| {"id": "po-consecutive_sum-t0-eval-000002", "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) = 140. 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) = 140. 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": "14, 8", "ref": "14, 8", "family": "po_consecutive_sum", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001339, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 140}, "seed": 342287044200000002, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 14, "valid_answers": ["14, 8", "17, 7", "26, 5"]}} | |
| {"id": "po-consecutive_sum-t0-eval-000008", "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) = 120. 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) = 120. 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, 15", "ref": "1, 15", "family": "po_consecutive_sum", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001667, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 120}, "seed": 342287044200000011, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 15, "valid_answers": ["1, 15", "22, 5", "39, 3"]}} | |
| {"id": "po-consecutive_sum-t0-eval-000005", "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) = 30. 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) = 30. 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, 5", "ref": "4, 5", "family": "po_consecutive_sum", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.014286, "solvable": true, "holdout": false, "meta": {"family": "po_consecutive_sum", "rung": "witness", "params": {"N": 30}, "seed": 342287044200000005, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["4, 5", "6, 4", "9, 3"]}} | |
| {"id": "po-crt_range-t0-eval-000004", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 796 <= x <= 1874 such that x \\equiv 0 \\pmod{9}, x \\equiv 9 \\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 796 <= x <= 1874 such that x \\equiv 0 \\pmod{9}, x \\equiv 9 \\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": "801", "ref": "801", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.010195, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[0, 9], [9, 11]], "L": 796, "U": 1874}, "seed": 401691043400000006, "version": "po-v0", "tier": 0, "n_raw": 11, "small": 801, "valid_answers": ["801", "900", "999", "1098", "1197", "1296", "1395", "1494", "1593", "1692", "1791"]}} | |
| {"id": "po-crt_range-t0-eval-000150", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 418 <= x <= 574 such that x \\equiv 4 \\pmod{11}, x \\equiv 0 \\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 418 <= x <= 574 such that x \\equiv 4 \\pmod{11}, x \\equiv 0 \\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": "455", "ref": "455", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.019108, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[4, 11], [0, 5]], "L": 418, "U": 574}, "seed": 401691043400000210, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 455, "valid_answers": ["455", "510", "565"]}} | |
| {"id": "po-crt_range-t0-eval-000208", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 568 <= x <= 1638 such that x \\equiv 6 \\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 568 <= x <= 1638 such that x \\equiv 6 \\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": "578", "ref": "578", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.010271, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[6, 11], [2, 9]], "L": 568, "U": 1638}, "seed": 401691043400000286, "version": "po-v0", "tier": 0, "n_raw": 11, "small": 578, "valid_answers": ["578", "677", "776", "875", "974", "1073", "1172", "1271", "1370", "1469", "1568"]}} | |
| {"id": "po-crt_range-t0-eval-000117", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 681 <= x <= 1856 such that x \\equiv 2 \\pmod{9}, x \\equiv 1 \\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 681 <= x <= 1856 such that x \\equiv 2 \\pmod{9}, x \\equiv 1 \\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": "749", "ref": "749", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.010204, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[2, 9], [1, 11]], "L": 681, "U": 1856}, "seed": 401691043400000161, "version": "po-v0", "tier": 0, "n_raw": 12, "small": 749, "valid_answers": ["749", "848", "947", "1046", "1145", "1244", "1343", "1442", "1541", "1640", "1739", "1838"]}} | |
| {"id": "po-crt_range-t0-eval-000166", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 202 <= x <= 375 such that x \\equiv 3 \\pmod{11}, 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 202 <= x <= 375 such that x \\equiv 3 \\pmod{11}, 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": "256", "ref": "256", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011494, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[3, 11], [4, 7]], "L": 202, "U": 375}, "seed": 401691043400000228, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 256, "valid_answers": ["256", "333"]}} | |
| {"id": "po-crt_range-t0-eval-000154", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 596 <= x <= 1201 such that x \\equiv 6 \\pmod{8}, 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 596 <= x <= 1201 such that x \\equiv 6 \\pmod{8}, 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": "606", "ref": "606", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.014851, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[6, 8], [3, 9]], "L": 596, "U": 1201}, "seed": 401691043400000215, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 606, "valid_answers": ["606", "678", "750", "822", "894", "966", "1038", "1110", "1182"]}} | |
| {"id": "po-crt_range-t0-eval-000077", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1722 <= x <= 1834 such that x \\equiv 3 \\pmod{8}, x \\equiv 0 \\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 1722 <= x <= 1834 such that x \\equiv 3 \\pmod{8}, x \\equiv 0 \\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": "1755", "ref": "1755", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.017699, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[3, 8], [0, 5]], "L": 1722, "U": 1834}, "seed": 401691043400000104, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 1755, "valid_answers": ["1755", "1795"]}} | |
| {"id": "po-crt_range-t0-eval-000192", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 545 <= x <= 1222 such that x \\equiv 4 \\pmod{5}, x \\equiv 2 \\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 545 <= x <= 1222 such that x \\equiv 4 \\pmod{5}, x \\equiv 2 \\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": "574", "ref": "574", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 12, "bucket": "9-20", "guess_rate": 0.017699, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[4, 5], [2, 11]], "L": 545, "U": 1222}, "seed": 401691043400000262, "version": "po-v0", "tier": 0, "n_raw": 12, "small": 574, "valid_answers": ["574", "629", "684", "739", "794", "849", "904", "959", "1014", "1069", "1124", "1179"]}} | |
| {"id": "po-crt_range-t0-eval-000094", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1596 <= x <= 2069 such that x \\equiv 3 \\pmod{9}, x \\equiv 7 \\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 1596 <= x <= 2069 such that x \\equiv 3 \\pmod{9}, x \\equiv 7 \\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": "1623", "ref": "1623", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.014768, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[3, 9], [7, 8]], "L": 1596, "U": 2069}, "seed": 401691043400000127, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 1623, "valid_answers": ["1623", "1695", "1767", "1839", "1911", "1983", "2055"]}} | |
| {"id": "po-crt_range-t0-eval-000101", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1215 <= x <= 1477 such that x \\equiv 2 \\pmod{5}, x \\equiv 9 \\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 1215 <= x <= 1477 such that x \\equiv 2 \\pmod{5}, x \\equiv 9 \\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": "1252", "ref": "1252", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.019011, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[2, 5], [9, 11]], "L": 1215, "U": 1477}, "seed": 401691043400000136, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 1252, "valid_answers": ["1252", "1307", "1362", "1417", "1472"]}} | |
| {"id": "po-crt_range-t0-eval-000206", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1266 <= x <= 1731 such that x \\equiv 4 \\pmod{11}, 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 1266 <= x <= 1731 such that x \\equiv 4 \\pmod{11}, 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": "1313", "ref": "1313", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.012876, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[4, 11], [4, 7]], "L": 1266, "U": 1731}, "seed": 401691043400000282, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 1313, "valid_answers": ["1313", "1390", "1467", "1544", "1621", "1698"]}} | |
| {"id": "po-crt_range-t0-eval-000067", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1831 <= x <= 2103 such that x \\equiv 5 \\pmod{8}, 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 1831 <= x <= 2103 such that x \\equiv 5 \\pmod{8}, 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": "1837", "ref": "1837", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.018315, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[5, 8], [3, 7]], "L": 1831, "U": 2103}, "seed": 401691043400000092, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 1837, "valid_answers": ["1837", "1893", "1949", "2005", "2061"]}} | |
| {"id": "po-crt_range-t0-eval-000044", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1719 <= x <= 2445 such that x \\equiv 5 \\pmod{8}, 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 1719 <= x <= 2445 such that x \\equiv 5 \\pmod{8}, 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": "1733", "ref": "1733", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.01238, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[5, 8], [6, 11]], "L": 1719, "U": 2445}, "seed": 401691043400000064, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 1733, "valid_answers": ["1733", "1821", "1909", "1997", "2085", "2173", "2261", "2349", "2437"]}} | |
| {"id": "po-crt_range-t0-eval-000162", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1622 <= x <= 2210 such that x \\equiv 3 \\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 1622 <= x <= 2210 such that x \\equiv 3 \\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": "1697", "ref": "1697", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.011885, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[3, 11], [3, 7]], "L": 1622, "U": 2210}, "seed": 401691043400000224, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 1697, "valid_answers": ["1697", "1774", "1851", "1928", "2005", "2082", "2159"]}} | |
| {"id": "po-crt_range-t0-eval-000212", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 1294 <= x <= 1552 such that x \\equiv 0 \\pmod{9}, 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 1294 <= x <= 1552 such that x \\equiv 0 \\pmod{9}, 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": "1314", "ref": "1314", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.015444, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[0, 9], [5, 7]], "L": 1294, "U": 1552}, "seed": 401691043400000290, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 1314, "valid_answers": ["1314", "1377", "1440", "1503"]}} | |
| {"id": "po-crt_range-t0-eval-000295", "task": "cx_po_crt_range", "source": "cx_po_crt_range", "split": "eval", "problem": "Find an integer x with 890 <= x <= 1522 such that x \\equiv 4 \\pmod{8}, x \\equiv 0 \\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 890 <= x <= 1522 such that x \\equiv 4 \\pmod{8}, x \\equiv 0 \\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": "924", "ref": "924", "family": "po_crt_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.011058, "solvable": true, "holdout": false, "meta": {"family": "po_crt_range", "rung": "witness", "params": {"cong": [[4, 8], [0, 11]], "L": 890, "U": 1522}, "seed": 401691043400000424, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 924, "valid_answers": ["924", "1012", "1100", "1188", "1276", "1364", "1452"]}} | |
| {"id": "po-diff_squares-t0-eval-000008", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 143. Several answers are possible; any one correct answer is 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 = 143. Several answers are possible; any one correct answer is accepted. Give 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, 1", "ref": "12, 1", "family": "po_diff_squares", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000375, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 143}, "seed": 194317368800000013, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 12, "valid_answers": ["12, 1", "72, 71"]}} | |
| {"id": "po-diff_squares-t0-eval-000013", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 391. Several answers are possible; any one correct answer is 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 = 391. Several answers are possible; any one correct answer is accepted. Give 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_diff_squares", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 5.2e-05, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 391}, "seed": 194317368800000020, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 20, "valid_answers": ["20, 3", "196, 195"]}} | |
| {"id": "po-diff_squares-t0-eval-000018", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 133. Several answers are possible; any one correct answer is 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 = 133. Several answers are possible; any one correct answer is accepted. Give 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_diff_squares", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000433, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 133}, "seed": 194317368800000028, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 13, "valid_answers": ["13, 6", "67, 66"]}} | |
| {"id": "po-diff_squares-t0-eval-000022", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 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 nonnegative integers x, y such that x^2 - y^2 = 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": "8, 5", "ref": "8, 5", "family": "po_diff_squares", "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_diff_squares", "rung": "witness", "params": {"N": 39}, "seed": 194317368800000037, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 8, "valid_answers": ["8, 5", "20, 19"]}} | |
| {"id": "po-diff_squares-t0-eval-000003", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 247. Several answers are possible; any one correct answer is 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 = 247. Several answers are possible; any one correct answer is accepted. Give 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, 3", "ref": "16, 3", "family": "po_diff_squares", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000128, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 247}, "seed": 194317368800000005, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 16, "valid_answers": ["16, 3", "124, 123"]}} | |
| {"id": "po-diff_squares-t0-eval-000019", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 55. Several answers are possible; any one correct answer is 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 = 55. Several answers are possible; any one correct answer is accepted. Give 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, 3", "ref": "8, 3", "family": "po_diff_squares", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002378, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 55}, "seed": 194317368800000030, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 8, "valid_answers": ["8, 3", "28, 27"]}} | |
| {"id": "po-diff_squares-t0-eval-000005", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 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 nonnegative integers x, y such that x^2 - y^2 = 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": "11, 8", "ref": "11, 8", "family": "po_diff_squares", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002222, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 57}, "seed": 194317368800000007, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 11, "valid_answers": ["11, 8", "29, 28"]}} | |
| {"id": "po-diff_squares-t0-eval-000031", "task": "cx_po_diff_squares", "source": "cx_po_diff_squares", "split": "eval", "problem": "Find nonnegative integers x, y such that x^2 - y^2 = 289. Several answers are possible; any one correct answer is 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 = 289. Several answers are possible; any one correct answer is accepted. Give 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, 0", "ref": "17, 0", "family": "po_diff_squares", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 9.4e-05, "solvable": true, "holdout": false, "meta": {"family": "po_diff_squares", "rung": "witness", "params": {"N": 289}, "seed": 194317368800000120, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 17, "valid_answers": ["17, 0", "145, 144"]}} | |
| {"id": "po-egyptian-t0-eval-000001", "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}{11}. 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}{11}. 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, 132", "ref": "12, 132", "family": "po_egyptian", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000228, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 1, "v": 11}, "seed": 186125617000000001, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 22, "valid_answers": ["12, 132", "22, 22"]}} | |
| {"id": "po-egyptian-t0-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{1}{18}. 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}{18}. 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": "19, 342", "ref": "19, 342", "family": "po_egyptian", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.000136, "solvable": true, "holdout": false, "meta": {"family": "po_egyptian", "rung": "witness", "params": {"u": 1, "v": 18}, "seed": 186125617000000002, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 36, "valid_answers": ["19, 342", "20, 180", "21, 126", "22, 99", "24, 72", "27, 54", "30, 45", "36, 36"]}} | |
| {"id": "po-gcd_lcm-t0-eval-000093", "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 1960. 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 1960. 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, 1960", "ref": "10, 1960", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 10, "L": 1960, "k": 2}, "seed": 43074069000000110, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 490, "valid_answers": ["10, 1960", "40, 490"]}} | |
| {"id": "po-gcd_lcm-t0-eval-000252", "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 40898. 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 40898. 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, 40898", "ref": "2, 40898", "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": 2, "L": 40898, "k": 2}, "seed": 43074069000000474, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 338, "valid_answers": ["2, 40898", "242, 338"]}} | |
| {"id": "po-gcd_lcm-t0-eval-000105", "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 396. 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 396. 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, 396", "ref": "4, 396", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.5e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 4, "L": 396, "k": 2}, "seed": 43074069000000123, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 44, "valid_answers": ["4, 396", "36, 44"]}} | |
| {"id": "po-gcd_lcm-t0-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 6 and whose least common multiple is 378. 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 378. 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, 378", "ref": "6, 378", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.8e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 6, "L": 378, "k": 2}, "seed": 43074069000000233, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 54, "valid_answers": ["6, 378", "42, 54"]}} | |
| {"id": "po-gcd_lcm-t0-eval-000091", "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 546. 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 546. 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, 546", "ref": "6, 546", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1.3e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 6, "L": 546, "k": 2}, "seed": 43074069000000108, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 78, "valid_answers": ["6, 546", "42, 78"]}} | |
| {"id": "po-gcd_lcm-t0-eval-000218", "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 1274. 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 1274. 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, 1274", "ref": "2, 1274", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 2, "L": 1274, "k": 2}, "seed": 43074069000000357, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 98, "valid_answers": ["2, 1274", "26, 98"]}} | |
| {"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 6 and whose least common multiple is 9438. 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 9438. 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, 9438", "ref": "6, 9438", "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": 9438, "k": 2}, "seed": 43074069000000551, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 726, "valid_answers": ["6, 9438", "78, 726"]}} | |
| {"id": "po-gcd_lcm-t0-eval-000250", "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 4563. 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 4563. 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, 4563", "ref": "3, 4563", "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": 4563, "k": 2}, "seed": 43074069000000460, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 507, "valid_answers": ["3, 4563", "27, 507"]}} | |
| {"id": "po-gcd_lcm-t0-eval-000128", "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 1690. 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 1690. 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, 1690", "ref": "2, 1690", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 2, "L": 1690, "k": 2}, "seed": 43074069000000156, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 338, "valid_answers": ["2, 1690", "10, 338"]}} | |
| {"id": "po-gcd_lcm-t0-eval-000029", "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 2156. 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 2156. 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, 2156", "ref": "4, 2156", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 1e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 4, "L": 2156, "k": 2}, "seed": 43074069000000031, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 196, "valid_answers": ["4, 2156", "44, 196"]}} | |
| {"id": "po-gcd_lcm-t0-eval-000261", "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 110. 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 110. 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, 110", "ref": "2, 110", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000328, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 2, "L": 110, "k": 2}, "seed": 43074069000000528, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 22, "valid_answers": ["2, 110", "10, 22"]}} | |
| {"id": "po-gcd_lcm-t0-eval-000030", "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 4719. 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 4719. 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, 4719", "ref": "3, 4719", "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": 4719, "k": 2}, "seed": 43074069000000032, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 363, "valid_answers": ["3, 4719", "39, 363"]}} | |
| {"id": "po-gcd_lcm-t0-eval-000147", "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 726. 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 726. 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, 726", "ref": "2, 726", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 8e-06, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 2, "L": 726, "k": 2}, "seed": 43074069000000186, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 242, "valid_answers": ["2, 726", "6, 242"]}} | |
| {"id": "po-gcd_lcm-t0-eval-000023", "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 225. 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 225. 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, 225", "ref": "3, 225", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 7.9e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 3, "L": 225, "k": 2}, "seed": 43074069000000025, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 75, "valid_answers": ["3, 225", "9, 75"]}} | |
| {"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 52. 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 52. 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, 52", "ref": "1, 52", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001451, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 1, "L": 52, "k": 2}, "seed": 43074069000000494, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 13, "valid_answers": ["1, 52", "4, 13"]}} | |
| {"id": "po-gcd_lcm-t0-eval-000066", "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 390. 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 390. 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, 390", "ref": "10, 390", "family": "po_gcd_lcm", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 2.6e-05, "solvable": true, "holdout": false, "meta": {"family": "po_gcd_lcm", "rung": "witness", "params": {"g": 10, "L": 390, "k": 2}, "seed": 43074069000000077, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 130, "valid_answers": ["10, 390", "30, 130"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000033", "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, 0), B = (-2, -1) 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, 0), B = (-2, -1) 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, 5", "ref": "-2, 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": [3, 0], "B": [-2, -1], "R": 6, "S2": 30}, "seed": 9453480200000605, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-2, 5", "3, -6", "3, 6"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000130", "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, -3), B = (2, -1) and C = (x, y) has area \\frac{33}{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, -3), B = (2, -1) and C = (x, y) has area \\frac{33}{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": 3, "bucket": "2-3", "guess_rate": 0.017751, "solvable": true, "holdout": false, "meta": {"family": "po_lattice_tri_area", "rung": "witness", "params": {"A": [-3, -3], "B": [2, -1], "R": 6, "S2": 33}, "seed": 9453480200001907, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-2, 4", "3, 6", "6, -6"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000028", "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 = (-2, -1) and C = (x, y) has area 12. Several answers are possible; any one correct answer is 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 = (-2, -1) and C = (x, y) has area 12. Several answers are possible; any one correct answer is accepted. Give 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": 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": [1, -2], "B": [-2, -1], "R": 6, "S2": 24}, "seed": 9453480200000527, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["1, 6", "4, 5"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000055", "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, 0) and C = (x, y) has area 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 -6 <= x, y <= 6 such that the triangle with vertices A = (3, 1), B = (-1, 0) and C = (x, y) has area 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": "-5, 6", "ref": "-5, 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": [3, 1], "B": [-1, 0], "R": 6, "S2": 28}, "seed": 9453480200000908, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 6, "valid_answers": ["-5, 6", "3, -6"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000295", "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 = (3, 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 = (3, 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": "-4, 1", "ref": "-4, 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": [3, 2], "R": 6, "S2": 25}, "seed": 9453480200004381, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-4, 1", "-1, 5", "4, -5"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000222", "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 = (-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 = (0, -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": "-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, -2], "B": [-3, 3], "R": 6, "S2": 29}, "seed": 9453480200003382, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-4, -5", "1, 6", "4, 1"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000006", "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, -2) 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, -2) 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": "4, -2", "ref": "4, -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": [-2, 3], "B": [-3, -2], "R": 6, "S2": 35}, "seed": 9453480200000077, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["4, -2", "5, 3"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000286", "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 = (-3, -3) 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 -6 <= x, y <= 6 such that the triangle with vertices A = (1, 3), B = (-3, -3) 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": "-5, 5", "ref": "-5, 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": [1, 3], "B": [-3, -3], "R": 6, "S2": 44}, "seed": 9453480200004245, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-5, 5", "3, -5", "5, -2"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000279", "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 = (2, -2) and C = (x, y) has area \\frac{23}{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 = (2, -2) and C = (x, y) has area \\frac{23}{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": [3, 2], "B": [2, -2], "R": 6, "S2": 23}, "seed": 9453480200004161, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["-4, -3", "-3, 1", "-2, 5"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000069", "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": "-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, -3], "B": [3, 3], "R": 6, "S2": 48}, "seed": 9453480200001064, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-6, 1", "-5, 3", "-4, 5"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000005", "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 = (1, 3) 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 -6 <= x, y <= 6 such that the triangle with vertices A = (-2, 1), B = (1, 3) 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": "0, -5", "ref": "0, -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": [-2, 1], "B": [1, 3], "R": 6, "S2": 22}, "seed": 9453480200000059, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["0, -5", "3, -3", "6, -1"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000091", "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, -2) and C = (x, y) has area 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 -6 <= x, y <= 6 such that the triangle with vertices A = (2, 3), B = (-1, -2) and C = (x, y) has area 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": "-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": [2, 3], "B": [-1, -2], "R": 6, "S2": 34}, "seed": 9453480200001426, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-6, 1", "-3, 6", "4, -5"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000132", "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, 1) 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 -6 <= x, y <= 6 such that the triangle with vertices A = (-2, -2), B = (3, 1) 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": "-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, -2], "B": [3, 1], "R": 6, "S2": 44}, "seed": 9453480200001938, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-5, 5", "6, -6"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000293", "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, 3), B = (-2, 2) and C = (x, y) has area 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 the triangle with vertices A = (3, 3), B = (-2, 2) and C = (x, y) has area 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": "0, -6", "ref": "0, -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": [3, 3], "B": [-2, 2], "R": 6, "S2": 42}, "seed": 9453480200004359, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["0, -6", "5, -5"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000197", "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, -3), B = (-2, 3) and C = (x, y) has area \\frac{43}{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, -3), B = (-2, 3) and C = (x, y) has area \\frac{43}{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, -2", "ref": "-5, -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, -3], "B": [-2, 3], "R": 6, "S2": 43}, "seed": 9453480200002995, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-5, -2", "6, 2"]}} | |
| {"id": "po-lattice_tri_area-t0-eval-000065", "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 = (2, -3) and C = (x, y) has area \\frac{43}{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, 3), B = (2, -3) and C = (x, y) has area \\frac{43}{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, 2", "ref": "-6, 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": [1, 3], "B": [2, -3], "R": 6, "S2": 43}, "seed": 9453480200001015, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-6, 2", "-5, -4"]}} | |
| {"id": "po-lin2_box-t0-eval-000220", "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 - 5y = 41. Several answers are possible; any one correct answer is 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 - 5y = 41. Several answers are possible; any one correct answer is accepted. Give 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, 3", "ref": "7, 3", "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": 8, "b": -5, "B": 20, "n": 41}, "seed": 20666920700000282, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 7, "valid_answers": ["7, 3", "12, 11", "17, 19"]}} | |
| {"id": "po-lin2_box-t0-eval-000029", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 3x - 3y = -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 0 <= x, y <= 20 such that 3x - 3y = -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": "0, 13", "ref": "0, 13", "family": "po_lin2_box", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.018141, "solvable": true, "holdout": false, "meta": {"family": "po_lin2_box", "rung": "witness", "params": {"a": 3, "b": -3, "B": 20, "n": -39}, "seed": 20666920700000035, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 13, "valid_answers": ["0, 13", "1, 14", "2, 15", "3, 16", "4, 17", "5, 18", "6, 19", "7, 20"]}} | |
| {"id": "po-lin2_box-t0-eval-000226", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 7x + 9y = 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 0 <= x, y <= 20 such that 7x + 9y = 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": "1, 15", "ref": "1, 15", "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": 7, "b": 9, "B": 20, "n": 142}, "seed": 20666920700000295, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 10, "valid_answers": ["1, 15", "10, 8", "19, 1"]}} | |
| {"id": "po-lin2_box-t0-eval-000045", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 7x - 5y = 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 <= 20 such that 7x - 5y = 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": 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": 7, "b": -5, "B": 20, "n": 9}, "seed": 20666920700000054, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["2, 1", "7, 8", "12, 15"]}} | |
| {"id": "po-lin2_box-t0-eval-000168", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 4x + 8y = 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 0 <= x, y <= 20 such that 4x + 8y = 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": "12, 20", "ref": "12, 20", "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": 4, "b": 8, "B": 20, "n": 208}, "seed": 20666920700000215, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 18, "valid_answers": ["12, 20", "14, 19", "16, 18", "18, 17", "20, 16"]}} | |
| {"id": "po-lin2_box-t0-eval-000019", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 3x + 2y = 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 integers x, y with 0 <= x, y <= 20 such that 3x + 2y = 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": "8, 20", "ref": "8, 20", "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": 3, "b": 2, "B": 20, "n": 64}, "seed": 20666920700000023, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 14, "valid_answers": ["8, 20", "10, 17", "12, 14", "14, 11", "16, 8", "18, 5", "20, 2"]}} | |
| {"id": "po-lin2_box-t0-eval-000107", "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 = -78. Several answers are possible; any one correct answer is 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 = -78. Several answers are possible; any one correct answer is accepted. Give 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_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": -78}, "seed": 20666920700000137, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 10, "valid_answers": ["6, 10", "15, 12"]}} | |
| {"id": "po-lin2_box-t0-eval-000027", "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 = 78. Several answers are possible; any one correct answer is 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 = 78. Several answers are possible; any one correct answer is accepted. Give 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, 17", "ref": "2, 17", "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": 4, "B": 20, "n": 78}, "seed": 20666920700000033, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 10, "valid_answers": ["2, 17", "6, 12", "10, 7", "14, 2"]}} | |
| {"id": "po-lin2_box-t0-eval-000182", "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 - 3y = 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 integers x, y with 0 <= x, y <= 20 such that 6x - 3y = 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": "16, 0", "ref": "16, 0", "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": -3, "B": 20, "n": 96}, "seed": 20666920700000231, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 16, "valid_answers": ["16, 0", "17, 2", "18, 4", "19, 6", "20, 8"]}} | |
| {"id": "po-lin2_box-t0-eval-000245", "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 = 135. Several answers are possible; any one correct answer is 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 = 135. Several answers are possible; any one correct answer is accepted. Give 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, 20", "ref": "11, 20", "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": 135}, "seed": 20666920700000319, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 15, "valid_answers": ["11, 20", "15, 15", "19, 10"]}} | |
| {"id": "po-lin2_box-t0-eval-000284", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 7x - 2y = 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 0 <= x, y <= 20 such that 7x - 2y = 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": "10, 5", "ref": "10, 5", "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": 7, "b": -2, "B": 20, "n": 60}, "seed": 20666920700000372, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 10, "valid_answers": ["10, 5", "12, 12", "14, 19"]}} | |
| {"id": "po-lin2_box-t0-eval-000124", "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 + 8y = 169. Several answers are possible; any one correct answer is 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 + 8y = 169. Several answers are possible; any one correct answer is accepted. Give 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, 18", "ref": "5, 18", "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": 5, "b": 8, "B": 20, "n": 169}, "seed": 20666920700000159, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 13, "valid_answers": ["5, 18", "13, 13"]}} | |
| {"id": "po-lin2_box-t0-eval-000091", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 3x + 8y = 164. Several answers are possible; any one correct answer is 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 3x + 8y = 164. Several answers are possible; any one correct answer is accepted. Give 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, 19", "ref": "4, 19", "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": 3, "b": 8, "B": 20, "n": 164}, "seed": 20666920700000114, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 16, "valid_answers": ["4, 19", "12, 16", "20, 13"]}} | |
| {"id": "po-lin2_box-t0-eval-000150", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 4x + 5y = 70. Several answers are possible; any one correct answer is 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 4x + 5y = 70. Several answers are possible; any one correct answer is accepted. Give 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, 14", "ref": "0, 14", "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": 4, "b": 5, "B": 20, "n": 70}, "seed": 20666920700000191, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 10, "valid_answers": ["0, 14", "5, 10", "10, 6", "15, 2"]}} | |
| {"id": "po-lin2_box-t0-eval-000298", "task": "cx_po_lin2_box", "source": "cx_po_lin2_box", "split": "eval", "problem": "Find integers x, y with 0 <= x, y <= 20 such that 3x + 7y = 135. Several answers are possible; any one correct answer is 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 3x + 7y = 135. Several answers are possible; any one correct answer is accepted. Give 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, 18", "ref": "3, 18", "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": 3, "b": 7, "B": 20, "n": 135}, "seed": 20666920700000387, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 15, "valid_answers": ["3, 18", "10, 15", "17, 12"]}} | |
| {"id": "po-lin2_box-t0-eval-000192", "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 + 4y = 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 9x + 4y = 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": "0, 19", "ref": "0, 19", "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": 9, "b": 4, "B": 20, "n": 76}, "seed": 20666920700000244, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 8, "valid_answers": ["0, 19", "4, 10", "8, 1"]}} | |
| {"id": "po-lin_congruence-t0-eval-000037", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 297 such that 34x \\equiv 162 \\pmod{298}. Several answers are possible; any 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 <= 297 such that 34x \\equiv 162 \\pmod{298}. Several answers are possible; any 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": "145", "ref": "145", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.006711, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 34, "b": 162, "m": 298, "M": 298}, "seed": 318040182700000065, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 145, "valid_answers": ["145", "294"]}} | |
| {"id": "po-lin_congruence-t0-eval-000160", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 615 such that 604x \\equiv 152 \\pmod{616}. Several answers are possible; any 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 <= 615 such that 604x \\equiv 152 \\pmod{616}. Several answers are possible; any 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_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.006494, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 604, "b": 152, "m": 616, "M": 616}, "seed": 318040182700000283, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 90, "valid_answers": ["90", "244", "398", "552"]}} | |
| {"id": "po-lin_congruence-t0-eval-000086", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 542 such that 501x \\equiv 93 \\pmod{543}. Several answers are possible; any 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 <= 542 such that 501x \\equiv 93 \\pmod{543}. Several answers are possible; any 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_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.005525, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 501, "b": 93, "m": 543, "M": 543}, "seed": 318040182700000152, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 140, "valid_answers": ["140", "321", "502"]}} | |
| {"id": "po-lin_congruence-t0-eval-000256", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 419 such that 411x \\equiv 318 \\pmod{420}. Several answers are possible; any 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 <= 419 such that 411x \\equiv 318 \\pmod{420}. Several answers are possible; any 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": "58", "ref": "58", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.007143, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 411, "b": 318, "m": 420, "M": 420}, "seed": 318040182700000426, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 58, "valid_answers": ["58", "198", "338"]}} | |
| {"id": "po-lin_congruence-t0-eval-000052", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 217 such that 34x \\equiv 38 \\pmod{218}. Several answers are possible; any 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 <= 217 such that 34x \\equiv 38 \\pmod{218}. Several answers are possible; any 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": "46", "ref": "46", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.009174, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 34, "b": 38, "m": 218, "M": 218}, "seed": 318040182700000091, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 46, "valid_answers": ["46", "155"]}} | |
| {"id": "po-lin_congruence-t0-eval-000022", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 255 such that 250x \\equiv 122 \\pmod{256}. Several answers are possible; any 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 <= 255 such that 250x \\equiv 122 \\pmod{256}. Several answers are possible; any 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": "65", "ref": "65", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.007812, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 250, "b": 122, "m": 256, "M": 256}, "seed": 318040182700000042, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 65, "valid_answers": ["65", "193"]}} | |
| {"id": "po-lin_congruence-t0-eval-000126", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 715 such that 436x \\equiv 660 \\pmod{716}. Several answers are possible; any 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 <= 715 such that 436x \\equiv 660 \\pmod{716}. Several answers are possible; any 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": 4, "bucket": "4-8", "guess_rate": 0.005587, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 436, "b": 660, "m": 716, "M": 716}, "seed": 318040182700000226, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 36, "valid_answers": ["36", "215", "394", "573"]}} | |
| {"id": "po-lin_congruence-t0-eval-000098", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 302 such that 168x \\equiv 168 \\pmod{303}. Several answers are possible; any 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 <= 302 such that 168x \\equiv 168 \\pmod{303}. Several answers are possible; any 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": "1", "ref": "1", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.009901, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 168, "b": 168, "m": 303, "M": 303}, "seed": 318040182700000177, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 1, "valid_answers": ["1", "102", "203"]}} | |
| {"id": "po-lin_congruence-t0-eval-000244", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 643 such that 104x \\equiv 548 \\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 104x \\equiv 548 \\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": "61", "ref": "61", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "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": 104, "b": 548, "m": 644, "M": 644}, "seed": 318040182700000411, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 61, "valid_answers": ["61", "222", "383", "544"]}} | |
| {"id": "po-lin_congruence-t0-eval-000284", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 375 such that 298x \\equiv 240 \\pmod{376}. Several answers are possible; any 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 <= 375 such that 298x \\equiv 240 \\pmod{376}. Several answers are possible; any 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": "156", "ref": "156", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.005319, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 298, "b": 240, "m": 376, "M": 376}, "seed": 318040182700000473, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 156, "valid_answers": ["156", "344"]}} | |
| {"id": "po-lin_congruence-t0-eval-000047", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 212 such that 111x \\equiv 141 \\pmod{213}. Several answers are possible; any 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 <= 212 such that 111x \\equiv 141 \\pmod{213}. Several answers are possible; any 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": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.014085, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 111, "b": 141, "m": 213, "M": 213}, "seed": 318040182700000082, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 55, "valid_answers": ["55", "126", "197"]}} | |
| {"id": "po-lin_congruence-t0-eval-000169", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 428 such that 102x \\equiv 297 \\pmod{429}. Several answers are possible; any 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 <= 428 such that 102x \\equiv 297 \\pmod{429}. Several answers are possible; any 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": "66", "ref": "66", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006993, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 102, "b": 297, "m": 429, "M": 429}, "seed": 318040182700000304, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 66, "valid_answers": ["66", "209", "352"]}} | |
| {"id": "po-lin_congruence-t0-eval-000225", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 189 such that 148x \\equiv 118 \\pmod{190}. Several answers are possible; any 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 <= 189 such that 148x \\equiv 118 \\pmod{190}. Several answers are possible; any 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": "56", "ref": "56", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.010526, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 148, "b": 118, "m": 190, "M": 190}, "seed": 318040182700000382, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 56, "valid_answers": ["56", "151"]}} | |
| {"id": "po-lin_congruence-t0-eval-000186", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 339 such that 308x \\equiv 228 \\pmod{340}. Several answers are possible; any 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 <= 339 such that 308x \\equiv 228 \\pmod{340}. Several answers are possible; any 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": "46", "ref": "46", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.011765, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 308, "b": 228, "m": 340, "M": 340}, "seed": 318040182700000326, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 46, "valid_answers": ["46", "131", "216", "301"]}} | |
| {"id": "po-lin_congruence-t0-eval-000078", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 715 such that 688x \\equiv 652 \\pmod{716}. Several answers are possible; any 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 <= 715 such that 688x \\equiv 652 \\pmod{716}. Several answers are possible; any 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_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.005587, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 688, "b": 652, "m": 716, "M": 716}, "seed": 318040182700000136, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 79, "valid_answers": ["79", "258", "437", "616"]}} | |
| {"id": "po-lin_congruence-t0-eval-000166", "task": "cx_po_lin_congruence", "source": "cx_po_lin_congruence", "split": "eval", "problem": "Find an integer x with 0 <= x <= 497 such that 39x \\equiv 450 \\pmod{498}. Several answers are possible; any 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 <= 497 such that 39x \\equiv 450 \\pmod{498}. Several answers are possible; any 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": "152", "ref": "152", "family": "po_lin_congruence", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006024, "solvable": true, "holdout": false, "meta": {"family": "po_lin_congruence", "rung": "witness", "params": {"a": 39, "b": 450, "m": 498, "M": 498}, "seed": 318040182700000300, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 152, "valid_answers": ["152", "318", "484"]}} | |
| {"id": "po-mat2_commute-t0-eval-000229", "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 -8 and commutes with \\begin{pmatrix} 3 & 2 \\\\ 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 -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -8 and commutes with \\begin{pmatrix} 3 & 2 \\\\ 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": "-4, -2, -2, 1", "ref": "-4, -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, 2, 2, -2], "D": -8}, "seed": 319369941700000237, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["-4, -2, -2, 1", "-1, -2, -2, 4", "1, 2, 2, -4", "4, 2, 2, -1"]}} | |
| {"id": "po-mat2_commute-t0-eval-000097", "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 -8 and commutes with \\begin{pmatrix} 1 & 2 \\\\ 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 -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -8 and commutes with \\begin{pmatrix} 1 & 2 \\\\ 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, -2, -3, 1", "ref": "-2, -2, -3, 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, 2, 3, -2], "D": -8}, "seed": 319369941700000101, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["-2, -2, -3, 1", "-1, -2, -3, 2", "1, 2, 3, -2", "2, 2, 3, -1"]}} | |
| {"id": "po-mat2_commute-t0-eval-000017", "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} -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 -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} -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": "-4, -2, -4, 2", "ref": "-4, -2, -4, 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, -1, -2, 1], "D": -16}, "seed": 319369941700000018, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["-4, -2, -4, 2", "-2, -2, -4, 4", "2, 2, 4, -4", "4, 2, 4, -2"]}} | |
| {"id": "po-mat2_commute-t0-eval-000194", "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} 2 & 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} 2 & 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": [2, 3, -2, 3], "D": 16}, "seed": 319369941700000200, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-4, 0, 0, -4", "4, 0, 0, 4"]}} | |
| {"id": "po-mat2_commute-t0-eval-000013", "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 11 and commutes with \\begin{pmatrix} -3 & 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 11 and commutes with \\begin{pmatrix} -3 & 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": "-3, -2, 1, -3", "ref": "-3, -2, 1, -3", "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, 2, -1, -3], "D": 11}, "seed": 319369941700000014, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 3, "valid_answers": ["-3, -2, 1, -3", "-3, 2, -1, -3", "3, -2, 1, 3", "3, 2, -1, 3"]}} | |
| {"id": "po-mat2_commute-t0-eval-000110", "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} 2 & 0 \\\\ -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 -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} 2 & 0 \\\\ -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": "-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": [2, 0, -3, -2], "D": 16}, "seed": 319369941700000114, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-4, 0, 0, -4", "4, 0, 0, 4"]}} | |
| {"id": "po-mat2_commute-t0-eval-000029", "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 24 and commutes with \\begin{pmatrix} -2 & 2 \\\\ -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 24 and commutes with \\begin{pmatrix} -2 & 2 \\\\ -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": "-4, 4, -4, -2", "ref": "-4, 4, -4, -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, -2, -1], "D": 24}, "seed": 319369941700000030, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["-4, 4, -4, -2", "-2, -4, 4, -4", "2, 4, -4, 4", "4, -4, 4, 2"]}} | |
| {"id": "po-mat2_commute-t0-eval-000216", "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} 1 & -1 \\\\ -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 -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} 1 & -1 \\\\ -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, 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": [1, -1, -3, 1], "D": 16}, "seed": 319369941700000224, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-4, 0, 0, -4", "4, 0, 0, 4"]}} | |
| {"id": "po-mat2_commute-t0-eval-000030", "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 0 and commutes with \\begin{pmatrix} -2 & 0 \\\\ -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 -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 0 and commutes with \\begin{pmatrix} -2 & 0 \\\\ -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": "-4, 0, -2, 0", "ref": "-4, 0, -2, 0", "family": "po_mat2_commute", "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_commute", "rung": "witness", "params": {"B": 4, "M": [-2, 0, -2, 2], "D": 0}, "seed": 319369941700000031, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 0, "valid_answers": ["-4, 0, -2, 0", "-2, 0, -1, 0", "0, 0, -2, 4", "0, 0, -1, 2", "0, 0, 0, 0", "0, 0, 1, -2", "0, 0, 2, -4", "2, 0, 1, 0", "4, 0, 2, 0"]}} | |
| {"id": "po-mat2_commute-t0-eval-000082", "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 0 and commutes with \\begin{pmatrix} 0 & 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 -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 0 and commutes with \\begin{pmatrix} 0 & 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": "-2, -3, 0, 0", "ref": "-2, -3, 0, 0", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000762, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 4, "M": [0, 3, 0, -2], "D": 0}, "seed": 319369941700000085, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 0, "valid_answers": ["-2, -3, 0, 0", "0, -3, 0, 2", "0, 0, 0, 0", "0, 3, 0, -2", "2, 3, 0, 0"]}} | |
| {"id": "po-mat2_commute-t0-eval-000008", "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 10 and commutes with \\begin{pmatrix} 2 & 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 10 and commutes with \\begin{pmatrix} 2 & 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, -3", "ref": "-4, 1, 2, -3", "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, 2, 3], "D": 10}, "seed": 319369941700000009, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["-4, 1, 2, -3", "-3, -1, -2, -4", "3, 1, 2, 4", "4, -1, -2, 3"]}} | |
| {"id": "po-mat2_commute-t0-eval-000188", "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 0 and commutes with \\begin{pmatrix} 0 & 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 -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 0 and commutes with \\begin{pmatrix} 0 & 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": "-3, 0, 1, 0", "ref": "-3, 0, 1, 0", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.000762, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 4, "M": [0, 0, -1, -3], "D": 0}, "seed": 319369941700000194, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 0, "valid_answers": ["-3, 0, 1, 0", "0, 0, -1, -3", "0, 0, 0, 0", "0, 0, 1, 3", "3, 0, -1, 0"]}} | |
| {"id": "po-mat2_commute-t0-eval-000148", "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} 1 & 0 \\\\ -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 -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} 1 & 0 \\\\ -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": "-2, 0, -4, -2", "ref": "-2, 0, -4, -2", "family": "po_mat2_commute", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 18, "bucket": "9-20", "guess_rate": 0.002743, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_commute", "rung": "witness", "params": {"B": 4, "M": [1, 0, -3, 1], "D": 4}, "seed": 319369941700000153, "version": "po-v0", "tier": 0, "n_raw": 18, "small": 2, "valid_answers": ["-2, 0, -4, -2", "-2, 0, -3, -2", "-2, 0, -2, -2", "-2, 0, -1, -2", "-2, 0, 0, -2", "-2, 0, 1, -2", "-2, 0, 2, -2", "-2, 0, 3, -2", "-2, 0, 4, -2", "2, 0, -4, 2", "2, 0, -3, 2", "2, 0, -2, 2", "2, 0, -1, 2", "2, 0, 0, 2", "2, 0, 1, 2", "2, 0, 2, 2", "2, 0, 3, 2", "2, 0, 4, 2"]}} | |
| {"id": "po-mat2_commute-t0-eval-000157", "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} 2 & 1 \\\\ 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 -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} 2 & 1 \\\\ 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": "-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": [2, 1, 3, -3], "D": 9}, "seed": 319369941700000162, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["-3, 0, 0, -3", "3, 0, 0, 3"]}} | |
| {"id": "po-mat2_commute-t0-eval-000251", "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 -3 and commutes with \\begin{pmatrix} 0 & 1 \\\\ 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 -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant -3 and commutes with \\begin{pmatrix} 0 & 1 \\\\ 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": "-1, -2, -2, -1", "ref": "-1, -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": [0, 1, 1, 0], "D": -3}, "seed": 319369941700000259, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 2, "valid_answers": ["-1, -2, -2, -1", "-1, 2, 2, -1", "1, -2, -2, 1", "1, 2, 2, 1"]}} | |
| {"id": "po-mat2_commute-t0-eval-000052", "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 1 and commutes with \\begin{pmatrix} 1 & 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 -4 <= a, b, c, d <= 4 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} has determinant 1 and commutes with \\begin{pmatrix} 1 & 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, 0, 0, -1", "ref": "-1, 0, 0, -1", "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, 1, 2, -2], "D": 1}, "seed": 319369941700000054, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 1, "valid_answers": ["-1, 0, 0, -1", "1, 0, 0, 1"]}} | |
| {"id": "po-mat2_nilpotent-t0-eval-000008", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "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 = 0 and A \\ne 0 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 = 0 and A \\ne 0 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, -2, 2, 2", "ref": "-2, -2, 2, 2", "family": "po_mat2_nilpotent", "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_nilpotent", "rung": "witness", "params": {"B": 3, "rel": [1, 0, 0, 0], "s": -2}, "seed": 92186124800000014, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["-2, -2, 2, 2", "-2, 2, -2, 2"]}} | |
| {"id": "po-mat2_nilpotent-t0-eval-000013", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "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 = 0 and A \\ne 0 and -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 -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -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": "-2, -2, 2, 2", "ref": "-2, -2, 2, 2", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001249, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 3, "rel": [0, 0, -2, 0], "s": -4}, "seed": 92186124800000024, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["-2, -2, 2, 2", "0, 0, 2, 0", "2, -2, 2, -2"]}} | |
| {"id": "po-mat2_nilpotent-t0-eval-000052", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "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 = 0 and A \\ne 0 and -2a = -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 = 0 and A \\ne 0 and -2a = -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, -3, 3, -3", "ref": "3, -3, 3, -3", "family": "po_mat2_nilpotent", "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_nilpotent", "rung": "witness", "params": {"B": 3, "rel": [-2, 0, 0, 0], "s": -6}, "seed": 92186124800000193, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["3, -3, 3, -3", "3, 3, -3, -3"]}} | |
| {"id": "po-mat2_nilpotent-t0-eval-000025", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "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 = 0 and A \\ne 0 and -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 -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and -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": "-2, -2, 2, 2", "ref": "-2, -2, 2, 2", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001249, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 3, "rel": [0, -2, 0, 0], "s": 4}, "seed": 92186124800000050, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["-2, -2, 2, 2", "0, -2, 0, 0", "2, -2, 2, -2"]}} | |
| {"id": "po-mat2_nilpotent-t0-eval-000037", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "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 = 0 and A \\ne 0 and -a = -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 = 0 and A \\ne 0 and -a = -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": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000833, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 3, "rel": [-1, 0, 0, 0], "s": -3}, "seed": 92186124800000086, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["3, -3, 3, -3", "3, 3, -3, -3"]}} | |
| {"id": "po-mat2_nilpotent-t0-eval-000010", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "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 = 0 and A \\ne 0 and 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 -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 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": "-3, 3, -3, 3", "ref": "-3, 3, -3, 3", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001249, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 3, "rel": [0, 2, 0, 0], "s": 6}, "seed": 92186124800000017, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["-3, 3, -3, 3", "0, 3, 0, 0", "3, 3, -3, -3"]}} | |
| {"id": "po-mat2_nilpotent-t0-eval-000033", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "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 = 0 and A \\ne 0 and 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 -3 <= a, b, c, d <= 3 such that the matrix A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} satisfies A^2 = 0 and A \\ne 0 and 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": "-3, 3, -3, 3", "ref": "-3, 3, -3, 3", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001249, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 3, "rel": [0, 0, 2, 0], "s": -6}, "seed": 92186124800000069, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["-3, 3, -3, 3", "0, 0, -3, 0", "3, 3, -3, -3"]}} | |
| {"id": "po-mat2_nilpotent-t0-eval-000009", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "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 = 0 and A \\ne 0 and -2a = 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 = 0 and A \\ne 0 and -2a = 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, -3, 3, 3", "ref": "-3, -3, 3, 3", "family": "po_mat2_nilpotent", "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_nilpotent", "rung": "witness", "params": {"B": 3, "rel": [-2, 0, 0, 0], "s": 6}, "seed": 92186124800000016, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["-3, -3, 3, 3", "-3, 3, -3, 3"]}} | |
| {"id": "po-mat2_nilpotent-t0-eval-000031", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "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 = 0 and A \\ne 0 and 2a = 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 = 0 and A \\ne 0 and 2a = 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, -3, 3, -3", "ref": "3, -3, 3, -3", "family": "po_mat2_nilpotent", "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_nilpotent", "rung": "witness", "params": {"B": 3, "rel": [2, 0, 0, 0], "s": 6}, "seed": 92186124800000066, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["3, -3, 3, -3", "3, 3, -3, -3"]}} | |
| {"id": "po-mat2_nilpotent-t0-eval-000016", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "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 = 0 and A \\ne 0 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 = 0 and A \\ne 0 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": "-2, -2, 2, 2", "ref": "-2, -2, 2, 2", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001249, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 3, "rel": [0, 1, 0, 0], "s": -2}, "seed": 92186124800000029, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["-2, -2, 2, 2", "0, -2, 0, 0", "2, -2, 2, -2"]}} | |
| {"id": "po-mat2_nilpotent-t0-eval-000036", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "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 = 0 and A \\ne 0 and -a = 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 = 0 and A \\ne 0 and -a = 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": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000833, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 3, "rel": [-1, 0, 0, 0], "s": 3}, "seed": 92186124800000080, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["-3, -3, 3, 3", "-3, 3, -3, 3"]}} | |
| {"id": "po-mat2_nilpotent-t0-eval-000035", "task": "cx_po_mat2_nilpotent", "source": "cx_po_mat2_nilpotent", "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 = 0 and A \\ne 0 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 = 0 and A \\ne 0 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": "-2, -2, 2, 2", "ref": "-2, -2, 2, 2", "family": "po_mat2_nilpotent", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.001249, "solvable": true, "holdout": false, "meta": {"family": "po_mat2_nilpotent", "rung": "witness", "params": {"B": 3, "rel": [0, 0, -1, 0], "s": -2}, "seed": 92186124800000077, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["-2, -2, 2, 2", "0, 0, 2, 0", "2, -2, 2, -2"]}} | |
| {"id": "po-mat3_det-t0-eval-000221", "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 \\\\ -1 & 2 & -1 \\\\ x & y & z \\end{pmatrix} is -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 -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -1 & -1 & -2 \\\\ -1 & 2 & -1 \\\\ x & y & z \\end{pmatrix} is -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, -4, -4", "ref": "-4, -4, -4", "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, -1, -2, -2, -1], "pos": [6, 7, 8], "D": -12}, "seed": 261600747200001608, "version": "po-v0", "tier": 0, "n_raw": 14, "small": 2, "valid_answers": ["-4, -4, -4", "-4, -1, -3", "-4, 2, -2", "-3, -3, -2", "-3, 0, -1", "-3, 3, 0", "-2, -2, 0", "-2, 1, 1", "-2, 4, 2", "-1, -4, 1", "-1, -1, 2", "-1, 2, 3", "0, -3, 3", "0, 0, 4"]}} | |
| {"id": "po-mat3_det-t0-eval-000069", "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 & -1 \\\\ 0 & 1 & 0 \\\\ x & y & z \\end{pmatrix} is 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 -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} 2 & 1 & -1 \\\\ 0 & 1 & 0 \\\\ x & y & z \\end{pmatrix} is 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": "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, -1, 0, 1, 0, -2, 0, -1], "pos": [6, 7, 8], "D": 11}, "seed": 261600747200000446, "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"]}} | |
| {"id": "po-mat3_det-t0-eval-000226", "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 & 2 \\\\ x & y & z \\\\ 2 & -1 & -2 \\end{pmatrix} is 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 -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -2 & 2 & 2 \\\\ x & y & z \\\\ 2 & -1 & -2 \\end{pmatrix} is 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, -4, 4", "ref": "4, -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, 2, 2, 0, 1, 2, -1, -2], "pos": [3, 4, 5], "D": 16}, "seed": 261600747200001681, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 4, "valid_answers": ["4, -4, 4", "4, -3, 4", "4, -2, 4", "4, -1, 4", "4, 0, 4", "4, 1, 4", "4, 2, 4", "4, 3, 4", "4, 4, 4"]}} | |
| {"id": "po-mat3_det-t0-eval-000189", "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 \\\\ x & y & z \\\\ 2 & 2 & 1 \\end{pmatrix} is 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 -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -1 & 1 & 2 \\\\ x & y & z \\\\ 2 & 2 & 1 \\end{pmatrix} is 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, -4, 2", "ref": "-4, -4, 2", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.015089, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [-1, 1, 2, -2, -1, 2, 2, 2, 1], "pos": [3, 4, 5], "D": 16}, "seed": 261600747200001378, "version": "po-v0", "tier": 0, "n_raw": 11, "small": 2, "valid_answers": ["-4, -4, 2", "-2, -2, 3", "-1, -3, 1", "0, -4, -1", "0, 0, 4", "1, -1, 2", "2, -2, 0", "3, -3, -2", "3, 1, 3", "4, -4, -4", "4, 0, 1"]}} | |
| {"id": "po-mat3_det-t0-eval-000262", "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 & -2 \\\\ x & y & z \\\\ 1 & 0 & -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 -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} 0 & -1 & -2 \\\\ x & y & z \\\\ 1 & 0 & -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": "1, -4, 4", "ref": "1, -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": [0, -1, -2, 2, 1, -1, 1, 0, -2], "pos": [3, 4, 5], "D": -14}, "seed": 261600747200001917, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 3, "valid_answers": ["1, -4, 4", "2, -4, 2", "2, -3, 4", "3, -4, 0", "3, -3, 2", "3, -2, 4", "4, -4, -2", "4, -3, 0", "4, -2, 2", "4, -1, 4"]}} | |
| {"id": "po-mat3_det-t0-eval-000037", "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 \\\\ 0 & 1 & 2 \\\\ 1 & 2 & -2 \\end{pmatrix} is -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 -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} x & y & z \\\\ 0 & 1 & 2 \\\\ 1 & 2 & -2 \\end{pmatrix} is -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": "-1, -4, 1", "ref": "-1, -4, 1", "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": [-1, -1, 1, 0, 1, 2, 1, 2, -2], "pos": [0, 1, 2], "D": -3}, "seed": 261600747200000223, "version": "po-v0", "tier": 0, "n_raw": 12, "small": 1, "valid_answers": ["-1, -4, 1", "-1, -3, 3", "0, -3, -3", "0, -2, -1", "0, -1, 1", "0, 0, 3", "1, 0, -3", "1, 1, -1", "1, 2, 1", "1, 3, 3", "2, 3, -3", "2, 4, -1"]}} | |
| {"id": "po-mat3_det-t0-eval-000016", "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 & 2 \\\\ -2 & -1 & -1 \\end{pmatrix} is 21. Several answers are possible; any one correct 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 & 2 \\\\ -2 & -1 & -1 \\end{pmatrix} is 21. Several answers are possible; any one correct 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": [-1, -2, 2, 1, -2, 2, -2, -1, -1], "pos": [0, 1, 2], "D": 21}, "seed": 261600747200000071, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 2, "valid_answers": ["-2, -3, -4", "0, -2, -3", "1, -4, -1", "1, 1, -4", "2, -1, -2", "3, -3, 0", "3, 2, -3", "4, 0, -1"]}} | |
| {"id": "po-mat3_det-t0-eval-000153", "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 & 1 \\\\ 1 & 0 & -1 \\\\ x & y & z \\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 -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} 2 & 1 & 1 \\\\ 1 & 0 & -1 \\\\ x & y & z \\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": "-4, 2, -4", "ref": "-4, 2, -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, 1, 1, 1, 0, -1, -1, 0, -1], "pos": [6, 7, 8], "D": 14}, "seed": 261600747200001085, "version": "po-v0", "tier": 0, "n_raw": 12, "small": 3, "valid_answers": ["-4, 2, -4", "-4, 3, -1", "-4, 4, 2", "-3, 3, -2", "-3, 4, 1", "-2, 3, -3", "-2, 4, 0", "-1, 3, -4", "-1, 4, -1", "0, 4, -2", "1, 4, -3", "2, 4, -4"]}} | |
| {"id": "po-mat3_det-t0-eval-000020", "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 \\\\ x & y & z \\\\ 0 & 1 & -2 \\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 \\\\ x & y & z \\\\ 0 & 1 & -2 \\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": "2, -4, -3", "ref": "2, -4, -3", "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": [1, 1, 2, 2, -1, -2, 0, 1, -2], "pos": [3, 4, 5], "D": 19}, "seed": 261600747200000088, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 3, "valid_answers": ["2, -4, -3", "3, -4, 1", "3, -3, -1", "3, -2, -3", "4, -3, 3", "4, -2, 1", "4, -1, -1", "4, 0, -3"]}} | |
| {"id": "po-mat3_det-t0-eval-000146", "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 \\\\ 2 & -1 & 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 -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} 1 & 2 & 1 \\\\ 2 & -1 & 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": "-2, 4, -4", "ref": "-2, 4, -4", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 11, "bucket": "9-20", "guess_rate": 0.015089, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [1, 2, 1, 2, -1, 1, -2, -1, 0], "pos": [6, 7, 8], "D": 18}, "seed": 261600747200001058, "version": "po-v0", "tier": 0, "n_raw": 11, "small": 2, "valid_answers": ["-2, 4, -4", "-1, 1, -4", "0, -2, -4", "0, 3, -3", "1, 0, -3", "2, -3, -3", "2, 2, -2", "3, -1, -2", "3, 4, -1", "4, -4, -2", "4, 1, -1"]}} | |
| {"id": "po-mat3_det-t0-eval-000159", "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 & 1 & -2 \\\\ x & y & z \\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 -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} 0 & -1 & -1 \\\\ -1 & 1 & -2 \\\\ x & y & z \\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": "2, 4, -4", "ref": "2, 4, -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": [0, -1, -1, -1, 1, -2, 0, 1, 0], "pos": [6, 7, 8], "D": 14}, "seed": 261600747200001131, "version": "po-v0", "tier": 0, "n_raw": 12, "small": 3, "valid_answers": ["2, 4, -4", "3, 1, -4", "3, 2, -3", "3, 3, -2", "3, 4, -1", "4, -2, -4", "4, -1, -3", "4, 0, -2", "4, 1, -1", "4, 2, 0", "4, 3, 1", "4, 4, 2"]}} | |
| {"id": "po-mat3_det-t0-eval-000125", "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 & 2 & 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 -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} -2 & -1 & 2 \\\\ x & y & z \\\\ 2 & 2 & 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": "-1, -4, 2", "ref": "-1, -4, 2", "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, -1, 2, 1, 1, 1, 2, 2, 2], "pos": [3, 4, 5], "D": 30}, "seed": 261600747200000864, "version": "po-v0", "tier": 0, "n_raw": 12, "small": 2, "valid_answers": ["-1, -4, 2", "0, -4, -1", "0, -3, 3", "1, -4, -4", "1, -3, 0", "1, -2, 4", "2, -3, -3", "2, -2, 1", "3, -2, -2", "3, -1, 2", "4, -1, -1", "4, 0, 3"]}} | |
| {"id": "po-mat3_det-t0-eval-000210", "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 \\\\ 2 & 1 & 1 \\\\ 1 & -2 & 2 \\end{pmatrix} is 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 -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} x & y & z \\\\ 2 & 1 & 1 \\\\ 1 & -2 & 2 \\end{pmatrix} is 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": "-4, -4, -3", "ref": "-4, -4, -3", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.017833, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [-2, 1, -1, 2, 1, 1, 1, -2, 2], "pos": [0, 1, 2], "D": 11}, "seed": 261600747200001520, "version": "po-v0", "tier": 0, "n_raw": 13, "small": 2, "valid_answers": ["-4, -4, -3", "-3, -1, -4", "-2, -3, -2", "-1, 0, -3", "0, -2, -1", "0, 3, -4", "1, -4, 1", "1, 1, -2", "2, -1, 0", "2, 4, -3", "3, -3, 2", "3, 2, -1", "4, 0, 1"]}} | |
| {"id": "po-mat3_det-t0-eval-000150", "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 & -1 \\\\ x & y & z \\\\ 2 & 1 & 0 \\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 -4 <= x, y, z <= 4 such that the determinant of \\begin{pmatrix} 1 & -1 & -1 \\\\ x & y & z \\\\ 2 & 1 & 0 \\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": "-4, -1, -4", "ref": "-4, -1, -4", "family": "po_mat3_det", "domain": "linalg", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 13, "bucket": "9-20", "guess_rate": 0.017833, "solvable": true, "holdout": false, "meta": {"family": "po_mat3_det", "rung": "witness", "params": {"B": 4, "M": [1, -1, -1, 0, 2, -2, 2, 1, 0], "pos": [3, 4, 5], "D": 14}, "seed": 261600747200001077, "version": "po-v0", "tier": 0, "n_raw": 13, "small": 3, "valid_answers": ["-4, -1, -4", "-4, 2, -2", "-3, 1, -3", "-3, 4, -1", "-2, 0, -4", "-2, 3, -2", "-1, 2, -3", "0, 1, -4", "0, 4, -2", "1, 3, -3", "2, 2, -4", "3, 4, -3", "4, 3, -4"]}} | |
| {"id": "po-mat3_det-t0-eval-000255", "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 & -1 & -2 \\\\ -2 & 0 & 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} x & y & z \\\\ -1 & -1 & -2 \\\\ -2 & 0 & 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": "-4, -4, 3", "ref": "-4, -4, 3", "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, -2, -1, -1, -2, -2, 0, 0], "pos": [0, 1, 2], "D": -22}, "seed": 261600747200001857, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 4, "valid_answers": ["-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"]}} | |
| {"id": "po-mat3_det-t0-eval-000091", "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 & 2 \\\\ x & y & z \\\\ 2 & 0 & -1 \\end{pmatrix} is -17. Several answers are possible; any one correct 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 & 2 \\\\ x & y & z \\\\ 2 & 0 & -1 \\end{pmatrix} is -17. Several answers are possible; any one correct 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, -3", "ref": "-3, 2, -3", "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, 2, -1, -2, 1, 2, 0, -1], "pos": [3, 4, 5], "D": -17}, "seed": 261600747200000624, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 3, "valid_answers": ["-3, 2, -3", "-3, 3, -1", "-3, 4, 1", "-1, 2, -4", "-1, 3, -2", "-1, 4, 0", "1, 3, -3", "1, 4, -1", "3, 3, -4", "3, 4, -2"]}} | |
| {"id": "po-order_mod-t0-eval-000005", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 100 whose multiplicative order modulo 101 is 4, that is, x^4 \\equiv 1 \\pmod{101} 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 <= 100 whose multiplicative order modulo 101 is 4, that is, x^4 \\equiv 1 \\pmod{101} 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": "10", "ref": "10", "family": "po_order_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.02, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 101, "k": 4}, "seed": 171199287000000041, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 10, "valid_answers": ["10", "91"]}} | |
| {"id": "po-order_mod-t0-eval-000026", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 162 whose multiplicative order modulo 163 is 3, that is, x^3 \\equiv 1 \\pmod{163} 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 <= 162 whose multiplicative order modulo 163 is 3, that is, x^3 \\equiv 1 \\pmod{163} 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": "58", "ref": "58", "family": "po_order_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.012346, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 163, "k": 3}, "seed": 171199287000000492, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 58, "valid_answers": ["58", "104"]}} | |
| {"id": "po-order_mod-t0-eval-000025", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 150 whose multiplicative order modulo 151 is 3, that is, x^3 \\equiv 1 \\pmod{151} 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 <= 150 whose multiplicative order modulo 151 is 3, that is, x^3 \\equiv 1 \\pmod{151} 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": "32", "ref": "32", "family": "po_order_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.013333, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 151, "k": 3}, "seed": 171199287000000465, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 32, "valid_answers": ["32", "118"]}} | |
| {"id": "po-order_mod-t0-eval-000024", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 180 whose multiplicative order modulo 181 is 4, that is, x^4 \\equiv 1 \\pmod{181} 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 <= 180 whose multiplicative order modulo 181 is 4, that is, x^4 \\equiv 1 \\pmod{181} 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": "19", "ref": "19", "family": "po_order_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011111, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 181, "k": 4}, "seed": 171199287000000402, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 19, "valid_answers": ["19", "162"]}} | |
| {"id": "po-order_mod-t0-eval-000004", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 102 whose multiplicative order modulo 103 is 6, that is, x^6 \\equiv 1 \\pmod{103} 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 <= 102 whose multiplicative order modulo 103 is 6, that is, x^6 \\equiv 1 \\pmod{103} 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": "47", "ref": "47", "family": "po_order_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.019608, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 103, "k": 6}, "seed": 171199287000000038, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 47, "valid_answers": ["47", "57"]}} | |
| {"id": "po-order_mod-t0-eval-000029", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 180 whose multiplicative order modulo 181 is 3, that is, x^3 \\equiv 1 \\pmod{181} 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 <= 180 whose multiplicative order modulo 181 is 3, that is, x^3 \\equiv 1 \\pmod{181} 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": "48", "ref": "48", "family": "po_order_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.011111, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 181, "k": 3}, "seed": 171199287000003226, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 48, "valid_answers": ["48", "132"]}} | |
| {"id": "po-order_mod-t0-eval-000027", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 108 whose multiplicative order modulo 109 is 3, that is, x^3 \\equiv 1 \\pmod{109} 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 <= 108 whose multiplicative order modulo 109 is 3, that is, x^3 \\equiv 1 \\pmod{109} 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": "45", "ref": "45", "family": "po_order_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.018519, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 109, "k": 3}, "seed": 171199287000000561, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 45, "valid_answers": ["45", "63"]}} | |
| {"id": "po-order_mod-t0-eval-000007", "task": "cx_po_order_mod", "source": "cx_po_order_mod", "split": "eval", "problem": "Find an integer x with 1 <= x <= 126 whose multiplicative order modulo 127 is 3, that is, x^3 \\equiv 1 \\pmod{127} 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 <= 126 whose multiplicative order modulo 127 is 3, that is, x^3 \\equiv 1 \\pmod{127} 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": "19", "ref": "19", "family": "po_order_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.015873, "solvable": true, "holdout": false, "meta": {"family": "po_order_mod", "rung": "witness", "params": {"n": 127, "k": 3}, "seed": 171199287000000057, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 19, "valid_answers": ["19", "107"]}} | |
| {"id": "po-parallelogram-t0-eval-000201", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (1, 1), (4, -4) and (3, 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 (1, 1), (4, -4) and (3, 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": "0, 10", "ref": "0, 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, 1], [4, -4], [3, 5]]}, "seed": 421451545000000215, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 6, "valid_answers": ["0, 10", "2, -8", "6, 0"]}} | |
| {"id": "po-parallelogram-t0-eval-000047", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-1, 0), (-3, 3) and (-5, -2) 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, 0), (-3, 3) and (-5, -2) 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, 1", "ref": "-7, 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": [[-1, 0], [-3, 3], [-5, -2]]}, "seed": 421451545000000052, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-7, 1", "-3, -5", "1, 5"]}} | |
| {"id": "po-parallelogram-t0-eval-000016", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (3, 0), (-2, 1) and (5, 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 (3, 0), (-2, 1) and (5, 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": "-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, 0], [-2, 1], [5, 0]]}, "seed": 421451545000000017, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 1, "valid_answers": ["-4, 1", "0, 1", "10, -1"]}} | |
| {"id": "po-parallelogram-t0-eval-000228", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-3, -3), (0, 2) 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 (-3, -3), (0, 2) 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": "-8, -2", "ref": "-8, -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": [[-3, -3], [0, 2], [5, 1]]}, "seed": 421451545000000242, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-8, -2", "2, -4", "8, 6"]}} | |
| {"id": "po-parallelogram-t0-eval-000206", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-4, 3), (-4, -1) and (5, 2) 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), (-4, -1) and (5, 2) 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": "-13, 0", "ref": "-13, 0", "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], [-4, -1], [5, 2]]}, "seed": 421451545000000220, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-13, 0", "5, -2", "5, 6"]}} | |
| {"id": "po-parallelogram-t0-eval-000062", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-5, -5), (1, -4) and (0, 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 (-5, -5), (1, -4) and (0, 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, 4", "ref": "-6, 4", "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, -5], [1, -4], [0, 5]]}, "seed": 421451545000000067, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 6, "valid_answers": ["-6, 4", "-4, -14", "6, 6"]}} | |
| {"id": "po-parallelogram-t0-eval-000230", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-4, 1), (-2, -3) and (-5, 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 (-4, 1), (-2, -3) and (-5, 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": "-7, 4", "ref": "-7, 4", "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], [-2, -3], [-5, 0]]}, "seed": 421451545000000245, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["-7, 4", "-3, -4", "-1, -2"]}} | |
| {"id": "po-parallelogram-t0-eval-000168", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (4, 5), (3, 1) and (-5, -2) 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, 5), (3, 1) and (-5, -2) 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, -6", "ref": "-6, -6", "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, 5], [3, 1], [-5, -2]]}, "seed": 421451545000000182, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-6, -6", "-4, 2", "12, 8"]}} | |
| {"id": "po-parallelogram-t0-eval-000282", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (4, 2), (-1, 4) and (5, -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, 2), (-1, 4) and (5, -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": "-2, 11", "ref": "-2, 11", "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, 2], [-1, 4], [5, -5]]}, "seed": 421451545000000297, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["-2, 11", "0, -3", "10, -7"]}} | |
| {"id": "po-parallelogram-t0-eval-000115", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-5, -1), (0, 4) and (4, -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 (-5, -1), (0, 4) and (4, -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": "-9, 4", "ref": "-9, 4", "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, -1], [0, 4], [4, -1]]}, "seed": 421451545000000125, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 6, "valid_answers": ["-9, 4", "-1, -6", "9, 4"]}} | |
| {"id": "po-parallelogram-t0-eval-000031", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-2, -3), (4, 5) and (-2, 2) 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, -3), (4, 5) and (-2, 2) 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, -6", "ref": "-8, -6", "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": [[-2, -3], [4, 5], [-2, 2]]}, "seed": 421451545000000036, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-8, -6", "4, 0", "4, 10"]}} | |
| {"id": "po-parallelogram-t0-eval-000065", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (5, 0), (2, -5) and (-2, 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 (5, 0), (2, -5) and (-2, 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": "-5, -5", "ref": "-5, -5", "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, 0], [2, -5], [-2, 0]]}, "seed": 421451545000000070, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-5, -5", "1, 5", "9, -5"]}} | |
| {"id": "po-parallelogram-t0-eval-000269", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-4, -1), (-5, 3) and (2, 2) 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), (-5, 3) and (2, 2) 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": "-11, 0", "ref": "-11, 0", "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], [-5, 3], [2, 2]]}, "seed": 421451545000000284, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["-11, 0", "1, 6", "3, -2"]}} | |
| {"id": "po-parallelogram-t0-eval-000159", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (2, -3), (2, -1) and (1, -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 (2, -3), (2, -1) and (1, -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, -6", "ref": "1, -6", "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, -3], [2, -1], [1, -4]]}, "seed": 421451545000000172, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["1, -6", "1, -2", "3, 0"]}} | |
| {"id": "po-parallelogram-t0-eval-000226", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (3, -5), (4, 4) 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 (3, -5), (4, 4) 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": "-1, -5", "ref": "-1, -5", "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], [4, 4], [0, 4]]}, "seed": 421451545000000240, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-1, -5", "1, 13", "7, -5"]}} | |
| {"id": "po-parallelogram-t0-eval-000089", "task": "cx_po_parallelogram", "source": "cx_po_parallelogram", "split": "eval", "problem": "The points (-3, -2), (-4, 4) and (4, -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 (-3, -2), (-4, 4) and (4, -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": "-11, 3", "ref": "-11, 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": [[-3, -2], [-4, 4], [4, -1]]}, "seed": 421451545000000095, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-11, 3", "3, 5", "5, -7"]}} | |
| {"id": "po-poly_critical-t0-eval-000220", "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) = 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(-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": "3, -6, -3", "ref": "3, -6, -3", "family": "po_poly_critical", "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_critical", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "val", "x": -1, "v": "6"}]}, "seed": 207778151600000514, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["3, -6, -3", "4, -4, -2", "5, -2, -1", "6, 0, 0"]}} | |
| {"id": "po-poly_critical-t0-eval-000007", "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) = 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(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": "-4, 4, 2", "ref": "-4, 4, 2", "family": "po_poly_critical", "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_critical", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": -1, "m": 1, "v": "0"}, {"t": "val", "x": 1, "v": "2"}]}, "seed": 207778151600000009, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 2, "valid_answers": ["-4, 4, 2", "-1, 2, 1", "2, 0, 0", "5, -2, -1"]}} | |
| {"id": "po-poly_critical-t0-eval-000266", "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) = 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'(2) = 0 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, -4, 1", "ref": "-1, -4, 1", "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": 2, "m": 1, "v": "0"}, {"t": "val", "x": -1, "v": "4"}]}, "seed": 207778151600000842, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-1, -4, 1", "4, 0, 0"]}} | |
| {"id": "po-poly_critical-t0-eval-000151", "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": 207778151600000237, "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"]}} | |
| {"id": "po-poly_critical-t0-eval-000183", "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": 207778151600000330, "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"]}} | |
| {"id": "po-poly_critical-t0-eval-000246", "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": 207778151600000632, "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"]}} | |
| {"id": "po-poly_critical-t0-eval-000191", "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) = -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(2) = -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, 0, -1", "ref": "-3, 0, -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": 0, "m": 1, "v": "0"}, {"t": "val", "x": 2, "v": "-7"}]}, "seed": 207778151600000353, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["-3, 0, -1", "1, 0, -2", "5, 0, -3"]}} | |
| {"id": "po-poly_critical-t0-eval-000054", "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(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 -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(2) = 0 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, -4, 1", "ref": "0, -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": 0, "v": "0"}]}, "seed": 207778151600000066, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 0, "valid_answers": ["0, -4, 1", "0, 0, 0", "0, 4, -1"]}} | |
| {"id": "po-poly_critical-t0-eval-000069", "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": "-5, 4, -1", "ref": "-5, 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": 207778151600000084, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["-5, 4, -1", "-2, 0, 0", "1, -4, 1"]}} | |
| {"id": "po-poly_critical-t0-eval-000060", "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) = 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(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 = ...}, 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": 2, "v": "1"}]}, "seed": 207778151600000072, "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"]}} | |
| {"id": "po-poly_critical-t0-eval-000076", "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) = -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) = 0 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, -6", "ref": "-3, 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": "-3"}]}, "seed": 207778151600000092, "version": "po-v0", "tier": 0, "n_raw": 13, "small": 3, "valid_answers": ["-3, 0, -6", "-3, 0, -5", "-3, 0, -4", "-3, 0, -3", "-3, 0, -2", "-3, 0, -1", "-3, 0, 0", "-3, 0, 1", "-3, 0, 2", "-3, 0, 3", "-3, 0, 4", "-3, 0, 5", "-3, 0, 6"]}} | |
| {"id": "po-poly_critical-t0-eval-000291", "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) = -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 -6 <= a, b, c <= 6 such that the polynomial p(x) = a + bx + cx^2 satisfies p'(0) = 0 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": "-6, 0, -3", "ref": "-6, 0, -3", "family": "po_poly_critical", "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_critical", "rung": "witness", "params": {"d": 2, "B": 6, "cons": [{"t": "der", "x": 0, "m": 1, "v": "0"}, {"t": "val", "x": -1, "v": "-9"}]}, "seed": 207778151600001387, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 5, "valid_answers": ["-6, 0, -3", "-5, 0, -4", "-4, 0, -5", "-3, 0, -6"]}} | |
| {"id": "po-poly_critical-t0-eval-000083", "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": 207778151600000100, "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"]}} | |
| {"id": "po-poly_critical-t0-eval-000002", "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) = 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(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, 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": 1, "v": "2"}]}, "seed": 207778151600000002, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 2, "valid_answers": ["-1, 6, -3", "0, 4, -2", "1, 2, -1", "2, 0, 0", "3, -2, 1", "4, -4, 2", "5, -6, 3"]}} | |
| {"id": "po-poly_critical-t0-eval-000171", "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) = 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 = ...}, 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) = 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 = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, 4, 2", "ref": "-4, 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": "12"}]}, "seed": 207778151600000296, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-4, 4, 2", "4, 2, 1"]}} | |
| {"id": "po-poly_critical-t0-eval-000248", "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) = -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 = ...}, 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) = -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 = ...}, 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": 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": "-14"}]}, "seed": 207778151600000659, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-6, -2, -1", "2, -4, -2"]}} | |
| {"id": "po-poly_integral-t0-eval-000030", "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{10}{3} 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}^{0} p(x)\\,dx = \\frac{10}{3} 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": "0, -4, 4", "ref": "0, -4, 4", "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": "10/3"}, {"t": "val", "x": 1, "v": "0"}]}, "seed": 143586649300000038, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["0, -4, 4", "5, 0, -5"]}} | |
| {"id": "po-poly_integral-t0-eval-000013", "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{7}{2} 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{7}{2} 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, 5, -3", "ref": "-5, 5, -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": "-7/2"}, {"t": "val", "x": 1, "v": "-3"}]}, "seed": 143586649300000016, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["-5, 5, -3", "-4, 1, 0", "-3, -3, 3"]}} | |
| {"id": "po-poly_integral-t0-eval-000004", "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{43}{6} 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}^{1} p(x)\\,dx = -\\frac{43}{6} 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_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": "-43/6"}, {"t": "val", "x": 0, "v": "-5"}]}, "seed": 143586649300000005, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["-5, -5, 1", "-5, -3, -2", "-5, -1, -5"]}} | |
| {"id": "po-poly_integral-t0-eval-000043", "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{13}{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 = ...}, 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{13}{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 = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -5, -1", "ref": "0, -5, -1", "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": 0, "v": "13/6"}, {"t": "val", "x": 0, "v": "0"}]}, "seed": 143586649300000053, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["0, -5, -1", "0, -3, 2", "0, -1, 5"]}} | |
| {"id": "po-poly_integral-t0-eval-000185", "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{3}{2} 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{3}{2} 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": "0, 3, 0", "ref": "0, 3, 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": "3/2"}, {"t": "val", "x": 1, "v": "3"}]}, "seed": 143586649300000291, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["0, 3, 0", "1, -1, 3"]}} | |
| {"id": "po-poly_integral-t0-eval-000250", "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{17}{3} 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 \\int_{0}^{1} p(x)\\,dx = -\\frac{17}{3} 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, -4, 1", "ref": "-4, -4, 1", "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": "-17/3"}, {"t": "val", "x": 0, "v": "-4"}]}, "seed": 143586649300000405, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-4, -4, 1", "-4, -2, -2", "-4, 0, -5"]}} | |
| {"id": "po-poly_integral-t0-eval-000164", "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 = -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 -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{2} p(x)\\,dx = -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, -1, 3", "ref": "-4, -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": "-2"}, {"t": "val", "x": 0, "v": "-4"}]}, "seed": 143586649300000257, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-4, -1, 3", "-4, 3, 0"]}} | |
| {"id": "po-poly_integral-t0-eval-000040", "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 = 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 \\int_{-1}^{1} p(x)\\,dx = 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, 3", "ref": "1, -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": -1, "hi": 1, "v": "4"}, {"t": "val", "x": 1, "v": "0"}]}, "seed": 143586649300000050, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["1, -4, 3", "2, -2, 0", "3, 0, -3"]}} | |
| {"id": "po-poly_integral-t0-eval-000179", "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{34}{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_{0}^{2} p(x)\\,dx = \\frac{34}{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, 5", "ref": "1, -2, 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": 2, "v": "34/3"}, {"t": "val", "x": 0, "v": "1"}]}, "seed": 143586649300000279, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["1, -2, 5", "1, 2, 2"]}} | |
| {"id": "po-poly_integral-t0-eval-000120", "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{4}{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 = ...}, 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{4}{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 = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "1, 5, -5", "ref": "1, 5, -5", "family": "po_poly_integral", "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_integral", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "-4/3"}, {"t": "val", "x": 1, "v": "1"}]}, "seed": 143586649300000187, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 5, "valid_answers": ["1, 5, -5", "2, 4, -5", "3, 3, -5", "4, 2, -5", "5, 1, -5"]}} | |
| {"id": "po-poly_integral-t0-eval-000104", "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{52}{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 -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{2} p(x)\\,dx = \\frac{52}{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": "1, 5, 2", "ref": "1, 5, 2", "family": "po_poly_integral", "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_integral", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "int", "lo": 0, "hi": 2, "v": "52/3"}, {"t": "val", "x": 1, "v": "8"}]}, "seed": 143586649300000157, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 3, "valid_answers": ["1, 5, 2", "2, 4, 2", "3, 3, 2", "4, 2, 2", "5, 1, 2"]}} | |
| {"id": "po-poly_integral-t0-eval-000145", "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 = -\\frac{4}{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 \\int_{-1}^{1} p(x)\\,dx = -\\frac{4}{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": "-2, 1, 4", "ref": "-2, 1, 4", "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": "-4/3"}, {"t": "val", "x": 1, "v": "3"}]}, "seed": 143586649300000225, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["-2, 1, 4", "-1, 3, 1", "0, 5, -2"]}} | |
| {"id": "po-poly_integral-t0-eval-000180", "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 = \\frac{4}{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}^{1} p(x)\\,dx = \\frac{4}{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, -5, 5", "ref": "-1, -5, 5", "family": "po_poly_integral", "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_integral", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "int", "lo": -1, "hi": 1, "v": "4/3"}, {"t": "val", "x": 0, "v": "-1"}]}, "seed": 143586649300000280, "version": "po-v0", "tier": 0, "n_raw": 11, "small": 5, "valid_answers": ["-1, -5, 5", "-1, -4, 5", "-1, -3, 5", "-1, -2, 5", "-1, -1, 5", "-1, 0, 5", "-1, 1, 5", "-1, 2, 5", "-1, 3, 5", "-1, 4, 5", "-1, 5, 5"]}} | |
| {"id": "po-poly_integral-t0-eval-000052", "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{10}{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_{0}^{1} p(x)\\,dx = \\frac{10}{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": 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": "10/3"}, {"t": "val", "x": 0, "v": "1"}]}, "seed": 143586649300000069, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["1, 2, 4", "1, 4, 1"]}} | |
| {"id": "po-poly_integral-t0-eval-000175", "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{13}{6} 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 -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{-1}^{0} p(x)\\,dx = \\frac{13}{6} 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, -3, -4", "ref": "2, -3, -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": "13/6"}, {"t": "val", "x": 0, "v": "2"}]}, "seed": 143586649300000272, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 2, "valid_answers": ["2, -3, -4", "2, -1, -1", "2, 1, 2", "2, 3, 5"]}} | |
| {"id": "po-poly_integral-t0-eval-000127", "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{13}{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 -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies \\int_{0}^{1} p(x)\\,dx = -\\frac{13}{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, 2, -1", "ref": "-5, 2, -1", "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": "-13/3"}, {"t": "val", "x": 1, "v": "-4"}]}, "seed": 143586649300000196, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-5, 2, -1", "-4, -2, 2"]}} | |
| {"id": "po-poly_roots-t0-eval-000018", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c with -7 <= a, b, c <= 7, not all zero, such that the polynomial p(x) = a + bx + cx^2 satisfies 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 = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -7 <= a, b, c <= 7, not all zero, such that the polynomial p(x) = a + bx + cx^2 satisfies 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 = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -5, -1", "ref": "-6, -5, -1", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 24, "bucket": null, "guess_rate": 0.007111, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 2, "B": 7, "cons": [{"t": "root", "x": -3, "v": "0"}]}, "seed": 140069831700000032, "version": "po-v0", "tier": 0, "n_raw": 24, "small": 3, "valid_answers": ["-6, -5, -1", "-6, -2, 0", "-6, 1, 1", "-6, 4, 2", "-6, 7, 3", "-3, -7, -2", "-3, -4, -1", "-3, -1, 0", "-3, 2, 1", "-3, 5, 2", "0, -6, -2", "0, -3, -1", "0, 3, 1", "0, 6, 2", "3, -5, -2", "3, -2, -1", "3, 1, 0", "3, 4, 1", "3, 7, 2", "6, -7, -3", "6, -4, -2", "6, -1, -1", "6, 2, 0", "6, 5, 1"]}} | |
| {"id": "po-poly_roots-t0-eval-000003", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c with -7 <= a, b, c <= 7, not all zero, such that the polynomial p(x) = a + bx + cx^2 satisfies 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 = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -7 <= a, b, c <= 7, not all zero, such that the polynomial p(x) = a + bx + cx^2 satisfies 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 = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -7, 2", "ref": "-4, -7, 2", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 10, "bucket": "9-20", "guess_rate": 0.002963, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 2, "B": 7, "cons": [{"t": "root", "x": 4, "v": "0"}]}, "seed": 140069831700000004, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 4, "valid_answers": ["-4, -7, 2", "-4, -3, 1", "-4, 1, 0", "-4, 5, -1", "0, -4, 1", "0, 4, -1", "4, -5, 1", "4, -1, 0", "4, 3, -1", "4, 7, -2"]}} | |
| {"id": "po-poly_roots-t0-eval-000019", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c with -8 <= a, b, c <= 8, not all zero, such that the polynomial p(x) = a + bx + cx^2 satisfies 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 = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -8 <= a, b, c <= 8, not all zero, such that the polynomial p(x) = a + bx + cx^2 satisfies 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 = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-6, -7, 3", "ref": "-6, -7, 3", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 28, "bucket": null, "guess_rate": 0.005699, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 2, "B": 8, "cons": [{"t": "root", "x": 3, "v": "0"}]}, "seed": 140069831700000037, "version": "po-v0", "tier": 0, "n_raw": 28, "small": 3, "valid_answers": ["-6, -7, 3", "-6, -4, 2", "-6, -1, 1", "-6, 2, 0", "-6, 5, -1", "-6, 8, -2", "-3, -8, 3", "-3, -5, 2", "-3, -2, 1", "-3, 1, 0", "-3, 4, -1", "-3, 7, -2", "0, -6, 2", "0, -3, 1", "0, 3, -1", "0, 6, -2", "3, -7, 2", "3, -4, 1", "3, -1, 0", "3, 2, -1", "3, 5, -2", "3, 8, -3", "6, -8, 2", "6, -5, 1", "6, -2, 0", "6, 1, -1", "6, 4, -2", "6, 7, -3"]}} | |
| {"id": "po-poly_roots-t0-eval-000033", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5, not all zero, such that the polynomial p(x) = a + bx + cx^2 satisfies 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 = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5, not all zero, such that the polynomial p(x) = a + bx + cx^2 satisfies 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 = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-4, -4, -1", "ref": "-4, -4, -1", "family": "po_poly_roots", "domain": "polynomial", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 26, "bucket": null, "guess_rate": 0.019534, "solvable": true, "holdout": false, "meta": {"family": "po_poly_roots", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "root", "x": -2, "v": "0"}]}, "seed": 140069831700000146, "version": "po-v0", "tier": 0, "n_raw": 26, "small": 2, "valid_answers": ["-4, -4, -1", "-4, -2, 0", "-4, 0, 1", "-4, 2, 2", "-4, 4, 3", "-2, -5, -2", "-2, -3, -1", "-2, -1, 0", "-2, 1, 1", "-2, 3, 2", "-2, 5, 3", "0, -4, -2", "0, -2, -1", "0, 2, 1", "0, 4, 2", "2, -5, -3", "2, -3, -2", "2, -1, -1", "2, 1, 0", "2, 3, 1", "2, 5, 2", "4, -4, -3", "4, -2, -2", "4, 0, -1", "4, 2, 0", "4, 4, 1"]}} | |
| {"id": "po-poly_roots-t0-eval-000021", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5, not all zero, such that the polynomial p(x) = a + bx + cx^2 satisfies 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 = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5, not all zero, such that the polynomial p(x) = a + bx + cx^2 satisfies 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 = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -4, 1", "ref": "-5, -4, 1", "family": "po_poly_roots", "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_roots", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "root", "x": 5, "v": "0"}]}, "seed": 140069831700000043, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 5, "valid_answers": ["-5, -4, 1", "-5, 1, 0", "0, -5, 1", "0, 5, -1", "5, -1, 0", "5, 4, -1"]}} | |
| {"id": "po-poly_roots-t0-eval-000001", "task": "cx_po_poly_roots", "source": "cx_po_poly_roots", "split": "eval", "problem": "Find integers a, b, c with -5 <= a, b, c <= 5, not all zero, such that the polynomial p(x) = a + bx + cx^2 satisfies 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 = ...}, with the dots replaced by the values.", "prompt": "Problem:\nFind integers a, b, c with -5 <= a, b, c <= 5, not all zero, such that the polynomial p(x) = a + bx + cx^2 satisfies 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 = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -5, 2", "ref": "-3, -5, 2", "family": "po_poly_roots", "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_roots", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "root", "x": 3, "v": "0"}]}, "seed": 140069831700000001, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 3, "valid_answers": ["-3, -5, 2", "-3, -2, 1", "-3, 1, 0", "-3, 4, -1", "0, -3, 1", "0, 3, -1", "3, -4, 1", "3, -1, 0", "3, 2, -1", "3, 5, -2"]}} | |
| {"id": "po-poly_values-t0-eval-000126", "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(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) = -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, -3, -5", "ref": "-1, -3, -5", "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": -1, "v": "-3"}, {"t": "val", "x": 0, "v": "-1"}]}, "seed": 358655629800000135, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 1, "valid_answers": ["-1, -3, -5", "-1, -2, -4", "-1, -1, -3", "-1, 0, -2", "-1, 1, -1", "-1, 2, 0", "-1, 3, 1", "-1, 4, 2", "-1, 5, 3"]}} | |
| {"id": "po-poly_values-t0-eval-000247", "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) = -5 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 -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(0) = -5 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": "-5, -2, -5", "ref": "-5, -2, -5", "family": "po_poly_values", "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_values", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "val", "x": 0, "v": "-5"}, {"t": "val", "x": -1, "v": "-8"}]}, "seed": 358655629800000278, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 5, "valid_answers": ["-5, -2, -5", "-5, -1, -4", "-5, 0, -3", "-5, 1, -2", "-5, 2, -1", "-5, 3, 0", "-5, 4, 1", "-5, 5, 2"]}} | |
| {"id": "po-poly_values-t0-eval-000133", "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) = 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 -5 <= a, b, c <= 5 such that the polynomial p(x) = a + bx + cx^2 satisfies p(-1) = 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": "-5, -1, 4", "ref": "-5, -1, 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": "0"}, {"t": "val", "x": 1, "v": "-2"}]}, "seed": 358655629800000142, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 1, "valid_answers": ["-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"]}} | |
| {"id": "po-poly_values-t0-eval-000271", "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) = 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(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 = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, -5, 2", "ref": "4, -5, 2", "family": "po_poly_values", "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_values", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "val", "x": 0, "v": "4"}, {"t": "val", "x": 1, "v": "1"}]}, "seed": 358655629800000308, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 4, "valid_answers": ["4, -5, 2", "4, -4, 1", "4, -3, 0", "4, -2, -1", "4, -1, -2", "4, 0, -3", "4, 1, -4", "4, 2, -5"]}} | |
| {"id": "po-poly_values-t0-eval-000122", "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) = -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 p(-1) = 3 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": "-5, -5, 3", "ref": "-5, -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": -1, "v": "3"}, {"t": "val", "x": 1, "v": "-7"}]}, "seed": 358655629800000131, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 5, "valid_answers": ["-5, -5, 3", "-4, -5, 2", "-3, -5, 1", "-2, -5, 0", "-1, -5, -1", "0, -5, -2", "1, -5, -3", "2, -5, -4", "3, -5, -5"]}} | |
| {"id": "po-poly_values-t0-eval-000229", "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) = -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) = -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 = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-5, -1, 3", "ref": "-5, -1, 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": 1, "v": "-3"}, {"t": "val", "x": -1, "v": "-1"}]}, "seed": 358655629800000256, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 1, "valid_answers": ["-5, -1, 3", "-4, -1, 2", "-3, -1, 1", "-2, -1, 0", "-1, -1, -1", "0, -1, -2", "1, -1, -3", "2, -1, -4", "3, -1, -5"]}} | |
| {"id": "po-poly_values-t0-eval-000171", "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) = -2 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) = -2 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, -5", "ref": "-2, -4, -5", "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": 0, "v": "-2"}, {"t": "val", "x": -1, "v": "-3"}]}, "seed": 358655629800000185, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 2, "valid_answers": ["-2, -4, -5", "-2, -3, -4", "-2, -2, -3", "-2, -1, -2", "-2, 0, -1", "-2, 1, 0", "-2, 2, 1", "-2, 3, 2", "-2, 4, 3", "-2, 5, 4"]}} | |
| {"id": "po-poly_values-t0-eval-000217", "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(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) = -3 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, -5, -2", "ref": "4, -5, -2", "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": 1, "v": "-3"}, {"t": "val", "x": 0, "v": "4"}]}, "seed": 358655629800000243, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["4, -5, -2", "4, -4, -3", "4, -3, -4", "4, -2, -5"]}} | |
| {"id": "po-poly_values-t0-eval-000044", "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) = 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) = 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 = ...}, 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": 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": 1, "v": "3"}, {"t": "val", "x": -1, "v": "1"}]}, "seed": 358655629800000045, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 1, "valid_answers": ["-3, 1, 5", "-2, 1, 4", "-1, 1, 3", "0, 1, 2", "1, 1, 1", "2, 1, 0", "3, 1, -1", "4, 1, -2", "5, 1, -3"]}} | |
| {"id": "po-poly_values-t0-eval-000208", "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, -5, 1", "ref": "0, -5, 1", "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": 358655629800000233, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 2, "valid_answers": ["0, -5, 1", "0, -4, 0", "0, -3, -1", "0, -2, -2", "0, -1, -3", "0, 0, -4", "0, 1, -5"]}} | |
| {"id": "po-poly_values-t0-eval-000188", "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) = 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 p(-1) = 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": "-5, -1, 5", "ref": "-5, -1, 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": "1"}, {"t": "val", "x": 1, "v": "-1"}]}, "seed": 358655629800000208, "version": "po-v0", "tier": 0, "n_raw": 11, "small": 1, "valid_answers": ["-5, -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, -1, -5"]}} | |
| {"id": "po-poly_values-t0-eval-000022", "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) = -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) = 1 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_values", "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_values", "rung": "witness", "params": {"d": 2, "B": 5, "cons": [{"t": "val", "x": 0, "v": "1"}, {"t": "val", "x": 1, "v": "-5"}]}, "seed": 358655629800000022, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 3, "valid_answers": ["1, -5, -1", "1, -4, -2", "1, -3, -3", "1, -2, -4", "1, -1, -5"]}} | |
| {"id": "po-poly_values-t0-eval-000095", "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) = -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) = 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 = ...}, 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": 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": "3"}, {"t": "val", "x": 1, "v": "-1"}]}, "seed": 358655629800000102, "version": "po-v0", "tier": 0, "n_raw": 10, "small": 2, "valid_answers": ["-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"]}} | |
| {"id": "po-poly_values-t0-eval-000031", "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) = -12 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) = -12 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_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": -1, "v": "-12"}, {"t": "val", "x": 0, "v": "-4"}]}, "seed": 358655629800000031, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-4, 3, -5", "-4, 4, -4", "-4, 5, -3"]}} | |
| {"id": "po-poly_values-t0-eval-000286", "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) = -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(0) = -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 = ...}, 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": 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": "-3"}, {"t": "val", "x": -1, "v": "-1"}]}, "seed": 358655629800000333, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 3, "valid_answers": ["-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"]}} | |
| {"id": "po-poly_values-t0-eval-000099", "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) = 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) = -1 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, -5, -1", "ref": "-1, -5, -1", "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": 0, "v": "-1"}, {"t": "val", "x": -1, "v": "3"}]}, "seed": 358655629800000106, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 2, "valid_answers": ["-1, -5, -1", "-1, -4, 0", "-1, -3, 1", "-1, -2, 2", "-1, -1, 3", "-1, 0, 4", "-1, 1, 5"]}} | |
| {"id": "po-recur_coef-t0-eval-000257", "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 = 2 and u_{n+2} = p u_{n+1} + q u_n has u_3 = -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 -5 <= p, q <= 5 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 = -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, -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": 2, "m": 3, "B": 5, "T": -14}, "seed": 71449463200001403, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["1, -4", "3, -4"]}} | |
| {"id": "po-recur_coef-t0-eval-000294", "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 = 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 = -1, 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": "-2, 3", "ref": "-2, 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": -1, "u1": 4, "m": 3, "B": 5, "T": 34}, "seed": 71449463200001852, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["-2, 3", "3, -2"]}} | |
| {"id": "po-recur_coef-t0-eval-000006", "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 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 75. 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 = 3 and u_{n+2} = p u_{n+1} + q u_n has u_3 = 75. 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": 1, "u1": 3, "m": 3, "B": 5, "T": 75}, "seed": 71449463200000018, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-5, 0", "5, 0"]}} | |
| {"id": "po-recur_coef-t0-eval-000046", "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 = 112. 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 = 112. 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, 3", "ref": "-5, 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": 0, "u1": 4, "m": 3, "B": 5, "T": 112}, "seed": 71449463200000173, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-5, 3", "5, 3"]}} | |
| {"id": "po-recur_coef-t0-eval-000149", "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 = -10. 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 = -10. 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, -5", "ref": "0, -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": 2, "m": 3, "B": 5, "T": -10}, "seed": 71449463200000678, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["0, -5", "1, -4"]}} | |
| {"id": "po-recur_coef-t0-eval-000209", "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_3 = 41. 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_3 = 41. 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": 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": 1, "m": 3, "B": 5, "T": 41}, "seed": 71449463200001074, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-4, 5", "5, -4"]}} | |
| {"id": "po-recur_coef-t0-eval-000217", "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 = 3 and u_{n+2} = p u_{n+1} + q 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{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 = 3 and u_{n+2} = p u_{n+1} + q 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{p = ..., q = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-2, 2", "ref": "-2, 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": -3, "u1": 3, "m": 3, "B": 5, "T": 30}, "seed": 71449463200001127, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["-2, 2", "4, 2"]}} | |
| {"id": "po-recur_coef-t0-eval-000052", "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 = 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 = 1, 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": "-2, -2", "ref": "-2, -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": 12}, "seed": 71449463200000208, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 2, "valid_answers": ["-2, -2", "0, 3"]}} | |
| {"id": "po-recur_coef-t0-eval-000263", "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 = 52. 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 = 52. 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, 4", "ref": "-4, 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": 1, "m": 3, "B": 5, "T": 52}, "seed": 71449463200001432, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-4, 4", "5, -3"]}} | |
| {"id": "po-recur_coef-t0-eval-000058", "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 = 66. 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 = 66. 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, -3", "ref": "-5, -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": 0, "u1": 3, "m": 3, "B": 5, "T": 66}, "seed": 71449463200000229, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-5, -3", "5, -3"]}} | |
| {"id": "po-recur_coef-t0-eval-000297", "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 = 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 = 0, 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": 0, "u1": 2, "m": 3, "B": 5, "T": 50}, "seed": 71449463200001871, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-5, 0", "5, 0"]}} | |
| {"id": "po-recur_coef-t0-eval-000293", "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 = 144. 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 = 144. 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": 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": 144}, "seed": 71449463200001844, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-5, -4", "4, 5"]}} | |
| {"id": "po-recur_coef-t0-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_3 = 10. 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 = 10. 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, 3", "ref": "-2, 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": 3, "u1": 4, "m": 3, "B": 5, "T": 10}, "seed": 71449463200001372, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["-2, 3", "3, -2"]}} | |
| {"id": "po-recur_coef-t0-eval-000291", "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_3 = 41. 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_3 = 41. 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": 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": 1, "m": 3, "B": 5, "T": 41}, "seed": 71449463200001828, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-3, -4", "5, 1"]}} | |
| {"id": "po-recur_coef-t0-eval-000090", "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 = 28. 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 = 28. 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, 4", "ref": "-4, 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": 3, "m": 3, "B": 5, "T": 28}, "seed": 71449463200000357, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-4, 4", "1, 5"]}} | |
| {"id": "po-recur_coef-t0-eval-000017", "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 = 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 -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 = 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": "-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": 2, "m": 3, "B": 5, "T": 13}, "seed": 71449463200000063, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 3, "valid_answers": ["-3, 5", "3, -1"]}} | |
| {"id": "po-recur_init-t0-eval-000054", "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 = 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} = -u_{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, 0", "ref": "-9, 0", "family": "po_recur_init", "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_recur_init", "rung": "witness", "params": {"p": -1, "q": 2, "m": 3, "B": 10, "T": 18}, "seed": 46805164700000086, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 4, "valid_answers": ["-9, 0", "-6, 2", "-3, 4", "0, 6", "3, 8", "6, 10"]}} | |
| {"id": "po-recur_init-t0-eval-000138", "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 = 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 -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 = 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": "5, 9", "ref": "5, 9", "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": 2, "m": 3, "B": 10, "T": 74}, "seed": 46805164700000238, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 8, "valid_answers": ["5, 9", "8, 7"]}} | |
| {"id": "po-recur_init-t0-eval-000098", "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 = -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 -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 = -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": "-10, -4", "ref": "-10, -4", "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": -40}, "seed": 46805164700000155, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 6, "valid_answers": ["-10, -4", "-5, -6", "0, -8", "5, -10"]}} | |
| {"id": "po-recur_init-t0-eval-000144", "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 = -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 -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 = -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": "-10, -1", "ref": "-10, -1", "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": -40}, "seed": 46805164700000244, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["-10, -1", "0, -4", "10, -7"]}} | |
| {"id": "po-recur_init-t0-eval-000227", "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 = 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 -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 = 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": "-10, 8", "ref": "-10, 8", "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": 2, "m": 3, "B": 10, "T": 4}, "seed": 46805164700000375, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 2, "valid_answers": ["-10, 8", "-7, 6", "-4, 4", "-1, 2", "2, 0", "5, -2", "8, -4"]}} | |
| {"id": "po-recur_init-t0-eval-000127", "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 = -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} = -2u_{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": "-7, -4", "ref": "-7, -4", "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": -6}, "seed": 46805164700000214, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 2, "valid_answers": ["-7, -4", "-2, -2", "3, 0", "8, 2"]}} | |
| {"id": "po-recur_init-t0-eval-000247", "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 = -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} = -u_{n+1} + 2u_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": "2, -10", "ref": "2, -10", "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": -1, "q": 2, "m": 3, "B": 10, "T": -34}, "seed": 46805164700000415, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 8, "valid_answers": ["2, -10", "5, -8", "8, -6"]}} | |
| {"id": "po-recur_init-t0-eval-000293", "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 = 48. Several answers are possible; any one correct answer is 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 = 48. Several answers are possible; any one correct answer is accepted. Give 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, 2", "ref": "-9, 2", "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": 2, "m": 3, "B": 10, "T": 48}, "seed": 46805164700000493, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 6, "valid_answers": ["-9, 2", "-6, 4", "-3, 6", "0, 8", "3, 10"]}} | |
| {"id": "po-recur_init-t0-eval-000048", "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 = 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 -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 = 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_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": -1, "m": 3, "B": 10, "T": 2}, "seed": 46805164700000074, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 1, "valid_answers": ["-8, 6", "-5, 4", "-2, 2", "1, 0", "4, -2", "7, -4", "10, -6"]}} | |
| {"id": "po-recur_init-t0-eval-000084", "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 = -25. Several answers are possible; any one correct answer is 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 = -25. Several answers are possible; any one correct answer is accepted. Give 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_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": -25}, "seed": 46805164700000134, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-5, -1", "5, -4"]}} | |
| {"id": "po-recur_init-t0-eval-000209", "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": 46805164700000347, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 8, "valid_answers": ["-10, 6", "-5, 8", "0, 10"]}} | |
| {"id": "po-recur_init-t0-eval-000249", "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 = -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 -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 = -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, -10", "ref": "1, -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": -1, "q": 2, "m": 3, "B": 10, "T": -32}, "seed": 46805164700000418, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 7, "valid_answers": ["1, -10", "4, -8", "7, -6", "10, -4"]}} | |
| {"id": "po-recur_init-t0-eval-000267", "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 = 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 -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 = 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": "-10, 5", "ref": "-10, 5", "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": 45}, "seed": 46805164700000446, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 7, "valid_answers": ["-10, 5", "-5, 7", "0, 9"]}} | |
| {"id": "po-recur_init-t0-eval-000169", "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 = 52. Several answers are possible; any one correct answer is 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 = 52. Several answers are possible; any one correct answer is accepted. Give 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_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": 52}, "seed": 46805164700000280, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-4, 5", "4, 8"]}} | |
| {"id": "po-recur_init-t0-eval-000242", "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} + 2u_n has u_3 = -43. Several answers are possible; any one correct answer is 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} + 2u_n has u_3 = -43. Several answers are possible; any one correct answer is accepted. Give 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_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": 2, "m": 3, "B": 10, "T": -43}, "seed": 46805164700000405, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-9, 1", "2, -5"]}} | |
| {"id": "po-recur_init-t0-eval-000297", "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 = 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 -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 = 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, -2", "ref": "-6, -2", "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": 2}, "seed": 46805164700000498, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 1, "valid_answers": ["-6, -2", "-1, 0", "4, 2", "9, 4"]}} | |
| {"id": "po-sfft-t0-eval-000017", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 6x - y = -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 positive integers x, y such that xy - 6x - y = -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, 2", "ref": "3, 2", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.004115, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -6, "b": -1, "c": -14}, "seed": 167733942800000029, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["3, 2", "5, 4", "9, 5"]}} | |
| {"id": "po-sfft-t0-eval-000011", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 4y = 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 + 4y = 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": "2, 6", "ref": "2, 6", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.003125, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 0, "b": 4, "c": 36}, "seed": 167733942800000016, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 5, "valid_answers": ["2, 6", "5, 4", "8, 3", "14, 2", "32, 1"]}} | |
| {"id": "po-sfft-t0-eval-000251", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 3x + 6y = -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 positive integers x, y such that xy - 3x + 6y = -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, 1", "ref": "4, 1", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002378, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -3, "b": 6, "c": -2}, "seed": 167733942800000437, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["4, 1", "14, 2"]}} | |
| {"id": "po-sfft-t0-eval-000001", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 2x + 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 positive integers x, y such that xy - 2x + 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, 5", "ref": "1, 5", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.001461, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -2, "b": 4, "c": 23}, "seed": 167733942800000001, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["1, 5", "11, 3"]}} | |
| {"id": "po-sfft-t0-eval-000159", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 6y = -10. Several answers are possible; any one correct answer is 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 - 6y = -10. Several answers are possible; any one correct answer is accepted. Give 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": 0.011719, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 0, "b": -6, "c": -10}, "seed": 167733942800000274, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["1, 2", "4, 5", "5, 10"]}} | |
| {"id": "po-sfft-t0-eval-000253", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 6x - 2y = -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 - 2y = -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": "1, 9", "ref": "1, 9", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.002449, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -6, "b": -2, "c": -15}, "seed": 167733942800000445, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["1, 9", "3, 3", "5, 5"]}} | |
| {"id": "po-sfft-t0-eval-000075", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 4x - 5y = -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 - 5y = -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, 5", "ref": "3, 5", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001538, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -4, "b": -5, "c": -22}, "seed": 167733942800000124, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 5, "valid_answers": ["3, 5", "4, 6", "6, 2", "7, 3"]}} | |
| {"id": "po-sfft-t0-eval-000023", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 6x + 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 - 6x + 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, 4", "ref": "1, 4", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.002959, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -6, "b": 2, "c": 6}, "seed": 167733942800000040, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["1, 4", "4, 5"]}} | |
| {"id": "po-sfft-t0-eval-000187", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 3x - y = 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 positive integers x, y such that xy + 3x - y = 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": "2, 17", "ref": "2, 17", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.004444, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 3, "b": -1, "c": 23}, "seed": 167733942800000319, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 5, "valid_answers": ["2, 17", "3, 7", "5, 2", "6, 1"]}} | |
| {"id": "po-sfft-t0-eval-000082", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 5x - 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 - 5x - 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, 8", "ref": "1, 8", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.002583, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -5, "b": -3, "c": -21}, "seed": 167733942800000133, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 5, "valid_answers": ["1, 8", "2, 11", "5, 2", "6, 3", "9, 4"]}} | |
| {"id": "po-sfft-t0-eval-000240", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 6y = -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 - 6y = -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": "2, 4", "ref": "2, 4", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.006198, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 0, "b": -6, "c": -16}, "seed": 167733942800000412, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 4, "valid_answers": ["2, 4", "4, 8", "5, 16"]}} | |
| {"id": "po-sfft-t0-eval-000147", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 6x - 6y = 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 + 6x - 6y = 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": 3, "bucket": "2-3", "guess_rate": 0.001109, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 6, "b": -6, "c": 4}, "seed": 167733942800000250, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["2, 2", "4, 10", "5, 26"]}} | |
| {"id": "po-sfft-t0-eval-000175", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 3x - 6y = -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 positive integers x, y such that xy + 3x - 6y = -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": "1, 1", "ref": "1, 1", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.004756, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 3, "b": -6, "c": -2}, "seed": 167733942800000299, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 1, "valid_answers": ["1, 1", "2, 2", "4, 7", "5, 17"]}} | |
| {"id": "po-sfft-t0-eval-000076", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 5x = 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 + 5x = 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, 25", "ref": "1, 25", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.003265, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 5, "b": 0, "c": 30}, "seed": 167733942800000125, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 5, "valid_answers": ["1, 25", "2, 10", "3, 5", "5, 1"]}} | |
| {"id": "po-sfft-t0-eval-000019", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy + 6x + 5y = 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 + 6x + 5y = 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, 4", "ref": "2, 4", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000305, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": 6, "b": 5, "c": 40}, "seed": 167733942800000032, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["2, 4", "5, 1"]}} | |
| {"id": "po-sfft-t0-eval-000207", "task": "cx_po_sfft", "source": "cx_po_sfft", "split": "eval", "problem": "Find positive integers x, y such that xy - 6x - 5y = -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 - 6x - 5y = -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": "2, 5", "ref": "2, 5", "family": "po_sfft", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.000865, "solvable": true, "holdout": false, "meta": {"family": "po_sfft", "rung": "witness", "params": {"a": -6, "b": -5, "c": -27}, "seed": 167733942800000345, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 4, "valid_answers": ["2, 5", "4, 3", "6, 9", "8, 7"]}} | |
| {"id": "po-sigma_inv-t0-eval-000031", "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 104. 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 104. 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": "63", "ref": "63", "family": "po_sigma_inv", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.019231, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 104}, "seed": 44459245700000249, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 63, "valid_answers": ["63", "103"]}} | |
| {"id": "po-sigma_inv-t0-eval-000025", "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 272. 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 272. 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": "201", "ref": "201", "family": "po_sigma_inv", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.007353, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 272}, "seed": 44459245700000142, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 201, "valid_answers": ["201", "271"]}} | |
| {"id": "po-sigma_inv-t0-eval-000004", "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 294. 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 294. 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": "164", "ref": "164", "family": "po_sigma_inv", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.010204, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 294}, "seed": 44459245700000009, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 164, "valid_answers": ["164", "194", "293"]}} | |
| {"id": "po-sigma_inv-t0-eval-000011", "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 114. 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 114. 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": "74", "ref": "74", "family": "po_sigma_inv", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.017544, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 114}, "seed": 44459245700000030, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 74, "valid_answers": ["74", "113"]}} | |
| {"id": "po-sigma_inv-t0-eval-000013", "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 399. 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 399. 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": "196", "ref": "196", "family": "po_sigma_inv", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.005013, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 399}, "seed": 44459245700000033, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 196, "valid_answers": ["196", "242"]}} | |
| {"id": "po-sigma_inv-t0-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 126. 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 126. 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": "68", "ref": "68", "family": "po_sigma_inv", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.015873, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 126}, "seed": 44459245700000226, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 68, "valid_answers": ["68", "82"]}} | |
| {"id": "po-sigma_inv-t0-eval-000015", "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 156. 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 156. 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": "99", "ref": "99", "family": "po_sigma_inv", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.012821, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 156}, "seed": 44459245700000041, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 99, "valid_answers": ["99", "125"]}} | |
| {"id": "po-sigma_inv-t0-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 132. 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 132. 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": "86", "ref": "86", "family": "po_sigma_inv", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.015152, "solvable": true, "holdout": false, "meta": {"family": "po_sigma_inv", "rung": "witness", "params": {"s": 132}, "seed": 44459245700000093, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 86, "valid_answers": ["86", "131"]}} | |
| {"id": "po-sqrt_mod-t0-eval-000002", "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 172 \\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 172 \\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": "56", "ref": "56", "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": 172}, "seed": 109058477000000003, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 56, "valid_answers": ["56", "113", "134", "191"]}} | |
| {"id": "po-sqrt_mod-t0-eval-000234", "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 210 \\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 210 \\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": "76", "ref": "76", "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": 210}, "seed": 109058477000000407, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 76, "valid_answers": ["76", "122", "131", "177"]}} | |
| {"id": "po-sqrt_mod-t0-eval-000077", "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 233 \\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 233 \\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": "47", "ref": "47", "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": 233}, "seed": 109058477000000104, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 47, "valid_answers": ["47", "86", "161", "200"]}} | |
| {"id": "po-sqrt_mod-t0-eval-000285", "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 49 \\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 49 \\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": "7", "ref": "7", "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": 49}, "seed": 109058477000000555, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 7, "valid_answers": ["7", "45", "202", "240"]}} | |
| {"id": "po-sqrt_mod-t0-eval-000125", "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 179 \\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 179 \\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": "31", "ref": "31", "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": 179}, "seed": 109058477000000187, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 31, "valid_answers": ["31", "54", "337", "360"]}} | |
| {"id": "po-sqrt_mod-t0-eval-000134", "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 220 \\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 220 \\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": "31", "ref": "31", "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": 220}, "seed": 109058477000000202, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 31, "valid_answers": ["31", "83", "164", "216"]}} | |
| {"id": "po-sqrt_mod-t0-eval-000222", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 322 such that x^2 \\equiv 321 \\pmod{323}. Several answers are possible; any 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 <= 322 such that x^2 \\equiv 321 \\pmod{323}. Several answers are possible; any 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": "44", "ref": "44", "family": "po_sqrt_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.012384, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 323, "a": 321}, "seed": 109058477000000372, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 44, "valid_answers": ["44", "146", "177", "279"]}} | |
| {"id": "po-sqrt_mod-t0-eval-000088", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 322 such that x^2 \\equiv 36 \\pmod{323}. Several answers are possible; any 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 <= 322 such that x^2 \\equiv 36 \\pmod{323}. Several answers are possible; any 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": "6", "ref": "6", "family": "po_sqrt_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.012384, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 323, "a": 36}, "seed": 109058477000000123, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 6, "valid_answers": ["6", "108", "215", "317"]}} | |
| {"id": "po-sqrt_mod-t0-eval-000128", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 208 such that x^2 \\equiv 92 \\pmod{209}. Several answers are possible; any 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 <= 208 such that x^2 \\equiv 92 \\pmod{209}. Several answers are possible; any 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_sqrt_mod", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.019139, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 209, "a": 92}, "seed": 109058477000000192, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 42, "valid_answers": ["42", "53", "156", "167"]}} | |
| {"id": "po-sqrt_mod-t0-eval-000102", "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 185 \\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 185 \\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": "47", "ref": "47", "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": 185}, "seed": 109058477000000145, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 47, "valid_answers": ["47", "91", "162", "206"]}} | |
| {"id": "po-sqrt_mod-t0-eval-000218", "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 196 \\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 196 \\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": "14", "ref": "14", "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": 196}, "seed": 109058477000000364, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 14, "valid_answers": ["14", "105", "116", "207"]}} | |
| {"id": "po-sqrt_mod-t0-eval-000278", "task": "cx_po_sqrt_mod", "source": "cx_po_sqrt_mod", "split": "eval", "problem": "Find an integer x with 0 <= x <= 322 such that x^2 \\equiv 308 \\pmod{323}. Several answers are possible; any 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 <= 322 such that x^2 \\equiv 308 \\pmod{323}. Several answers are possible; any 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": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.012384, "solvable": true, "holdout": false, "meta": {"family": "po_sqrt_mod", "rung": "witness", "params": {"m": 323, "a": 308}, "seed": 109058477000000538, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 40, "valid_answers": ["40", "74", "249", "283"]}} | |
| {"id": "po-sqrt_mod-t0-eval-000085", "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 202 \\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 202 \\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": "61", "ref": "61", "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": 202}, "seed": 109058477000000114, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 61, "valid_answers": ["61", "146", "245", "330"]}} | |
| {"id": "po-sqrt_mod-t0-eval-000228", "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 120 \\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 120 \\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": "35", "ref": "35", "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": 120}, "seed": 109058477000000389, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 35, "valid_answers": ["35", "69", "152", "186"]}} | |
| {"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 <= 298 such that x^2 \\equiv 62 \\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 62 \\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": "19", "ref": "19", "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": 62}, "seed": 109058477000000013, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 19, "valid_answers": ["19", "111", "188", "280"]}} | |
| {"id": "po-sqrt_mod-t0-eval-000035", "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 123 \\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 123 \\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": "79", "ref": "79", "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": 123}, "seed": 109058477000000049, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 79, "valid_answers": ["79", "174", "263", "358"]}} | |
| {"id": "po-tau_range-t0-eval-000223", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 9838 <= n <= 10302 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 9838 <= n <= 10302 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": "9872", "ref": "9872", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.012903, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 9838, "U": 10302, "k": 10}, "seed": 193876035800000730, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 9872, "valid_answers": ["9872", "9904", "10096", "10256", "10287", "10288"]}} | |
| {"id": "po-tau_range-t0-eval-000094", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 14501 <= n <= 15041 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 14501 <= n <= 15041 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": "14512", "ref": "14512", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.011091, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 14501, "U": 15041, "k": 10}, "seed": 193876035800000337, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 14512, "valid_answers": ["14512", "14576", "14661", "14704", "14864", "14992"]}} | |
| {"id": "po-tau_range-t0-eval-000272", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 13722 <= n <= 14267 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 13722 <= n <= 14267 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": "13744", "ref": "13744", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.012821, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 13722, "U": 14267, "k": 10}, "seed": 193876035800000884, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 13744, "valid_answers": ["13744", "13808", "14013", "14032", "14096", "14128", "14192"]}} | |
| {"id": "po-tau_range-t0-eval-000089", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 7193 <= n <= 7690 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 7193 <= n <= 7690 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": "7203", "ref": "7203", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.014056, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 7193, "U": 7690, "k": 10}, "seed": 193876035800000317, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 7203, "valid_answers": ["7203", "7209", "7312", "7376", "7408", "7472", "7664"]}} | |
| {"id": "po-tau_range-t0-eval-000014", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 4877 <= n <= 5239 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 4877 <= n <= 5239 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": "4912", "ref": "4912", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.013774, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 4877, "U": 5239, "k": 10}, "seed": 193876035800000051, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 4912, "valid_answers": ["4912", "4941", "4976", "5008", "5072"]}} | |
| {"id": "po-tau_range-t0-eval-000159", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 13833 <= n <= 14181 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 13833 <= n <= 14181 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": "14013", "ref": "14013", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.011461, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 13833, "U": 14181, "k": 10}, "seed": 193876035800000534, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 14013, "valid_answers": ["14013", "14032", "14096", "14128"]}} | |
| {"id": "po-tau_range-t0-eval-000117", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 12323 <= n <= 12738 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 12323 <= n <= 12738 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": "12368", "ref": "12368", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 3, "bucket": "2-3", "guess_rate": 0.007212, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 12323, "U": 12738, "k": 10}, "seed": 193876035800000390, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 12368, "valid_answers": ["12368", "12592", "12717"]}} | |
| {"id": "po-tau_range-t0-eval-000021", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 9225 <= n <= 9741 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 9225 <= n <= 9741 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": "9232", "ref": "9232", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.011605, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 9225, "U": 9741, "k": 10}, "seed": 193876035800000065, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 9232, "valid_answers": ["9232", "9392", "9488", "9584", "9616", "9712"]}} | |
| {"id": "po-tau_range-t0-eval-000202", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 12910 <= n <= 13439 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 12910 <= n <= 13439 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": "12944", "ref": "12944", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.015094, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 12910, "U": 13439, "k": 10}, "seed": 193876035800000666, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 12944, "valid_answers": ["12944", "12976", "13136", "13168", "13203", "13232", "13264", "13424"]}} | |
| {"id": "po-tau_range-t0-eval-000004", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 6567 <= n <= 7141 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 6567 <= n <= 7141 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": "6704", "ref": "6704", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 8, "bucket": "4-8", "guess_rate": 0.013913, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 6567, "U": 7141, "k": 10}, "seed": 193876035800000018, "version": "po-v0", "tier": 0, "n_raw": 8, "small": 6704, "valid_answers": ["6704", "6723", "6736", "6875", "6896", "6928", "7024", "7088"]}} | |
| {"id": "po-tau_range-t0-eval-000041", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 3376 <= n <= 3700 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 3376 <= n <= 3700 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": 5, "bucket": "4-8", "guess_rate": 0.015385, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 3376, "U": 3700, "k": 10}, "seed": 193876035800000160, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 3376, "valid_answers": ["3376", "3483", "3568", "3632", "3664"]}} | |
| {"id": "po-tau_range-t0-eval-000002", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 3194 <= n <= 3708 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 3194 <= n <= 3708 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": "3321", "ref": "3321", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.01165, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 3194, "U": 3708, "k": 10}, "seed": 193876035800000013, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 3321, "valid_answers": ["3321", "3376", "3483", "3568", "3632", "3664"]}} | |
| {"id": "po-tau_range-t0-eval-000167", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 8705 <= n <= 9122 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 8705 <= n <= 9122 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": "8752", "ref": "8752", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 5, "bucket": "4-8", "guess_rate": 0.011962, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 8705, "U": 9122, "k": 10}, "seed": 193876035800000553, "version": "po-v0", "tier": 0, "n_raw": 5, "small": 8752, "valid_answers": ["8752", "8829", "8912", "9008", "9104"]}} | |
| {"id": "po-tau_range-t0-eval-000147", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 2224 <= n <= 2658 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 2224 <= n <= 2658 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": "2224", "ref": "2224", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 7, "bucket": "4-8", "guess_rate": 0.016092, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 2224, "U": 2658, "k": 10}, "seed": 193876035800000485, "version": "po-v0", "tier": 0, "n_raw": 7, "small": 2224, "valid_answers": ["2224", "2349", "2384", "2416", "2511", "2512", "2608"]}} | |
| {"id": "po-tau_range-t0-eval-000083", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 4303 <= n <= 4655 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 4303 <= n <= 4655 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": "4304", "ref": "4304", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.016997, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 4303, "U": 4655, "k": 10}, "seed": 193876035800000301, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 4304, "valid_answers": ["4304", "4336", "4375", "4432", "4496", "4528"]}} | |
| {"id": "po-tau_range-t0-eval-000270", "task": "cx_po_tau_range", "source": "cx_po_tau_range", "split": "eval", "problem": "Find an integer n with 11107 <= n <= 11566 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 11107 <= n <= 11566 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": "11216", "ref": "11216", "family": "po_tau_range", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.008696, "solvable": true, "holdout": false, "meta": {"family": "po_tau_range", "rung": "witness", "params": {"L": 11107, "U": 11566, "k": 10}, "seed": 193876035800000876, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 11216, "valid_answers": ["11216", "11259", "11344", "11504"]}} | |
| {"id": "po-totient_inv-t0-eval-000013", "task": "cx_po_totient_inv", "source": "cx_po_totient_inv", "split": "eval", "problem": "Find a positive integer n with \\varphi(n) = 42, 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) = 42, 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": "43", "ref": "43", "family": "po_totient_inv", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 4, "bucket": "4-8", "guess_rate": 0.001133, "solvable": true, "holdout": false, "meta": {"family": "po_totient_inv", "rung": "witness", "params": {"m": 42}, "seed": 55511482900000034, "version": "po-v0", "tier": 0, "n_raw": 4, "small": 43, "valid_answers": ["43", "49", "86", "98"]}} | |
| {"id": "po-totient_inv-t0-eval-000007", "task": "cx_po_totient_inv", "source": "cx_po_totient_inv", "split": "eval", "problem": "Find a positive integer n with \\varphi(n) = 46, 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) = 46, 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": "47", "ref": "47", "family": "po_totient_inv", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000472, "solvable": true, "holdout": false, "meta": {"family": "po_totient_inv", "rung": "witness", "params": {"m": 46}, "seed": 55511482900000010, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 47, "valid_answers": ["47", "94"]}} | |
| {"id": "po-totient_inv-t0-eval-000001", "task": "cx_po_totient_inv", "source": "cx_po_totient_inv", "split": "eval", "problem": "Find a positive integer n with \\varphi(n) = 16, 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) = 16, 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": "17", "ref": "17", "family": "po_totient_inv", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 6, "bucket": "4-8", "guess_rate": 0.011696, "solvable": true, "holdout": false, "meta": {"family": "po_totient_inv", "rung": "witness", "params": {"m": 16}, "seed": 55511482900000002, "version": "po-v0", "tier": 0, "n_raw": 6, "small": 17, "valid_answers": ["17", "32", "34", "40", "48", "60"]}} | |
| {"id": "po-totient_inv-t0-eval-000000", "task": "cx_po_totient_inv", "source": "cx_po_totient_inv", "split": "eval", "problem": "Find a positive integer n with \\varphi(n) = 60, 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) = 60, 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": "61", "ref": "61", "family": "po_totient_inv", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 9, "bucket": "9-20", "guess_rate": 0.00125, "solvable": true, "holdout": false, "meta": {"family": "po_totient_inv", "rung": "witness", "params": {"m": 60}, "seed": 55511482900000000, "version": "po-v0", "tier": 0, "n_raw": 9, "small": 61, "valid_answers": ["61", "77", "93", "99", "122", "124", "154", "186", "198"]}} | |
| {"id": "po-totient_inv-t0-eval-000002", "task": "cx_po_totient_inv", "source": "cx_po_totient_inv", "split": "eval", "problem": "Find a positive integer n with \\varphi(n) = 58, 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) = 58, 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": "59", "ref": "59", "family": "po_totient_inv", "domain": "number", "difficulty": 0, "tier": 0, "rung": "witness", "n_valid": 2, "bucket": "2-3", "guess_rate": 0.000297, "solvable": true, "holdout": false, "meta": {"family": "po_totient_inv", "rung": "witness", "params": {"m": 58}, "seed": 55511482900000003, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 59, "valid_answers": ["59", "118"]}} | |
| {"id": "po-vec_dot-t0-eval-000012", "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) = -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 -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-3, -1) = -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": "4, 5", "ref": "4, 5", "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": [[-3, -1]], "s": [-17]}, "seed": 205845576100000055, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["4, 5", "5, 2", "6, -1"]}} | |
| {"id": "po-vec_dot-t0-eval-000205", "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, 4) = 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 -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (3, 4) = 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": "-4, 4", "ref": "-4, 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": [[3, 4]], "s": [4]}, "seed": 205845576100000985, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 1, "valid_answers": ["-4, 4", "0, 1", "4, -2"]}} | |
| {"id": "po-vec_dot-t0-eval-000193", "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) = 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 (4, -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 = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "0, -3", "ref": "0, -3", "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": [9]}, "seed": 205845576100000943, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["0, -3", "3, 1", "6, 5"]}} | |
| {"id": "po-vec_dot-t0-eval-000036", "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, -2) = 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 -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (4, -2) = 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": "5, -5", "ref": "5, -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": [[4, -2]], "s": [30]}, "seed": 205845576100000143, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["5, -5", "6, -3"]}} | |
| {"id": "po-vec_dot-t0-eval-000014", "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) = -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 v = (x, y) satisfies v \\cdot (2, -3) = -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, 1", "ref": "-6, 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": [-15]}, "seed": 205845576100000061, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["-6, 1", "-3, 3", "0, 5"]}} | |
| {"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 (3, 4) = -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 -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (3, 4) = -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": "-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, 4]], "s": [-22]}, "seed": 205845576100001011, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 4, "valid_answers": ["-6, -1", "-2, -4"]}} | |
| {"id": "po-vec_dot-t0-eval-000148", "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, -1) = -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 -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (4, -1) = -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": "-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, -1]], "s": [-23]}, "seed": 205845576100000719, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 5, "valid_answers": ["-6, -1", "-5, 3"]}} | |
| {"id": "po-vec_dot-t0-eval-000284", "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, 2) = -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 -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (-2, 2) = -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": "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": [[-2, 2]], "s": [-22]}, "seed": 205845576100001392, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 6, "valid_answers": ["5, -6", "6, -5"]}} | |
| {"id": "po-vec_dot-t0-eval-000150", "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, 1) = 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 -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (4, 1) = 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": "-1, 5", "ref": "-1, 5", "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, 1]], "s": [1]}, "seed": 205845576100000725, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 1, "valid_answers": ["-1, 5", "0, 1", "1, -3"]}} | |
| {"id": "po-vec_dot-t0-eval-000072", "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, -2) = 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 (3, -2) = 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": "1, -4", "ref": "1, -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": [[3, -2]], "s": [11]}, "seed": 205845576100000338, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 3, "valid_answers": ["1, -4", "3, -1", "5, 2"]}} | |
| {"id": "po-vec_dot-t0-eval-000135", "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) = -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 -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (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 = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "-3, -1", "ref": "-3, -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": [-7]}, "seed": 205845576100000633, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 2, "valid_answers": ["-3, -1", "1, -2", "5, -3"]}} | |
| {"id": "po-vec_dot-t0-eval-000119", "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, -2) = -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 integers x, y with -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (1, -2) = -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": "-6, 5", "ref": "-6, 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, -2]], "s": [-16]}, "seed": 205845576100000532, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 6, "valid_answers": ["-6, 5", "-4, 6"]}} | |
| {"id": "po-vec_dot-t0-eval-000186", "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, -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 = ...}, 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, -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 = ...}, with the dots replaced by the values.\n\n<think>\n", "answer": "4, -6", "ref": "4, -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, -1]], "s": [22]}, "seed": 205845576100000916, "version": "po-v0", "tier": 0, "n_raw": 3, "small": 5, "valid_answers": ["4, -6", "5, -2", "6, 2"]}} | |
| {"id": "po-vec_dot-t0-eval-000182", "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, 3) = -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 (1, 3) = -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": "-6, -5", "ref": "-6, -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, 3]], "s": [-21]}, "seed": 205845576100000903, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 6, "valid_answers": ["-6, -5", "-3, -6"]}} | |
| {"id": "po-vec_dot-t0-eval-000166", "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) = -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 (-1, 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 = ...}, 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": [[-1, 4]], "s": [-26]}, "seed": 205845576100000800, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 6, "valid_answers": ["2, -6", "6, -5"]}} | |
| {"id": "po-vec_dot-t0-eval-000001", "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, 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 -6 <= x, y <= 6 such that v = (x, y) satisfies v \\cdot (2, 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": "-6, -5", "ref": "-6, -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": [[2, 4]], "s": [-32]}, "seed": 205845576100000007, "version": "po-v0", "tier": 0, "n_raw": 2, "small": 6, "valid_answers": ["-6, -5", "-4, -6"]}} | |