 12, "w": [-9, 10, 28]}, "seed": 43056419102000838, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 10, "valid_answers": ["4, 12, -3", "6, 11, -2", "8, 10, -1", "10, 9, 0", "12, 8, 1"]}, "cell": "vec_cross/2", "stage": "B"}
+{"id": "po-vec_cross-t2-eval-001510", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -12 <= x, y, z <= 12 such that (0, 0, -3) \\times (x, y, z) = (-9, -21, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -12 <= x, y, z <= 12 such that (0, 0, -3) \\times (x, y, z) = (-9, -21, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "7, -3, -12", "ref": "7, -3, -12", "family": "po_vec_cross", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 25, "bucket": null, "guess_rate": 0.0016, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [0, 0, -3], "B": 12, "w": [-9, -21, 0]}, "seed": 43056419102001553, "version": "po-v0", "tier": 2, "n_raw": 25, "small": 7, "valid_answers": ["7, -3, -12", "7, -3, -11", "7, -3, -10", "7, -3, -9", "7, -3, -8", "7, -3, -7", "7, -3, -6", "7, -3, -5", "7, -3, -4", "7, -3, -3", "7, -3, -2", "7, -3, -1", "7, -3, 0", "7, -3, 1", "7, -3, 2", "7, -3, 3", "7, -3, 4", "7, -3, 5", "7, -3, 6", "7, -3, 7", "7, -3, 8", "7, -3, 9", "7, -3, 10", "7, -3, 11", "7, -3, 12"]}, "cell": "vec_cross/2", "stage": "B"}
+{"id": "po-vec_cross-t3-eval-001031", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -20 <= x, y, z <= 20 such that (-3, 4, -3) \\times (x, y, z) = (-15, -42, -41). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -20 <= x, y, z <= 20 such that (-3, 4, -3) \\times (x, y, z) = (-15, -42, -41). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 19, -18", "ref": "-4, 19, -18", "family": "po_vec_cross", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000131, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-3, 4, -3], "B": 20, "w": [-15, -42, -41]}, "seed": 43056419103001042, "version": "po-v0", "tier": 3, "n_raw": 9, "small": 8, "valid_answers": ["-4, 19, -18", "-1, 15, -15", "2, 11, -12", "5, 7, -9", "8, 3, -6", "11, -1, -3", "14, -5, 0", "17, -9, 3", "20, -13, 6"]}, "cell": "vec_cross/3", "stage": "C"}
+{"id": "po-vec_cross-t3-eval-000558", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -20 <= x, y, z <= 20 such that (-3, 0, -2) \\times (x, y, z) = (26, 11, -39). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -20 <= x, y, z <= 20 such that (-3, 0, -2) \\times (x, y, z) = (26, 11, -39). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-19, 13, -9", "ref": "-19, 13, -9", "family": "po_vec_cross", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.000203, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-3, 0, -2], "B": 20, "w": [26, 11, -39]}, "seed": 43056419103000563, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 13, "valid_answers": ["-19, 13, -9", "-16, 13, -7", "-13, 13, -5", "-10, 13, -3", "-7, 13, -1", "-4, 13, 1", "-1, 13, 3", "2, 13, 5", "5, 13, 7", "8, 13, 9", "11, 13, 11", "14, 13, 13", "17, 13, 15", "20, 13, 17"]}, "cell": "vec_cross/3", "stage": "C"}
+{"id": "po-vec_cross-t3-eval-001893", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -20 <= x, y, z <= 20 such that (-1, 2, -2) \\times (x, y, z) = (-44, -2, 20). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -20 <= x, y, z <= 20 such that (-1, 2, -2) \\times (x, y, z) = (-44, -2, 20). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -2, -20", "ref": "-9, -2, -20", "family": "po_vec_cross", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.000145, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-1, 2, -2], "B": 20, "w": [-44, -2, 20]}, "seed": 43056419103001921, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 12, "valid_answers": ["-9, -2, -20", "-8, -4, -18", "-7, -6, -16", "-6, -8, -14", "-5, -10, -12", "-4, -12, -10", "-3, -14, -8", "-2, -16, -6", "-1, -18, -4", "0, -20, -2"]}, "cell": "vec_cross/3", "stage": "C"}
+{"id": "po-vec_cross-t3-eval-000872", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -20 <= x, y, z <= 20 such that (3, -1, -3) \\times (x, y, z) = (-37, 42, -51). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -20 <= x, y, z <= 20 such that (3, -1, -3) \\times (x, y, z) = (-37, 42, -51). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-18, -11, 4", "ref": "-18, -11, 4", "family": "po_vec_cross", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000131, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [3, -1, -3], "B": 20, "w": [-37, 42, -51]}, "seed": 43056419103000882, "version": "po-v0", "tier": 3, "n_raw": 9, "small": 13, "valid_answers": ["-18, -11, 4", "-15, -12, 1", "-12, -13, -2", "-9, -14, -5", "-6, -15, -8", "-3, -16, -11", "0, -17, -14", "3, -18, -17", "6, -19, -20"]}, "cell": "vec_cross/3", "stage": "C"}
+{"id": "po-vec_cross-t3-eval-002403", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -20 <= x, y, z <= 20 such that (2, -2, -3) \\times (x, y, z) = (74, 68, 4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -20 <= x, y, z <= 20 such that (2, -2, -3) \\times (x, y, z) = (74, 68, 4). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-18, 20, -7", "ref": "-18, 20, -7", "family": "po_vec_cross", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 7.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [2, -2, -3], "B": 20, "w": [74, 68, 4]}, "seed": 43056419103002438, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 16, "valid_answers": ["-18, 20, -7", "-16, 18, -10", "-14, 16, -13", "-12, 14, -16", "-10, 12, -19"]}, "cell": "vec_cross/3", "stage": "C"}
+{"id": "po-vec_cross-t3-eval-000899", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -20 <= x, y, z <= 20 such that (-2, 2, 0) \\times (x, y, z) = (-12, -12, -52). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -20 <= x, y, z <= 20 such that (-2, 2, 0) \\times (x, y, z) = (-12, -12, -52). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, 20, -6", "ref": "6, 20, -6", "family": "po_vec_cross", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 15, "bucket": "9-20", "guess_rate": 0.000218, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-2, 2, 0], "B": 20, "w": [-12, -12, -52]}, "seed": 43056419103000909, "version": "po-v0", "tier": 3, "n_raw": 15, "small": 13, "valid_answers": ["6, 20, -6", "7, 19, -6", "8, 18, -6", "9, 17, -6", "10, 16, -6", "11, 15, -6", "12, 14, -6", "13, 13, -6", "14, 12, -6", "15, 11, -6", "16, 10, -6", "17, 9, -6", "18, 8, -6", "19, 7, -6", "20, 6, -6"]}, "cell": "vec_cross/3", "stage": "C"}
+{"id": "po-vec_cross-t3-eval-002172", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -20 <= x, y, z <= 20 such that (-1, -1, -3) \\times (x, y, z) = (-11, -43, 18). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -20 <= x, y, z <= 20 such that (-1, -1, -3) \\times (x, y, z) = (-11, -43, 18). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "8, -10, -19", "ref": "8, -10, -19", "family": "po_vec_cross", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.000189, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-1, -1, -3], "B": 20, "w": [-11, -43, 18]}, "seed": 43056419103002202, "version": "po-v0", "tier": 3, "n_raw": 13, "small": 11, "valid_answers": ["8, -10, -19", "9, -9, -16", "10, -8, -13", "11, -7, -10", "12, -6, -7", "13, -5, -4", "14, -4, -1", "15, -3, 2", "16, -2, 5", "17, -1, 8", "18, 0, 11", "19, 1, 14", "20, 2, 17"]}, "cell": "vec_cross/3", "stage": "C"}
+{"id": "po-vec_cross-t3-eval-001947", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -20 <= x, y, z <= 20 such that (4, -4, 1) \\times (x, y, z) = (-17, -10, 28). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -20 <= x, y, z <= 20 such that (4, -4, 1) \\times (x, y, z) = (-17, -10, 28). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 17, 0", "ref": "-10, 17, 0", "family": "po_vec_cross", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000116, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [4, -4, 1], "B": 20, "w": [-17, -10, 28]}, "seed": 43056419103001976, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 5, "valid_answers": ["-10, 17, 0", "-6, 13, 1", "-2, 9, 2", "2, 5, 3", "6, 1, 4", "10, -3, 5", "14, -7, 6", "18, -11, 7"]}, "cell": "vec_cross/3", "stage": "C"}
+{"id": "po-vec_cross-t3-eval-000278", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -20 <= x, y, z <= 20 such that (-1, -3, 0) \\times (x, y, z) = (-27, 9, -56). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -20 <= x, y, z <= 20 such that (-1, -3, 0) \\times (x, y, z) = (-27, 9, -56). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-20, -4, 9", "ref": "-20, -4, 9", "family": "po_vec_cross", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.000131, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [-1, -3, 0], "B": 20, "w": [-27, 9, -56]}, "seed": 43056419103000281, "version": "po-v0", "tier": 3, "n_raw": 9, "small": 14, "valid_answers": ["-20, -4, 9", "-19, -1, 9", "-18, 2, 9", "-17, 5, 9", "-16, 8, 9", "-15, 11, 9", "-14, 14, 9", "-13, 17, 9", "-12, 20, 9"]}, "cell": "vec_cross/3", "stage": "C"}
+{"id": "po-vec_cross-t3-eval-000565", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -20 <= x, y, z <= 20 such that (2, 3, 1) \\times (x, y, z) = (27, 0, -54). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -20 <= x, y, z <= 20 such that (2, 3, 1) \\times (x, y, z) = (27, 0, -54). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, -18, 3", "ref": "6, -18, 3", "family": "po_vec_cross", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000116, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [2, 3, 1], "B": 20, "w": [27, 0, -54]}, "seed": 43056419103000570, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 12, "valid_answers": ["6, -18, 3", "8, -15, 4", "10, -12, 5", "12, -9, 6", "14, -6, 7", "16, -3, 8", "18, 0, 9", "20, 3, 10"]}, "cell": "vec_cross/3", "stage": "C"}
+{"id": "po-vec_cross-t3-eval-000715", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -20 <= x, y, z <= 20 such that (0, 3, -2) \\times (x, y, z) = (46, -12, -18). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -20 <= x, y, z <= 20 such that (0, 3, -2) \\times (x, y, z) = (46, -12, -18). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "6, -7, 20", "ref": "6, -7, 20", "family": "po_vec_cross", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.000145, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [0, 3, -2], "B": 20, "w": [46, -12, -18]}, "seed": 43056419103000724, "version": "po-v0", "tier": 3, "n_raw": 10, "small": 10, "valid_answers": ["6, -7, 20", "6, -4, 18", "6, -1, 16", "6, 2, 14", "6, 5, 12", "6, 8, 10", "6, 11, 8", "6, 14, 6", "6, 17, 4", "6, 20, 2"]}, "cell": "vec_cross/3", "stage": "C"}
+{"id": "po-vec_cross-t3-eval-002322", "task": "cx_po_vec_cross", "source": "cx_po_vec_cross", "split": "eval", "problem": "Find integers x, y, z with -20 <= x, y, z <= 20 such that (0, 1, -2) \\times (x, y, z) = (-45, 0, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -20 <= x, y, z <= 20 such that (0, 1, -2) \\times (x, y, z) = (-45, 0, 0). Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -20, -5", "ref": "0, -20, -5", "family": "po_vec_cross", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000116, "solvable": true, "holdout": false, "meta": {"family": "po_vec_cross", "rung": "witness", "params": {"u": [0, 1, -2], "B": 20, "w": [-45, 0, 0]}, "seed": 43056419103002356, "version": "po-v0", "tier": 3, "n_raw": 8, "small": 15, "valid_answers": ["0, -20, -5", "0, -19, -7", "0, -18, -9", "0, -17, -11", "0, -16, -13", "0, -15, -15", "0, -14, -17", "0, -13, -19"]}, "cell": "vec_cross/3", "stage": "C"}
+{"id": "po-vec_dot-t0-eval-000301", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-1, -4) = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-1, -4) = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -1", "ref": "-5, -1", "family": "po_vec_dot", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017751, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 2, "B": 6, "U": [[-1, -4]], "s": [9]}, "seed": 205845576200001546, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["-5, -1", "-1, -2", "3, -3"]}, "cell": "vec_dot/0", "stage": "A"}
+{"id": "po-vec_dot-t0-eval-001129", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (1, -4) = -24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (1, -4) = -24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 5", "ref": "-4, 5", "family": "po_vec_dot", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 2, "B": 6, "U": [[1, -4]], "s": [-24]}, "seed": 205845576200014036, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-4, 5", "0, 6"]}, "cell": "vec_dot/0", "stage": "A"}
+{"id": "po-vec_dot-t0-eval-001176", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (2, 3) = -19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (2, 3) = -19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -3", "ref": "-5, -3", "family": "po_vec_dot", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 2, "B": 6, "U": [[2, 3]], "s": [-19]}, "seed": 205845576200018701, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-5, -3", "-2, -5"]}, "cell": "vec_dot/0", "stage": "A"}
+{"id": "po-vec_dot-t0-eval-000731", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-4, -3) = -26. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-4, -3) = -26. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 6", "ref": "2, 6", "family": "po_vec_dot", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 2, "B": 6, "U": [[-4, -3]], "s": [-26]}, "seed": 205845576200004918, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["2, 6", "5, 2"]}, "cell": "vec_dot/0", "stage": "A"}
+{"id": "po-vec_dot-t0-eval-001006", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (4, 3) = 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (4, 3) = 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 6", "ref": "0, 6", "family": "po_vec_dot", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017751, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 2, "B": 6, "U": [[4, 3]], "s": [18]}, "seed": 205845576200009493, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["0, 6", "3, 2", "6, -2"]}, "cell": "vec_dot/0", "stage": "A"}
+{"id": "po-vec_dot-t0-eval-001182", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-3, 1) = 19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-3, 1) = 19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 1", "ref": "-6, 1", "family": "po_vec_dot", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 2, "B": 6, "U": [[-3, 1]], "s": [19]}, "seed": 205845576200019280, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-6, 1", "-5, 4"]}, "cell": "vec_dot/0", "stage": "A"}
+{"id": "po-vec_dot-t0-eval-000607", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (4, -3) = -27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (4, -3) = -27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, 1", "ref": "-6, 1", "family": "po_vec_dot", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 2, "B": 6, "U": [[4, -3]], "s": [-27]}, "seed": 205845576200003652, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-6, 1", "-3, 5"]}, "cell": "vec_dot/0", "stage": "A"}
+{"id": "po-vec_dot-t0-eval-000484", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-2, 3) = 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-2, 3) = 11. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 1", "ref": "-4, 1", "family": "po_vec_dot", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017751, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 2, "B": 6, "U": [[-2, 3]], "s": [11]}, "seed": 205845576200002688, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["-4, 1", "-1, 3", "2, 5"]}, "cell": "vec_dot/0", "stage": "A"}
+{"id": "po-vec_dot-t0-eval-000627", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-1, 4) = 19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-1, 4) = 19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 4", "ref": "-3, 4", "family": "po_vec_dot", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017751, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 2, "B": 6, "U": [[-1, 4]], "s": [19]}, "seed": 205845576200003817, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-3, 4", "1, 5", "5, 6"]}, "cell": "vec_dot/0", "stage": "A"}
+{"id": "po-vec_dot-t0-eval-000954", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-3, 1) = -21. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-3, 1) = -21. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "5, -6", "ref": "5, -6", "family": "po_vec_dot", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 2, "B": 6, "U": [[-3, 1]], "s": [-21]}, "seed": 205845576200008316, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 6, "valid_answers": ["5, -6", "6, -3"]}, "cell": "vec_dot/0", "stage": "A"}
+{"id": "po-vec_dot-t0-eval-000209", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-4, -3) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-4, -3) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 4", "ref": "-5, 4", "family": "po_vec_dot", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.017751, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 2, "B": 6, "U": [[-4, -3]], "s": [8]}, "seed": 205845576200001020, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["-5, 4", "-2, 0", "1, -4"]}, "cell": "vec_dot/0", "stage": "A"}
+{"id": "po-vec_dot-t0-eval-001178", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-4, 3) = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-4, 3) = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 0", "ref": "-5, 0", "family": "po_vec_dot", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011834, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 2, "B": 6, "U": [[-4, 3]], "s": [20]}, "seed": 205845576200018989, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-5, 0", "-2, 4"]}, "cell": "vec_dot/0", "stage": "A"}
+{"id": "po-vec_dot-t1-eval-000653", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (2, 1, 0) = -2 and v \\cdot (4, 2, 4) = -12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (2, 1, 0) = -2 and v \\cdot (4, 2, 4) = -12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 6, -2", "ref": "-4, 6, -2", "family": "po_vec_dot", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.003186, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 6, "U": [[2, 1, 0], [4, 2, 4]], "s": [-2, -12]}, "seed": 205845576201001189, "version": "po-v0", "tier": 1, "n_raw": 7, "small": 2, "valid_answers": ["-4, 6, -2", "-3, 4, -2", "-2, 2, -2", "-1, 0, -2", "0, -2, -2", "1, -4, -2", "2, -6, -2"]}, "cell": "vec_dot/1", "stage": "B"}
+{"id": "po-vec_dot-t1-eval-001939", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (1, 1, 0) = 3 and v \\cdot (-2, 1, -1) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (1, 1, 0) = 3 and v \\cdot (-2, 1, -1) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 2, 6", "ref": "1, 2, 6", "family": "po_vec_dot", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002276, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 6, "U": [[1, 1, 0], [-2, 1, -1]], "s": [3, -6]}, "seed": 205845576201003565, "version": "po-v0", "tier": 1, "n_raw": 5, "small": 3, "valid_answers": ["1, 2, 6", "2, 1, 3", "3, 0, 0", "4, -1, -3", "5, -2, -6"]}, "cell": "vec_dot/1", "stage": "B"}
+{"id": "po-vec_dot-t1-eval-001735", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (-3, 2, 1) = -8 and v \\cdot (2, -2, -4) = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (-3, 2, 1) = -8 and v \\cdot (2, -2, -4) = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -2, -4", "ref": "0, -2, -4", "family": "po_vec_dot", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00091, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 6, "U": [[-3, 2, 1], [2, -2, -4]], "s": [-8, 20]}, "seed": 205845576201003190, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["0, -2, -4", "3, 3, -5"]}, "cell": "vec_dot/1", "stage": "B"}
+{"id": "po-vec_dot-t1-eval-002449", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (-4, 2, 0) = 10 and v \\cdot (1, -3, 2) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (-4, 2, 0) = 10 and v \\cdot (1, -3, 2) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -1, -4", "ref": "-3, -1, -4", "family": "po_vec_dot", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00091, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 6, "U": [[-4, 2, 0], [1, -3, 2]], "s": [10, -8]}, "seed": 205845576201004443, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 3, "valid_answers": ["-3, -1, -4", "-1, 3, 1"]}, "cell": "vec_dot/1", "stage": "B"}
+{"id": "po-vec_dot-t1-eval-002259", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (-2, 4, 2) = 0 and v \\cdot (-2, -3, -1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (-2, 4, 2) = 0 and v \\cdot (-2, -3, -1) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 0, -2", "ref": "-2, 0, -2", "family": "po_vec_dot", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00091, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 6, "U": [[-2, 4, 2], [-2, -3, -1]], "s": [0, 6]}, "seed": 205845576201004111, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 2, "valid_answers": ["-2, 0, -2", "-1, -3, 5"]}, "cell": "vec_dot/1", "stage": "B"}
+{"id": "po-vec_dot-t1-eval-001135", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (-3, 1, -3) = -2 and v \\cdot (2, 4, 2) = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (-3, 1, -3) = -2 and v \\cdot (2, 4, 2) = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 4, 6", "ref": "-4, 4, 6", "family": "po_vec_dot", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.005007, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 6, "U": [[-3, 1, -3], [2, 4, 2]], "s": [-2, 20]}, "seed": 205845576201002091, "version": "po-v0", "tier": 1, "n_raw": 11, "small": 4, "valid_answers": ["-4, 4, 6", "-3, 4, 5", "-2, 4, 4", "-1, 4, 3", "0, 4, 2", "1, 4, 1", "2, 4, 0", "3, 4, -1", "4, 4, -2", "5, 4, -3", "6, 4, -4"]}, "cell": "vec_dot/1", "stage": "B"}
+{"id": "po-vec_dot-t1-eval-001841", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (3, -1, -3) = 18 and v \\cdot (-3, -3, 0) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (3, -1, -3) = 18 and v \\cdot (-3, -3, 0) = 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, 0, -6", "ref": "0, 0, -6", "family": "po_vec_dot", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001365, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 6, "U": [[3, -1, -3], [-3, -3, 0]], "s": [18, 0]}, "seed": 205845576201003372, "version": "po-v0", "tier": 1, "n_raw": 3, "small": 3, "valid_answers": ["0, 0, -6", "3, -3, -2", "6, -6, 2"]}, "cell": "vec_dot/1", "stage": "B"}
+{"id": "po-vec_dot-t1-eval-001749", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (-2, -3, 4) = -25 and v \\cdot (-4, -3, -1) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (-2, -3, 4) = -25 and v \\cdot (-4, -3, -1) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 5, -3", "ref": "-1, 5, -3", "family": "po_vec_dot", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00091, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 6, "U": [[-2, -3, 4], [-4, -3, -1]], "s": [-25, -8]}, "seed": 205845576201003214, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 5, "valid_answers": ["-1, 5, -3", "4, -1, -5"]}, "cell": "vec_dot/1", "stage": "B"}
+{"id": "po-vec_dot-t1-eval-002067", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (4, 1, 2) = -1 and v \\cdot (4, 2, 0) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (4, 1, 2) = -1 and v \\cdot (4, 2, 0) = 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, 5, -1", "ref": "-1, 5, -1", "family": "po_vec_dot", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.002731, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 6, "U": [[4, 1, 2], [4, 2, 0]], "s": [-1, 6]}, "seed": 205845576201003782, "version": "po-v0", "tier": 1, "n_raw": 6, "small": 3, "valid_answers": ["-1, 5, -1", "0, 3, -2", "1, 1, -3", "2, -1, -4", "3, -3, -5", "4, -5, -6"]}, "cell": "vec_dot/1", "stage": "B"}
+{"id": "po-vec_dot-t1-eval-001381", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (-1, -1, 4) = 11 and v \\cdot (-2, 1, 0) = -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (-1, -1, 4) = 11 and v \\cdot (-2, 1, 0) = -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -5, 2", "ref": "2, -5, 2", "family": "po_vec_dot", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00091, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 6, "U": [[-1, -1, 4], [-2, 1, 0]], "s": [11, -9]}, "seed": 205845576201002546, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 5, "valid_answers": ["2, -5, 2", "6, 3, 5"]}, "cell": "vec_dot/1", "stage": "B"}
+{"id": "po-vec_dot-t1-eval-001402", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (1, -2, 0) = 2 and v \\cdot (2, -1, 4) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (1, -2, 0) = 2 and v \\cdot (2, -1, 4) = 7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -3, 3", "ref": "-4, -3, 3", "family": "po_vec_dot", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.00091, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 6, "U": [[1, -2, 0], [2, -1, 4]], "s": [2, 7]}, "seed": 205845576201002590, "version": "po-v0", "tier": 1, "n_raw": 2, "small": 4, "valid_answers": ["-4, -3, 3", "4, 1, 0"]}, "cell": "vec_dot/1", "stage": "B"}
+{"id": "po-vec_dot-t1-eval-000590", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (2, 0, 3) = 4 and v \\cdot (4, 0, 1) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -6 <= x, y, z <= 6 such that v = (x, y, z) satisfies v \\cdot (2, 0, 3) = 4 and v \\cdot (4, 0, 1) = 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -6, 0", "ref": "2, -6, 0", "family": "po_vec_dot", "domain": "linalg", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.005917, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 6, "U": [[2, 0, 3], [4, 0, 1]], "s": [4, 8]}, "seed": 205845576201001071, "version": "po-v0", "tier": 1, "n_raw": 13, "small": 2, "valid_answers": ["2, -6, 0", "2, -5, 0", "2, -4, 0", "2, -3, 0", "2, -2, 0", "2, -1, 0", "2, 0, 0", "2, 1, 0", "2, 2, 0", "2, 3, 0", "2, 4, 0", "2, 5, 0", "2, 6, 0"]}, "cell": "vec_dot/1", "stage": "B"}
+{"id": "po-vec_dot-t2-eval-001293", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (-4, -2, 4) = -12 and v \\cdot (-2, 1, -3) = 29. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (-4, -2, 4) = -12 and v \\cdot (-2, 1, -3) = 29. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 10, -3", "ref": "-5, 10, -3", "family": "po_vec_dot", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000216, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 10, "U": [[-4, -2, 4], [-2, 1, -3]], "s": [-12, 29]}, "seed": 205845576202001773, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 7, "valid_answers": ["-5, 10, -3", "-4, 0, -7"]}, "cell": "vec_dot/2", "stage": "B"}
+{"id": "po-vec_dot-t2-eval-002270", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (2, 0, -3) = 12 and v \\cdot (-2, 0, -3) = 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (2, 0, -3) = 12 and v \\cdot (-2, 0, -3) = 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -10, -6", "ref": "-3, -10, -6", "family": "po_vec_dot", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.002268, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 10, "U": [[2, 0, -3], [-2, 0, -3]], "s": [12, 24]}, "seed": 205845576202003123, "version": "po-v0", "tier": 2, "n_raw": 21, "small": 6, "valid_answers": ["-3, -10, -6", "-3, -9, -6", "-3, -8, -6", "-3, -7, -6", "-3, -6, -6", "-3, -5, -6", "-3, -4, -6", "-3, -3, -6", "-3, -2, -6", "-3, -1, -6", "-3, 0, -6", "-3, 1, -6", "-3, 2, -6", "-3, 3, -6", "-3, 4, -6", "-3, 5, -6", "-3, 6, -6", "-3, 7, -6", "-3, 8, -6", "-3, 9, -6", "-3, 10, -6"]}, "cell": "vec_dot/2", "stage": "B"}
+{"id": "po-vec_dot-t2-eval-000769", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (4, 1, -1) = -7 and v \\cdot (-4, -4, 4) = 40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (4, 1, -1) = -7 and v \\cdot (-4, -4, 4) = 40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, -10, 1", "ref": "1, -10, 1", "family": "po_vec_dot", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.00108, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 10, "U": [[4, 1, -1], [-4, -4, 4]], "s": [-7, 40]}, "seed": 205845576202001060, "version": "po-v0", "tier": 2, "n_raw": 10, "small": 6, "valid_answers": ["1, -10, 1", "1, -9, 2", "1, -8, 3", "1, -7, 4", "1, -6, 5", "1, -5, 6", "1, -4, 7", "1, -3, 8", "1, -2, 9", "1, -1, 10"]}, "cell": "vec_dot/2", "stage": "B"}
+{"id": "po-vec_dot-t2-eval-000313", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (3, 3, 3) = 24 and v \\cdot (-4, -2, 2) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (3, 3, 3) = 24 and v \\cdot (-4, -2, 2) = -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 9, 1", "ref": "-2, 9, 1", "family": "po_vec_dot", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000756, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 10, "U": [[3, 3, 3], [-4, -2, 2]], "s": [24, -8]}, "seed": 205845576202000442, "version": "po-v0", "tier": 2, "n_raw": 7, "small": 3, "valid_answers": ["-2, 9, 1", "0, 6, 2", "2, 3, 3", "4, 0, 4", "6, -3, 5", "8, -6, 6", "10, -9, 7"]}, "cell": "vec_dot/2", "stage": "B"}
+{"id": "po-vec_dot-t2-eval-001234", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (1, 3, 0) = -18 and v \\cdot (-1, -2, 0) = 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (1, 3, 0) = -18 and v \\cdot (-1, -2, 0) = 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -3, -10", "ref": "-9, -3, -10", "family": "po_vec_dot", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 21, "bucket": null, "guess_rate": 0.002268, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 10, "U": [[1, 3, 0], [-1, -2, 0]], "s": [-18, 15]}, "seed": 205845576202001691, "version": "po-v0", "tier": 2, "n_raw": 21, "small": 9, "valid_answers": ["-9, -3, -10", "-9, -3, -9", "-9, -3, -8", "-9, -3, -7", "-9, -3, -6", "-9, -3, -5", "-9, -3, -4", "-9, -3, -3", "-9, -3, -2", "-9, -3, -1", "-9, -3, 0", "-9, -3, 1", "-9, -3, 2", "-9, -3, 3", "-9, -3, 4", "-9, -3, 5", "-9, -3, 6", "-9, -3, 7", "-9, -3, 8", "-9, -3, 9", "-9, -3, 10"]}, "cell": "vec_dot/2", "stage": "B"}
+{"id": "po-vec_dot-t2-eval-000401", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (1, -1, -4) = 24 and v \\cdot (2, -4, -4) = 26. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (1, -1, -4) = 24 and v \\cdot (2, -4, -4) = 26. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -3, -7", "ref": "-7, -3, -7", "family": "po_vec_dot", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000324, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 10, "U": [[1, -1, -4], [2, -4, -4]], "s": [24, 26]}, "seed": 205845576202000568, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 5, "valid_answers": ["-7, -3, -7", "-1, -1, -6", "5, 1, -5"]}, "cell": "vec_dot/2", "stage": "B"}
+{"id": "po-vec_dot-t2-eval-000918", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (-2, 1, -3) = 20 and v \\cdot (1, -4, 4) = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (-2, 1, -3) = 20 and v \\cdot (1, -4, 4) = -7. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -3, -3", "ref": "-7, -3, -3", "family": "po_vec_dot", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000216, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 10, "U": [[-2, 1, -3], [1, -4, 4]], "s": [20, -7]}, "seed": 205845576202001260, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 7, "valid_answers": ["-7, -3, -3", "1, -8, -10"]}, "cell": "vec_dot/2", "stage": "B"}
+{"id": "po-vec_dot-t2-eval-002366", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (-4, 4, 1) = -48 and v \\cdot (-2, 3, 0) = -31. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (-4, 4, 1) = -48 and v \\cdot (-2, 3, 0) = -31. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, -9, -4", "ref": "2, -9, -4", "family": "po_vec_dot", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.000324, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 10, "U": [[-4, 4, 1], [-2, 3, 0]], "s": [-48, -31]}, "seed": 205845576202003265, "version": "po-v0", "tier": 2, "n_raw": 3, "small": 7, "valid_answers": ["2, -9, -4", "5, -7, 0", "8, -5, 4"]}, "cell": "vec_dot/2", "stage": "B"}
+{"id": "po-vec_dot-t2-eval-000655", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (-2, 0, -4) = 0 and v \\cdot (0, 4, -2) = 32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (-2, 0, -4) = 0 and v \\cdot (0, 4, -2) = 32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 10, 4", "ref": "-8, 10, 4", "family": "po_vec_dot", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.00054, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 10, "U": [[-2, 0, -4], [0, 4, -2]], "s": [0, 32]}, "seed": 205845576202000901, "version": "po-v0", "tier": 2, "n_raw": 5, "small": 7, "valid_answers": ["-8, 10, 4", "-4, 9, 2", "0, 8, 0", "4, 7, -2", "8, 6, -4"]}, "cell": "vec_dot/2", "stage": "B"}
+{"id": "po-vec_dot-t2-eval-001851", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (3, 0, 0) = 27 and v \\cdot (4, -4, 4) = 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (3, 0, 0) = 27 and v \\cdot (4, -4, 4) = 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "9, -10, -8", "ref": "9, -10, -8", "family": "po_vec_dot", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 19, "bucket": "9-20", "guess_rate": 0.002052, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 10, "U": [[3, 0, 0], [4, -4, 4]], "s": [27, 44]}, "seed": 205845576202002519, "version": "po-v0", "tier": 2, "n_raw": 19, "small": 9, "valid_answers": ["9, -10, -8", "9, -9, -7", "9, -8, -6", "9, -7, -5", "9, -6, -4", "9, -5, -3", "9, -4, -2", "9, -3, -1", "9, -2, 0", "9, -1, 1", "9, 0, 2", "9, 1, 3", "9, 2, 4", "9, 3, 5", "9, 4, 6", "9, 5, 7", "9, 6, 8", "9, 7, 9", "9, 8, 10"]}, "cell": "vec_dot/2", "stage": "B"}
+{"id": "po-vec_dot-t2-eval-000503", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (4, -2, -4) = 18 and v \\cdot (-4, 4, 1) = -19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (4, -2, -4) = 18 and v \\cdot (-4, 4, 1) = -19. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-1, -5, -3", "ref": "-1, -5, -3", "family": "po_vec_dot", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000216, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 10, "U": [[4, -2, -4], [-4, 4, 1]], "s": [18, -19]}, "seed": 205845576202000704, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 5, "valid_answers": ["-1, -5, -3", "6, 1, 1"]}, "cell": "vec_dot/2", "stage": "B"}
+{"id": "po-vec_dot-t2-eval-002357", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (3, -4, 4) = 19 and v \\cdot (0, -1, -3) = -26. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z with -10 <= x, y, z <= 10 such that v = (x, y, z) satisfies v \\cdot (3, -4, 4) = 19 and v \\cdot (0, -1, -3) = -26. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-7, -1, 9", "ref": "-7, -1, 9", "family": "po_vec_dot", "domain": "linalg", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000216, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 3, "B": 10, "U": [[3, -4, 4], [0, -1, -3]], "s": [19, -26]}, "seed": 205845576202003250, "version": "po-v0", "tier": 2, "n_raw": 2, "small": 9, "valid_answers": ["-7, -1, 9", "9, 8, 6"]}, "cell": "vec_dot/2", "stage": "B"}
+{"id": "po-vec_dot-t3-eval-002458", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (2, 2, -1, 2) = -22 and v \\cdot (-4, 1, 0, -3) = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (2, 2, -1, 2) = -22 and v \\cdot (-4, 1, 0, -3) = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -6, 4, 3", "ref": "-6, -6, 4, 3", "family": "po_vec_dot", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.000245, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 4, "B": 6, "U": [[2, 2, -1, 2], [-4, 1, 0, -3]], "s": [-22, 9]}, "seed": 205845576203002599, "version": "po-v0", "tier": 3, "n_raw": 7, "small": 5, "valid_answers": ["-6, -6, 4, 3", "-5, -5, 6, 2", "-3, -6, 2, -1", "-2, -5, 4, -2", "-1, -4, 6, -3", "0, -6, 0, -5", "1, -5, 2, -6"]}, "cell": "vec_dot/3", "stage": "B"}
+{"id": "po-vec_dot-t3-eval-000905", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (-1, 1, -4, 0) = 19 and v \\cdot (2, -2, 4, -1) = -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (-1, 1, -4, 0) = 19 and v \\cdot (2, -2, 4, -1) = -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -3, -4, 0", "ref": "-6, -3, -4, 0", "family": "po_vec_dot", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 28, "bucket": null, "guess_rate": 0.00098, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 4, "B": 6, "U": [[-1, 1, -4, 0], [2, -2, 4, -1]], "s": [19, -22]}, "seed": 205845576203000963, "version": "po-v0", "tier": 3, "n_raw": 28, "small": 4, "valid_answers": ["-6, -3, -4, 0", "-6, 1, -3, -4", "-5, -6, -5, 4", "-5, -2, -4, 0", "-5, 2, -3, -4", "-4, -5, -5, 4", "-4, -1, -4, 0", "-4, 3, -3, -4", "-3, -4, -5, 4", "-3, 0, -4, 0", "-3, 4, -3, -4", "-2, -3, -5, 4", "-2, 1, -4, 0", "-2, 5, -3, -4", "-1, -2, -5, 4", "-1, 2, -4, 0", "-1, 6, -3, -4", "0, -1, -5, 4", "0, 3, -4, 0", "1, 0, -5, 4", "1, 4, -4, 0", "2, 1, -5, 4", "2, 5, -4, 0", "3, 2, -5, 4", "3, 6, -4, 0", "4, 3, -5, 4", "5, 4, -5, 4", "6, 5, -5, 4"]}, "cell": "vec_dot/3", "stage": "B"}
+{"id": "po-vec_dot-t3-eval-001316", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (0, 2, 4, 2) = 18 and v \\cdot (1, 4, 4, 4) = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (0, 2, 4, 2) = 18 and v \\cdot (1, 4, 4, 4) = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -3, 3, 6", "ref": "-4, -3, 3, 6", "family": "po_vec_dot", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 34, "bucket": null, "guess_rate": 0.00119, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 4, "B": 6, "U": [[0, 2, 4, 2], [1, 4, 4, 4]], "s": [18, 20]}, "seed": 205845576203001387, "version": "po-v0", "tier": 3, "n_raw": 34, "small": 4, "valid_answers": ["-4, -3, 3, 6", "-4, -2, 3, 5", "-4, -1, 3, 4", "-4, 0, 3, 3", "-4, 1, 3, 2", "-4, 2, 3, 1", "-4, 3, 3, 0", "-4, 4, 3, -1", "-4, 5, 3, -2", "-4, 6, 3, -3", "0, -5, 4, 6", "0, -4, 4, 5", "0, -3, 4, 4", "0, -2, 4, 3", "0, -1, 4, 2", "0, 0, 4, 1", "0, 1, 4, 0", "0, 2, 4, -1", "0, 3, 4, -2", "0, 4, 4, -3", "0, 5, 4, -4", "0, 6, 4, -5", "4, -6, 5, 5", "4, -5, 5, 4", "4, -4, 5, 3", "4, -3, 5, 2", "4, -2, 5, 1", "4, -1, 5, 0", "4, 0, 5, -1", "4, 1, 5, -2", "4, 2, 5, -3", "4, 3, 5, -4", "4, 4, 5, -5", "4, 5, 5, -6"]}, "cell": "vec_dot/3", "stage": "B"}
+{"id": "po-vec_dot-t3-eval-000625", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (1, 3, -3, -2) = -10 and v \\cdot (-1, 2, 2, -4) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (1, 3, -3, -2) = -10 and v \\cdot (-1, 2, 2, -4) = -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -4, -2, -1", "ref": "-6, -4, -2, -1", "family": "po_vec_dot", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.00042, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 4, "B": 6, "U": [[1, 3, -3, -2], [-1, 2, 2, -4]], "s": [-10, -2]}, "seed": 205845576203000666, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 2, "valid_answers": ["-6, -4, -2, -1", "-6, 0, 0, 2", "-6, 4, 2, 5", "-2, -5, -1, -2", "-2, -1, 1, 1", "-2, 3, 3, 4", "2, -6, 0, -3", "2, -2, 2, 0", "2, 2, 4, 3", "2, 6, 6, 6", "6, -3, 3, -1", "6, 1, 5, 2"]}, "cell": "vec_dot/3", "stage": "B"}
+{"id": "po-vec_dot-t3-eval-000289", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (-1, -3, 2, 3) = 8 and v \\cdot (0, -1, 1, 3) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (-1, -3, 2, 3) = 8 and v \\cdot (0, -1, 1, 3) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -5, -5, -1", "ref": "-6, -5, -5, -1", "family": "po_vec_dot", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 0.00063, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 4, "B": 6, "U": [[-1, -3, 2, 3], [0, -1, 1, 3]], "s": [8, -3]}, "seed": 205845576203000306, "version": "po-v0", "tier": 3, "n_raw": 18, "small": 3, "valid_answers": ["-6, -5, -5, -1", "-6, -2, 1, -2", "-5, -6, -6, -1", "-5, -3, 0, -2", "-5, 0, 6, -3", "-4, -4, -1, -2", "-4, -1, 5, -3", "-3, -5, -2, -2", "-3, -2, 4, -3", "-2, -6, -3, -2", "-2, -3, 3, -3", "-1, -4, 2, -3", "0, -5, 1, -3", "1, -6, 0, -3", "1, -3, 6, -4", "2, -4, 5, -4", "3, -5, 4, -4", "4, -6, 3, -4"]}, "cell": "vec_dot/3", "stage": "B"}
+{"id": "po-vec_dot-t3-eval-001741", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (3, -2, 0, -3) = -9 and v \\cdot (-4, 3, -3, 2) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (3, -2, 0, -3) = -9 and v \\cdot (-4, 3, -3, 2) = -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -6, 4, 2", "ref": "-5, -6, 4, 2", "family": "po_vec_dot", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.000385, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 4, "B": 6, "U": [[3, -2, 0, -3], [-4, 3, -3, 2]], "s": [-9, -6]}, "seed": 205845576203001842, "version": "po-v0", "tier": 3, "n_raw": 11, "small": 4, "valid_answers": ["-5, -6, 4, 2", "-4, -3, 5, 1", "-3, 0, 6, 0", "-2, -6, 2, 5", "-1, -3, 3, 4", "0, 0, 4, 3", "1, 3, 5, 2", "2, 6, 6, 1", "3, 0, 2, 6", "4, 3, 3, 5", "5, 6, 4, 4"]}, "cell": "vec_dot/3", "stage": "B"}
+{"id": "po-vec_dot-t3-eval-001094", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (3, 3, 1, -2) = 9 and v \\cdot (-4, 2, 0, -4) = -24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (3, 3, 1, -2) = 9 and v \\cdot (-4, 2, 0, -4) = -24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "2, 4, 3, 6", "ref": "2, 4, 3, 6", "family": "po_vec_dot", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.00049, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 4, "B": 6, "U": [[3, 3, 1, -2], [-4, 2, 0, -4]], "s": [9, -24]}, "seed": 205845576203001156, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 4, "valid_answers": ["2, 4, 3, 6", "3, 0, 6, 3", "3, 2, 2, 4", "3, 4, -2, 5", "3, 6, -6, 6", "4, -2, 5, 1", "4, 0, 1, 2", "4, 2, -3, 3", "5, -4, 4, -1", "5, -2, 0, 0", "5, 0, -4, 1", "6, -6, 3, -3", "6, -4, -1, -2", "6, -2, -5, -1"]}, "cell": "vec_dot/3", "stage": "B"}
+{"id": "po-vec_dot-t3-eval-002491", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (2, 0, -2, 4) = 28 and v \\cdot (2, 0, -3, -1) = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (2, 0, -2, 4) = 28 and v \\cdot (2, 0, -3, -1) = 9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -6, -6, 5", "ref": "-2, -6, -6, 5", "family": "po_vec_dot", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 26, "bucket": null, "guess_rate": 0.00091, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 4, "B": 6, "U": [[2, 0, -2, 4], [2, 0, -3, -1]], "s": [28, 9]}, "seed": 205845576203002633, "version": "po-v0", "tier": 3, "n_raw": 26, "small": 5, "valid_answers": ["-2, -6, -6, 5", "-2, -5, -6, 5", "-2, -4, -6, 5", "-2, -3, -6, 5", "-2, -2, -6, 5", "-2, -1, -6, 5", "-2, 0, -6, 5", "-2, 1, -6, 5", "-2, 2, -6, 5", "-2, 3, -6, 5", "-2, 4, -6, 5", "-2, 5, -6, 5", "-2, 6, -6, 5", "5, -6, -1, 4", "5, -5, -1, 4", "5, -4, -1, 4", "5, -3, -1, 4", "5, -2, -1, 4", "5, -1, -1, 4", "5, 0, -1, 4", "5, 1, -1, 4", "5, 2, -1, 4", "5, 3, -1, 4", "5, 4, -1, 4", "5, 5, -1, 4", "5, 6, -1, 4"]}, "cell": "vec_dot/3", "stage": "B"}
+{"id": "po-vec_dot-t3-eval-001103", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (0, -2, 3, 1) = 20 and v \\cdot (2, 4, 2, -2) = -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (0, -2, 3, 1) = 20 and v \\cdot (2, 4, 2, -2) = -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -6, 4, -4", "ref": "-6, -6, 4, -4", "family": "po_vec_dot", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 14, "bucket": "9-20", "guess_rate": 0.00049, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 4, "B": 6, "U": [[0, -2, 3, 1], [2, 4, 2, -2]], "s": [20, -20]}, "seed": 205845576203001166, "version": "po-v0", "tier": 3, "n_raw": 14, "small": 4, "valid_answers": ["-6, -6, 4, -4", "-6, -5, 4, -2", "-6, -4, 4, 0", "-6, -3, 4, 2", "-6, -2, 4, 4", "-6, -1, 4, 6", "-2, -6, 3, -1", "-2, -5, 3, 1", "-2, -4, 3, 3", "-2, -3, 3, 5", "2, -6, 2, 2", "2, -5, 2, 4", "2, -4, 2, 6", "6, -6, 1, 5"]}, "cell": "vec_dot/3", "stage": "B"}
+{"id": "po-vec_dot-t3-eval-001265", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (3, 2, 1, 1) = 27 and v \\cdot (-2, -2, 1, 3) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (3, 2, 1, 1) = 27 and v \\cdot (-2, -2, 1, 3) = -3. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, 4, 4, 3", "ref": "4, 4, 4, 3", "family": "po_vec_dot", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000175, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 4, "B": 6, "U": [[3, 2, 1, 1], [-2, -2, 1, 3]], "s": [27, -3]}, "seed": 205845576203001336, "version": "po-v0", "tier": 3, "n_raw": 5, "small": 4, "valid_answers": ["4, 4, 4, 3", "4, 5, 0, 5", "6, 1, 5, 2", "6, 2, 1, 4", "6, 3, -3, 6"]}, "cell": "vec_dot/3", "stage": "B"}
+{"id": "po-vec_dot-t3-eval-001343", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (-2, 4, 0, -3) = -2 and v \\cdot (2, 2, -1, 2) = -10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (-2, 4, 0, -3) = -2 and v \\cdot (2, 2, -1, 2) = -10. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -2, -2, 2", "ref": "-6, -2, -2, 2", "family": "po_vec_dot", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.00042, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 4, "B": 6, "U": [[-2, 4, 0, -3], [2, 2, -1, 2]], "s": [-2, -10]}, "seed": 205845576203001418, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 2, "valid_answers": ["-6, -2, -2, 2", "-5, -3, -6, 0", "-4, -1, 4, 2", "-3, -2, 0, 0", "-2, -3, -4, -2", "-1, -1, 6, 0", "0, -2, 2, -2", "1, -3, -2, -4", "2, -4, -6, -6", "3, -2, 4, -4", "4, -3, 0, -6", "6, -2, 6, -6"]}, "cell": "vec_dot/3", "stage": "B"}
+{"id": "po-vec_dot-t3-eval-001264", "task": "cx_po_vec_dot", "source": "cx_po_vec_dot", "split": "eval", "problem": "Find integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (4, -3, -3, 0) = 18 and v \\cdot (4, -2, -2, 2) = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers x, y, z, w with -6 <= x, y, z, w <= 6 such that v = (x, y, z, w) satisfies v \\cdot (4, -3, -3, 0) = 18 and v \\cdot (4, -2, -2, 2) = 20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{x = ..., y = ..., z = ..., w = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -6, -4, 6", "ref": "-3, -6, -4, 6", "family": "po_vec_dot", "domain": "linalg", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 32, "bucket": null, "guess_rate": 0.00112, "solvable": true, "holdout": false, "meta": {"family": "po_vec_dot", "rung": "witness", "params": {"n": 4, "B": 6, "U": [[4, -3, -3, 0], [4, -2, -2, 2]], "s": [18, 20]}, "seed": 205845576203001335, "version": "po-v0", "tier": 3, "n_raw": 32, "small": 3, "valid_answers": ["-3, -6, -4, 6", "-3, -5, -5, 6", "-3, -4, -6, 6", "0, -6, 0, 4", "0, -5, -1, 4", "0, -4, -2, 4", "0, -3, -3, 4", "0, -2, -4, 4", "0, -1, -5, 4", "0, 0, -6, 4", "3, -6, 4, 2", "3, -5, 3, 2", "3, -4, 2, 2", "3, -3, 1, 2", "3, -2, 0, 2", "3, -1, -1, 2", "3, 0, -2, 2", "3, 1, -3, 2", "3, 2, -4, 2", "3, 3, -5, 2", "3, 4, -6, 2", "6, -4, 6, 0", "6, -3, 5, 0", "6, -2, 4, 0", "6, -1, 3, 0", "6, 0, 2, 0", "6, 1, 1, 0", "6, 2, 0, 0", "6, 3, -1, 0", "6, 4, -2, 0", "6, 5, -3, 0", "6, 6, -4, 0"]}, "cell": "vec_dot/3", "stage": "B"}
+{"id": "po-vieta3-t1-eval-000030", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -9 and constant term 60. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -9 and constant term 60. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, 2, 10", "ref": "-3, 2, 10", "family": "po_vieta3", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7e-06, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 9, "P": -60}, "seed": 52979190401000219, "version": "po-v0", "tier": 1, "n_raw": 12, "small": 6, "valid_answers": ["-3, 2, 10", "-2, 5, 6"]}, "cell": "vieta3/1", "stage": "B"}
+{"id": "po-vieta3-t1-eval-000059", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 10 and constant term -70. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 10 and constant term -70. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-14, -1, 5", "ref": "-14, -1, 5", "family": "po_vieta3", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 4e-06, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": -10, "P": 70}, "seed": 52979190401000495, "version": "po-v0", "tier": 1, "n_raw": 12, "small": 7, "valid_answers": ["-14, -1, 5", "-7, -5, 2"]}, "cell": "vieta3/1", "stage": "B"}
+{"id": "po-vieta3-t1-eval-000082", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -2 and constant term 90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -2 and constant term 90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, 1, 10", "ref": "-9, 1, 10", "family": "po_vieta3", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2e-06, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 2, "P": -90}, "seed": 52979190401000971, "version": "po-v0", "tier": 1, "n_raw": 12, "small": 6, "valid_answers": ["-9, 1, 10", "-6, 3, 5"]}, "cell": "vieta3/1", "stage": "B"}
+{"id": "po-vieta3-t1-eval-000021", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 5 and constant term -45. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 5 and constant term -45. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-9, -1, 5", "ref": "-9, -1, 5", "family": "po_vieta3", "domain": "polynomial", "difficulty": 1, "tier": 1, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": -5, "P": 45}, "seed": 52979190401000141, "version": "po-v0", "tier": 1, "n_raw": 12, "small": 5, "valid_answers": ["-9, -1, 5", "-5, -3, 3"]}, "cell": "vieta3/1", "stage": "B"}
+{"id": "po-vieta3-t2-eval-000466", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -15 and constant term 840. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -15 and constant term 840. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-15, 2, 28", "ref": "-15, 2, 28", "family": "po_vieta3", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 15, "P": -840}, "seed": 52979190402004124, "version": "po-v0", "tier": 2, "n_raw": 18, "small": 12, "valid_answers": ["-15, 2, 28", "-10, 4, 21", "-7, 10, 12"]}, "cell": "vieta3/2", "stage": "C"}
+{"id": "po-vieta3-t2-eval-000191", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -9 and constant term 648. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -9 and constant term 648. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-12, 3, 18", "ref": "-12, 3, 18", "family": "po_vieta3", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 9, "P": -648}, "seed": 52979190402001003, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 12, "valid_answers": ["-12, 3, 18", "-9, 6, 12"]}, "cell": "vieta3/2", "stage": "C"}
+{"id": "po-vieta3-t2-eval-000289", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 8 and constant term -800. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 8 and constant term -800. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-32, -1, 25", "ref": "-32, -1, 25", "family": "po_vieta3", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": -8, "P": 800}, "seed": 52979190402001669, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 10, "valid_answers": ["-32, -1, 25", "-10, -8, 10"]}, "cell": "vieta3/2", "stage": "C"}
+{"id": "po-vieta3-t2-eval-000495", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -11 and constant term 560. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -11 and constant term 560. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-8, 5, 14", "ref": "-8, 5, 14", "family": "po_vieta3", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 11, "P": -560}, "seed": 52979190402005139, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 10, "valid_answers": ["-8, 5, 14", "-7, 8, 10"]}, "cell": "vieta3/2", "stage": "C"}
+{"id": "po-vieta3-t2-eval-000000", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -21 and constant term -168. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -21 and constant term -168. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -1, 28", "ref": "-6, -1, 28", "family": "po_vieta3", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 21, "P": 168}, "seed": 52979190402000004, "version": "po-v0", "tier": 2, "n_raw": 18, "small": 12, "valid_answers": ["-6, -1, 28", "2, 7, 12", "3, 4, 14"]}, "cell": "vieta3/2", "stage": "C"}
+{"id": "po-vieta3-t2-eval-000072", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -3 and constant term 360. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -3 and constant term 360. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-18, 1, 20", "ref": "-18, 1, 20", "family": "po_vieta3", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 3, "P": -360}, "seed": 52979190402000306, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 10, "valid_answers": ["-18, 1, 20", "-10, 4, 9"]}, "cell": "vieta3/2", "stage": "C"}
+{"id": "po-vieta3-t2-eval-000440", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 16 and constant term -1152. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 16 and constant term -1152. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-32, -2, 18", "ref": "-32, -2, 18", "family": "po_vieta3", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": -16, "P": 1152}, "seed": 52979190402003616, "version": "po-v0", "tier": 2, "n_raw": 15, "small": 12, "valid_answers": ["-32, -2, 18", "-24, -4, 12", "-12, -12, 8"]}, "cell": "vieta3/2", "stage": "C"}
+{"id": "po-vieta3-t2-eval-000556", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -4 and constant term 576. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -4 and constant term 576. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-16, 2, 18", "ref": "-16, 2, 18", "family": "po_vieta3", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 4, "P": -576}, "seed": 52979190402012200, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 12, "valid_answers": ["-16, 2, 18", "-12, 4, 12"]}, "cell": "vieta3/2", "stage": "C"}
+{"id": "po-vieta3-t2-eval-000333", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 10 and constant term -90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 10 and constant term -90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-15, -1, 6", "ref": "-15, -1, 6", "family": "po_vieta3", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2e-06, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": -10, "P": 90}, "seed": 52979190402002101, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 10, "valid_answers": ["-15, -1, 6", "-10, -3, 3"]}, "cell": "vieta3/2", "stage": "C"}
+{"id": "po-vieta3-t2-eval-000557", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -36 and constant term -1728. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -36 and constant term -1728. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-27, -1, 64", "ref": "-27, -1, 64", "family": "po_vieta3", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 36, "P": 1728}, "seed": 52979190402012971, "version": "po-v0", "tier": 2, "n_raw": 16, "small": 12, "valid_answers": ["-27, -1, 64", "-16, -2, 54", "-6, -6, 48", "12, 12, 12"]}, "cell": "vieta3/2", "stage": "C"}
+{"id": "po-vieta3-t2-eval-000392", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -6 and constant term 924. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -6 and constant term 924. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-28, 1, 33", "ref": "-28, 1, 33", "family": "po_vieta3", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 6, "P": -924}, "seed": 52979190402002870, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 12, "valid_answers": ["-28, 1, 33", "-12, 7, 11"]}, "cell": "vieta3/2", "stage": "C"}
+{"id": "po-vieta3-t2-eval-000039", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 13 and constant term -280. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 13 and constant term -280. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-14, -4, 5", "ref": "-14, -4, 5", "family": "po_vieta3", "domain": "polynomial", "difficulty": 2, "tier": 2, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": -13, "P": 280}, "seed": 52979190402000146, "version": "po-v0", "tier": 2, "n_raw": 12, "small": 10, "valid_answers": ["-14, -4, 5", "-10, -7, 4"]}, "cell": "vieta3/2", "stage": "C"}
+{"id": "po-vieta3-t3-eval-001558", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -13 and constant term 2880. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -13 and constant term 2880. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-48, 1, 60", "ref": "-48, 1, 60", "family": "po_vieta3", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 13, "P": -2880}, "seed": 52979190403008592, "version": "po-v0", "tier": 3, "n_raw": 18, "small": 16, "valid_answers": ["-48, 1, 60", "-16, 9, 20", "-15, 12, 16"]}, "cell": "vieta3/3", "stage": "B"}
+{"id": "po-vieta3-t3-eval-002075", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 24 and constant term -2800. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 24 and constant term -2800. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-50, -2, 28", "ref": "-50, -2, 28", "family": "po_vieta3", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": -24, "P": 2800}, "seed": 52979190403015842, "version": "po-v0", "tier": 3, "n_raw": 18, "small": 20, "valid_answers": ["-50, -2, 28", "-35, -5, 16", "-20, -14, 10"]}, "cell": "vieta3/3", "stage": "B"}
+{"id": "po-vieta3-t3-eval-000083", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -1 and constant term -960. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -1 and constant term -960. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-30, -1, 32", "ref": "-30, -1, 32", "family": "po_vieta3", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 1, "P": 960}, "seed": 52979190403000308, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 20, "valid_answers": ["-30, -1, 32", "-16, -3, 20"]}, "cell": "vieta3/3", "stage": "B"}
+{"id": "po-vieta3-t3-eval-001087", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 21 and constant term -1944. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 21 and constant term -1944. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-36, -3, 18", "ref": "-36, -3, 18", "family": "po_vieta3", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": -21, "P": 1944}, "seed": 52979190403004978, "version": "po-v0", "tier": 3, "n_raw": 18, "small": 18, "valid_answers": ["-36, -3, 18", "-27, -6, 12", "-18, -12, 9"]}, "cell": "vieta3/3", "stage": "B"}
+{"id": "po-vieta3-t3-eval-000237", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 12 and constant term -5120. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 12 and constant term -5120. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-40, -4, 32", "ref": "-40, -4, 32", "family": "po_vieta3", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": -12, "P": 5120}, "seed": 52979190403000797, "version": "po-v0", "tier": 3, "n_raw": 9, "small": 20, "valid_answers": ["-40, -4, 32", "-16, -16, 20"]}, "cell": "vieta3/3", "stage": "B"}
+{"id": "po-vieta3-t3-eval-000691", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -21 and constant term 240. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -21 and constant term 240. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, 2, 24", "ref": "-5, 2, 24", "family": "po_vieta3", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 21, "P": -240}, "seed": 52979190403002816, "version": "po-v0", "tier": 3, "n_raw": 18, "small": 15, "valid_answers": ["-5, 2, 24", "-3, 4, 20", "-2, 8, 15"]}, "cell": "vieta3/3", "stage": "B"}
+{"id": "po-vieta3-t3-eval-000507", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -13 and constant term 420. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -13 and constant term 420. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-10, 2, 21", "ref": "-10, 2, 21", "family": "po_vieta3", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 13, "P": -420}, "seed": 52979190403001963, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 14, "valid_answers": ["-10, 2, 21", "-6, 5, 14"]}, "cell": "vieta3/3", "stage": "B"}
+{"id": "po-vieta3-t3-eval-001990", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 10 and constant term -2880. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 10 and constant term -2880. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-30, -4, 24", "ref": "-30, -4, 24", "family": "po_vieta3", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": -10, "P": 2880}, "seed": 52979190403014395, "version": "po-v0", "tier": 3, "n_raw": 18, "small": 20, "valid_answers": ["-30, -4, 24", "-24, -6, 20", "-20, -8, 18"]}, "cell": "vieta3/3", "stage": "B"}
+{"id": "po-vieta3-t3-eval-000866", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -32 and constant term 2160. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -32 and constant term 2160. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-16, 3, 45", "ref": "-16, 3, 45", "family": "po_vieta3", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 32, "P": -2160}, "seed": 52979190403003690, "version": "po-v0", "tier": 3, "n_raw": 18, "small": 20, "valid_answers": ["-16, 3, 45", "-10, 6, 36", "-6, 18, 20"]}, "cell": "vieta3/3", "stage": "B"}
+{"id": "po-vieta3-t3-eval-000444", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -18 and constant term -42. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient -18 and constant term -42. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, -1, 21", "ref": "-2, -1, 21", "family": "po_vieta3", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": 18, "P": 42}, "seed": 52979190403001652, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 14, "valid_answers": ["-2, -1, 21", "1, 3, 14"]}, "cell": "vieta3/3", "stage": "B"}
+{"id": "po-vieta3-t3-eval-000281", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 24 and constant term -1680. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 24 and constant term -1680. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-42, -2, 20", "ref": "-42, -2, 20", "family": "po_vieta3", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": -24, "P": 1680}, "seed": 52979190403000968, "version": "po-v0", "tier": 3, "n_raw": 18, "small": 16, "valid_answers": ["-42, -2, 20", "-28, -6, 10", "-16, -15, 7"]}, "cell": "vieta3/3", "stage": "B"}
+{"id": "po-vieta3-t3-eval-000445", "task": "cx_po_vieta3", "source": "cx_po_vieta3", "split": "eval", "problem": "Find integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 6 and constant term -1144. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c such that the cubic (x - a)(x - b)(x - c) has x^2-coefficient 6 and constant term -1144. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \\boxed{a = ..., b = ..., c = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-26, -2, 22", "ref": "-26, -2, 22", "family": "po_vieta3", "domain": "polynomial", "difficulty": 3, "tier": 3, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.0, "solvable": true, "holdout": false, "meta": {"family": "po_vieta3", "rung": "witness", "params": {"S": -6, "P": 1144}, "seed": 52979190403001653, "version": "po-v0", "tier": 3, "n_raw": 12, "small": 13, "valid_answers": ["-26, -2, 22", "-11, -8, 13"]}, "cell": "vieta3/3", "stage": "B"}