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Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"442d0fdf4b0f415878e9d70d03998b48e1a60f56f84f901a821491dcd72bace6","repetition":3,"request_hash":"4426394d19c0e8ade212e6cd01ebd8917e3cb884e6a9c0de2488b536ab8d49d9","request_id":"req_8e2adb6d5f4142aba57322c2e7562a83","sent_at":"2026-10-05T05:54:35.123262+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-allocation","trial_id":"66659a94d5b4e9c03341a2cef19dbb5421a3034c6cab9330d51f93d2e0055e17","usage":{"input_tokens":199,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":44,"output_tokens_details":{"reasoning_tokens":19},"total_tokens":243}},{"answer":"P(D|F) = 0.02·0.95 / (0.02·0.95 + 0.98·0.04) = 0.019 / (0.019 + 0.0392) = 0.019/0.0582 = 190/582 = 95/291.\n\nCompute 95/291: 291·0.326 = 94.866, remainder 0.134. 0.134/291 = 0.00046048...\nSo 0.32646048...\n\nMore digits: 95/291 long division.\n950/291 = 3 (873), r77\n770/291 = 2 (582), r188\n1880/291 = 6 (1746), r134\n1340/291 = 4 (1164), r176\n1760/291 = 6 (1746), r14\n140 → 0, r140\n1400/291 = 4 (1164), r236\n2360/291 = 8 (2328), r32\n320/291 = 1, r29\n290 → 0, r290\n2900/291 = 9 (2619), r281\n2810/291 = 9 (2619), r191\n1910/291 = 6 (1746), r164\n\nDigits: 0.3264604810996...\n\n{\"probability\": 0.3264604810996}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.004076,"finished_at":"2026-10-05T05:41:49.793652+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"c8bd20f0fe2059526059d08d24ef45ef38c362868f2635c9983d69b69a4fe976","latency_seconds":3.763627,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"A defect affects 2% of units. A detector flags 95% of defective units and 4% of nondefective units. Given a flag, calculate the probability the unit is defective, as a decimal from 0 to 1. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: probability. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"1ac23b25e7d5c2c72c7db00db9f91c0e84d8360b64a71771ffc8c5f3c6273915","repetition":1,"request_hash":"cd1309fff38bf1885e01989b99b566f17ee038eacd188f8410bd04547fb5da57","request_id":"req_011CfiZBPJj3TH5QYgikhUay","sent_at":"2026-10-05T05:41:46.030025+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-bayes","trial_id":"3820c01605df5471fa1b14b7c0631003780b045a6aed257c7b8e160f545c2b80","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":233,"output_tokens":361,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"P(D|F) = 0.02*0.95 / (0.02*0.95 + 0.98*0.04) = 0.019 / (0.019 + 0.0392) = 0.019/0.0582 = 190/582 = 95/291.\n\nCompute 95/291: 291*0.3 = 87.3, remainder 7.7 → 0.02646... Let's do long division.\n95/291 = 0.326460481...\n291*0.3 = 87.3; remainder 7.7\n7.7/291 = 0.026460...\n291*0.026 = 7.566; remainder 0.134\n0.134/291 = 0.00046048...\n291*0.00046 = 0.13386; remainder 0.00014\n0.00014/291 = 4.8110e-7\nTotal: 0.3 + 0.026 + 0.00046 + 0.00000048110 = 0.3264604811\n\nMore digits: 0.00014/291 = 4.81099656e-7. So 0.326460481099656...\n\n{\"probability\": 0.3264604810997}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.003806,"finished_at":"2026-10-05T05:49:43.352534+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"c8bd20f0fe2059526059d08d24ef45ef38c362868f2635c9983d69b69a4fe976","latency_seconds":3.291166,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"A defect affects 2% of units. A detector flags 95% of defective units and 4% of nondefective units. Given a flag, calculate the probability the unit is defective, as a decimal from 0 to 1. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: probability. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"8d8ae29add57885926936503b6e3b25f0ba8e40449f292340737847cc6c58bf6","repetition":2,"request_hash":"cd1309fff38bf1885e01989b99b566f17ee038eacd188f8410bd04547fb5da57","request_id":"req_011CfiZnKqq83qkG8DzD7MCe","sent_at":"2026-10-05T05:49:40.061368+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-bayes","trial_id":"194bd6ccb70795930ed8b60c5cd52d92b91848c7a68d3c8da859d905bae58671","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":233,"output_tokens":334,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"P(defective | flag) = (0.02 × 0.95) / (0.02 × 0.95 + 0.98 × 0.04)\n\nThe numerator is 0.019 and the second term is 0.0392, so the denominator is 0.0582.\n\n0.019 / 0.0582 = 190/582 = 95/291\n\nLong division of 95/291:\n- 950/291 = 3 remainder 77, giving digit 3\n- 770/291 = 2 remainder 188, giving digit 2\n- 1880/291 = 6 remainder 134 (1746), giving digit 6\n- 1340/291 = 4 remainder 176 (1164), giving digit 4\n- 1760/291 = 6 remainder 14 (1746), giving digit 6\n- 140/291 = 0, giving digit 0\n- 1400/291 = 4 remainder 236 (1164), giving digit 4\n- 2360/291 = 8 remainder 32 (2328), giving digit 8\n- 320/291 = 1 remainder 29, giving digit 1\n- 290/291 = 0, giving digit 0\n- 2900/291 = 9 remainder 281 (2619), giving digit 9\n- 2810/291 = 9 remainder 191 (2619), giving digit 9\n\nSo 95/291 = 0.326460481099...\n\n{\"probability\": 0.326460481099}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.004886,"finished_at":"2026-10-05T05:51:06.749120+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"c8bd20f0fe2059526059d08d24ef45ef38c362868f2635c9983d69b69a4fe976","latency_seconds":3.486374,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"A defect affects 2% of units. A detector flags 95% of defective units and 4% of nondefective units. Given a flag, calculate the probability the unit is defective, as a decimal from 0 to 1. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: probability. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"22fb2cfabf84793fff93012a96109a2b4c82edd0b8fdbd6a53afa087a2ea153e","repetition":3,"request_hash":"cd1309fff38bf1885e01989b99b566f17ee038eacd188f8410bd04547fb5da57","request_id":"req_011CfiZtTc5CggBfqxBPMtKu","sent_at":"2026-10-05T05:51:03.262746+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-bayes","trial_id":"c4edc7640f630b603d0de159b31c5458413fccf5f1a0e1f9279bf71add64bb65","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":233,"output_tokens":442,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"{\"probability\": 0.32646048109965636}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.004247,"finished_at":"2026-10-05T05:41:55.358687+00:00","grade":{"checks":[{"field":"probability","passed":true}],"passed":true,"reason":"correct"},"input_hash":"c8bd20f0fe2059526059d08d24ef45ef38c362868f2635c9983d69b69a4fe976","latency_seconds":5.544168,"model_id":"gemini-3.8-flash","output_cap_reached":false,"prompt":"A defect affects 2% of units. A detector flags 95% of defective units and 4% of nondefective units. Given a flag, calculate the probability the unit is defective, as a decimal from 0 to 1. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: probability. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"f5025cc546e5380a86b7e64aae3dab48d35cdbca4355599751f043e7fe1fa061","repetition":1,"request_hash":"e5c404743beaadef936cec54237ec20232d9eb44759172baabf1c9df026a0315","request_id":null,"sent_at":"2026-10-05T05:41:49.814519+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Gemini Developer API, paid Standard"},"state":"responded","task_id":"math-bayes","trial_id":"6806a5393b0474731a9300507d4abb6f769fb78716e467365b264c5e0ecd9f7e","usage":{"candidatesTokenCount":24,"promptTokenCount":162,"promptTokensDetails":[{"modality":"TEXT","tokenCount":162}],"serviceTier":"standard","thoughtsTokenCount":1076,"totalTokenCount":1262}},{"answer":"{\"probability\": 0.326460481099656}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.003872,"finished_at":"2026-10-05T05:49:48.526952+00:00","grade":{"checks":[{"field":"probability","passed":true}],"passed":true,"reason":"correct"},"input_hash":"c8bd20f0fe2059526059d08d24ef45ef38c362868f2635c9983d69b69a4fe976","latency_seconds":5.161833,"model_id":"gemini-3.8-flash","output_cap_reached":false,"prompt":"A defect affects 2% of units. A detector flags 95% of defective units and 4% of nondefective units. Given a flag, calculate the probability the unit is defective, as a decimal from 0 to 1. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: probability. Numeric fields must be finite numbers. 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A detector flags 95% of defective units and 4% of nondefective units. Given a flag, calculate the probability the unit is defective, as a decimal from 0 to 1. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: probability. Numeric fields must be finite numbers. 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A detector flags 95% of defective units and 4% of nondefective units. Given a flag, calculate the probability the unit is defective, as a decimal from 0 to 1. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: probability. Numeric fields must be finite numbers. 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A detector flags 95% of defective units and 4% of nondefective units. Given a flag, calculate the probability the unit is defective, as a decimal from 0 to 1. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: probability. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"99d9e2974aac1b4df09e83be534070946677aa909515196418b8c4e9189b3f27","repetition":2,"request_hash":"480156e1316e1a2df3782c288e63bc183555c77085b5fc0862297c00c17ded65","request_id":"req_13c2e3d2e4924a568944f69c6b848f51","sent_at":"2026-10-05T05:49:48.556252+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-bayes","trial_id":"6228de2c1d5ead3126eb6abfb2b3c0f722a026e35790aa281554e1f0867de874","usage":{"input_tokens":157,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":61,"output_tokens_details":{"reasoning_tokens":42},"total_tokens":218}},{"answer":"{\"probability\":0.32646048109965636}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.000824,"finished_at":"2026-10-05T05:51:03.250951+00:00","grade":{"checks":[{"field":"probability","passed":true}],"passed":true,"reason":"correct"},"input_hash":"c8bd20f0fe2059526059d08d24ef45ef38c362868f2635c9983d69b69a4fe976","latency_seconds":2.605576,"model_id":"gpt-6.1-sol","output_cap_reached":false,"prompt":"A defect affects 2% of units. A detector flags 95% of defective units and 4% of nondefective units. Given a flag, calculate the probability the unit is defective, as a decimal from 0 to 1. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: probability. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"459c9104fd37e105f83047448bc7d48c0890122256fde81098de3d096fac9c5f","repetition":3,"request_hash":"480156e1316e1a2df3782c288e63bc183555c77085b5fc0862297c00c17ded65","request_id":"req_93da05ba2bbc44d7959927b1693a8556","sent_at":"2026-10-05T05:51:00.645375+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-bayes","trial_id":"55ac5e9688082ea4a14cad649d519eddae9f32d211781e8219dce9711b7e3b46","usage":{"input_tokens":157,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":51,"output_tokens_details":{"reasoning_tokens":32},"total_tokens":208}},{"answer":"Price 40. Variable costs: 17 + 3 + 0.025·40 + 0.30 = 17 + 3 + 1 + 0.3 = 21.3.\n\nContribution per unit = 40 − 21.3 = 18.7.\n\nBreak-even units = 1870 / 18.7 = 100 exactly.\n\nAt 100 units: profit = 100 · 18.7 − 1870 = 1870 − 1870 = 0.\n\n{\"profit\": 0, \"units\": 100}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.002172,"finished_at":"2026-10-05T05:44:54.996764+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"6621a52acc46e3d976876748e998815a29d9dc21048e0945718ad9d90d3c976a","latency_seconds":1.762,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"A product sells for $40. Variable costs are $17 production, $3 shipping, and a payment fee of 2.5% of sale price plus $0.30. Fixed monthly costs are $1870. Return the minimum whole units needed to break even and profit at that unit count. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. These fields require exact integer-valued numbers (1 and 1.0 are equivalent): units. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: profit, units. Numeric fields must be finite numbers. 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Variable costs are $17 production, $3 shipping, and a payment fee of 2.5% of sale price plus $0.30. Fixed monthly costs are $1870. Return the minimum whole units needed to break even and profit at that unit count. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. These fields require exact integer-valued numbers (1 and 1.0 are equivalent): units. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: profit, units. Numeric fields must be finite numbers. 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Variable costs are $17 production, $3 shipping, and a payment fee of 2.5% of sale price plus $0.30. Fixed monthly costs are $1870. Return the minimum whole units needed to break even and profit at that unit count. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. These fields require exact integer-valued numbers (1 and 1.0 are equivalent): units. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: profit, units. Numeric fields must be finite numbers. 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Variable costs are $17 production, $3 shipping, and a payment fee of 2.5% of sale price plus $0.30. Fixed monthly costs are $1870. Return the minimum whole units needed to break even and profit at that unit count. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. These fields require exact integer-valued numbers (1 and 1.0 are equivalent): units. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: profit, units. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"4fd60ee0fbb0bcf6e1854a0c780a81c44dc6739016f1d140c4b765a4803d7d93","repetition":2,"request_hash":"65493125a33d321365c28d0010603fcbd06be67c7c1b1d7f24c655ffbc9bd006","request_id":null,"sent_at":"2026-10-05T05:47:50.191947+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Gemini Developer API, paid Standard"},"state":"responded","task_id":"math-break-even","trial_id":"85726139667263eff172bb4bb224f349fe79670bd28eeb7a025c33d5857aa4b2","usage":{"candidatesTokenCount":14,"promptTokenCount":199,"promptTokensDetails":[{"modality":"TEXT","tokenCount":199}],"serviceTier":"standard","thoughtsTokenCount":584,"totalTokenCount":797}},{"answer":"{\"profit\": 0, \"units\": 100}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.002077,"finished_at":"2026-10-05T05:53:11.547640+00:00","grade":{"checks":[{"field":"profit","passed":true},{"field":"units","passed":true}],"passed":true,"reason":"correct"},"input_hash":"6621a52acc46e3d976876748e998815a29d9dc21048e0945718ad9d90d3c976a","latency_seconds":2.128158,"model_id":"gemini-3.8-flash","output_cap_reached":false,"prompt":"A product sells for $40. 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Variable costs are $17 production, $3 shipping, and a payment fee of 2.5% of sale price plus $0.30. Fixed monthly costs are $1870. Return the minimum whole units needed to break even and profit at that unit count. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. These fields require exact integer-valued numbers (1 and 1.0 are equivalent): units. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: profit, units. Numeric fields must be finite numbers. 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Variable costs are $17 production, $3 shipping, and a payment fee of 2.5% of sale price plus $0.30. Fixed monthly costs are $1870. Return the minimum whole units needed to break even and profit at that unit count. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. These fields require exact integer-valued numbers (1 and 1.0 are equivalent): units. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: profit, units. Numeric fields must be finite numbers. 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Contribution = 800 + 1500 = 2300.\n\nOther vertices: A = 40, B = 0 gives 1600. A = 0, B = 40 gives 2000. So the LP optimum is at the integer point (20, 30) with value 2300.\n\nTie check: the objective ratio 40:50 differs from both constraint ratios (3:2 and 2:4), so the optimum is unique.\n\n{\"a\": 20, \"b\": 30, \"contribution\": 2300}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.003214,"finished_at":"2026-10-05T05:43:37.952621+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"1069c216d2bceb802535e22e6a66f4c7541774ac3804baa9a70d4d43896faabc","latency_seconds":2.734215,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"Product A uses 3 labour hours and 2 machine hours and earns $40 contribution. B uses 2 labour and 4 machine hours and earns $50. Weekly limits are 120 labour and 160 machine hours. Products must be whole units. Maximize contribution; if tied choose more A. Return A, B and contribution. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. These fields require exact integer-valued numbers (1 and 1.0 are equivalent): a, b. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: a, b, contribution. Numeric fields must be finite numbers. 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Subtracting that from the first gives 2A ≤ 40 (since 3A + 2B − (A + 2B) = 2A ≤ 120 − 80), so A = 20 and B = 30. Contribution there is 800 + 1500 = 2300.\n\nCheck the other vertices:\n- A = 40, B = 0 gives 1600.\n- A = 0, B = 40 gives 2000.\n\nThe vertex (20, 30) is integer and best among the vertices, so it is optimal for the LP and therefore for the integer problem.\n\nCheck for ties: the objective ratio 40:50 is not parallel to either constraint (3:2 or 2:4 = 1:2). The objective 4:5 differs from both, so the optimum is unique.\n\n{\"a\": 20, \"b\": 30, \"contribution\": 2300}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.004054,"finished_at":"2026-10-05T05:47:42.888385+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"1069c216d2bceb802535e22e6a66f4c7541774ac3804baa9a70d4d43896faabc","latency_seconds":3.555645,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"Product A uses 3 labour hours and 2 machine hours and earns $40 contribution. B uses 2 labour and 4 machine hours and earns $50. Weekly limits are 120 labour and 160 machine hours. Products must be whole units. Maximize contribution; if tied choose more A. Return A, B and contribution. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. These fields require exact integer-valued numbers (1 and 1.0 are equivalent): a, b. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: a, b, contribution. Numeric fields must be finite numbers. 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From 3A + 2B = 120 and A + 2B = 80, subtracting gives 2A = 40, so A = 20 and B = 30. The contribution there is 800 + 1500 = 2300.\n\nThe other vertices are worse:\n- (40, 0) gives 1600.\n- (0, 40) gives 2000.\n\nThe vertex (20, 30) is integer, so it is the integer optimum. Check feasibility: labour is 60 + 60 = 120 and machine is 40 + 120 = 160, both within limits.\n\nTies: the objective 40A + 50B has slope ratio 4:5, while the constraint slopes are 3:2 and 1:2, so there are no ties along an edge.\n\n{\"a\":20,\"b\":30,\"contribution\":2300}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.003794,"finished_at":"2026-10-05T05:51:16.264916+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"1069c216d2bceb802535e22e6a66f4c7541774ac3804baa9a70d4d43896faabc","latency_seconds":3.228587,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"Product A uses 3 labour hours and 2 machine hours and earns $40 contribution. 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B uses 2 labour and 4 machine hours and earns $50. Weekly limits are 120 labour and 160 machine hours. Products must be whole units. Maximize contribution; if tied choose more A. Return A, B and contribution. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. These fields require exact integer-valued numbers (1 and 1.0 are equivalent): a, b. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: a, b, contribution. Numeric fields must be finite numbers. 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Calculate the monthly payment and total interest, without intermediate rounding. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: interest, payment. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"5c4864aed0f9f056f75492ca7845d0d4f702013e270f01f80fe43f8e367f48d3","repetition":2,"request_hash":"73e922de08f313569f1240234e4e683ca4710a5812b97cd50cf35910a848276f","request_id":"req_b7837355f52d4ccc96dec46d1b98f099","sent_at":"2026-10-05T05:46:35.570663+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-loan","trial_id":"74bd504280d04ab898f0d85fc0c4d31e16a1317294288ad7461a65e9353c8822","usage":{"input_tokens":149,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":395,"output_tokens_details":{"reasoning_tokens":368},"total_tokens":544}},{"answer":"{\"interest\":1557.1600003002552,\"payment\":564.8816666791773}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.005058,"finished_at":"2026-10-05T05:51:34.479528+00:00","grade":{"checks":[{"field":"interest","passed":true},{"field":"payment","passed":true}],"passed":true,"reason":"correct"},"input_hash":"dd46d1e73914a3f87735654a3473b6b07e7aee3f72a7a1b43b9138c9395d9bd7","latency_seconds":12.397981,"model_id":"gpt-6.1-sol","output_cap_reached":false,"prompt":"A $12000 loan charges a nominal annual rate of 12%, compounded monthly, repaid with 24 equal end-of-month payments. Calculate the monthly payment and total interest, without intermediate rounding. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: interest, payment. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"284f2906657a37c4284365a36d635a094e81e4d85b6da97e32d84d38f1c85ed4","repetition":3,"request_hash":"73e922de08f313569f1240234e4e683ca4710a5812b97cd50cf35910a848276f","request_id":"req_92e5db5b81414f73ace27d70c6cd628e","sent_at":"2026-10-05T05:51:22.081547+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-loan","trial_id":"9395fc5b9006c07e0385a2d4613c05175c3dcc75536ab13cf9e2a77fce372ffa","usage":{"input_tokens":149,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":476,"output_tokens_details":{"reasoning_tokens":448},"total_tokens":625}},{"answer":"Revenue: 210×30 = 6300; 30×24 = 720; total 7020.\n\nCost of goods: 240×18 = 4320.\n\nFees: 3% of 7020 = 210.6; per item fee 0.20×240 sold items = 48 (all 240 are sold). Total fees = 258.6.\n\nShipping: 2.50×240 = 600.\n\nProfit = 7020 − 4320 − 258.6 − 600 = 1841.4.\n\nMargin = 1841.4/7020 = 0.262307692...×100 = 26.2307692307...%\n\nCheck: 7020×0.2623 = 1841.346; remainder 0.054/7020 = 0.00000769, so 0.26230769. Margin is 26.230769230769...% (repeating 230769).\n\n{\"margin_percent\": 26.230769230769, \"profit\": 1841.4, \"revenue\": 7020}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.003482,"finished_at":"2026-10-05T05:44:08.308467+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"3f50c3f6adda92d493ef987eda30e04af127872c995655637c55ee5836c4fe0c","latency_seconds":2.837064,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"A shop buys 240 items at $18 each. It sells 210 at $30 and 30 at $24. Platform fees are 3% of revenue plus $0.20 per sold item. Shipping costs $2.50 per sold item. Calculate revenue, profit after these costs, and profit margin as a percentage of revenue. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: margin_percent, profit, revenue. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"06fe67254284bbc4e1d5703f21992d5940a201e4eb4cf3310e62b7c2d1a507bc","repetition":1,"request_hash":"1f9e6630013712352e18c9e12f133764cc272b7d693bb019837770e89c905617","request_id":"req_011CfiZMfYZbJBxf9ctDMucG","sent_at":"2026-10-05T05:44:05.471403+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-margin","trial_id":"d951609af55a7c1134cc682e31004b5d5ff05a3e7e55ef8f660d617415b378c0","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":281,"output_tokens":292,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"Revenue: 210×30 = 6300; 30×24 = 720; total 7020.\n\nCost of goods: 240×18 = 4320.\n\nPlatform fees: 3% of 7020 = 210.60; plus 0.20×240 sold items = 48 (all 240 are sold). Total 258.60.\n\nShipping: 2.50×240 = 600.\n\nProfit: 7020 − 4320 − 258.60 − 600 = 1841.40.\n\nMargin: 1841.40/7020 = 0.262307692307... → 26.2307692307...%\n\nCheck: 7020×0.2623 = 1841.346; the remainder is 0.054/7020 = 0.0000076923, so 0.2623076923 and the margin is 26.23076923076923%.\n\n{\"margin_percent\": 26.230769230769, \"profit\": 1841.4, \"revenue\": 7020}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.003412,"finished_at":"2026-10-05T05:49:17.510620+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"3f50c3f6adda92d493ef987eda30e04af127872c995655637c55ee5836c4fe0c","latency_seconds":2.634334,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"A shop buys 240 items at $18 each. It sells 210 at $30 and 30 at $24. Platform fees are 3% of revenue plus $0.20 per sold item. Shipping costs $2.50 per sold item. Calculate revenue, profit after these costs, and profit margin as a percentage of revenue. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: margin_percent, profit, revenue. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"d82817736937403c7f57d1336aef2aac520f4cd531820d05f67d5aeb7ff27981","repetition":2,"request_hash":"1f9e6630013712352e18c9e12f133764cc272b7d693bb019837770e89c905617","request_id":"req_011CfiZkUE4jTYHV4KYLnss1","sent_at":"2026-10-05T05:49:14.876286+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-margin","trial_id":"7c05350474057fa64c068ac226f33e8c3b38fd53fde3a9a1f1de824bedd6c9c0","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":281,"output_tokens":285,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"Revenue: 210×30 = 6300; 30×24 = 720; total 7020.\n\nCost of goods: 240×18 = 4320.\n\nFees: 3% × 7020 = 210.6; per item 0.20×240 = 48 (sold items: 210+30 = 240, so all items are sold).\n\nShipping: 2.50×240 = 600.\n\nProfit = 7020 − 4320 − 210.6 − 48 − 600 = 1841.4.\n\nMargin = 1841.4/7020 × 100 = 26.230769230...% (1841.4/7020 = 0.26230769...; check: 7020×0.2623 = 1841.346, remainder 0.054/7020 = 0.0000076923, so 0.2623076923). Margin percent = 26.23076923076923.\n\n{\"margin_percent\": 26.23076923076923, \"profit\": 1841.4, \"revenue\": 7020}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.003542,"finished_at":"2026-10-05T05:52:45.111400+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"3f50c3f6adda92d493ef987eda30e04af127872c995655637c55ee5836c4fe0c","latency_seconds":2.761921,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"A shop buys 240 items at $18 each. It sells 210 at $30 and 30 at $24. Platform fees are 3% of revenue plus $0.20 per sold item. Shipping costs $2.50 per sold item. Calculate revenue, profit after these costs, and profit margin as a percentage of revenue. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: margin_percent, profit, revenue. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"a73795fc8a5800e14211e4d4d6c7fb39327499c4b959869e5eb2cd15a79f42d2","repetition":3,"request_hash":"1f9e6630013712352e18c9e12f133764cc272b7d693bb019837770e89c905617","request_id":"req_011Cfia1mHQ6r2yUtxu2v8cH","sent_at":"2026-10-05T05:52:42.349479+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-margin","trial_id":"9cb3bb24117a2f5157daba817cf9fbc85a6f015b697b049a670172425a967820","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":281,"output_tokens":298,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"{\"margin_percent\": 26.23076923076923, \"profit\": 1841.4, \"revenue\": 7020}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.004471,"finished_at":"2026-10-05T05:44:12.194757+00:00","grade":{"checks":[{"field":"margin_percent","passed":true},{"field":"profit","passed":true},{"field":"revenue","passed":true}],"passed":true,"reason":"correct"},"input_hash":"3f50c3f6adda92d493ef987eda30e04af127872c995655637c55ee5836c4fe0c","latency_seconds":3.865852,"model_id":"gemini-3.8-flash","output_cap_reached":false,"prompt":"A shop buys 240 items at $18 each. It sells 210 at $30 and 30 at $24. Platform fees are 3% of revenue plus $0.20 per sold item. Shipping costs $2.50 per sold item. Calculate revenue, profit after these costs, and profit margin as a percentage of revenue. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: margin_percent, profit, revenue. Numeric fields must be finite numbers. 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It sells 210 at $30 and 30 at $24. Platform fees are 3% of revenue plus $0.20 per sold item. Shipping costs $2.50 per sold item. Calculate revenue, profit after these costs, and profit margin as a percentage of revenue. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: margin_percent, profit, revenue. Numeric fields must be finite numbers. 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It sells 210 at $30 and 30 at $24. Platform fees are 3% of revenue plus $0.20 per sold item. Shipping costs $2.50 per sold item. Calculate revenue, profit after these costs, and profit margin as a percentage of revenue. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: margin_percent, profit, revenue. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"b7958337656ccedb864631aac3965641507bd6ce873bcc17bf73f5398700d520","repetition":3,"request_hash":"932444e739500b4da36b2be20e96cc185ecfd256d313919f4c2b4df6dc0596c2","request_id":null,"sent_at":"2026-10-05T05:52:34.464418+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Gemini Developer API, paid Standard"},"state":"responded","task_id":"math-margin","trial_id":"0cbee9a5cdbeaab068c166bab4a6aee6ddec148dd4d00cad29bacf1d03e76775","usage":{"candidatesTokenCount":44,"promptTokenCount":196,"promptTokensDetails":[{"modality":"TEXT","tokenCount":196}],"serviceTier":"standard","thoughtsTokenCount":1372,"totalTokenCount":1612}},{"answer":"{\"margin_percent\":26.23076923076923,\"profit\":1841.4,\"revenue\":7020}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.001442,"finished_at":"2026-10-05T05:44:05.424312+00:00","grade":{"checks":[{"field":"margin_percent","passed":true},{"field":"profit","passed":true},{"field":"revenue","passed":true}],"passed":true,"reason":"correct"},"input_hash":"3f50c3f6adda92d493ef987eda30e04af127872c995655637c55ee5836c4fe0c","latency_seconds":3.343253,"model_id":"gpt-6.1-sol","output_cap_reached":false,"prompt":"A shop buys 240 items at $18 each. It sells 210 at $30 and 30 at $24. Platform fees are 3% of revenue plus $0.20 per sold item. Shipping costs $2.50 per sold item. Calculate revenue, profit after these costs, and profit margin as a percentage of revenue. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: margin_percent, profit, revenue. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"1948105292d5762be71e7ba87f44755fe24f4f22f39d028d9a83b085db5b0522","repetition":1,"request_hash":"8a4afd300b651e8ee22dd79feb5dadf9aa1e54e3ed3dcf13165abb7d9b7699f5","request_id":"req_688fbd36207e4b19a42f093676368a43","sent_at":"2026-10-05T05:44:02.081059+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-margin","trial_id":"db59bd761ecdd3ac0aac09cdb3d2e0935c1206198b5c39196b97543627f4a11a","usage":{"input_tokens":181,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":108,"output_tokens_details":{"reasoning_tokens":77},"total_tokens":289}},{"answer":"{\"margin_percent\":26.23076923076923,\"profit\":1841.4,\"revenue\":7020}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.001822,"finished_at":"2026-10-05T05:49:24.817678+00:00","grade":{"checks":[{"field":"margin_percent","passed":true},{"field":"profit","passed":true},{"field":"revenue","passed":true}],"passed":true,"reason":"correct"},"input_hash":"3f50c3f6adda92d493ef987eda30e04af127872c995655637c55ee5836c4fe0c","latency_seconds":4.113846,"model_id":"gpt-6.1-sol","output_cap_reached":false,"prompt":"A shop buys 240 items at $18 each. 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Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"67500d0e84fa51511c0a12d4d62cb4a491921514b5b9b77a7de7f4d44cd74b7c","repetition":2,"request_hash":"8a4afd300b651e8ee22dd79feb5dadf9aa1e54e3ed3dcf13165abb7d9b7699f5","request_id":"req_875ad2dd41fc4bb98a56be4be840db33","sent_at":"2026-10-05T05:49:20.703832+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-margin","trial_id":"65ef3dac45fbc6ef5ebd818c2c42f936b66bd834b517b45e5b8aba64c7f2b79f","usage":{"input_tokens":181,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":146,"output_tokens_details":{"reasoning_tokens":115},"total_tokens":327}},{"answer":"{\"margin_percent\":26.23076923076923,\"profit\":1841.4,\"revenue\":7020}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.001452,"finished_at":"2026-10-05T05:52:42.322275+00:00","grade":{"checks":[{"field":"margin_percent","passed":true},{"field":"profit","passed":true},{"field":"revenue","passed":true}],"passed":true,"reason":"correct"},"input_hash":"3f50c3f6adda92d493ef987eda30e04af127872c995655637c55ee5836c4fe0c","latency_seconds":3.100744,"model_id":"gpt-6.1-sol","output_cap_reached":false,"prompt":"A shop buys 240 items at $18 each. It sells 210 at $30 and 30 at $24. Platform fees are 3% of revenue plus $0.20 per sold item. Shipping costs $2.50 per sold item. Calculate revenue, profit after these costs, and profit margin as a percentage of revenue. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: margin_percent, profit, revenue. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"364f397b68908cb7b776d8294f640934631279f3870d458d9e7a3f2192e758b7","repetition":3,"request_hash":"8a4afd300b651e8ee22dd79feb5dadf9aa1e54e3ed3dcf13165abb7d9b7699f5","request_id":"req_6a4ca5e37d09439594b7a1c1d43e5a27","sent_at":"2026-10-05T05:52:39.221531+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-margin","trial_id":"0455cba68d5554f7337ab76fbf6d3774616ae832fb3cccdc839db886eff14e5e","usage":{"input_tokens":181,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":109,"output_tokens_details":{"reasoning_tokens":78},"total_tokens":290}},{"answer":"Let x be litres of the 20% stock and y litres of the 60% stock.\n\nx + y = 80 and 0.20x + 0.60y = 0.35 × 80 = 28.\n\nSubstituting x = 80 − y: 16 + 0.40y = 28, so y = 30 and x = 50.\n\n{\"litres_20\": 50, \"litres_60\": 30}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.001638,"finished_at":"2026-10-05T05:41:59.262595+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"fb11dee229c1c2b0315b3909ef0d51d873e4826a8790c8458080025875f0e8b3","latency_seconds":1.797313,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"Make exactly 80 litres of a 35% concentrate from 20% and 60% stocks, with additive volumes. How many litres of each stock are needed? Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: litres_20, litres_60. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"0c06cb2128f954a8dd41616321bbc59dab298852c491ef9752c5d506d1df3fb0","repetition":1,"request_hash":"6e4321feda2d0a681339494c0216140cfef4289db236549015f628d57201d56b","request_id":"req_011CfiZCEAa4caUXseRHq3gF","sent_at":"2026-10-05T05:41:57.465282+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-mix","trial_id":"101f707c948cd72b80f81885b39a9078e75329621d8d335b2f8cd072803b5492","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":224,"output_tokens":119,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"Let x be the litres of the 20% stock and y the litres of the 60% stock.\n\nVolume: x + y = 80.\nConcentrate: 0.20x + 0.60y = 0.35 × 80 = 28.\n\nSubstitute x = 80 − y: 0.20(80 − y) + 0.60y = 28, so 16 + 0.40y = 28, giving y = 30 and x = 50.\n\n{\"litres_20\": 50, \"litres_60\": 30}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.001978,"finished_at":"2026-10-05T05:47:15.392835+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"fb11dee229c1c2b0315b3909ef0d51d873e4826a8790c8458080025875f0e8b3","latency_seconds":1.801083,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"Make exactly 80 litres of a 35% concentrate from 20% and 60% stocks, with additive volumes. How many litres of each stock are needed? Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: litres_20, litres_60. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"8f8a3789a701f3f7aaaf6e954897fc5866e65b12934181cdce4479610519fe8c","repetition":2,"request_hash":"6e4321feda2d0a681339494c0216140cfef4289db236549015f628d57201d56b","request_id":"req_011CfiZbXkPmkEqzQGsm5uHJ","sent_at":"2026-10-05T05:47:13.591752+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-mix","trial_id":"d231912e366bfb9c6680722b63ea27589588c9b0a2f88815bf2dcec5a366d0b5","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":224,"output_tokens":153,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"Let x be the litres of the 20% stock and y the litres of the 60% stock.\n\nVolume: x + y = 80.\nConcentrate: 0.20x + 0.60y = 0.35 × 80 = 28.\n\nSubstitute x = 80 − y:\n0.20(80 − y) + 0.60y = 28\n16 + 0.40y = 28\ny = 30, so x = 50.\n\n{\"litres_20\": 50, \"litres_60\": 30}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.001938,"finished_at":"2026-10-05T05:54:33.070212+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"fb11dee229c1c2b0315b3909ef0d51d873e4826a8790c8458080025875f0e8b3","latency_seconds":1.956099,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"Make exactly 80 litres of a 35% concentrate from 20% and 60% stocks, with additive volumes. How many litres of each stock are needed? Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: litres_20, litres_60. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"86433a1fabc271198626589c2373682742024d67bdf89bbb980f17540403bf04","repetition":3,"request_hash":"6e4321feda2d0a681339494c0216140cfef4289db236549015f628d57201d56b","request_id":"req_011Cfia9nMNJqAKU1wKx5322","sent_at":"2026-10-05T05:54:31.114113+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-mix","trial_id":"be1cb80186cb9e95f56517f2902259438419e97d2b3497bc189463c0d294161e","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":224,"output_tokens":149,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"{\"litres_20\": 50, \"litres_60\": 30}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.001854,"finished_at":"2026-10-05T05:42:01.575985+00:00","grade":{"checks":[{"field":"litres_20","passed":true},{"field":"litres_60","passed":true}],"passed":true,"reason":"correct"},"input_hash":"fb11dee229c1c2b0315b3909ef0d51d873e4826a8790c8458080025875f0e8b3","latency_seconds":2.289042,"model_id":"gemini-3.8-flash","output_cap_reached":false,"prompt":"Make exactly 80 litres of a 35% concentrate from 20% and 60% stocks, with additive volumes. How many litres of each stock are needed? Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: litres_20, litres_60. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"95a9ce56f05f6f909eadf8f72b0b6c9d9d84fc52c7814fb921dbdbc2faf22bf0","repetition":1,"request_hash":"3b7b4c7377c5290d36868b80c2e591cebe56bed9ba0a0a4e13ac941d27adff72","request_id":null,"sent_at":"2026-10-05T05:41:59.286943+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Gemini Developer API, paid Standard"},"state":"responded","task_id":"math-mix","trial_id":"81b85cc2021f74107cd2d1fdfe14a75cd122e611db136e7495443d91fb5d0fb9","usage":{"candidatesTokenCount":22,"promptTokenCount":157,"promptTokensDetails":[{"modality":"TEXT","tokenCount":157}],"serviceTier":"standard","thoughtsTokenCount":441,"totalTokenCount":620}},{"answer":"{\"litres_20\": 50, \"litres_60\": 30}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.001434,"finished_at":"2026-10-05T05:47:17.193130+00:00","grade":{"checks":[{"field":"litres_20","passed":true},{"field":"litres_60","passed":true}],"passed":true,"reason":"correct"},"input_hash":"fb11dee229c1c2b0315b3909ef0d51d873e4826a8790c8458080025875f0e8b3","latency_seconds":1.787829,"model_id":"gemini-3.8-flash","output_cap_reached":false,"prompt":"Make exactly 80 litres of a 35% concentrate from 20% and 60% stocks, with additive volumes. How many litres of each stock are needed? Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: litres_20, litres_60. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"30866187d95a8c3f073929ddfc4508b6a172d14609b14f729d0be220d4b8b1cd","repetition":2,"request_hash":"3b7b4c7377c5290d36868b80c2e591cebe56bed9ba0a0a4e13ac941d27adff72","request_id":null,"sent_at":"2026-10-05T05:47:15.405301+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Gemini Developer API, paid Standard"},"state":"responded","task_id":"math-mix","trial_id":"6505f3d92220d92a344ec4da949c3fcff8bb0d2e45b1eb667e3f2db61a3308ff","usage":{"candidatesTokenCount":22,"promptTokenCount":157,"promptTokensDetails":[{"modality":"TEXT","tokenCount":157}],"serviceTier":"standard","thoughtsTokenCount":329,"totalTokenCount":508}},{"answer":"{\"litres_20\": 50, \"litres_60\": 30}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.001678,"finished_at":"2026-10-05T05:54:27.527599+00:00","grade":{"checks":[{"field":"litres_20","passed":true},{"field":"litres_60","passed":true}],"passed":true,"reason":"correct"},"input_hash":"fb11dee229c1c2b0315b3909ef0d51d873e4826a8790c8458080025875f0e8b3","latency_seconds":1.657897,"model_id":"gemini-3.8-flash","output_cap_reached":false,"prompt":"Make exactly 80 litres of a 35% concentrate from 20% and 60% stocks, with additive volumes. How many litres of each stock are needed? Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: litres_20, litres_60. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"56a77c8bb8c5d1d70949d30e844a562ad674383fe8a2bfd68769b7ed3e6a9400","repetition":3,"request_hash":"3b7b4c7377c5290d36868b80c2e591cebe56bed9ba0a0a4e13ac941d27adff72","request_id":null,"sent_at":"2026-10-05T05:54:25.869702+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Gemini Developer API, paid Standard"},"state":"responded","task_id":"math-mix","trial_id":"ee92c66b04e3534f0c78cc56a76e2e17047be41cbc848ed1ca2a1fa9cb7c70a3","usage":{"candidatesTokenCount":22,"promptTokenCount":157,"promptTokensDetails":[{"modality":"TEXT","tokenCount":157}],"serviceTier":"standard","thoughtsTokenCount":394,"totalTokenCount":573}},{"answer":"{\"litres_20\":50,\"litres_60\":30}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.000634,"finished_at":"2026-10-05T05:41:57.448746+00:00","grade":{"checks":[{"field":"litres_20","passed":true},{"field":"litres_60","passed":true}],"passed":true,"reason":"correct"},"input_hash":"fb11dee229c1c2b0315b3909ef0d51d873e4826a8790c8458080025875f0e8b3","latency_seconds":2.06311,"model_id":"gpt-6.1-sol","output_cap_reached":false,"prompt":"Make exactly 80 litres of a 35% concentrate from 20% and 60% stocks, with additive volumes. How many litres of each stock are needed? Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: litres_20, litres_60. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"e3f3fec296c2ef3259010a3f081eff9255a570c399e635f709b23e76dd5a999f","repetition":1,"request_hash":"3142e3186b1c8d66758eede4e2365caeaded9b8db15707df96d52a5a60e66c52","request_id":"req_c21f462d9d264b4fa5a5d81e18c0ecb3","sent_at":"2026-10-05T05:41:55.385636+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-mix","trial_id":"775a88e183a390a160b2a4d9f136344b9198c0fdd7dfeac722daea6afaffb4de","usage":{"input_tokens":147,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":34,"output_tokens_details":{"reasoning_tokens":13},"total_tokens":181}},{"answer":"{\"litres_20\":50,\"litres_60\":30}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.000674,"finished_at":"2026-10-05T05:47:19.564576+00:00","grade":{"checks":[{"field":"litres_20","passed":true},{"field":"litres_60","passed":true}],"passed":true,"reason":"correct"},"input_hash":"fb11dee229c1c2b0315b3909ef0d51d873e4826a8790c8458080025875f0e8b3","latency_seconds":2.348922,"model_id":"gpt-6.1-sol","output_cap_reached":false,"prompt":"Make exactly 80 litres of a 35% concentrate from 20% and 60% stocks, with additive volumes. How many litres of each stock are needed? Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: litres_20, litres_60. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"ce0344cb85d6d09077c202ea5ec3114b1b6e012ba188ca2c405598bc63930265","repetition":2,"request_hash":"3142e3186b1c8d66758eede4e2365caeaded9b8db15707df96d52a5a60e66c52","request_id":"req_6d64020a982346a18336085d931dad6c","sent_at":"2026-10-05T05:47:17.215654+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-mix","trial_id":"a8f61c3c3893a4dcea756a91bb5bb6334a8c71e119e5ed65ededf673f0bb11e5","usage":{"input_tokens":147,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":38,"output_tokens_details":{"reasoning_tokens":17},"total_tokens":185}},{"answer":"{\"litres_20\":50,\"litres_60\":30}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.000904,"finished_at":"2026-10-05T05:54:31.090730+00:00","grade":{"checks":[{"field":"litres_20","passed":true},{"field":"litres_60","passed":true}],"passed":true,"reason":"correct"},"input_hash":"fb11dee229c1c2b0315b3909ef0d51d873e4826a8790c8458080025875f0e8b3","latency_seconds":3.550422,"model_id":"gpt-6.1-sol","output_cap_reached":false,"prompt":"Make exactly 80 litres of a 35% concentrate from 20% and 60% stocks, with additive volumes. How many litres of each stock are needed? Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: litres_20, litres_60. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"b23b65f6536c4ac3d153a052c80f2757b42884784b3faae6246fc9f57bf2d62c","repetition":3,"request_hash":"3142e3186b1c8d66758eede4e2365caeaded9b8db15707df96d52a5a60e66c52","request_id":"req_a24a78a4a5d74f8fb4aed0a74fcb4e90","sent_at":"2026-10-05T05:54:27.540308+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-mix","trial_id":"290eafdc68cb2add2fafcbffae1b6807ad09df3b12e06c7beed7244e7bf21932","usage":{"input_tokens":147,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":61,"output_tokens_details":{"reasoning_tokens":40},"total_tokens":208}},{"answer":"NPV = -1000 + 400 × (1/1.1 + 1/1.21 + 1/1.331)\n\nThe annuity factor is 1/1.1 + 1/1.21 + 1/1.331 = (1.21×1.1 + 1.1×1.1... let me compute over the common denominator 1.331:\n1.21 + 1.1 + 1 = 3.31, so the factor is 3.31/1.331 = 2.486851990984...\n\nThen 400 × 3.31/1.331 = 1324/1.331 = 994.7407963936...\n\nCheck the division: 1.331 × 994.74 = 1323.98..., since 1.331×994 = 1323.014 and 1.331×0.74 = 0.98494, the total is 1323.99894. The remainder is 1324 − 1323.99894 = 0.00106, and 0.00106/1.331 = 0.000796394. So the value is 994.740796394...\n\nMore digits: 0.00106/1.331 = 0.00079639368...\n1.331 × 0.000796 = 0.001059476; the remainder is 0.000000524, and 0.000000524/1.331 = 3.9369e-7. So the value is 0.00079639369, and 1324/1.331 = 994.74079639369.\n\nNPV = -5.25920360631\n\nCheck: 1000 − 994.74079639369 = 5.25920360631.\n\n```json\n{\"npv\": -5.25920360631}\n```","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.005116,"finished_at":"2026-10-05T05:43:58.005836+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"0b0a34c37f4e55e2cbf712c6699c04373d6c02a0b2982671f234008ec7ac43a1","latency_seconds":4.658371,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"Pay $1000 now, then receive $400 at each year end for three years. Use a 10% annual discount rate. Return net present value without rounding intermediate values. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: npv. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"db0899da9a9aa65d2226c14d1b4b8a89c8969c3a01e08d141042f95c4ea426ae","repetition":1,"request_hash":"dc73ca65bba189f7801d108c7145ba16699e7f36212f9245ad6f12a9fa35bfa0","request_id":"req_011CfiZLme92m8rCaoKtAXbN","sent_at":"2026-10-05T05:43:53.347465+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-npv","trial_id":"ec86fcbd1683da3ef05e8262b33d5a372eeb7b010e32dc471aacdf0b2519e87c","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":218,"output_tokens":468,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"NPV = -1000 + 400 × (1/1.1 + 1/1.21 + 1/1.331)\n\nThe annuity factor is (1 − 1.1^-3)/0.1. Since 1.1^3 = 1.331, 1/1.331 = 0.7513148009015778.\n\nThe annuity factor is (1 − 0.7513148009015778)/0.1 = 2.486851990984222.\n\nMultiplying by 400 gives 994.7407963936888.\n\nSo NPV = -5.2592036063112.\n\nExact check: 1/1.331 = 1000/1331. The annuity factor is 10 × (1 − 1000/1331) = 10 × 331/1331 = 3310/1331. Then 400 × 3310/1331 = 1324000/1331 = 994.74079639369...\n\nDivision check: 1331 × 994 = 1,323,014. The remainder is 986, and 986/1331 = 0.74079639...\n- 1331 × 0.74 = 984.94, remainder 1.06\n- 1.06/1331 = 0.000796393688...\n\nSo 994.740796393688, and NPV = -5.259203606312.\n\n{\"npv\": -5.259203606312}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.004306,"finished_at":"2026-10-05T05:45:56.926909+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"0b0a34c37f4e55e2cbf712c6699c04373d6c02a0b2982671f234008ec7ac43a1","latency_seconds":3.925314,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"Pay $1000 now, then receive $400 at each year end for three years. Use a 10% annual discount rate. Return net present value without rounding intermediate values. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: npv. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"a6849c3141d7dd615b56f3c75cfb1a6d6ce360d2581088b46cdb18ec07bcb2e0","repetition":2,"request_hash":"dc73ca65bba189f7801d108c7145ba16699e7f36212f9245ad6f12a9fa35bfa0","request_id":"req_011CfiZVb9c9yT94VjUUhm31","sent_at":"2026-10-05T05:45:53.001595+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-npv","trial_id":"005ab1abdf51ec14817ef997d984540a69f6bf21cbfb6af186a1e4a3f3729d52","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":218,"output_tokens":387,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"NPV = -1000 + 400 × (1/1.1 + 1/1.21 + 1/1.331)\n\nThe annuity factor is 1/1.1 + 1/1.21 + 1/1.331 = (1.21×1.1 + 1.1×1.1 ... ) computed over the common denominator 1.331:\n- 1/1.1 = 1.21/1.331\n- 1/1.21 = 1.1/1.331\n- 1/1.331 = 1/1.331\n\nThe sum is 3.31/1.331 = 2.486851990984222...\n\nCheck: 1.331 × 2.486851991 = 3.31000000. Good.\n\nMultiplying by 400 gives 994.7407963936889, since 3.31 × 400 = 1324 and 1324/1.331 = 994.74079639368...\n\nLong division of 1324/1.331 (that is, 1324000/1331):\n- 1331 × 994 = 1,323,014, remainder 986.\n- 9860/1331 = 7 (9317), remainder 543.\n- 5430/1331 = 4 (5324), remainder 106.\n- 1060/1331 = 0, remainder 1060.\n- 10600/1331 = 7 (9317), remainder 1283.\n- 12830/1331 = 9 (11979), remainder 851.\n- 8510/1331 = 6 (7986), remainder 524.\n- 5240/1331 = 3 (3993), remainder 1247.\n- 12470/1331 = 9 (11979), remainder 491.\n- 4910/1331 = 3 (3993), remainder 917.\n- 9170/1331 = 6 (7986), remainder 1184.\n- 11840/1331 = 8 (10648), remainder 1192.\n\nThis gives 994.74079639368..., so NPV = -5.259203606311...\n\nSubtracting 1000 from 994.74079639368 gives -5.259203606320, using digits 994.740796393680. The digits so far are 994.74079639368, so NPV = -5.25920360632 (approximately).\n\n```json\n{\"npv\": -5.25920360632}\n```","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.007276,"finished_at":"2026-10-05T05:52:59.707604+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"0b0a34c37f4e55e2cbf712c6699c04373d6c02a0b2982671f234008ec7ac43a1","latency_seconds":5.87701,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"Pay $1000 now, then receive $400 at each year end for three years. Use a 10% annual discount rate. Return net present value without rounding intermediate values. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: npv. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"a6bbfd608b7e9da752b071724f9b708821ccf9912ad8ac78f989932a566ee4b3","repetition":3,"request_hash":"dc73ca65bba189f7801d108c7145ba16699e7f36212f9245ad6f12a9fa35bfa0","request_id":"req_011Cfia2cSrmEheReZwwXjnG","sent_at":"2026-10-05T05:52:53.830594+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-npv","trial_id":"dffbdc7c5c1445f0cbe64d42f36313e84de8e2a1ba79c6aa2a64e12b413fea7b","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":218,"output_tokens":684,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"{\"npv\": -5.259196093163035}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.004476,"finished_at":"2026-10-05T05:44:02.059855+00:00","grade":{"checks":[{"field":"npv","passed":false}],"passed":false,"reason":"answer_mismatch"},"input_hash":"0b0a34c37f4e55e2cbf712c6699c04373d6c02a0b2982671f234008ec7ac43a1","latency_seconds":4.02746,"model_id":"gemini-3.8-flash","output_cap_reached":false,"prompt":"Pay $1000 now, then receive $400 at each year end for three years. Use a 10% annual discount rate. Return net present value without rounding intermediate values. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: npv. Numeric fields must be finite numbers. 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Use a 10% annual discount rate. Return net present value without rounding intermediate values. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: npv. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"49f05413a0cd82f2345bd1ee8b41f33e835555624bd6b6ac82cafcddfefef123","repetition":2,"request_hash":"b2113e057db8b8c0281c4242cd858abe02235fde0a4fc89b0a9ec8d6aff7c43a","request_id":null,"sent_at":"2026-10-05T05:45:56.942913+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Gemini Developer API, paid Standard"},"state":"responded","task_id":"math-npv","trial_id":"243af3d34bf8def6903b5bc257ffc2b8cdc2d68e8ba2b384d1616a7629e264e8","usage":{"candidatesTokenCount":23,"promptTokenCount":153,"promptTokensDetails":[{"modality":"TEXT","tokenCount":153}],"serviceTier":"standard","thoughtsTokenCount":1063,"totalTokenCount":1239}},{"answer":"{\"npv\": -5.259203606311044}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.003989,"finished_at":"2026-10-05T05:52:49.055754+00:00","grade":{"checks":[{"field":"npv","passed":true}],"passed":true,"reason":"correct"},"input_hash":"0b0a34c37f4e55e2cbf712c6699c04373d6c02a0b2982671f234008ec7ac43a1","latency_seconds":3.930244,"model_id":"gemini-3.8-flash","output_cap_reached":false,"prompt":"Pay $1000 now, then receive $400 at each year end for three years. Use a 10% annual discount rate. Return net present value without rounding intermediate values. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: npv. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"d5ccfe44e6b8c7931b930ea3b59b4f8ac866307ce33d3fdaf6721410b98ca53c","repetition":3,"request_hash":"b2113e057db8b8c0281c4242cd858abe02235fde0a4fc89b0a9ec8d6aff7c43a","request_id":null,"sent_at":"2026-10-05T05:52:45.125510+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Gemini Developer API, paid Standard"},"state":"responded","task_id":"math-npv","trial_id":"26ed5771136258553fb3ca550b6366ef6676024b06157387f2ea0a25a254bf0b","usage":{"candidatesTokenCount":23,"promptTokenCount":153,"promptTokensDetails":[{"modality":"TEXT","tokenCount":153}],"serviceTier":"standard","thoughtsTokenCount":1010,"totalTokenCount":1186}},{"answer":"{\"npv\":-5.259203606311044}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.001768,"finished_at":"2026-10-05T05:43:53.333291+00:00","grade":{"checks":[{"field":"npv","passed":true}],"passed":true,"reason":"correct"},"input_hash":"0b0a34c37f4e55e2cbf712c6699c04373d6c02a0b2982671f234008ec7ac43a1","latency_seconds":6.321913,"model_id":"gpt-6.1-sol","output_cap_reached":false,"prompt":"Pay $1000 now, then receive $400 at each year end for three years. Use a 10% annual discount rate. Return net present value without rounding intermediate values. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: npv. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"67d1f640293f38fce23533dbb2472486ba6e9fe7a11d3e7b899d2ac82b80b1ef","repetition":1,"request_hash":"aece3bff68244af6830ee7c0be351b071a6e81a0c72de6962f551fc17c074ca4","request_id":"req_cc42731b02cd44eaac6a6d8f65fcb2ba","sent_at":"2026-10-05T05:43:47.011378+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-npv","trial_id":"f55029be6a607bca12d1285d6141f6e271ad15d8511fbcbd8282dcaa8fd8c9a3","usage":{"input_tokens":144,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":148,"output_tokens_details":{"reasoning_tokens":129},"total_tokens":292}},{"answer":"{\"npv\":-5.259203606311045}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.001558,"finished_at":"2026-10-05T05:46:05.406321+00:00","grade":{"checks":[{"field":"npv","passed":true}],"passed":true,"reason":"correct"},"input_hash":"0b0a34c37f4e55e2cbf712c6699c04373d6c02a0b2982671f234008ec7ac43a1","latency_seconds":4.283193,"model_id":"gpt-6.1-sol","output_cap_reached":false,"prompt":"Pay $1000 now, then receive $400 at each year end for three years. Use a 10% annual discount rate. Return net present value without rounding intermediate values. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: npv. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"e8b312103347023f721b2df377f2ab5d916c93a31578e854a12fc8b8e40169e4","repetition":2,"request_hash":"aece3bff68244af6830ee7c0be351b071a6e81a0c72de6962f551fc17c074ca4","request_id":"req_6e42714db6e84eaeb1c534c5c32fc35f","sent_at":"2026-10-05T05:46:01.123128+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-npv","trial_id":"0b81451555639b694b5fb48c917ce82bec2eb4bf0abc0e4b8131e3a7e988c49c","usage":{"input_tokens":144,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":127,"output_tokens_details":{"reasoning_tokens":108},"total_tokens":271}},{"answer":"{\"npv\":-5.259203606311045}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.001798,"finished_at":"2026-10-05T05:52:53.817915+00:00","grade":{"checks":[{"field":"npv","passed":true}],"passed":true,"reason":"correct"},"input_hash":"0b0a34c37f4e55e2cbf712c6699c04373d6c02a0b2982671f234008ec7ac43a1","latency_seconds":4.738992,"model_id":"gpt-6.1-sol","output_cap_reached":false,"prompt":"Pay $1000 now, then receive $400 at each year end for three years. Use a 10% annual discount rate. Return net present value without rounding intermediate values. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: npv. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"d2d22912f330acee0b5df171bbd05972336033cffc19caf72e1b2103a9ebcc3e","repetition":3,"request_hash":"aece3bff68244af6830ee7c0be351b071a6e81a0c72de6962f551fc17c074ca4","request_id":"req_0a2eb006801447d4848e2ebbfb773682","sent_at":"2026-10-05T05:52:49.078923+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-npv","trial_id":"4d79236bae4a76e0117d8e62dfa3f1f1acacb2fa555da3a49d345db7f92a5279","usage":{"input_tokens":144,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":151,"output_tokens_details":{"reasoning_tokens":132},"total_tokens":295}},{"answer":"p = 0.98^3 = 0.941192\nq = 1 - p = 0.058808\n\nSuccess probability with one retry: 1 - q^2.\nq^2 = 0.058808^2 = 0.003458620864 (check: 0.058808^2 = 0.058808 × 0.058808. 58808^2 = 58808×58808. 58808×58000 = 3,410,864,000; 58808×808 = 47,516,864; total 3,458,380,864. Hmm, recompute: 58808×58000 = 58808×58×1000 = 3,410,864×1000 = 3,410,864,000. 58808×808 = 58808×800 = 47,046,400 plus 58808×8 = 470,464, giving 47,516,864. Sum = 3,458,380,864.)\nSo q^2 = 3,458,380,864 × 10^-12 = 0.003458380864.\n\nSuccess probability = 1 - 0.003458380864 = 0.996541619136.\n\nExpected number of attempts = 1 + q = 1.058808 (the second attempt happens only if the first fails).\n\n{\"expected_attempts\": 1.058808, \"success_probability\": 0.996541619136}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.004332,"finished_at":"2026-10-05T05:44:46.953528+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"615401fb820259e3fd133d0353e1b4b23b9b91d2cec100347a56015b51a3c070","latency_seconds":3.522207,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"Each of three independent services succeeds with probability 0.98. A workflow requires all three. It retries the complete workflow once if the first attempt fails; attempts are independent. Calculate eventual success probability and expected number of complete attempts. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: expected_attempts, success_probability. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"b7b9c2f63b14f92d82d153e56c8e6f4833eaa3792470ba6dc8e669f51247510c","repetition":1,"request_hash":"d828731f86541d95fcbda6cdf09edc72e55aa7a57e76a042fb7bf65a4b3264fc","request_id":"req_011CfiZQTsAaeDmB2JFLWDiy","sent_at":"2026-10-05T05:44:43.431321+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-queue","trial_id":"5ba39eb4c12059db5d8196f14a9d551adc4c285c0ff8f2c99151436230b6075d","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":251,"output_tokens":383,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"p = 0.98^3 = 0.941192\nq = 1 - p = 0.058808\n\nSuccess probability with one retry: 1 - q^2 = p + q·p = 0.941192 + 0.058808·0.941192.\nq^2 = 0.058808^2 = 0.003458380864 (check: 0.058808^2 = 0.0034583809 approx; exact: 58808^2 = 3,458,380,864, so q^2 = 3,458,380,864e-12 = 0.003458380864).\nSuccess = 0.996541619136.\n\nExpected number of complete attempts = 1 + q = 1.058808 (the second attempt happens only if the first fails).\n\n{\"expected_attempts\": 1.058808, \"success_probability\": 0.996541619136}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.002972,"finished_at":"2026-10-05T05:47:56.650610+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"615401fb820259e3fd133d0353e1b4b23b9b91d2cec100347a56015b51a3c070","latency_seconds":2.624248,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"Each of three independent services succeeds with probability 0.98. A workflow requires all three. It retries the complete workflow once if the first attempt fails; attempts are independent. Calculate eventual success probability and expected number of complete attempts. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: expected_attempts, success_probability. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"6ed789b79ab8d94a08986450212418c08035229d9771154679725b03c6e66fa1","repetition":2,"request_hash":"d828731f86541d95fcbda6cdf09edc72e55aa7a57e76a042fb7bf65a4b3264fc","request_id":"req_011CfiZeWZnGBV6u5VvsFbEx","sent_at":"2026-10-05T05:47:54.026362+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-queue","trial_id":"141ddb119fad6f040da6040d7158a28ae7b433fe7bce0b0834872f45bf6df779","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":251,"output_tokens":247,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"p = 0.98^3 = 0.941192\nq = 1 - p = 0.058808\n\nSuccess probability with one retry: 1 - q^2.\nq^2 = 0.058808^2 = 0.003458620864 (check: 0.058808^2 = 58808^2 × 10^-12; 58808^2 = 58808×58808 = 58808×58000 = 3,410,864,000; 58808×808 = 47,516,864; total 3,458,380,864. So q^2 = 0.003458380864.)\n\nRecheck: 58808×58000 = 58808×58 ×1000 = 3,410,864×1000 = 3,410,864,000. Correct. 58808×808 = 58808×800 = 47,046,400, plus 58808×8 = 470,464, total 47,516,864. Sum = 3,458,380,864. So q^2 = 0.003458380864.\n\nSuccess probability = 1 - 0.003458380864 = 0.996541619136.\n\nExpected number of attempts = 1 + q = 1.058808 (the second attempt happens only if the first fails).\n\n{\"expected_attempts\": 1.058808, \"success_probability\": 0.996541619136}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.004322,"finished_at":"2026-10-05T05:54:16.476240+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"615401fb820259e3fd133d0353e1b4b23b9b91d2cec100347a56015b51a3c070","latency_seconds":3.613292,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"Each of three independent services succeeds with probability 0.98. A workflow requires all three. It retries the complete workflow once if the first attempt fails; attempts are independent. Calculate eventual success probability and expected number of complete attempts. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: expected_attempts, success_probability. Numeric fields must be finite numbers. 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A workflow requires all three. It retries the complete workflow once if the first attempt fails; attempts are independent. Calculate eventual success probability and expected number of complete attempts. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: expected_attempts, success_probability. Numeric fields must be finite numbers. 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A workflow requires all three. It retries the complete workflow once if the first attempt fails; attempts are independent. Calculate eventual success probability and expected number of complete attempts. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: expected_attempts, success_probability. Numeric fields must be finite numbers. 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A workflow requires all three. It retries the complete workflow once if the first attempt fails; attempts are independent. Calculate eventual success probability and expected number of complete attempts. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: expected_attempts, success_probability. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"9a5073c868675d4b588ff744ab1bcfbb4f88f0d6b3f8515e6d3131b09be21c44","repetition":3,"request_hash":"7b51bf36d1f8cc02106b4e46cf5fdb1c6084af960e6d3ed83f1abfec27ac2e3a","request_id":null,"sent_at":"2026-10-05T05:54:04.557350+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Gemini Developer API, paid Standard"},"state":"responded","task_id":"math-queue","trial_id":"4a7c1094b793d57518229fd82f8b2d251eba5054d134c609cc2290c5366a9592","usage":{"candidatesTokenCount":36,"promptTokenCount":165,"promptTokensDetails":[{"modality":"TEXT","tokenCount":165}],"serviceTier":"standard","thoughtsTokenCount":964,"totalTokenCount":1165}},{"answer":"{\"expected_attempts\":1.058808,\"success_probability\":0.996541619136}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.001276,"finished_at":"2026-10-05T05:44:43.404196+00:00","grade":{"checks":[{"field":"expected_attempts","passed":true},{"field":"success_probability","passed":true}],"passed":true,"reason":"correct"},"input_hash":"615401fb820259e3fd133d0353e1b4b23b9b91d2cec100347a56015b51a3c070","latency_seconds":3.193968,"model_id":"gpt-6.1-sol","output_cap_reached":false,"prompt":"Each of three independent services succeeds with probability 0.98. A workflow requires all three. It retries the complete workflow once if the first attempt fails; attempts are independent. Calculate eventual success probability and expected number of complete attempts. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: expected_attempts, success_probability. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"e04c90fde6e57632df835f21b9ff2e72a0394ec2de93c97a05720734af8490dd","repetition":1,"request_hash":"a05bc75f8402ac9b6681fd7b4b6a29cd0005ac8e15eb4237e9a01d49e4db68d5","request_id":"req_1d1a3fe19cfd4491806cbdc5675e2707","sent_at":"2026-10-05T05:44:40.210228+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-queue","trial_id":"8bb3db4b065d74a077e8824603cbc8595714088052494b50490d066147776aae","usage":{"input_tokens":158,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":96,"output_tokens_details":{"reasoning_tokens":70},"total_tokens":254}},{"answer":"{\"expected_attempts\":1.058808,\"success_probability\":0.996541619136}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.001296,"finished_at":"2026-10-05T05:48:07.080907+00:00","grade":{"checks":[{"field":"expected_attempts","passed":true},{"field":"success_probability","passed":true}],"passed":true,"reason":"correct"},"input_hash":"615401fb820259e3fd133d0353e1b4b23b9b91d2cec100347a56015b51a3c070","latency_seconds":6.741261,"model_id":"gpt-6.1-sol","output_cap_reached":false,"prompt":"Each of three independent services succeeds with probability 0.98. A workflow requires all three. It retries the complete workflow once if the first attempt fails; attempts are independent. Calculate eventual success probability and expected number of complete attempts. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: expected_attempts, success_probability. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"41279bc54bf5896903ef039f31ac2179af614fc5c23a5dc05b6b0376de41b7a5","repetition":2,"request_hash":"a05bc75f8402ac9b6681fd7b4b6a29cd0005ac8e15eb4237e9a01d49e4db68d5","request_id":"req_45bdbf84515f443ea987a59df4bfeb7b","sent_at":"2026-10-05T05:48:00.339646+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-queue","trial_id":"d894fb282579db408f26918e299365229ddb80d5f1b62417b19db8523756cd6a","usage":{"input_tokens":158,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":98,"output_tokens_details":{"reasoning_tokens":72},"total_tokens":256}},{"answer":"{\"expected_attempts\":1.058808,\"success_probability\":0.996541619136}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.001256,"finished_at":"2026-10-05T05:54:12.836912+00:00","grade":{"checks":[{"field":"expected_attempts","passed":true},{"field":"success_probability","passed":true}],"passed":true,"reason":"correct"},"input_hash":"615401fb820259e3fd133d0353e1b4b23b9b91d2cec100347a56015b51a3c070","latency_seconds":4.961586,"model_id":"gpt-6.1-sol","output_cap_reached":false,"prompt":"Each of three independent services succeeds with probability 0.98. A workflow requires all three. It retries the complete workflow once if the first attempt fails; attempts are independent. Calculate eventual success probability and expected number of complete attempts. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: expected_attempts, success_probability. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"628b929a8115faf4e7cbea25ee2ef1f68c8e0d83dba07742c6e3f762e4491140","repetition":3,"request_hash":"a05bc75f8402ac9b6681fd7b4b6a29cd0005ac8e15eb4237e9a01d49e4db68d5","request_id":"req_cf0e6d8878d94867b3926c9226050e0e","sent_at":"2026-10-05T05:54:07.875326+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"OpenAI direct API, Standard global"},"state":"responded","task_id":"math-queue","trial_id":"76f5e6d30682b7159d84192d72b31dcd5ce1753f027ad7506223b4ebefbf0aa3","usage":{"input_tokens":158,"input_tokens_details":{"cache_write_tokens":0,"cached_tokens":0},"output_tokens":94,"output_tokens_details":{"reasoning_tokens":68},"total_tokens":252}},{"answer":"Mean = (4+7+7+10+12)/5 = 40/5 = 8.\n\nDeviations: -4, -1, -1, 2, 4. Squares: 16, 1, 1, 4, 16, sum = 38.\n\nSample variance = 38/4 = 9.5.\n\nStandard error = sqrt(9.5/5) = sqrt(1.9) ≈ 1.378404875209022.\n\nCheck: 1.3784^2 = 1.89998... and 1.378404875^2 ≈ 1.9. Good.\n\n{\"mean\": 8, \"sample_variance\": 9.5, \"standard_error\": 1.378404875209022}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.002576,"finished_at":"2026-10-05T05:44:37.908064+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"f58eb3e082bfb42cb37914e7c0d238de465097b2e65e0911ea26181d64397f31","latency_seconds":2.154747,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"For observations 4, 7, 7, 10, 12 calculate the arithmetic mean, unbiased sample variance (denominator n-1), and standard error of the mean. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: mean, sample_variance, standard_error. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"f68cff406f4dd427e63be9162ccabcb16cde9ef9636ca3580f15b5fd4fd6967a","repetition":1,"request_hash":"dbad1666e2193be7c5492b5c34fcf6574d7f2b139ba726327cfea8f6b05d2050","request_id":"req_011CfiZPtwmMuZAfS25pYCkk","sent_at":"2026-10-05T05:44:35.753317+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-variance","trial_id":"e3049313e34cdc6d94ab9b9a1d06b231cce4e52b11b9d85814ea5828d8628eba","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":233,"output_tokens":211,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"Mean = (4+7+7+10+12)/5 = 40/5 = 8.\n\nThe squared deviations from the mean are 16, 1, 1, 4, 16, which sum to 38.\n\nSample variance = 38/4 = 9.5.\n\nStandard error = sqrt(9.5/5) = sqrt(1.9) ≈ 1.378404875209022.\n\nCheck: 1.3784² = 1.89998..., which is consistent with sqrt(1.9). The more precise value is 1.37840487520902.\n\n{\"mean\": 8, \"sample_variance\": 9.5, \"standard_error\": 1.378404875209022}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.002526,"finished_at":"2026-10-05T05:49:53.563474+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"f58eb3e082bfb42cb37914e7c0d238de465097b2e65e0911ea26181d64397f31","latency_seconds":2.286563,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"For observations 4, 7, 7, 10, 12 calculate the arithmetic mean, unbiased sample variance (denominator n-1), and standard error of the mean. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: mean, sample_variance, standard_error. Numeric fields must be finite numbers. Use each requested key once, no extra keys, no numeric strings, booleans, Markdown or explanation.","raw_response_hash":"c42497e968b3ac7118c2868416d3713626ff3c4fe3a25fc677f0bd3ab8b1205d","repetition":2,"request_hash":"dbad1666e2193be7c5492b5c34fcf6574d7f2b139ba726327cfea8f6b05d2050","request_id":"req_011CfiZo9ynZ3nRqYU91gC5x","sent_at":"2026-10-05T05:49:51.276911+00:00","settings":{"effort":"medium","max_output_tokens":8192,"surface":"Claude direct API, Standard global"},"state":"responded","task_id":"math-variance","trial_id":"e176007f4e560e409351f3898d7e1fd02eccaaefb5845ccad201a7117bd216c5","usage":{"cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":0},"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"inference_geo":"global","input_tokens":233,"output_tokens":206,"output_tokens_details":{"thinking_tokens":0},"service_tier":"standard"}},{"answer":"Mean = (4+7+7+10+12)/5 = 40/5 = 8.\n\nDeviations: -4, -1, -1, 2, 4. Squares: 16, 1, 1, 4, 16. Sum = 38.\n\nSample variance = 38/4 = 9.5.\n\nStandard error = sqrt(9.5/5) = sqrt(1.9) ≈ 1.378404875209022.\n\nCheck: 1.3784² = 1.89998656; the remainder is 0.00001344, and dividing by 2·1.3784 gives 0.00000487, so the value is about 1.37840487. This agrees with the figure above.\n\n{\"mean\": 8, \"sample_variance\": 9.5, \"standard_error\": 1.378404875209022}","billed_net_usd":null,"enabled_tools":[],"estimated_net_usd":0.002916,"finished_at":"2026-10-05T05:50:21.440600+00:00","grade":{"checks":[],"passed":false,"reason":"invalid_artifact"},"input_hash":"f58eb3e082bfb42cb37914e7c0d238de465097b2e65e0911ea26181d64397f31","latency_seconds":2.694416,"model_id":"claude-sonnet-5-5","output_cap_reached":false,"prompt":"For observations 4, 7, 7, 10, 12 calculate the arithmetic mean, unbiased sample variance (denominator n-1), and standard error of the mean. Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: mean, sample_variance, standard_error. Numeric fields must be finite numbers. 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Return exact values when representable as terminating decimals. For nonterminating values, provide at least 10 significant digits; do not round intermediate values or final answers to cents or two decimal places. All other numeric fields use absolute tolerance 0.000001 or relative tolerance 0.00000001. Return exactly one JSON object with these keys: mean, sample_variance, standard_error. Numeric fields must be finite numbers. 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