[ { "family_id": "magnitude_estimation_001202", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10100000000000000 * 9800000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 9800000000000000?", "options": [ "98980000000000000000000000000000", "98980000000000000000000000000020", "98979999999999999999999999999990", "98980000000000000000000000000010" ], "answer": "98980000000000000000000000000000", "correct_value": 98980000000000000000000000000000, "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9800000000000000≈10000000000000000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10100000000000000 * 5001825339182240", "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 5001825339182240?", "options": [ "50518435925740623999999999999990", "50518435925740624000000000000010", "50518435925740624000000000000000", "50518435925740624000000000000020" ], "answer": "50518435925740624000000000000000", "correct_value": 50518435925740624000000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5155500612349868 * 6723920597574282", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5155500612349868 * 6723920597574282?", "options": [ "34665176758186101218092702894786", "34665176758186101218092702894796", "34665176758186101218092702894776", "34665176758186101218092702894766" ], "answer": "34665176758186101218092702894776", "correct_value": 34665176758186101218092702894776, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001203", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10200000000000000 * 9900000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 9900000000000000?", "options": [ "100979999999999999999999999999990", "100980000000000000000000000000020", "100980000000000000000000000000010", "100980000000000000000000000000000" ], "answer": "100980000000000000000000000000000", "correct_value": 100980000000000000000000000000000, "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 9900000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10200000000000000 * 5152184862388030", "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 5152184862388030?", "options": [ "52552285596357906000000000000000", "52552285596357905999999999999990", "52552285596357906000000000000010", "52552285596357906000000000000020" ], "answer": "52552285596357906000000000000000", "correct_value": 52552285596357906000000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5552630215968205 * 6748424206165293", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5552630215968205 * 6748424206165293?", "options": [ "37471504157324653254658262509055", "37471504157324653254658262509065", "37471504157324653254658262509085", "37471504157324653254658262509075" ], "answer": "37471504157324653254658262509065", "correct_value": 37471504157324653254658262509065, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001204", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9900000000000000 * 10100000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.99, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 10100000000000000?", "options": [ "99990000000000000000000000000000", "99990000000000000000000000000020", "99990000000000000000000000000010", "99989999999999999999999999999990" ], "answer": "99990000000000000000000000000000", "correct_value": 99990000000000000000000000000000, "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 10100000000000000≈10000000000000000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9900000000000000 * 4842111086770157", "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 4842111086770157?", "options": [ "47936899759024554300000000000010", "47936899759024554299999999999990", "47936899759024554300000000000020", "47936899759024554300000000000000" ], "answer": "47936899759024554300000000000000", "correct_value": 47936899759024554300000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "5356067257309484 * 5112654749188299", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5356067257309484 * 5112654749188299?", "options": [ "27383722700055280443823634527726", "27383722700055280443823634527716", "27383722700055280443823634527706", "27383722700055280443823634527736" ], "answer": "27383722700055280443823634527716", "correct_value": 27383722700055280443823634527716, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001205", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9800000000000000 * 10100000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 10100000000000000?", "options": [ "98980000000000000000000000000020", "98979999999999999999999999999990", "98980000000000000000000000000000", "98980000000000000000000000000010" ], "answer": "98980000000000000000000000000000", "correct_value": 98980000000000000000000000000000, "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10100000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10200000000000000 * 5120910310714107", "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 5120910310714107?", "options": [ "52233285169283891400000000000010", "52233285169283891400000000000020", "52233285169283891400000000000000", "52233285169283891399999999999990" ], "answer": "52233285169283891400000000000000", "correct_value": 52233285169283891400000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5812476740312957 * 4132091761466112", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5812476740312957 * 4132091761466112?", "options": [ "24017687252360571339628030013204", "24017687252360571339628030013194", "24017687252360571339628030013174", "24017687252360571339628030013184" ], "answer": "24017687252360571339628030013184", "correct_value": 24017687252360571339628030013184, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001206", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10100000000000000 * 9800000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 9800000000000000?", "options": [ "98980000000000000000000000000020", "98980000000000000000000000000000", "98980000000000000000000000000010", "98979999999999999999999999999990" ], "answer": "98980000000000000000000000000000", "correct_value": 98980000000000000000000000000000, "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9800000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9900000000000000 * 5103161774968639", "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 5103161774968639?", "options": [ "50521301572189526100000000000010", "50521301572189526099999999999990", "50521301572189526100000000000000", "50521301572189526100000000000020" ], "answer": "50521301572189526100000000000000", "correct_value": 50521301572189526100000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4152024030649896 * 5280123037233386", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 4152024030649896 * 5280123037233386?", "options": [ "21923197735381134231671408627876", "21923197735381134231671408627866", "21923197735381134231671408627846", "21923197735381134231671408627856" ], "answer": "21923197735381134231671408627856", "correct_value": 21923197735381134231671408627856, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001207", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9800000000000000 * 9800000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.98, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 9800000000000000?", "options": [ "96040000000000000000000000000000", "96039999999999999999999999999990", "96040000000000000000000000000020", "96040000000000000000000000000010" ], "answer": "96040000000000000000000000000000", "correct_value": 96040000000000000000000000000000, "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 9800000000000000≈10000000000000000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10100000000000000 * 5066956740204353", "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 5066956740204353?", "options": [ "51176263076063965300000000000010", "51176263076063965299999999999990", "51176263076063965300000000000000", "51176263076063965300000000000020" ], "answer": "51176263076063965300000000000000", "correct_value": 51176263076063965300000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6760244877354416 * 5958069320222046", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 6760244877354416 * 5958069320222046?", "options": [ "40278007600953594070154158655136", "40278007600953594070154158655146", "40278007600953594070154158655126", "40278007600953594070154158655156" ], "answer": "40278007600953594070154158655136", "correct_value": 40278007600953594070154158655136, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001208", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10100000000000000 * 9900000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.99, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 9900000000000000?", "options": [ "99990000000000000000000000000020", "99990000000000000000000000000000", "99989999999999999999999999999990", "99990000000000000000000000000010" ], "answer": "99990000000000000000000000000000", "correct_value": 99990000000000000000000000000000, "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9900000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9800000000000000 * 5073883247960602", "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 5073883247960602?", "options": [ "49724055830013899600000000000000", "49724055830013899600000000000010", "49724055830013899599999999999990", "49724055830013899600000000000020" ], "answer": "49724055830013899600000000000000", "correct_value": 49724055830013899600000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5516961977124086 * 5045411496661417", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5516961977124086 * 5045411496661417?", "options": [ "27835343386025764962916137589852", "27835343386025764962916137589872", "27835343386025764962916137589882", "27835343386025764962916137589862" ], "answer": "27835343386025764962916137589862", "correct_value": 27835343386025764962916137589862, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001209", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10200000000000000 * 10200000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.98, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 10200000000000000?", "options": [ "104039999999999999999999999999990", "104040000000000000000000000000010", "104040000000000000000000000000020", "104040000000000000000000000000000" ], "answer": "104040000000000000000000000000000", "correct_value": 104040000000000000000000000000000, "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 10200000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10100000000000000 * 4854921440655604", "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 4854921440655604?", "options": [ "49034706550621600400000000000020", "49034706550621600400000000000000", "49034706550621600399999999999990", "49034706550621600400000000000010" ], "answer": "49034706550621600400000000000000", "correct_value": 49034706550621600400000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6404689492378734 * 4383592137668171", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 6404689492378734 * 4383592137668171?", "options": [ "28075546502997367571226749075514", "28075546502997367571226749075524", "28075546502997367571226749075534", "28075546502997367571226749075504" ], "answer": "28075546502997367571226749075514", "correct_value": 28075546502997367571226749075514, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001210", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9900000000000000 * 9800000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 9800000000000000?", "options": [ "97020000000000000000000000000000", "97020000000000000000000000000010", "97020000000000000000000000000020", "97019999999999999999999999999990" ], "answer": "97020000000000000000000000000000", "correct_value": 97020000000000000000000000000000, "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 9800000000000000≈10000000000000000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10100000000000000 * 5129201688292491", "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 5129201688292491?", "options": [ "51804937051754159100000000000000", "51804937051754159100000000000010", "51804937051754159100000000000020", "51804937051754159099999999999990" ], "answer": "51804937051754159100000000000000", "correct_value": 51804937051754159100000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6891108841929046 * 4030300335533585", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 6891108841929046 * 4030300335533585?", "options": [ "27773238277825088451451120009900", "27773238277825088451451120009920", "27773238277825088451451120009910", "27773238277825088451451120009930" ], "answer": "27773238277825088451451120009910", "correct_value": 27773238277825088451451120009910, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001211", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10200000000000000 * 10100000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 10100000000000000?", "options": [ "103020000000000000000000000000010", "103019999999999999999999999999990", "103020000000000000000000000000020", "103020000000000000000000000000000" ], "answer": "103020000000000000000000000000000", "correct_value": 103020000000000000000000000000000, "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 10100000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9800000000000000 * 4906744557477605", "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 4906744557477605?", "options": [ "48086096663280529000000000000020", "48086096663280528999999999999990", "48086096663280529000000000000000", "48086096663280529000000000000010" ], "answer": "48086096663280529000000000000000", "correct_value": 48086096663280529000000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5822628859159496 * 4057342524693544", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5822628859159496 * 4057342524693544?", "options": [ "23624399675775679328704817493814", "23624399675775679328704817493844", "23624399675775679328704817493834", "23624399675775679328704817493824" ], "answer": "23624399675775679328704817493824", "correct_value": 23624399675775679328704817493824, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001212", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10200000000000000 * 9900000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 9900000000000000?", "options": [ "100980000000000000000000000000020", "100979999999999999999999999999990", "100980000000000000000000000000000", "100980000000000000000000000000010" ], "answer": "100980000000000000000000000000000", "correct_value": 100980000000000000000000000000000, "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 9900000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10200000000000000 * 4993047752579264", "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 4993047752579264?", "options": [ "50929087076308492800000000000010", "50929087076308492800000000000000", "50929087076308492800000000000020", "50929087076308492799999999999990" ], "answer": "50929087076308492800000000000000", "correct_value": 50929087076308492800000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4202610641184814 * 6629331028845125", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 4202610641184814 * 6629331028845125?", "options": [ "27860497125761193450740107931760", "27860497125761193450740107931740", "27860497125761193450740107931770", "27860497125761193450740107931750" ], "answer": "27860497125761193450740107931750", "correct_value": 27860497125761193450740107931750, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001213", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10200000000000000 * 9900000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 9900000000000000?", "options": [ "100980000000000000000000000000020", "100980000000000000000000000000000", "100979999999999999999999999999990", "100980000000000000000000000000010" ], "answer": "100980000000000000000000000000000", "correct_value": 100980000000000000000000000000000, "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 9900000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10200000000000000 * 4806801361212255", "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 4806801361212255?", "options": [ "49029373884365001000000000000020", "49029373884365000999999999999990", "49029373884365001000000000000000", "49029373884365001000000000000010" ], "answer": "49029373884365001000000000000000", "correct_value": 49029373884365001000000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5290781332893331 * 6742394815088882", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5290781332893331 * 6742394815088882?", "options": [ "35672536626669039108908990045932", "35672536626669039108908990045962", "35672536626669039108908990045952", "35672536626669039108908990045942" ], "answer": "35672536626669039108908990045942", "correct_value": 35672536626669039108908990045942, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001214", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10200000000000000 * 9800000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.98, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 9800000000000000?", "options": [ "99960000000000000000000000000000", "99960000000000000000000000000010", "99960000000000000000000000000020", "99959999999999999999999999999990" ], "answer": "99960000000000000000000000000000", "correct_value": 99960000000000000000000000000000, "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 9800000000000000≈10000000000000000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9900000000000000 * 4897509021467765", "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 4897509021467765?", "options": [ "48485339312530873499999999999990", "48485339312530873500000000000020", "48485339312530873500000000000000", "48485339312530873500000000000010" ], "answer": "48485339312530873500000000000000", "correct_value": 48485339312530873500000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5862411509202670 * 4343246576561364", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5862411509202670 * 4343246576561364?", "options": [ "25461898717738435742009967641900", "25461898717738435742009967641870", "25461898717738435742009967641890", "25461898717738435742009967641880" ], "answer": "25461898717738435742009967641880", "correct_value": 25461898717738435742009967641880, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001215", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10100000000000000 * 9800000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 9800000000000000?", "options": [ "98980000000000000000000000000010", "98980000000000000000000000000000", "98980000000000000000000000000020", "98979999999999999999999999999990" ], "answer": "98980000000000000000000000000000", "correct_value": 98980000000000000000000000000000, "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9800000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10200000000000000 * 5092523396627801", "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 5092523396627801?", "options": [ "51943738645603570200000000000010", "51943738645603570200000000000020", "51943738645603570200000000000000", "51943738645603570199999999999990" ], "answer": "51943738645603570200000000000000", "correct_value": 51943738645603570200000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5727248378323872 * 4166809038465592", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5727248378323872 * 4166809038465592?", "options": [ "23864350308337314167715704212214", "23864350308337314167715704212234", "23864350308337314167715704212244", "23864350308337314167715704212224" ], "answer": "23864350308337314167715704212224", "correct_value": 23864350308337314167715704212224, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001216", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9800000000000000 * 10100000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 10100000000000000?", "options": [ "98980000000000000000000000000010", "98980000000000000000000000000000", "98979999999999999999999999999990", "98980000000000000000000000000020" ], "answer": "98980000000000000000000000000000", "correct_value": 98980000000000000000000000000000, "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10100000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10100000000000000 * 5053602910249821", "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 5053602910249821?", "options": [ "51041389393523192100000000000000", "51041389393523192099999999999990", "51041389393523192100000000000010", "51041389393523192100000000000020" ], "answer": "51041389393523192100000000000000", "correct_value": 51041389393523192100000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5647170076319609 * 6712414650260605", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5647170076319609 * 6712414650260605?", "options": [ "37906147152801042291611121703465", "37906147152801042291611121703435", "37906147152801042291611121703455", "37906147152801042291611121703445" ], "answer": "37906147152801042291611121703445", "correct_value": 37906147152801042291611121703445, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001217", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10200000000000000 * 10200000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.98, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 10200000000000000?", "options": [ "104040000000000000000000000000020", "104040000000000000000000000000010", "104039999999999999999999999999990", "104040000000000000000000000000000" ], "answer": "104040000000000000000000000000000", "correct_value": 104040000000000000000000000000000, "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 10200000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10200000000000000 * 4822535086594521", "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 4822535086594521?", "options": [ "49189857883264114200000000000010", "49189857883264114200000000000020", "49189857883264114199999999999990", "49189857883264114200000000000000" ], "answer": "49189857883264114200000000000000", "correct_value": 49189857883264114200000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "4527955898884954 * 5244305882721060", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 4527955898884954 * 5244305882721060?", "options": [ "23745985757223889383949412931250", "23745985757223889383949412931260", "23745985757223889383949412931230", "23745985757223889383949412931240" ], "answer": "23745985757223889383949412931240", "correct_value": 23745985757223889383949412931240, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001218", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9900000000000000 * 9800000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 9800000000000000?", "options": [ "97020000000000000000000000000010", "97019999999999999999999999999990", "97020000000000000000000000000020", "97020000000000000000000000000000" ], "answer": "97020000000000000000000000000000", "correct_value": 97020000000000000000000000000000, "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 9800000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10200000000000000 * 5044852726630192", "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 5044852726630192?", "options": [ "51457497811627958400000000000010", "51457497811627958399999999999990", "51457497811627958400000000000000", "51457497811627958400000000000020" ], "answer": "51457497811627958400000000000000", "correct_value": 51457497811627958400000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6313687022006600 * 6449904410416486", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 6313687022006600 * 6449904410416486?", "options": [ "40722677769229698642153440807600", "40722677769229698642153440807610", "40722677769229698642153440807620", "40722677769229698642153440807590" ], "answer": "40722677769229698642153440807600", "correct_value": 40722677769229698642153440807600, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001219", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9800000000000000 * 10200000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.98, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 10200000000000000?", "options": [ "99960000000000000000000000000020", "99960000000000000000000000000010", "99959999999999999999999999999990", "99960000000000000000000000000000" ], "answer": "99960000000000000000000000000000", "correct_value": 99960000000000000000000000000000, "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10200000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9800000000000000 * 4980221007934411", "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 4980221007934411?", "options": [ "48806165877757227800000000000000", "48806165877757227799999999999990", "48806165877757227800000000000010", "48806165877757227800000000000020" ], "answer": "48806165877757227800000000000000", "correct_value": 48806165877757227800000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4930653271891821 * 4917490342333151", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 4930653271891821 * 4917490342333151?", "options": [ "24246439845921381904476814057971", "24246439845921381904476814057991", "24246439845921381904476814057961", "24246439845921381904476814057981" ], "answer": "24246439845921381904476814057971", "correct_value": 24246439845921381904476814057971, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001220", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9900000000000000 * 9900000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.99, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 9900000000000000?", "options": [ "98010000000000000000000000000020", "98010000000000000000000000000010", "98009999999999999999999999999990", "98010000000000000000000000000000" ], "answer": "98010000000000000000000000000000", "correct_value": 98010000000000000000000000000000, "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 9900000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10200000000000000 * 4800975125696054", "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 4800975125696054?", "options": [ "48969946282099750800000000000010", "48969946282099750800000000000020", "48969946282099750799999999999990", "48969946282099750800000000000000" ], "answer": "48969946282099750800000000000000", "correct_value": 48969946282099750800000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "5945567960139418 * 4191725138415944", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5945567960139418 * 4191725138415944?", "options": [ "24922186680676803734901914080602", "24922186680676803734901914080612", "24922186680676803734901914080592", "24922186680676803734901914080582" ], "answer": "24922186680676803734901914080592", "correct_value": 24922186680676803734901914080592, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001221", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9800000000000000 * 9800000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.98, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 9800000000000000?", "options": [ "96039999999999999999999999999990", "96040000000000000000000000000010", "96040000000000000000000000000020", "96040000000000000000000000000000" ], "answer": "96040000000000000000000000000000", "correct_value": 96040000000000000000000000000000, "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 9800000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10100000000000000 * 4945815364780484", "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 4945815364780484?", "options": [ "49952735184282888400000000000010", "49952735184282888399999999999990", "49952735184282888400000000000020", "49952735184282888400000000000000" ], "answer": "49952735184282888400000000000000", "correct_value": 49952735184282888400000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "5755494055002924 * 4451531537784221", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5755494055002924 * 4451531537784221?", "options": [ "25620763301375108116522636062194", "25620763301375108116522636062224", "25620763301375108116522636062214", "25620763301375108116522636062204" ], "answer": "25620763301375108116522636062204", "correct_value": 25620763301375108116522636062204, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001222", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10100000000000000 * 9800000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 9800000000000000?", "options": [ "98979999999999999999999999999990", "98980000000000000000000000000010", "98980000000000000000000000000000", "98980000000000000000000000000020" ], "answer": "98980000000000000000000000000000", "correct_value": 98980000000000000000000000000000, "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9800000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9900000000000000 * 5030392877535389", "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 5030392877535389?", "options": [ "49800889487600351100000000000010", "49800889487600351100000000000020", "49800889487600351099999999999990", "49800889487600351100000000000000" ], "answer": "49800889487600351100000000000000", "correct_value": 49800889487600351100000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "5329769258417834 * 6048338598006816", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5329769258417834 * 6048338598006816?", "options": [ "32236249124158749501022107956534", "32236249124158749501022107956564", "32236249124158749501022107956544", "32236249124158749501022107956554" ], "answer": "32236249124158749501022107956544", "correct_value": 32236249124158749501022107956544, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001223", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10100000000000000 * 9900000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.99, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 9900000000000000?", "options": [ "99990000000000000000000000000010", "99990000000000000000000000000000", "99990000000000000000000000000020", "99989999999999999999999999999990" ], "answer": "99990000000000000000000000000000", "correct_value": 99990000000000000000000000000000, "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9900000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10100000000000000 * 4923131470611700", "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 4923131470611700?", "options": [ "49723627853178170000000000000000", "49723627853178170000000000000020", "49723627853178169999999999999990", "49723627853178170000000000000010" ], "answer": "49723627853178170000000000000000", "correct_value": 49723627853178170000000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4619638249980770 * 6884881938188582", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 4619638249980770 * 6884881938188582?", "options": [ "31805663948257712840790133568160", "31805663948257712840790133568130", "31805663948257712840790133568140", "31805663948257712840790133568150" ], "answer": "31805663948257712840790133568140", "correct_value": 31805663948257712840790133568140, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001224", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9800000000000000 * 10100000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 10100000000000000?", "options": [ "98980000000000000000000000000010", "98980000000000000000000000000000", "98980000000000000000000000000020", "98979999999999999999999999999990" ], "answer": "98980000000000000000000000000000", "correct_value": 98980000000000000000000000000000, "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10100000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10100000000000000 * 5104882698796544", "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 5104882698796544?", "options": [ "51559315257845094400000000000020", "51559315257845094400000000000010", "51559315257845094399999999999990", "51559315257845094400000000000000" ], "answer": "51559315257845094400000000000000", "correct_value": 51559315257845094400000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "5745083024855275 * 5838254583182977", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5745083024855275 * 5838254583182977?", "options": [ "33541257300628030237114268653695", "33541257300628030237114268653675", "33541257300628030237114268653665", "33541257300628030237114268653685" ], "answer": "33541257300628030237114268653675", "correct_value": 33541257300628030237114268653675, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001225", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9900000000000000 * 10100000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.99, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 10100000000000000?", "options": [ "99990000000000000000000000000020", "99990000000000000000000000000010", "99990000000000000000000000000000", "99989999999999999999999999999990" ], "answer": "99990000000000000000000000000000", "correct_value": 99990000000000000000000000000000, "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 10100000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10200000000000000 * 4924573409349543", "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 4924573409349543?", "options": [ "50230648775365338599999999999990", "50230648775365338600000000000020", "50230648775365338600000000000010", "50230648775365338600000000000000" ], "answer": "50230648775365338600000000000000", "correct_value": 50230648775365338600000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "5356516177011934 * 4584296011508358", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5356516177011934 * 4584296011508358?", "options": [ "24555855745855806786308706744372", "24555855745855806786308706744362", "24555855745855806786308706744382", "24555855745855806786308706744392" ], "answer": "24555855745855806786308706744372", "correct_value": 24555855745855806786308706744372, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001226", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9800000000000000 * 9900000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 9900000000000000?", "options": [ "97020000000000000000000000000000", "97020000000000000000000000000010", "97020000000000000000000000000020", "97019999999999999999999999999990" ], "answer": "97020000000000000000000000000000", "correct_value": 97020000000000000000000000000000, "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 9900000000000000≈10000000000000000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9800000000000000 * 4886334506404108", "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 4886334506404108?", "options": [ "47886078162760258399999999999990", "47886078162760258400000000000010", "47886078162760258400000000000020", "47886078162760258400000000000000" ], "answer": "47886078162760258400000000000000", "correct_value": 47886078162760258400000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "5782567748443046 * 5954438778181177", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5782567748443046 * 5954438778181177?", "options": [ "34431945638789090503797453745162", "34431945638789090503797453745132", "34431945638789090503797453745152", "34431945638789090503797453745142" ], "answer": "34431945638789090503797453745142", "correct_value": 34431945638789090503797453745142, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001227", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9900000000000000 * 9800000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 9800000000000000?", "options": [ "97020000000000000000000000000020", "97020000000000000000000000000000", "97020000000000000000000000000010", "97019999999999999999999999999990" ], "answer": "97020000000000000000000000000000", "correct_value": 97020000000000000000000000000000, "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 9800000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10100000000000000 * 5164167204012696", "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 5164167204012696?", "options": [ "52158088760528229600000000000020", "52158088760528229600000000000000", "52158088760528229600000000000010", "52158088760528229599999999999990" ], "answer": "52158088760528229600000000000000", "correct_value": 52158088760528229600000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4532621438566095 * 4742685695159983", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 4532621438566095 * 4742685695159983?", "options": [ "21496798858262882444117144576395", "21496798858262882444117144576375", "21496798858262882444117144576405", "21496798858262882444117144576385" ], "answer": "21496798858262882444117144576385", "correct_value": 21496798858262882444117144576385, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001228", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9800000000000000 * 9900000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 9900000000000000?", "options": [ "97020000000000000000000000000010", "97019999999999999999999999999990", "97020000000000000000000000000000", "97020000000000000000000000000020" ], "answer": "97020000000000000000000000000000", "correct_value": 97020000000000000000000000000000, "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 9900000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10200000000000000 * 5126783907362554", "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 5126783907362554?", "options": [ "52293195855098050800000000000010", "52293195855098050800000000000000", "52293195855098050800000000000020", "52293195855098050799999999999990" ], "answer": "52293195855098050800000000000000", "correct_value": 52293195855098050800000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6105391114893111 * 6690701216482219", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 6105391114893111 * 6690701216482219?", "options": [ "40849347759515069075755617093319", "40849347759515069075755617093299", "40849347759515069075755617093309", "40849347759515069075755617093329" ], "answer": "40849347759515069075755617093309", "correct_value": 40849347759515069075755617093309, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001229", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9900000000000000 * 10200000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 10200000000000000?", "options": [ "100980000000000000000000000000020", "100980000000000000000000000000010", "100979999999999999999999999999990", "100980000000000000000000000000000" ], "answer": "100980000000000000000000000000000", "correct_value": 100980000000000000000000000000000, "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 10200000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10100000000000000 * 5074043480065477", "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 5074043480065477?", "options": [ "51247839148661317700000000000010", "51247839148661317699999999999990", "51247839148661317700000000000000", "51247839148661317700000000000020" ], "answer": "51247839148661317700000000000000", "correct_value": 51247839148661317700000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5329807492629888 * 5448721080235767", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5329807492629888 * 5448721080235767?", "options": [ "29040634438691008106753910804086", "29040634438691008106753910804096", "29040634438691008106753910804106", "29040634438691008106753910804116" ], "answer": "29040634438691008106753910804096", "correct_value": 29040634438691008106753910804096, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001230", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9800000000000000 * 10100000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 10100000000000000?", "options": [ "98980000000000000000000000000000", "98980000000000000000000000000010", "98979999999999999999999999999990", "98980000000000000000000000000020" ], "answer": "98980000000000000000000000000000", "correct_value": 98980000000000000000000000000000, "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10100000000000000≈10000000000000000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9800000000000000 * 4865539774409013", "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 4865539774409013?", "options": [ "47682289789208327400000000000020", "47682289789208327399999999999990", "47682289789208327400000000000010", "47682289789208327400000000000000" ], "answer": "47682289789208327400000000000000", "correct_value": 47682289789208327400000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6510477638092508 * 6840064374314141", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 6510477638092508 * 6840064374314141?", "options": [ "44532086152085437242818010555618", "44532086152085437242818010555638", "44532086152085437242818010555648", "44532086152085437242818010555628" ], "answer": "44532086152085437242818010555628", "correct_value": 44532086152085437242818010555628, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001231", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10100000000000000 * 10100000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.99, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 10100000000000000?", "options": [ "102010000000000000000000000000000", "102010000000000000000000000000020", "102009999999999999999999999999990", "102010000000000000000000000000010" ], "answer": "102010000000000000000000000000000", "correct_value": 102010000000000000000000000000000, "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 10100000000000000≈10000000000000000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9800000000000000 * 4871645584344437", "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 4871645584344437?", "options": [ "47742126726575482599999999999990", "47742126726575482600000000000010", "47742126726575482600000000000000", "47742126726575482600000000000020" ], "answer": "47742126726575482600000000000000", "correct_value": 47742126726575482600000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4555507416800086 * 6776608281358877", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 4555507416800086 * 6776608281358877?", "options": [ "30870889286479248144331130463442", "30870889286479248144331130463432", "30870889286479248144331130463422", "30870889286479248144331130463412" ], "answer": "30870889286479248144331130463422", "correct_value": 30870889286479248144331130463422, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001232", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10200000000000000 * 10200000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.98, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 10200000000000000?", "options": [ "104040000000000000000000000000020", "104040000000000000000000000000010", "104039999999999999999999999999990", "104040000000000000000000000000000" ], "answer": "104040000000000000000000000000000", "correct_value": 104040000000000000000000000000000, "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 10200000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9900000000000000 * 5016077832363676", "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 5016077832363676?", "options": [ "49659170540400392400000000000020", "49659170540400392400000000000010", "49659170540400392400000000000000", "49659170540400392399999999999990" ], "answer": "49659170540400392400000000000000", "correct_value": 49659170540400392400000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6016010535027170 * 5334562348016950", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 6016010535027170 * 5334562348016950?", "options": [ "32092783285429247617563870531510", "32092783285429247617563870531500", "32092783285429247617563870531520", "32092783285429247617563870531490" ], "answer": "32092783285429247617563870531500", "correct_value": 32092783285429247617563870531500, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001233", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10200000000000000 * 10200000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.98, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 10200000000000000?", "options": [ "104040000000000000000000000000010", "104040000000000000000000000000020", "104039999999999999999999999999990", "104040000000000000000000000000000" ], "answer": "104040000000000000000000000000000", "correct_value": 104040000000000000000000000000000, "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 10200000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9900000000000000 * 4883591223167229", "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 4883591223167229?", "options": [ "48347553109355567100000000000010", "48347553109355567099999999999990", "48347553109355567100000000000000", "48347553109355567100000000000020" ], "answer": "48347553109355567100000000000000", "correct_value": 48347553109355567100000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4305574056969059 * 5037627339029452", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 4305574056969059 * 5037627339029452?", "options": [ "21689877579603282862629853725658", "21689877579603282862629853725688", "21689877579603282862629853725678", "21689877579603282862629853725668" ], "answer": "21689877579603282862629853725668", "correct_value": 21689877579603282862629853725668, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001234", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9800000000000000 * 10100000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 10100000000000000?", "options": [ "98980000000000000000000000000020", "98980000000000000000000000000010", "98980000000000000000000000000000", "98979999999999999999999999999990" ], "answer": "98980000000000000000000000000000", "correct_value": 98980000000000000000000000000000, "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10100000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10100000000000000 * 4818275652129337", "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 4818275652129337?", "options": [ "48664584086506303700000000000000", "48664584086506303700000000000010", "48664584086506303699999999999990", "48664584086506303700000000000020" ], "answer": "48664584086506303700000000000000", "correct_value": 48664584086506303700000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5361084824088603 * 4942101398729140", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5361084824088603 * 4942101398729140?", "options": [ "26495024807833850350802957991420", "26495024807833850350802957991410", "26495024807833850350802957991440", "26495024807833850350802957991430" ], "answer": "26495024807833850350802957991420", "correct_value": 26495024807833850350802957991420, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001235", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10100000000000000 * 10200000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 10200000000000000?", "options": [ "103019999999999999999999999999990", "103020000000000000000000000000010", "103020000000000000000000000000000", "103020000000000000000000000000020" ], "answer": "103020000000000000000000000000000", "correct_value": 103020000000000000000000000000000, "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 10200000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10200000000000000 * 4953293039806411", "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 4953293039806411?", "options": [ "50523589006025392200000000000000", "50523589006025392200000000000010", "50523589006025392199999999999990", "50523589006025392200000000000020" ], "answer": "50523589006025392200000000000000", "correct_value": 50523589006025392200000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6105607364251042 * 4245060455227103", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 6105607364251042 * 4245060455227103?", "options": [ "25918672377125480835987614391326", "25918672377125480835987614391336", "25918672377125480835987614391316", "25918672377125480835987614391346" ], "answer": "25918672377125480835987614391326", "correct_value": 25918672377125480835987614391326, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001236", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9800000000000000 * 9800000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.98, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 9800000000000000?", "options": [ "96040000000000000000000000000000", "96040000000000000000000000000020", "96040000000000000000000000000010", "96039999999999999999999999999990" ], "answer": "96040000000000000000000000000000", "correct_value": 96040000000000000000000000000000, "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 9800000000000000≈10000000000000000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9800000000000000 * 4867979432292460", "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 4867979432292460?", "options": [ "47706198436466108000000000000010", "47706198436466108000000000000000", "47706198436466108000000000000020", "47706198436466107999999999999990" ], "answer": "47706198436466108000000000000000", "correct_value": 47706198436466108000000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4554192021933899 * 5058578672725689", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 4554192021933899 * 5058578672725689?", "options": [ "23037738633652304729407317231411", "23037738633652304729407317231431", "23037738633652304729407317231401", "23037738633652304729407317231421" ], "answer": "23037738633652304729407317231411", "correct_value": 23037738633652304729407317231411, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001237", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10100000000000000 * 10200000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 10200000000000000?", "options": [ "103020000000000000000000000000010", "103020000000000000000000000000000", "103020000000000000000000000000020", "103019999999999999999999999999990" ], "answer": "103020000000000000000000000000000", "correct_value": 103020000000000000000000000000000, "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 10200000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9800000000000000 * 4950420671806159", "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 4950420671806159?", "options": [ "48514122583700358200000000000020", "48514122583700358199999999999990", "48514122583700358200000000000000", "48514122583700358200000000000010" ], "answer": "48514122583700358200000000000000", "correct_value": 48514122583700358200000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6153933374897656 * 6134924508923020", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 6153933374897656 * 6134924508923020?", "options": [ "37753916687938985369851282441110", "37753916687938985369851282441130", "37753916687938985369851282441140", "37753916687938985369851282441120" ], "answer": "37753916687938985369851282441120", "correct_value": 37753916687938985369851282441120, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001238", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9900000000000000 * 9800000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 9800000000000000?", "options": [ "97020000000000000000000000000010", "97019999999999999999999999999990", "97020000000000000000000000000020", "97020000000000000000000000000000" ], "answer": "97020000000000000000000000000000", "correct_value": 97020000000000000000000000000000, "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 9800000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9800000000000000 * 4892875286646204", "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 4892875286646204?", "options": [ "47950177809132799200000000000010", "47950177809132799200000000000020", "47950177809132799199999999999990", "47950177809132799200000000000000" ], "answer": "47950177809132799200000000000000", "correct_value": 47950177809132799200000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "4977342321974566 * 4646971614284825", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 4977342321974566 * 4646971614284825?", "options": [ "23129568484794328158825929760960", "23129568484794328158825929760970", "23129568484794328158825929760950", "23129568484794328158825929760940" ], "answer": "23129568484794328158825929760950", "correct_value": 23129568484794328158825929760950, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001239", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9900000000000000 * 10200000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 10200000000000000?", "options": [ "100979999999999999999999999999990", "100980000000000000000000000000000", "100980000000000000000000000000010", "100980000000000000000000000000020" ], "answer": "100980000000000000000000000000000", "correct_value": 100980000000000000000000000000000, "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 10200000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9900000000000000 * 4981853464827535", "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 4981853464827535?", "options": [ "49320349301792596500000000000010", "49320349301792596500000000000000", "49320349301792596499999999999990", "49320349301792596500000000000020" ], "answer": "49320349301792596500000000000000", "correct_value": 49320349301792596500000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4737986033067062 * 5777948164299855", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 4737986033067062 * 5777948164299855?", "options": [ "27375837702238182973719491876030", "27375837702238182973719491876010", "27375837702238182973719491876020", "27375837702238182973719491876000" ], "answer": "27375837702238182973719491876010", "correct_value": 27375837702238182973719491876010, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001240", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10100000000000000 * 9900000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.99, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 9900000000000000?", "options": [ "99990000000000000000000000000020", "99990000000000000000000000000000", "99990000000000000000000000000010", "99989999999999999999999999999990" ], "answer": "99990000000000000000000000000000", "correct_value": 99990000000000000000000000000000, "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9900000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9900000000000000 * 4823594971824780", "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 4823594971824780?", "options": [ "47753590221065322000000000000020", "47753590221065322000000000000000", "47753590221065322000000000000010", "47753590221065321999999999999990" ], "answer": "47753590221065322000000000000000", "correct_value": 47753590221065322000000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4615464719270934 * 4097685519414294", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 4615464719270934 * 4097685519414294?", "options": [ "18912722945524065829810378330616", "18912722945524065829810378330586", "18912722945524065829810378330606", "18912722945524065829810378330596" ], "answer": "18912722945524065829810378330596", "correct_value": 18912722945524065829810378330596, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001241", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9800000000000000 * 10200000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.98, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 10200000000000000?", "options": [ "99959999999999999999999999999990", "99960000000000000000000000000020", "99960000000000000000000000000000", "99960000000000000000000000000010" ], "answer": "99960000000000000000000000000000", "correct_value": 99960000000000000000000000000000, "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10200000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10200000000000000 * 5000648950652713", "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 5000648950652713?", "options": [ "51006619296657672600000000000010", "51006619296657672600000000000000", "51006619296657672600000000000020", "51006619296657672599999999999990" ], "answer": "51006619296657672600000000000000", "correct_value": 51006619296657672600000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4317963557893896 * 6661580945381115", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 4317963557893896 * 6661580945381115?", "options": [ "28764463760116022606778452174030", "28764463760116022606778452174040", "28764463760116022606778452174060", "28764463760116022606778452174050" ], "answer": "28764463760116022606778452174040", "correct_value": 28764463760116022606778452174040, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001242", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10200000000000000 * 10100000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 10100000000000000?", "options": [ "103019999999999999999999999999990", "103020000000000000000000000000010", "103020000000000000000000000000000", "103020000000000000000000000000020" ], "answer": "103020000000000000000000000000000", "correct_value": 103020000000000000000000000000000, "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 10100000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9800000000000000 * 4905882100252471", "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 4905882100252471?", "options": [ "48077644582474215800000000000000", "48077644582474215799999999999990", "48077644582474215800000000000010", "48077644582474215800000000000020" ], "answer": "48077644582474215800000000000000", "correct_value": 48077644582474215800000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5457400268454454 * 6559819562684101", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5457400268454454 * 6559819562684101?", "options": [ "35799561042404991836279108435864", "35799561042404991836279108435874", "35799561042404991836279108435844", "35799561042404991836279108435854" ], "answer": "35799561042404991836279108435854", "correct_value": 35799561042404991836279108435854, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001243", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9800000000000000 * 9800000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.98, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 9800000000000000?", "options": [ "96039999999999999999999999999990", "96040000000000000000000000000010", "96040000000000000000000000000020", "96040000000000000000000000000000" ], "answer": "96040000000000000000000000000000", "correct_value": 96040000000000000000000000000000, "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 9800000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10200000000000000 * 4839293691081513", "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 4839293691081513?", "options": [ "49360795649031432600000000000010", "49360795649031432600000000000020", "49360795649031432600000000000000", "49360795649031432599999999999990" ], "answer": "49360795649031432600000000000000", "correct_value": 49360795649031432600000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5947788105771199 * 6816630932267362", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5947788105771199 * 6816630932267362?", "options": [ "40543876380371855341662667307058", "40543876380371855341662667307028", "40543876380371855341662667307038", "40543876380371855341662667307048" ], "answer": "40543876380371855341662667307038", "correct_value": 40543876380371855341662667307038, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001244", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10200000000000000 * 10200000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.98, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 10200000000000000?", "options": [ "104040000000000000000000000000010", "104040000000000000000000000000020", "104040000000000000000000000000000", "104039999999999999999999999999990" ], "answer": "104040000000000000000000000000000", "correct_value": 104040000000000000000000000000000, "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 10200000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9900000000000000 * 5119816128141137", "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 5119816128141137?", "options": [ "50686179668597256300000000000000", "50686179668597256300000000000010", "50686179668597256299999999999990", "50686179668597256300000000000020" ], "answer": "50686179668597256300000000000000", "correct_value": 50686179668597256300000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5345398198724517 * 6153362299073483", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5345398198724517 * 6153362299073483?", "options": [ "32892171749566748690621456682731", "32892171749566748690621456682701", "32892171749566748690621456682721", "32892171749566748690621456682711" ], "answer": "32892171749566748690621456682711", "correct_value": 32892171749566748690621456682711, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001245", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9900000000000000 * 10100000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.99, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 10100000000000000?", "options": [ "99990000000000000000000000000020", "99990000000000000000000000000010", "99989999999999999999999999999990", "99990000000000000000000000000000" ], "answer": "99990000000000000000000000000000", "correct_value": 99990000000000000000000000000000, "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 10100000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10100000000000000 * 4913911818799656", "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 4913911818799656?", "options": [ "49630509369876525600000000000020", "49630509369876525600000000000000", "49630509369876525600000000000010", "49630509369876525599999999999990" ], "answer": "49630509369876525600000000000000", "correct_value": 49630509369876525600000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6838050545676369 * 4381969857009705", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 6838050545676369 * 4381969857009705?", "options": [ "29964131371862613915755022161135", "29964131371862613915755022161155", "29964131371862613915755022161165", "29964131371862613915755022161145" ], "answer": "29964131371862613915755022161145", "correct_value": 29964131371862613915755022161145, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "magnitude_estimation_001246", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10100000000000000 * 9800000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 9800000000000000?", "options": [ "98980000000000000000000000000000", "98980000000000000000000000000020", "98979999999999999999999999999990", "98980000000000000000000000000010" ], "answer": "98980000000000000000000000000000", "correct_value": 98980000000000000000000000000000, "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9800000000000000≈10000000000000000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10100000000000000 * 5156959225765561", "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 5156959225765561?", "options": [ "52085288180232166100000000000000", "52085288180232166100000000000010", "52085288180232166099999999999990", "52085288180232166100000000000020" ], "answer": "52085288180232166100000000000000", "correct_value": 52085288180232166100000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4752119667718159 * 6042125643626516", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 4752119667718159 * 6042125643626516?", "options": [ "28712904105901806796391347104064", "28712904105901806796391347104054", "28712904105901806796391347104044", "28712904105901806796391347104034" ], "answer": "28712904105901806796391347104044", "correct_value": 28712904105901806796391347104044, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001247", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9800000000000000 * 10200000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.98, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 10200000000000000?", "options": [ "99959999999999999999999999999990", "99960000000000000000000000000020", "99960000000000000000000000000000", "99960000000000000000000000000010" ], "answer": "99960000000000000000000000000000", "correct_value": 99960000000000000000000000000000, "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10200000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9800000000000000 * 4801815449158777", "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 4801815449158777?", "options": [ "47057791401756014600000000000010", "47057791401756014600000000000020", "47057791401756014600000000000000", "47057791401756014599999999999990" ], "answer": "47057791401756014600000000000000", "correct_value": 47057791401756014600000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5817385105311973 * 5181621934063465", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5817385105311973 * 5181621934063465?", "options": [ "30143490260578619555324406366455", "30143490260578619555324406366445", "30143490260578619555324406366465", "30143490260578619555324406366435" ], "answer": "30143490260578619555324406366445", "correct_value": 30143490260578619555324406366445, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001248", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10100000000000000 * 10200000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 10200000000000000?", "options": [ "103020000000000000000000000000000", "103019999999999999999999999999990", "103020000000000000000000000000010", "103020000000000000000000000000020" ], "answer": "103020000000000000000000000000000", "correct_value": 103020000000000000000000000000000, "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 10200000000000000≈10000000000000000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9900000000000000 * 5035269051252501", "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 9900000000000000 * 5035269051252501?", "options": [ "49849163607399759900000000000020", "49849163607399759900000000000000", "49849163607399759900000000000010", "49849163607399759899999999999990" ], "answer": "49849163607399759900000000000000", "correct_value": 49849163607399759900000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6586488355273688 * 6439527846202457", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 6586488355273688 * 6439527846202457?", "options": [ "42413875172473135500059693051426", "42413875172473135500059693051416", "42413875172473135500059693051406", "42413875172473135500059693051436" ], "answer": "42413875172473135500059693051416", "correct_value": 42413875172473135500059693051416, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001249", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10100000000000000 * 10200000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 10200000000000000?", "options": [ "103019999999999999999999999999990", "103020000000000000000000000000010", "103020000000000000000000000000020", "103020000000000000000000000000000" ], "answer": "103020000000000000000000000000000", "correct_value": 103020000000000000000000000000000, "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 10200000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9800000000000000 * 4971993418120652", "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 4971993418120652?", "options": [ "48725535497582389600000000000000", "48725535497582389600000000000020", "48725535497582389599999999999990", "48725535497582389600000000000010" ], "answer": "48725535497582389600000000000000", "correct_value": 48725535497582389600000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5965904767560386 * 4974928785303299", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 5965904767560386 * 4974928785303299?", "options": [ "29679951358514351487207307513404", "29679951358514351487207307513434", "29679951358514351487207307513414", "29679951358514351487207307513424" ], "answer": "29679951358514351487207307513414", "correct_value": 29679951358514351487207307513414, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001250", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10100000000000000 * 10200000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.985, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 10200000000000000?", "options": [ "103020000000000000000000000000010", "103020000000000000000000000000020", "103019999999999999999999999999990", "103020000000000000000000000000000" ], "answer": "103020000000000000000000000000000", "correct_value": 103020000000000000000000000000000, "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 10200000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10200000000000000 * 5041863505507249", "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.794, "pure_math": { "question": "Which of the following is the best estimate for 10200000000000000 * 5041863505507249?", "options": [ "51427007756173939800000000000020", "51427007756173939799999999999990", "51427007756173939800000000000010", "51427007756173939800000000000000" ], "answer": "51427007756173939800000000000000", "correct_value": 51427007756173939800000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6773845913961462 * 6532756337663620", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 6773845913961462 * 6532756337663620?", "options": [ "44251884824788557279707799412430", "44251884824788557279707799412450", "44251884824788557279707799412440", "44251884824788557279707799412460" ], "answer": "44251884824788557279707799412440", "correct_value": 44251884824788557279707799412440, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "magnitude_estimation_001251", "category": "magnitude_estimation", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9800000000000000 * 9800000000000000", "ns_shortcut": "Round both to 10000000000000000 and multiply.", "shortcut_strength": 0.98, "pure_math": { "question": "Which of the following is the best estimate for 9800000000000000 * 9800000000000000?", "options": [ "96040000000000000000000000000020", "96040000000000000000000000000010", "96039999999999999999999999999990", "96040000000000000000000000000000" ], "answer": "96040000000000000000000000000000", "correct_value": 96040000000000000000000000000000, "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 9800000000000000≈10000000000000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10100000000000000 * 5039428072712825", "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", "shortcut_strength": 0.7969999999999999, "pure_math": { "question": "Which of the following is the best estimate for 10100000000000000 * 5039428072712825?", "options": [ "50898223534399532500000000000010", "50898223534399532499999999999990", "50898223534399532500000000000020", "50898223534399532500000000000000" ], "answer": "50898223534399532500000000000000", "correct_value": 50898223534399532500000000000000, "explanation": "One number rounds cleanly, the other less so." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6664231850608989 * 6391661009179223", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Which of the following is the best estimate for 6664231850608989 * 6391661009179223?", "options": [ "42595510875667771521171595835547", "42595510875667771521171595835537", "42595510875667771521171595835557", "42595510875667771521171595835567" ], "answer": "42595510875667771521171595835547", "correct_value": 42595510875667771521171595835547, "explanation": "Both numbers far from any round target; no clean rounding shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001252", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999999 * 9011938623869753", "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 9999999999999999 * 9011938623869753?", "options": [ "90119386238697520988061376130247", "90119386238697520988061376130267", "90119386238697520988061376130237", "90119386238697520988061376130257" ], "answer": "90119386238697520988061376130247", "correct_value": 90119386238697520988061376130247, "explanation": "Distributive: (10000000000000000-1)*9011938623869753 = 10000000000000000*9011938623869753 - 1*9011938623869753." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999956 * 8644745744233812", "ns_shortcut": "Use (10000000000000000-44)*m, correction is 44*8644745744233812.", "shortcut_strength": 0.999999999999912, "pure_math": { "question": "What is 9999999999999956 * 8644745744233812?", "options": [ "86447457442337739631187253712292", "86447457442337739631187253712262", "86447457442337739631187253712282", "86447457442337739631187253712272" ], "answer": "86447457442337739631187253712272", "correct_value": 86447457442337739631187253712272, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "4923371277053531 * 6368259785730583", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4923371277053531 * 6368259785730583?", "options": [ "31353307313881026117254434838593", "31353307313881026117254434838583", "31353307313881026117254434838573", "31353307313881026117254434838563" ], "answer": "31353307313881026117254434838573", "correct_value": 31353307313881026117254434838573, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001253", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 4494188112978724", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 4494188112978724?", "options": [ "44941881129787244494188112978744", "44941881129787244494188112978714", "44941881129787244494188112978734", "44941881129787244494188112978724" ], "answer": "44941881129787244494188112978724", "correct_value": 44941881129787244494188112978724, "explanation": "Distributive: (10000000000000000+1)*4494188112978724 = 10000000000000000*4494188112978724 + 1*4494188112978724." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000028 * 8515507261070221", "ns_shortcut": "Use (10000000000000000+28)*m, correction is 28*8515507261070221.", "shortcut_strength": 0.999999999999944, "pure_math": { "question": "What is 10000000000000028 * 8515507261070221?", "options": [ "85155072610702448434203309966198", "85155072610702448434203309966178", "85155072610702448434203309966208", "85155072610702448434203309966188" ], "answer": "85155072610702448434203309966188", "correct_value": 85155072610702448434203309966188, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "5377865884745713 * 6589366975414545", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5377865884745713 * 6589366975414545?", "options": [ "35436731859151924928220086595575", "35436731859151924928220086595585", "35436731859151924928220086595595", "35436731859151924928220086595605" ], "answer": "35436731859151924928220086595585", "correct_value": 35436731859151924928220086595585, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001254", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 4858514932636808", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 4858514932636808?", "options": [ "48585149326368084858514932636808", "48585149326368084858514932636818", "48585149326368084858514932636828", "48585149326368084858514932636798" ], "answer": "48585149326368084858514932636808", "correct_value": 48585149326368084858514932636808, "explanation": "Distributive: (10000000000000000+1)*4858514932636808 = 10000000000000000*4858514932636808 + 1*4858514932636808." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000027 * 2733255024386121", "ns_shortcut": "Use (10000000000000000+27)*m, correction is 27*2733255024386121.", "shortcut_strength": 0.999999999999946, "pure_math": { "question": "What is 10000000000000027 * 2733255024386121?", "options": [ "27332550243861283797885658425267", "27332550243861283797885658425277", "27332550243861283797885658425257", "27332550243861283797885658425287" ], "answer": "27332550243861283797885658425267", "correct_value": 27332550243861283797885658425267, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4016698708572911 * 6796267905713954", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4016698708572911 * 6796267905713954?", "options": [ "27298560519996761491501919100104", "27298560519996761491501919100114", "27298560519996761491501919100094", "27298560519996761491501919100084" ], "answer": "27298560519996761491501919100094", "correct_value": 27298560519996761491501919100094, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001255", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000002 * 2074831903814983", "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 10000000000000002 * 2074831903814983?", "options": [ "20748319038149834149663807629976", "20748319038149834149663807629956", "20748319038149834149663807629986", "20748319038149834149663807629966" ], "answer": "20748319038149834149663807629966", "correct_value": 20748319038149834149663807629966, "explanation": "Distributive: (10000000000000000+2)*2074831903814983 = 10000000000000000*2074831903814983 + 2*2074831903814983." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999957 * 8254254351744488", "ns_shortcut": "Use (10000000000000000-43)*m, correction is 43*8254254351744488.", "shortcut_strength": 0.999999999999914, "pure_math": { "question": "What is 9999999999999957 * 8254254351744488?", "options": [ "82542543517444525067062874987006", "82542543517444525067062874987026", "82542543517444525067062874987016", "82542543517444525067062874987036" ], "answer": "82542543517444525067062874987016", "correct_value": 82542543517444525067062874987016, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4596314980696829 * 4445658723356322", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4596314980696829 * 4445658723356322?", "options": [ "20433647789228202608841222502938", "20433647789228202608841222502958", "20433647789228202608841222502948", "20433647789228202608841222502928" ], "answer": "20433647789228202608841222502938", "correct_value": 20433647789228202608841222502938, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001256", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 5049805118679571", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 5049805118679571?", "options": [ "50498051186795715049805118679571", "50498051186795715049805118679591", "50498051186795715049805118679581", "50498051186795715049805118679561" ], "answer": "50498051186795715049805118679571", "correct_value": 50498051186795715049805118679571, "explanation": "Distributive: (10000000000000000+1)*5049805118679571 = 10000000000000000*5049805118679571 + 1*5049805118679571." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999953 * 7683288106032672", "ns_shortcut": "Use (10000000000000000-47)*m, correction is 47*7683288106032672.", "shortcut_strength": 0.999999999999906, "pure_math": { "question": "What is 9999999999999953 * 7683288106032672?", "options": [ "76832881060326358885459016464406", "76832881060326358885459016464436", "76832881060326358885459016464416", "76832881060326358885459016464426" ], "answer": "76832881060326358885459016464416", "correct_value": 76832881060326358885459016464416, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5585558016500916 * 4775288695808696", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5585558016500916 * 4775288695808696?", "options": [ "26672652055980466057656844765536", "26672652055980466057656844765526", "26672652055980466057656844765546", "26672652055980466057656844765556" ], "answer": "26672652055980466057656844765536", "correct_value": 26672652055980466057656844765536, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001257", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 1061684344876105", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 1061684344876105?", "options": [ "10616843448761051061684344876095", "10616843448761051061684344876125", "10616843448761051061684344876115", "10616843448761051061684344876105" ], "answer": "10616843448761051061684344876105", "correct_value": 10616843448761051061684344876105, "explanation": "Distributive: (10000000000000000+1)*1061684344876105 = 10000000000000000*1061684344876105 + 1*1061684344876105." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000018 * 5039884088307084", "ns_shortcut": "Use (10000000000000000+18)*m, correction is 18*5039884088307084.", "shortcut_strength": 0.999999999999964, "pure_math": { "question": "What is 10000000000000018 * 5039884088307084?", "options": [ "50398840883070930717913589527522", "50398840883070930717913589527512", "50398840883070930717913589527502", "50398840883070930717913589527532" ], "answer": "50398840883070930717913589527512", "correct_value": 50398840883070930717913589527512, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5609646615275373 * 5867744144269920", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5609646615275373 * 5867744144269920?", "options": [ "32915971078205646682560840680160", "32915971078205646682560840680180", "32915971078205646682560840680170", "32915971078205646682560840680150" ], "answer": "32915971078205646682560840680160", "correct_value": 32915971078205646682560840680160, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001258", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 2386261281789032", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 2386261281789032?", "options": [ "23862612817890322386261281789032", "23862612817890322386261281789052", "23862612817890322386261281789022", "23862612817890322386261281789042" ], "answer": "23862612817890322386261281789032", "correct_value": 23862612817890322386261281789032, "explanation": "Distributive: (10000000000000000+1)*2386261281789032 = 10000000000000000*2386261281789032 + 1*2386261281789032." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999956 * 8439388374464827", "ns_shortcut": "Use (10000000000000000-44)*m, correction is 44*8439388374464827.", "shortcut_strength": 0.999999999999912, "pure_math": { "question": "What is 9999999999999956 * 8439388374464827?", "options": [ "84393883744647898666911523547622", "84393883744647898666911523547612", "84393883744647898666911523547602", "84393883744647898666911523547632" ], "answer": "84393883744647898666911523547612", "correct_value": 84393883744647898666911523547612, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4907883938296077 * 5830873817096810", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4907883938296077 * 5830873817096810?", "options": [ "28617251953160571217182352214380", "28617251953160571217182352214390", "28617251953160571217182352214370", "28617251953160571217182352214360" ], "answer": "28617251953160571217182352214370", "correct_value": 28617251953160571217182352214370, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001259", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 5232951492619627", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 5232951492619627?", "options": [ "52329514926196275232951492619637", "52329514926196275232951492619617", "52329514926196275232951492619647", "52329514926196275232951492619627" ], "answer": "52329514926196275232951492619627", "correct_value": 52329514926196275232951492619627, "explanation": "Distributive: (10000000000000000+1)*5232951492619627 = 10000000000000000*5232951492619627 + 1*5232951492619627." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000028 * 8115604350974979", "ns_shortcut": "Use (10000000000000000+28)*m, correction is 28*8115604350974979.", "shortcut_strength": 0.999999999999944, "pure_math": { "question": "What is 10000000000000028 * 8115604350974979?", "options": [ "81156043509750017236921827299432", "81156043509750017236921827299412", "81156043509750017236921827299422", "81156043509750017236921827299402" ], "answer": "81156043509750017236921827299412", "correct_value": 81156043509750017236921827299412, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5343421197924214 * 4693024188092021", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5343421197924214 * 4693024188092021?", "options": [ "25076804929021978654942416096494", "25076804929021978654942416096484", "25076804929021978654942416096514", "25076804929021978654942416096504" ], "answer": "25076804929021978654942416096494", "correct_value": 25076804929021978654942416096494, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001260", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999998 * 1407322558019713", "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 9999999999999998 * 1407322558019713?", "options": [ "14073225580197127185354883960584", "14073225580197127185354883960574", "14073225580197127185354883960564", "14073225580197127185354883960594" ], "answer": "14073225580197127185354883960574", "correct_value": 14073225580197127185354883960574, "explanation": "Distributive: (10000000000000000-2)*1407322558019713 = 10000000000000000*1407322558019713 - 2*1407322558019713." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999966 * 2813108782065125", "ns_shortcut": "Use (10000000000000000-34)*m, correction is 34*2813108782065125.", "shortcut_strength": 0.999999999999932, "pure_math": { "question": "What is 9999999999999966 * 2813108782065125?", "options": [ "28131087820651154354301409785750", "28131087820651154354301409785770", "28131087820651154354301409785740", "28131087820651154354301409785760" ], "answer": "28131087820651154354301409785750", "correct_value": 28131087820651154354301409785750, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4311482825325165 * 5211984849428471", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4311482825325165 * 5211984849428471?", "options": [ "22471383164165818836074983772705", "22471383164165818836074983772725", "22471383164165818836074983772735", "22471383164165818836074983772715" ], "answer": "22471383164165818836074983772715", "correct_value": 22471383164165818836074983772715, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "structural_shortcuts_001261", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999998 * 6470993008833395", "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 9999999999999998 * 6470993008833395?", "options": [ "64709930088333937058013982333210", "64709930088333937058013982333220", "64709930088333937058013982333230", "64709930088333937058013982333200" ], "answer": "64709930088333937058013982333210", "correct_value": 64709930088333937058013982333210, "explanation": "Distributive: (10000000000000000-2)*6470993008833395 = 10000000000000000*6470993008833395 - 2*6470993008833395." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000018 * 5655073340204744", "ns_shortcut": "Use (10000000000000000+18)*m, correction is 18*5655073340204744.", "shortcut_strength": 0.999999999999964, "pure_math": { "question": "What is 10000000000000018 * 5655073340204744?", "options": [ "56550733402047541791320123685382", "56550733402047541791320123685402", "56550733402047541791320123685412", "56550733402047541791320123685392" ], "answer": "56550733402047541791320123685392", "correct_value": 56550733402047541791320123685392, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "4688405905327996 * 5210233974431528", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4688405905327996 * 5210233974431528?", "options": [ "24427691733865330796050483457908", "24427691733865330796050483457898", "24427691733865330796050483457878", "24427691733865330796050483457888" ], "answer": "24427691733865330796050483457888", "correct_value": 24427691733865330796050483457888, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "structural_shortcuts_001262", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000002 * 3434660663634979", "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 10000000000000002 * 3434660663634979?", "options": [ "34346606636349796869321327269948", "34346606636349796869321327269978", "34346606636349796869321327269968", "34346606636349796869321327269958" ], "answer": "34346606636349796869321327269958", "correct_value": 34346606636349796869321327269958, "explanation": "Distributive: (10000000000000000+2)*3434660663634979 = 10000000000000000*3434660663634979 + 2*3434660663634979." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999963 * 1752597361076109", "ns_shortcut": "Use (10000000000000000-37)*m, correction is 37*1752597361076109.", "shortcut_strength": 0.999999999999926, "pure_math": { "question": "What is 9999999999999963 * 1752597361076109?", "options": [ "17525973610761025153897640183967", "17525973610761025153897640183987", "17525973610761025153897640183957", "17525973610761025153897640183977" ], "answer": "17525973610761025153897640183967", "correct_value": 17525973610761025153897640183967, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4063775899240359 * 5449566808476649", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4063775899240359 * 5449566808476649?", "options": [ "22145818257587607539001089877001", "22145818257587607539001089876991", "22145818257587607539001089877011", "22145818257587607539001089876981" ], "answer": "22145818257587607539001089876991", "correct_value": 22145818257587607539001089876991, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001263", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 1169684179928312", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 1169684179928312?", "options": [ "11696841799283121169684179928322", "11696841799283121169684179928312", "11696841799283121169684179928302", "11696841799283121169684179928332" ], "answer": "11696841799283121169684179928312", "correct_value": 11696841799283121169684179928312, "explanation": "Distributive: (10000000000000000+1)*1169684179928312 = 10000000000000000*1169684179928312 + 1*1169684179928312." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999984 * 3784988998153142", "ns_shortcut": "Use (10000000000000000-16)*m, correction is 16*3784988998153142.", "shortcut_strength": 0.999999999999968, "pure_math": { "question": "What is 9999999999999984 * 3784988998153142?", "options": [ "37849889981531359440176029549738", "37849889981531359440176029549748", "37849889981531359440176029549718", "37849889981531359440176029549728" ], "answer": "37849889981531359440176029549728", "correct_value": 37849889981531359440176029549728, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6778683712221519 * 5217164744843272", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6778683712221519 * 5217164744843272?", "options": [ "35365509679845425016298600770158", "35365509679845425016298600770168", "35365509679845425016298600770188", "35365509679845425016298600770178" ], "answer": "35365509679845425016298600770168", "correct_value": 35365509679845425016298600770168, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001264", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999999 * 1810859629489210", "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 9999999999999999 * 1810859629489210?", "options": [ "18108596294892098189140370510800", "18108596294892098189140370510790", "18108596294892098189140370510780", "18108596294892098189140370510810" ], "answer": "18108596294892098189140370510790", "correct_value": 18108596294892098189140370510790, "explanation": "Distributive: (10000000000000000-1)*1810859629489210 = 10000000000000000*1810859629489210 - 1*1810859629489210." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999976 * 1637621934169284", "ns_shortcut": "Use (10000000000000000-24)*m, correction is 24*1637621934169284.", "shortcut_strength": 0.999999999999952, "pure_math": { "question": "What is 9999999999999976 * 1637621934169284?", "options": [ "16376219341692800697073579937194", "16376219341692800697073579937184", "16376219341692800697073579937174", "16376219341692800697073579937204" ], "answer": "16376219341692800697073579937184", "correct_value": 16376219341692800697073579937184, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5405931106176310 * 4184057188058838", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5405931106176310 * 4184057188058838?", "options": [ "22618724902947855225241481727800", "22618724902947855225241481727770", "22618724902947855225241481727780", "22618724902947855225241481727790" ], "answer": "22618724902947855225241481727780", "correct_value": 22618724902947855225241481727780, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001265", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000002 * 3573383218644172", "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 10000000000000002 * 3573383218644172?", "options": [ "35733832186441727146766437288344", "35733832186441727146766437288364", "35733832186441727146766437288354", "35733832186441727146766437288334" ], "answer": "35733832186441727146766437288344", "correct_value": 35733832186441727146766437288344, "explanation": "Distributive: (10000000000000000+2)*3573383218644172 = 10000000000000000*3573383218644172 + 2*3573383218644172." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999959 * 5851488878202253", "ns_shortcut": "Use (10000000000000000-41)*m, correction is 41*5851488878202253.", "shortcut_strength": 0.999999999999918, "pure_math": { "question": "What is 9999999999999959 * 5851488878202253?", "options": [ "58514888782022290088955993707637", "58514888782022290088955993707647", "58514888782022290088955993707617", "58514888782022290088955993707627" ], "answer": "58514888782022290088955993707627", "correct_value": 58514888782022290088955993707627, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6312530937697899 * 5915658060960045", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6312530937697899 * 5915658060960045?", "options": [ "37342774526652247828498119445465", "37342774526652247828498119445455", "37342774526652247828498119445445", "37342774526652247828498119445475" ], "answer": "37342774526652247828498119445455", "correct_value": 37342774526652247828498119445455, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001266", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999999 * 5511347764236148", "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 9999999999999999 * 5511347764236148?", "options": [ "55113477642361474488652235763852", "55113477642361474488652235763842", "55113477642361474488652235763872", "55113477642361474488652235763862" ], "answer": "55113477642361474488652235763852", "correct_value": 55113477642361474488652235763852, "explanation": "Distributive: (10000000000000000-1)*5511347764236148 = 10000000000000000*5511347764236148 - 1*5511347764236148." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999983 * 8737523485418743", "ns_shortcut": "Use (10000000000000000-17)*m, correction is 17*8737523485418743.", "shortcut_strength": 0.999999999999966, "pure_math": { "question": "What is 9999999999999983 * 8737523485418743?", "options": [ "87375234854187281462100747881379", "87375234854187281462100747881369", "87375234854187281462100747881359", "87375234854187281462100747881389" ], "answer": "87375234854187281462100747881369", "correct_value": 87375234854187281462100747881369, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6569533359666274 * 4716853321647638", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6569533359666274 * 4716853321647638?", "options": [ "30987525249216831414583500360802", "30987525249216831414583500360812", "30987525249216831414583500360832", "30987525249216831414583500360822" ], "answer": "30987525249216831414583500360812", "correct_value": 30987525249216831414583500360812, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001267", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999999 * 9029922271973114", "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 9999999999999999 * 9029922271973114?", "options": [ "90299222719731130970077728026906", "90299222719731130970077728026896", "90299222719731130970077728026886", "90299222719731130970077728026876" ], "answer": "90299222719731130970077728026886", "correct_value": 90299222719731130970077728026886, "explanation": "Distributive: (10000000000000000-1)*9029922271973114 = 10000000000000000*9029922271973114 - 1*9029922271973114." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000047 * 3849886884253673", "ns_shortcut": "Use (10000000000000000+47)*m, correction is 47*3849886884253673.", "shortcut_strength": 0.999999999999906, "pure_math": { "question": "What is 10000000000000047 * 3849886884253673?", "options": [ "38498868842536910944683559922651", "38498868842536910944683559922621", "38498868842536910944683559922641", "38498868842536910944683559922631" ], "answer": "38498868842536910944683559922631", "correct_value": 38498868842536910944683559922631, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "5317461807224890 * 6325453899000957", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5317461807224890 * 6325453899000957?", "options": [ "33635359521299355631295624219750", "33635359521299355631295624219720", "33635359521299355631295624219730", "33635359521299355631295624219740" ], "answer": "33635359521299355631295624219730", "correct_value": 33635359521299355631295624219730, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001268", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999999 * 1022409849974956", "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 9999999999999999 * 1022409849974956?", "options": [ "10224098499749558977590150025064", "10224098499749558977590150025044", "10224098499749558977590150025054", "10224098499749558977590150025034" ], "answer": "10224098499749558977590150025044", "correct_value": 10224098499749558977590150025044, "explanation": "Distributive: (10000000000000000-1)*1022409849974956 = 10000000000000000*1022409849974956 - 1*1022409849974956." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000022 * 8193255182143768", "ns_shortcut": "Use (10000000000000000+22)*m, correction is 22*8193255182143768.", "shortcut_strength": 0.999999999999956, "pure_math": { "question": "What is 10000000000000022 * 8193255182143768?", "options": [ "81932551821437860251614007162886", "81932551821437860251614007162896", "81932551821437860251614007162906", "81932551821437860251614007162916" ], "answer": "81932551821437860251614007162896", "correct_value": 81932551821437860251614007162896, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4406801110325705 * 6435918484949007", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4406801110325705 * 6435918484949007?", "options": [ "28361812725439013171138086324955", "28361812725439013171138086324935", "28361812725439013171138086324925", "28361812725439013171138086324945" ], "answer": "28361812725439013171138086324935", "correct_value": 28361812725439013171138086324935, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001269", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000002 * 4258930231094123", "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 10000000000000002 * 4258930231094123?", "options": [ "42589302310941238517860462188246", "42589302310941238517860462188256", "42589302310941238517860462188236", "42589302310941238517860462188266" ], "answer": "42589302310941238517860462188246", "correct_value": 42589302310941238517860462188246, "explanation": "Distributive: (10000000000000000+2)*4258930231094123 = 10000000000000000*4258930231094123 + 2*4258930231094123." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000046 * 4832404155096685", "ns_shortcut": "Use (10000000000000000+46)*m, correction is 46*4832404155096685.", "shortcut_strength": 0.999999999999908, "pure_math": { "question": "What is 10000000000000046 * 4832404155096685?", "options": [ "48324041550967072290591134447530", "48324041550967072290591134447510", "48324041550967072290591134447500", "48324041550967072290591134447520" ], "answer": "48324041550967072290591134447510", "correct_value": 48324041550967072290591134447510, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4601309318092579 * 4991077208697493", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4601309318092579 * 4991077208697493?", "options": [ "22965490067699274121043779204467", "22965490067699274121043779204457", "22965490067699274121043779204447", "22965490067699274121043779204437" ], "answer": "22965490067699274121043779204447", "correct_value": 22965490067699274121043779204447, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001270", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999998 * 2364631286113537", "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 9999999999999998 * 2364631286113537?", "options": [ "23646312861135365270737427772946", "23646312861135365270737427772916", "23646312861135365270737427772926", "23646312861135365270737427772936" ], "answer": "23646312861135365270737427772926", "correct_value": 23646312861135365270737427772926, "explanation": "Distributive: (10000000000000000-2)*2364631286113537 = 10000000000000000*2364631286113537 - 2*2364631286113537." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999965 * 1258108965332056", "ns_shortcut": "Use (10000000000000000-35)*m, correction is 35*1258108965332056.", "shortcut_strength": 0.99999999999993, "pure_math": { "question": "What is 9999999999999965 * 1258108965332056?", "options": [ "12581089653320515966186213378040", "12581089653320515966186213378030", "12581089653320515966186213378060", "12581089653320515966186213378050" ], "answer": "12581089653320515966186213378040", "correct_value": 12581089653320515966186213378040, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6826851018508091 * 6017709655352069", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6826851018508091 * 6017709655352069?", "options": [ "41082007289726245517553730090269", "41082007289726245517553730090289", "41082007289726245517553730090299", "41082007289726245517553730090279" ], "answer": "41082007289726245517553730090279", "correct_value": 41082007289726245517553730090279, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "structural_shortcuts_001271", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000002 * 5022872460994310", "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 10000000000000002 * 5022872460994310?", "options": [ "50228724609943110045744921988620", "50228724609943110045744921988630", "50228724609943110045744921988610", "50228724609943110045744921988640" ], "answer": "50228724609943110045744921988620", "correct_value": 50228724609943110045744921988620, "explanation": "Distributive: (10000000000000000+2)*5022872460994310 = 10000000000000000*5022872460994310 + 2*5022872460994310." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999959 * 8086675349306765", "ns_shortcut": "Use (10000000000000000-41)*m, correction is 41*8086675349306765.", "shortcut_strength": 0.999999999999918, "pure_math": { "question": "What is 9999999999999959 * 8086675349306765?", "options": [ "80866753493067318446310678422625", "80866753493067318446310678422635", "80866753493067318446310678422655", "80866753493067318446310678422645" ], "answer": "80866753493067318446310678422635", "correct_value": 80866753493067318446310678422635, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5413565279875586 * 6672200672390778", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5413565279875586 * 6672200672390778?", "options": [ "36120393900417255198533013745918", "36120393900417255198533013745898", "36120393900417255198533013745928", "36120393900417255198533013745908" ], "answer": "36120393900417255198533013745908", "correct_value": 36120393900417255198533013745908, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "structural_shortcuts_001272", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000002 * 2892692224834830", "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 10000000000000002 * 2892692224834830?", "options": [ "28926922248348305785384449669670", "28926922248348305785384449669660", "28926922248348305785384449669680", "28926922248348305785384449669650" ], "answer": "28926922248348305785384449669660", "correct_value": 28926922248348305785384449669660, "explanation": "Distributive: (10000000000000000+2)*2892692224834830 = 10000000000000000*2892692224834830 + 2*2892692224834830." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999965 * 8131686281351558", "ns_shortcut": "Use (10000000000000000-35)*m, correction is 35*8131686281351558.", "shortcut_strength": 0.99999999999993, "pure_math": { "question": "What is 9999999999999965 * 8131686281351558?", "options": [ "81316862813515295390980152695480", "81316862813515295390980152695490", "81316862813515295390980152695470", "81316862813515295390980152695460" ], "answer": "81316862813515295390980152695470", "correct_value": 81316862813515295390980152695470, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6616732695770404 * 5664502989890281", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6616732695770404 * 5664502989890281?", "options": [ "37480502138496232516860727043544", "37480502138496232516860727043534", "37480502138496232516860727043524", "37480502138496232516860727043514" ], "answer": "37480502138496232516860727043524", "correct_value": 37480502138496232516860727043524, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001273", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999999 * 3374066818926509", "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 9999999999999999 * 3374066818926509?", "options": [ "33740668189265086625933181073511", "33740668189265086625933181073491", "33740668189265086625933181073501", "33740668189265086625933181073481" ], "answer": "33740668189265086625933181073491", "correct_value": 33740668189265086625933181073491, "explanation": "Distributive: (10000000000000000-1)*3374066818926509 = 10000000000000000*3374066818926509 - 1*3374066818926509." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999971 * 9965938273520212", "ns_shortcut": "Use (10000000000000000-29)*m, correction is 29*9965938273520212.", "shortcut_strength": 0.999999999999942, "pure_math": { "question": "What is 9999999999999971 * 9965938273520212?", "options": [ "99659382735201830987790067913852", "99659382735201830987790067913872", "99659382735201830987790067913862", "99659382735201830987790067913842" ], "answer": "99659382735201830987790067913852", "correct_value": 99659382735201830987790067913852, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4357016573459543 * 5070933563660812", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4357016573459543 * 5070933563660812?", "options": [ "22094141579782420457282656528926", "22094141579782420457282656528916", "22094141579782420457282656528936", "22094141579782420457282656528906" ], "answer": "22094141579782420457282656528916", "correct_value": 22094141579782420457282656528916, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001274", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999999 * 9427623974760600", "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 9999999999999999 * 9427623974760600?", "options": [ "94276239747605990572376025239390", "94276239747605990572376025239410", "94276239747605990572376025239420", "94276239747605990572376025239400" ], "answer": "94276239747605990572376025239400", "correct_value": 94276239747605990572376025239400, "explanation": "Distributive: (10000000000000000-1)*9427623974760600 = 10000000000000000*9427623974760600 - 1*9427623974760600." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999975 * 9923137506073334", "ns_shortcut": "Use (10000000000000000-25)*m, correction is 25*9923137506073334.", "shortcut_strength": 0.99999999999995, "pure_math": { "question": "What is 9999999999999975 * 9923137506073334?", "options": [ "99231375060733091921562348166660", "99231375060733091921562348166640", "99231375060733091921562348166670", "99231375060733091921562348166650" ], "answer": "99231375060733091921562348166650", "correct_value": 99231375060733091921562348166650, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6952845194881645 * 6300183042433254", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6952845194881645 * 6300183042433254?", "options": [ "43804197393456873018127342222850", "43804197393456873018127342222820", "43804197393456873018127342222830", "43804197393456873018127342222840" ], "answer": "43804197393456873018127342222830", "correct_value": 43804197393456873018127342222830, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001275", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999998 * 4619003504800785", "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 9999999999999998 * 4619003504800785?", "options": [ "46190035048007840761992990398440", "46190035048007840761992990398430", "46190035048007840761992990398450", "46190035048007840761992990398420" ], "answer": "46190035048007840761992990398430", "correct_value": 46190035048007840761992990398430, "explanation": "Distributive: (10000000000000000-2)*4619003504800785 = 10000000000000000*4619003504800785 - 2*4619003504800785." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999978 * 5937253207598311", "ns_shortcut": "Use (10000000000000000-22)*m, correction is 22*5937253207598311.", "shortcut_strength": 0.999999999999956, "pure_math": { "question": "What is 9999999999999978 * 5937253207598311?", "options": [ "59372532075982979380429432837168", "59372532075982979380429432837178", "59372532075982979380429432837158", "59372532075982979380429432837148" ], "answer": "59372532075982979380429432837158", "correct_value": 59372532075982979380429432837158, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4660835605318799 * 6498762965001284", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4660835605318799 * 6498762965001284?", "options": [ "30289665817805152472396264337936", "30289665817805152472396264337916", "30289665817805152472396264337906", "30289665817805152472396264337926" ], "answer": "30289665817805152472396264337916", "correct_value": 30289665817805152472396264337916, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001276", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999999 * 9154116851230910", "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 9999999999999999 * 9154116851230910?", "options": [ "91541168512309090845883148769090", "91541168512309090845883148769100", "91541168512309090845883148769110", "91541168512309090845883148769080" ], "answer": "91541168512309090845883148769090", "correct_value": 91541168512309090845883148769090, "explanation": "Distributive: (10000000000000000-1)*9154116851230910 = 10000000000000000*9154116851230910 - 1*9154116851230910." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000044 * 6587606027809705", "ns_shortcut": "Use (10000000000000000+44)*m, correction is 44*6587606027809705.", "shortcut_strength": 0.999999999999912, "pure_math": { "question": "What is 10000000000000044 * 6587606027809705?", "options": [ "65876060278097339854665223627020", "65876060278097339854665223627010", "65876060278097339854665223627040", "65876060278097339854665223627030" ], "answer": "65876060278097339854665223627020", "correct_value": 65876060278097339854665223627020, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5894767962307215 * 5476745008448897", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5894767962307215 * 5476745008448897?", "options": [ "32284141013530515567608541891855", "32284141013530515567608541891865", "32284141013530515567608541891875", "32284141013530515567608541891845" ], "answer": "32284141013530515567608541891855", "correct_value": 32284141013530515567608541891855, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001277", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999999 * 9776748474662196", "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 9999999999999999 * 9776748474662196?", "options": [ "97767484746621950223251525337804", "97767484746621950223251525337824", "97767484746621950223251525337794", "97767484746621950223251525337814" ], "answer": "97767484746621950223251525337804", "correct_value": 97767484746621950223251525337804, "explanation": "Distributive: (10000000000000000-1)*9776748474662196 = 10000000000000000*9776748474662196 - 1*9776748474662196." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999965 * 6994733727541508", "ns_shortcut": "Use (10000000000000000-35)*m, correction is 35*6994733727541508.", "shortcut_strength": 0.99999999999993, "pure_math": { "question": "What is 9999999999999965 * 6994733727541508?", "options": [ "69947337275414835184319536047230", "69947337275414835184319536047240", "69947337275414835184319536047210", "69947337275414835184319536047220" ], "answer": "69947337275414835184319536047220", "correct_value": 69947337275414835184319536047220, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "4457391424358193 * 4769253407246610", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4457391424358193 * 4769253407246610?", "options": [ "21258429238052132052774524975750", "21258429238052132052774524975720", "21258429238052132052774524975730", "21258429238052132052774524975740" ], "answer": "21258429238052132052774524975730", "correct_value": 21258429238052132052774524975730, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001278", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 6693680691938313", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 6693680691938313?", "options": [ "66936806919383136693680691938303", "66936806919383136693680691938323", "66936806919383136693680691938333", "66936806919383136693680691938313" ], "answer": "66936806919383136693680691938313", "correct_value": 66936806919383136693680691938313, "explanation": "Distributive: (10000000000000000+1)*6693680691938313 = 10000000000000000*6693680691938313 + 1*6693680691938313." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999950 * 5072867841864335", "ns_shortcut": "Use (10000000000000000-50)*m, correction is 50*5072867841864335.", "shortcut_strength": 0.9999999999999, "pure_math": { "question": "What is 9999999999999950 * 5072867841864335?", "options": [ "50728678418643096356607906783270", "50728678418643096356607906783260", "50728678418643096356607906783240", "50728678418643096356607906783250" ], "answer": "50728678418643096356607906783250", "correct_value": 50728678418643096356607906783250, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "4421601049445143 * 6154705742013148", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4421601049445143 * 6154705742013148?", "options": [ "27213653367911382746709210740164", "27213653367911382746709210740154", "27213653367911382746709210740184", "27213653367911382746709210740174" ], "answer": "27213653367911382746709210740164", "correct_value": 27213653367911382746709210740164, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001279", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 1402731624738029", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 1402731624738029?", "options": [ "14027316247380291402731624738019", "14027316247380291402731624738029", "14027316247380291402731624738039", "14027316247380291402731624738049" ], "answer": "14027316247380291402731624738029", "correct_value": 14027316247380291402731624738029, "explanation": "Distributive: (10000000000000000+1)*1402731624738029 = 10000000000000000*1402731624738029 + 1*1402731624738029." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000026 * 6975611998330910", "ns_shortcut": "Use (10000000000000000+26)*m, correction is 26*6975611998330910.", "shortcut_strength": 0.999999999999948, "pure_math": { "question": "What is 10000000000000026 * 6975611998330910?", "options": [ "69756119983309281365911956603660", "69756119983309281365911956603680", "69756119983309281365911956603670", "69756119983309281365911956603650" ], "answer": "69756119983309281365911956603660", "correct_value": 69756119983309281365911956603660, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6177028245978908 * 4231515706557818", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6177028245978908 * 4231515706557818?", "options": [ "26138192042711038088844510502754", "26138192042711038088844510502764", "26138192042711038088844510502734", "26138192042711038088844510502744" ], "answer": "26138192042711038088844510502744", "correct_value": 26138192042711038088844510502744, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "structural_shortcuts_001280", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 1443330178587762", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 1443330178587762?", "options": [ "14433301785877621443330178587762", "14433301785877621443330178587782", "14433301785877621443330178587752", "14433301785877621443330178587772" ], "answer": "14433301785877621443330178587762", "correct_value": 14433301785877621443330178587762, "explanation": "Distributive: (10000000000000000+1)*1443330178587762 = 10000000000000000*1443330178587762 + 1*1443330178587762." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000047 * 2536309613708211", "ns_shortcut": "Use (10000000000000000+47)*m, correction is 47*2536309613708211.", "shortcut_strength": 0.999999999999906, "pure_math": { "question": "What is 10000000000000047 * 2536309613708211?", "options": [ "25363096137082229206551844285917", "25363096137082229206551844285937", "25363096137082229206551844285927", "25363096137082229206551844285907" ], "answer": "25363096137082229206551844285917", "correct_value": 25363096137082229206551844285917, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4168769638901135 * 6877213979510415", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4168769638901135 * 6877213979510415?", "options": [ "28669520838009470376205887821025", "28669520838009470376205887821045", "28669520838009470376205887821035", "28669520838009470376205887821015" ], "answer": "28669520838009470376205887821025", "correct_value": 28669520838009470376205887821025, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001281", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999998 * 8948678252428046", "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 9999999999999998 * 8948678252428046?", "options": [ "89486782524280442102643495143928", "89486782524280442102643495143898", "89486782524280442102643495143908", "89486782524280442102643495143918" ], "answer": "89486782524280442102643495143908", "correct_value": 89486782524280442102643495143908, "explanation": "Distributive: (10000000000000000-2)*8948678252428046 = 10000000000000000*8948678252428046 - 2*8948678252428046." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999954 * 5150263370449804", "ns_shortcut": "Use (10000000000000000-46)*m, correction is 46*5150263370449804.", "shortcut_strength": 0.999999999999908, "pure_math": { "question": "What is 9999999999999954 * 5150263370449804?", "options": [ "51502633704497803087884959309006", "51502633704497803087884959309036", "51502633704497803087884959309026", "51502633704497803087884959309016" ], "answer": "51502633704497803087884959309016", "correct_value": 51502633704497803087884959309016, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "5855542967303095 * 6130333741911067", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5855542967303095 * 6130333741911067?", "options": [ "35896432629668215016820323852375", "35896432629668215016820323852355", "35896432629668215016820323852365", "35896432629668215016820323852385" ], "answer": "35896432629668215016820323852365", "correct_value": 35896432629668215016820323852365, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001282", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000002 * 8712188127172187", "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 10000000000000002 * 8712188127172187?", "options": [ "87121881271721887424376254344394", "87121881271721887424376254344374", "87121881271721887424376254344384", "87121881271721887424376254344364" ], "answer": "87121881271721887424376254344374", "correct_value": 87121881271721887424376254344374, "explanation": "Distributive: (10000000000000000+2)*8712188127172187 = 10000000000000000*8712188127172187 + 2*8712188127172187." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000031 * 2834947370025633", "ns_shortcut": "Use (10000000000000000+31)*m, correction is 31*2834947370025633.", "shortcut_strength": 0.999999999999938, "pure_math": { "question": "What is 10000000000000031 * 2834947370025633?", "options": [ "28349473700256417883368470794633", "28349473700256417883368470794643", "28349473700256417883368470794613", "28349473700256417883368470794623" ], "answer": "28349473700256417883368470794623", "correct_value": 28349473700256417883368470794623, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "4984619018219265 * 6245536713628459", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4984619018219265 * 6245536713628459?", "options": [ "31131621081739064124947006062655", "31131621081739064124947006062625", "31131621081739064124947006062635", "31131621081739064124947006062645" ], "answer": "31131621081739064124947006062635", "correct_value": 31131621081739064124947006062635, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001283", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000002 * 5080314992659676", "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 10000000000000002 * 5080314992659676?", "options": [ "50803149926596770160629985319342", "50803149926596770160629985319352", "50803149926596770160629985319372", "50803149926596770160629985319362" ], "answer": "50803149926596770160629985319352", "correct_value": 50803149926596770160629985319352, "explanation": "Distributive: (10000000000000000+2)*5080314992659676 = 10000000000000000*5080314992659676 + 2*5080314992659676." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000016 * 4912166597522484", "ns_shortcut": "Use (10000000000000000+16)*m, correction is 16*4912166597522484.", "shortcut_strength": 0.999999999999968, "pure_math": { "question": "What is 10000000000000016 * 4912166597522484?", "options": [ "49121665975224918594665560359764", "49121665975224918594665560359734", "49121665975224918594665560359754", "49121665975224918594665560359744" ], "answer": "49121665975224918594665560359744", "correct_value": 49121665975224918594665560359744, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6520037536829451 * 4615662249162375", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6520037536829451 * 4615662249162375?", "options": [ "30094291121865335227137981106115", "30094291121865335227137981106145", "30094291121865335227137981106135", "30094291121865335227137981106125" ], "answer": "30094291121865335227137981106125", "correct_value": 30094291121865335227137981106125, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "structural_shortcuts_001284", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 2474276053813833", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 2474276053813833?", "options": [ "24742760538138332474276053813823", "24742760538138332474276053813853", "24742760538138332474276053813833", "24742760538138332474276053813843" ], "answer": "24742760538138332474276053813833", "correct_value": 24742760538138332474276053813833, "explanation": "Distributive: (10000000000000000+1)*2474276053813833 = 10000000000000000*2474276053813833 + 1*2474276053813833." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000040 * 7511057170397543", "ns_shortcut": "Use (10000000000000000+40)*m, correction is 40*7511057170397543.", "shortcut_strength": 0.99999999999992, "pure_math": { "question": "What is 10000000000000040 * 7511057170397543?", "options": [ "75110571703975730442286815901740", "75110571703975730442286815901730", "75110571703975730442286815901710", "75110571703975730442286815901720" ], "answer": "75110571703975730442286815901720", "correct_value": 75110571703975730442286815901720, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6420880883391922 * 6625948362483753", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6420880883391922 * 6625948362483753?", "options": [ "42544425175013938969915256443286", "42544425175013938969915256443266", "42544425175013938969915256443276", "42544425175013938969915256443256" ], "answer": "42544425175013938969915256443266", "correct_value": 42544425175013938969915256443266, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001285", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000002 * 2867228057095123", "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 10000000000000002 * 2867228057095123?", "options": [ "28672280570951235734456114190246", "28672280570951235734456114190236", "28672280570951235734456114190256", "28672280570951235734456114190266" ], "answer": "28672280570951235734456114190246", "correct_value": 28672280570951235734456114190246, "explanation": "Distributive: (10000000000000000+2)*2867228057095123 = 10000000000000000*2867228057095123 + 2*2867228057095123." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999954 * 4832819212624722", "ns_shortcut": "Use (10000000000000000-46)*m, correction is 46*4832819212624722.", "shortcut_strength": 0.999999999999908, "pure_math": { "question": "What is 9999999999999954 * 4832819212624722?", "options": [ "48328192126246997690316219262778", "48328192126246997690316219262798", "48328192126246997690316219262808", "48328192126246997690316219262788" ], "answer": "48328192126246997690316219262788", "correct_value": 48328192126246997690316219262788, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6577421179497447 * 6453424290600615", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6577421179497447 * 6453424290600615?", "options": [ "42446889609279772284511489129915", "42446889609279772284511489129905", "42446889609279772284511489129895", "42446889609279772284511489129925" ], "answer": "42446889609279772284511489129905", "correct_value": 42446889609279772284511489129905, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001286", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000002 * 7671846297191454", "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 10000000000000002 * 7671846297191454?", "options": [ "76718462971914555343692594382928", "76718462971914555343692594382898", "76718462971914555343692594382918", "76718462971914555343692594382908" ], "answer": "76718462971914555343692594382908", "correct_value": 76718462971914555343692594382908, "explanation": "Distributive: (10000000000000000+2)*7671846297191454 = 10000000000000000*7671846297191454 + 2*7671846297191454." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999964 * 8478435942327698", "ns_shortcut": "Use (10000000000000000-36)*m, correction is 36*8478435942327698.", "shortcut_strength": 0.999999999999928, "pure_math": { "question": "What is 9999999999999964 * 8478435942327698?", "options": [ "84784359423276674776306076202882", "84784359423276674776306076202862", "84784359423276674776306076202892", "84784359423276674776306076202872" ], "answer": "84784359423276674776306076202872", "correct_value": 84784359423276674776306076202872, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6313136572019354 * 4423533873620911", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6313136572019354 * 4423533873620911?", "options": [ "27926373475122612372647151111504", "27926373475122612372647151111494", "27926373475122612372647151111514", "27926373475122612372647151111484" ], "answer": "27926373475122612372647151111494", "correct_value": 27926373475122612372647151111494, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001287", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 9993094115021722", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 9993094115021722?", "options": [ "99930941150217229993094115021742", "99930941150217229993094115021732", "99930941150217229993094115021712", "99930941150217229993094115021722" ], "answer": "99930941150217229993094115021722", "correct_value": 99930941150217229993094115021722, "explanation": "Distributive: (10000000000000000+1)*9993094115021722 = 10000000000000000*9993094115021722 + 1*9993094115021722." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000023 * 5323077888782658", "ns_shortcut": "Use (10000000000000000+23)*m, correction is 23*5323077888782658.", "shortcut_strength": 0.999999999999954, "pure_math": { "question": "What is 10000000000000023 * 5323077888782658?", "options": [ "53230778887826702430791442001134", "53230778887826702430791442001154", "53230778887826702430791442001144", "53230778887826702430791442001124" ], "answer": "53230778887826702430791442001134", "correct_value": 53230778887826702430791442001134, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6660722383924499 * 4916011565770359", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6660722383924499 * 4916011565770359?", "options": [ "32744188275758354605788628125141", "32744188275758354605788628125161", "32744188275758354605788628125151", "32744188275758354605788628125131" ], "answer": "32744188275758354605788628125141", "correct_value": 32744188275758354605788628125141, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001288", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999998 * 9973186274612113", "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 9999999999999998 * 9973186274612113?", "options": [ "99731862746121110053627450775764", "99731862746121110053627450775774", "99731862746121110053627450775794", "99731862746121110053627450775784" ], "answer": "99731862746121110053627450775774", "correct_value": 99731862746121110053627450775774, "explanation": "Distributive: (10000000000000000-2)*9973186274612113 = 10000000000000000*9973186274612113 - 2*9973186274612113." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999980 * 5168202248385586", "ns_shortcut": "Use (10000000000000000-20)*m, correction is 20*5168202248385586.", "shortcut_strength": 0.99999999999996, "pure_math": { "question": "What is 9999999999999980 * 5168202248385586?", "options": [ "51682022483855756635955032288280", "51682022483855756635955032288300", "51682022483855756635955032288290", "51682022483855756635955032288270" ], "answer": "51682022483855756635955032288280", "correct_value": 51682022483855756635955032288280, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4682576869666766 * 6857153346748540", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4682576869666766 * 6857153346748540?", "options": [ "32109147653242766471919397021650", "32109147653242766471919397021640", "32109147653242766471919397021630", "32109147653242766471919397021660" ], "answer": "32109147653242766471919397021640", "correct_value": 32109147653242766471919397021640, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001289", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000002 * 8378430433899730", "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 10000000000000002 * 8378430433899730?", "options": [ "83784304338997316756860867799450", "83784304338997316756860867799480", "83784304338997316756860867799460", "83784304338997316756860867799470" ], "answer": "83784304338997316756860867799460", "correct_value": 83784304338997316756860867799460, "explanation": "Distributive: (10000000000000000+2)*8378430433899730 = 10000000000000000*8378430433899730 + 2*8378430433899730." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999957 * 5846527593448002", "ns_shortcut": "Use (10000000000000000-43)*m, correction is 43*5846527593448002.", "shortcut_strength": 0.999999999999914, "pure_math": { "question": "What is 9999999999999957 * 5846527593448002?", "options": [ "58465275934479768599313481735924", "58465275934479768599313481735934", "58465275934479768599313481735914", "58465275934479768599313481735904" ], "answer": "58465275934479768599313481735914", "correct_value": 58465275934479768599313481735914, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5871192653641243 * 5961350132853739", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5871192653641243 * 5961350132853739?", "options": [ "35000235105794120383621097157587", "35000235105794120383621097157597", "35000235105794120383621097157567", "35000235105794120383621097157577" ], "answer": "35000235105794120383621097157577", "correct_value": 35000235105794120383621097157577, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "structural_shortcuts_001290", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 4550244648356241", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 4550244648356241?", "options": [ "45502446483562414550244648356261", "45502446483562414550244648356241", "45502446483562414550244648356231", "45502446483562414550244648356251" ], "answer": "45502446483562414550244648356241", "correct_value": 45502446483562414550244648356241, "explanation": "Distributive: (10000000000000000+1)*4550244648356241 = 10000000000000000*4550244648356241 + 1*4550244648356241." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000026 * 4237294704401376", "ns_shortcut": "Use (10000000000000000+26)*m, correction is 26*4237294704401376.", "shortcut_strength": 0.999999999999948, "pure_math": { "question": "What is 10000000000000026 * 4237294704401376?", "options": [ "42372947044013870169662314435776", "42372947044013870169662314435766", "42372947044013870169662314435796", "42372947044013870169662314435786" ], "answer": "42372947044013870169662314435776", "correct_value": 42372947044013870169662314435776, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6763941457901719 * 6450938499450458", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6763941457901719 * 6450938499450458?", "options": [ "43633770358807258396623273537312", "43633770358807258396623273537322", "43633770358807258396623273537292", "43633770358807258396623273537302" ], "answer": "43633770358807258396623273537302", "correct_value": 43633770358807258396623273537302, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "structural_shortcuts_001291", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000002 * 4813441212099304", "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 10000000000000002 * 4813441212099304?", "options": [ "48134412120993049626882424198608", "48134412120993049626882424198628", "48134412120993049626882424198598", "48134412120993049626882424198618" ], "answer": "48134412120993049626882424198608", "correct_value": 48134412120993049626882424198608, "explanation": "Distributive: (10000000000000000+2)*4813441212099304 = 10000000000000000*4813441212099304 + 2*4813441212099304." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000030 * 5024193653620045", "ns_shortcut": "Use (10000000000000000+30)*m, correction is 30*5024193653620045.", "shortcut_strength": 0.99999999999994, "pure_math": { "question": "What is 10000000000000030 * 5024193653620045?", "options": [ "50241936536200600725809608601360", "50241936536200600725809608601350", "50241936536200600725809608601370", "50241936536200600725809608601340" ], "answer": "50241936536200600725809608601350", "correct_value": 50241936536200600725809608601350, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6037158240001338 * 4325011403484697", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6037158240001338 * 4325011403484697?", "options": [ "26110778232647390072711142524576", "26110778232647390072711142524596", "26110778232647390072711142524586", "26110778232647390072711142524606" ], "answer": "26110778232647390072711142524586", "correct_value": 26110778232647390072711142524586, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001292", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 4635354010835348", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 4635354010835348?", "options": [ "46353540108353484635354010835338", "46353540108353484635354010835368", "46353540108353484635354010835358", "46353540108353484635354010835348" ], "answer": "46353540108353484635354010835348", "correct_value": 46353540108353484635354010835348, "explanation": "Distributive: (10000000000000000+1)*4635354010835348 = 10000000000000000*4635354010835348 + 1*4635354010835348." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000019 * 3443640307357999", "ns_shortcut": "Use (10000000000000000+19)*m, correction is 19*3443640307357999.", "shortcut_strength": 0.999999999999962, "pure_math": { "question": "What is 10000000000000019 * 3443640307357999?", "options": [ "34436403073580055429165839802001", "34436403073580055429165839801971", "34436403073580055429165839801981", "34436403073580055429165839801991" ], "answer": "34436403073580055429165839801981", "correct_value": 34436403073580055429165839801981, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6062293240837032 * 5955826575703993", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6062293240837032 * 5955826575703993?", "options": [ "36105967193487882435221984668766", "36105967193487882435221984668786", "36105967193487882435221984668776", "36105967193487882435221984668796" ], "answer": "36105967193487882435221984668776", "correct_value": 36105967193487882435221984668776, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001293", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999998 * 5950225159180903", "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 9999999999999998 * 5950225159180903?", "options": [ "59502251591809018099549681638214", "59502251591809018099549681638204", "59502251591809018099549681638194", "59502251591809018099549681638184" ], "answer": "59502251591809018099549681638194", "correct_value": 59502251591809018099549681638194, "explanation": "Distributive: (10000000000000000-2)*5950225159180903 = 10000000000000000*5950225159180903 - 2*5950225159180903." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999955 * 2346224016906136", "ns_shortcut": "Use (10000000000000000-45)*m, correction is 45*2346224016906136.", "shortcut_strength": 0.99999999999991, "pure_math": { "question": "What is 9999999999999955 * 2346224016906136?", "options": [ "23462240169061254419919239223890", "23462240169061254419919239223880", "23462240169061254419919239223870", "23462240169061254419919239223900" ], "answer": "23462240169061254419919239223880", "correct_value": 23462240169061254419919239223880, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5717637492083665 * 6757737914354763", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5717637492083665 * 6757737914354763?", "options": [ "38638295660790064060178887246385", "38638295660790064060178887246395", "38638295660790064060178887246405", "38638295660790064060178887246415" ], "answer": "38638295660790064060178887246395", "correct_value": 38638295660790064060178887246395, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001294", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999999 * 7172716200124277", "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 9999999999999999 * 7172716200124277?", "options": [ "71727162001242762827283799875723", "71727162001242762827283799875743", "71727162001242762827283799875733", "71727162001242762827283799875713" ], "answer": "71727162001242762827283799875723", "correct_value": 71727162001242762827283799875723, "explanation": "Distributive: (10000000000000000-1)*7172716200124277 = 10000000000000000*7172716200124277 - 1*7172716200124277." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999951 * 8516256153203202", "ns_shortcut": "Use (10000000000000000-49)*m, correction is 49*8516256153203202.", "shortcut_strength": 0.999999999999902, "pure_math": { "question": "What is 9999999999999951 * 8516256153203202?", "options": [ "85162561532031602703448493043102", "85162561532031602703448493043112", "85162561532031602703448493043122", "85162561532031602703448493043092" ], "answer": "85162561532031602703448493043102", "correct_value": 85162561532031602703448493043102, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5456357042405382 * 5506557711119340", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5456357042405382 * 5506557711119340?", "options": [ "30045744946477671889441260287890", "30045744946477671889441260287870", "30045744946477671889441260287880", "30045744946477671889441260287900" ], "answer": "30045744946477671889441260287880", "correct_value": 30045744946477671889441260287880, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001295", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 3599012612734735", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 3599012612734735?", "options": [ "35990126127347353599012612734735", "35990126127347353599012612734745", "35990126127347353599012612734725", "35990126127347353599012612734755" ], "answer": "35990126127347353599012612734735", "correct_value": 35990126127347353599012612734735, "explanation": "Distributive: (10000000000000000+1)*3599012612734735 = 10000000000000000*3599012612734735 + 1*3599012612734735." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000023 * 2347019034476577", "ns_shortcut": "Use (10000000000000000+23)*m, correction is 23*2347019034476577.", "shortcut_strength": 0.999999999999954, "pure_math": { "question": "What is 10000000000000023 * 2347019034476577?", "options": [ "23470190344765823981437792961291", "23470190344765823981437792961261", "23470190344765823981437792961271", "23470190344765823981437792961281" ], "answer": "23470190344765823981437792961271", "correct_value": 23470190344765823981437792961271, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6111385739331078 * 5086269191456967", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6111385739331078 * 5086269191456967?", "options": [ "31084153003069120587362802720446", "31084153003069120587362802720416", "31084153003069120587362802720426", "31084153003069120587362802720436" ], "answer": "31084153003069120587362802720426", "correct_value": 31084153003069120587362802720426, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001296", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 4404454257196990", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 4404454257196990?", "options": [ "44044542571969904404454257196980", "44044542571969904404454257197010", "44044542571969904404454257197000", "44044542571969904404454257196990" ], "answer": "44044542571969904404454257196990", "correct_value": 44044542571969904404454257196990, "explanation": "Distributive: (10000000000000000+1)*4404454257196990 = 10000000000000000*4404454257196990 + 1*4404454257196990." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999959 * 9565690484335143", "ns_shortcut": "Use (10000000000000000-41)*m, correction is 41*9565690484335143.", "shortcut_strength": 0.999999999999918, "pure_math": { "question": "What is 9999999999999959 * 9565690484335143?", "options": [ "95656904843351037806690142259127", "95656904843351037806690142259137", "95656904843351037806690142259147", "95656904843351037806690142259157" ], "answer": "95656904843351037806690142259137", "correct_value": 95656904843351037806690142259137, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4515465193548196 * 6649740480831356", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4515465193548196 * 6649740480831356?", "options": [ "30026671687322432853621534033776", "30026671687322432853621534033786", "30026671687322432853621534033796", "30026671687322432853621534033766" ], "answer": "30026671687322432853621534033776", "correct_value": 30026671687322432853621534033776, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001297", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 1572206525590862", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 1572206525590862?", "options": [ "15722065255908621572206525590852", "15722065255908621572206525590862", "15722065255908621572206525590882", "15722065255908621572206525590872" ], "answer": "15722065255908621572206525590862", "correct_value": 15722065255908621572206525590862, "explanation": "Distributive: (10000000000000000+1)*1572206525590862 = 10000000000000000*1572206525590862 + 1*1572206525590862." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000017 * 5128191308201640", "ns_shortcut": "Use (10000000000000000+17)*m, correction is 17*5128191308201640.", "shortcut_strength": 0.999999999999966, "pure_math": { "question": "What is 10000000000000017 * 5128191308201640?", "options": [ "51281913082016487179252239427880", "51281913082016487179252239427890", "51281913082016487179252239427870", "51281913082016487179252239427900" ], "answer": "51281913082016487179252239427880", "correct_value": 51281913082016487179252239427880, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6231744209687154 * 5961076271859974", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6231744209687154 * 5961076271859974?", "options": [ "37147902540666880037904234574006", "37147902540666880037904234574016", "37147902540666880037904234573996", "37147902540666880037904234573986" ], "answer": "37147902540666880037904234573996", "correct_value": 37147902540666880037904234573996, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001298", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9999999999999998 * 5758831058390411", "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 9999999999999998 * 5758831058390411?", "options": [ "57588310583904098482337883219188", "57588310583904098482337883219178", "57588310583904098482337883219168", "57588310583904098482337883219198" ], "answer": "57588310583904098482337883219178", "correct_value": 57588310583904098482337883219178, "explanation": "Distributive: (10000000000000000-2)*5758831058390411 = 10000000000000000*5758831058390411 - 2*5758831058390411." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000026 * 1000305577513003", "ns_shortcut": "Use (10000000000000000+26)*m, correction is 26*1000305577513003.", "shortcut_strength": 0.999999999999948, "pure_math": { "question": "What is 10000000000000026 * 1000305577513003?", "options": [ "10003055775130056007945015338098", "10003055775130056007945015338078", "10003055775130056007945015338088", "10003055775130056007945015338068" ], "answer": "10003055775130056007945015338078", "correct_value": 10003055775130056007945015338078, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4928328456374384 * 4379690494815326", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4928328456374384 * 4379690494815326?", "options": [ "21584553295710777636927597009194", "21584553295710777636927597009174", "21584553295710777636927597009184", "21584553295710777636927597009204" ], "answer": "21584553295710777636927597009184", "correct_value": 21584553295710777636927597009184, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001299", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 4901325936247151", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 4901325936247151?", "options": [ "49013259362471514901325936247151", "49013259362471514901325936247161", "49013259362471514901325936247171", "49013259362471514901325936247141" ], "answer": "49013259362471514901325936247151", "correct_value": 49013259362471514901325936247151, "explanation": "Distributive: (10000000000000000+1)*4901325936247151 = 10000000000000000*4901325936247151 + 1*4901325936247151." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000022 * 8169869467556913", "ns_shortcut": "Use (10000000000000000+22)*m, correction is 22*8169869467556913.", "shortcut_strength": 0.999999999999956, "pure_math": { "question": "What is 10000000000000022 * 8169869467556913?", "options": [ "81698694675569309737128286252076", "81698694675569309737128286252106", "81698694675569309737128286252086", "81698694675569309737128286252096" ], "answer": "81698694675569309737128286252086", "correct_value": 81698694675569309737128286252086, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6054126300933369 * 4204178868261372", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6054126300933369 * 4204178868261372?", "options": [ "25452629860169457725373848522268", "25452629860169457725373848522278", "25452629860169457725373848522288", "25452629860169457725373848522258" ], "answer": "25452629860169457725373848522268", "correct_value": 25452629860169457725373848522268, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001300", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000002 * 2160793514766419", "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "What is 10000000000000002 * 2160793514766419?", "options": [ "21607935147664194321587029532828", "21607935147664194321587029532838", "21607935147664194321587029532848", "21607935147664194321587029532858" ], "answer": "21607935147664194321587029532838", "correct_value": 21607935147664194321587029532838, "explanation": "Distributive: (10000000000000000+2)*2160793514766419 = 10000000000000000*2160793514766419 + 2*2160793514766419." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9999999999999970 * 2843490753480827", "ns_shortcut": "Use (10000000000000000-30)*m, correction is 30*2843490753480827.", "shortcut_strength": 0.99999999999994, "pure_math": { "question": "What is 9999999999999970 * 2843490753480827?", "options": [ "28434907534808184695277395575200", "28434907534808184695277395575210", "28434907534808184695277395575180", "28434907534808184695277395575190" ], "answer": "28434907534808184695277395575190", "correct_value": 28434907534808184695277395575190, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6603921942806788 * 4891363152584986", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6603921942806788 * 4891363152584986?", "options": [ "32302180453592576160310547684958", "32302180453592576160310547684968", "32302180453592576160310547684988", "32302180453592576160310547684978" ], "answer": "32302180453592576160310547684968", "correct_value": 32302180453592576160310547684968, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "structural_shortcuts_001301", "category": "structural_shortcuts", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10000000000000001 * 3371397224183750", "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "What is 10000000000000001 * 3371397224183750?", "options": [ "33713972241837503371397224183740", "33713972241837503371397224183750", "33713972241837503371397224183770", "33713972241837503371397224183760" ], "answer": "33713972241837503371397224183750", "correct_value": 33713972241837503371397224183750, "explanation": "Distributive: (10000000000000000+1)*3371397224183750 = 10000000000000000*3371397224183750 + 1*3371397224183750." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "10000000000000033 * 4901458774180398", "ns_shortcut": "Use (10000000000000000+33)*m, correction is 33*4901458774180398.", "shortcut_strength": 0.999999999999934, "pure_math": { "question": "What is 10000000000000033 * 4901458774180398?", "options": [ "49014587741804141748139547953144", "49014587741804141748139547953154", "49014587741804141748139547953134", "49014587741804141748139547953124" ], "answer": "49014587741804141748139547953134", "correct_value": 49014587741804141748139547953134, "explanation": "Distributive still works but correction term is larger." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4833249658143153 * 6261721011777479", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4833249658143153 * 6261721011777479?", "options": [ "30264460939561298496851163451287", "30264460939561298496851163451307", "30264460939561298496851163451297", "30264460939561298496851163451277" ], "answer": "30264460939561298496851163451287", "correct_value": 30264460939561298496851163451287, "explanation": "Neither factor near a power of 10; no clean decomposition." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "relative_distance_001302", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 3156830228486078/6566646687092828 or 4548715704741623/8842405860316785?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.033683566889417926, "pure_math": { "question": "Which is larger: 3156830228486078/6566646687092828 or 4548715704741623/8842405860316785?", "options": [ "826917412147113/1931820504787809", "2032708981357408/3456041580915706", "3156830228486078/6566646687092828", "4548715704741623/8842405860316785" ], "answer": "4548715704741623/8842405860316785", "correct_value": 0.5144205973575005, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 3372175200288107/6861628584742188 or 2434697431897746/4835965254004392?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.012002297163120124, "pure_math": { "question": "Which is larger: 3372175200288107/6861628584742188 or 2434697431897746/4835965254004392?", "options": [ "4349766789706744/9811428863908402", "2434697431897746/4835965254004392", "1101892974897070/2293131082977623", "3372175200288107/6861628584742188" ], "answer": "2434697431897746/4835965254004392", "correct_value": 0.5034563533891625, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 3794324659519859/6898772108217924 or 5297536326472213/9133683321503815?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 3794324659519859/6898772108217924 or 5297536326472213/9133683321503815?", "options": [ "3843524577931784/7577230160805086", "3969910162369072/7203693675667891", "5297536326472213/9133683321503815", "3794324659519859/6898772108217924" ], "answer": "5297536326472213/9133683321503815", "correct_value": 0.5800000000000001, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001303", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 1628148390326898/3343200627235451 or 2259459937403775/4373538981659057?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.029617612703456753, "pure_math": { "question": "Which is larger: 1628148390326898/3343200627235451 or 2259459937403775/4373538981659057?", "options": [ "2259459937403775/4373538981659057", "1706426004228856/3802526850057808", "1324042127974059/2584332021447318", "1628148390326898/3343200627235451" ], "answer": "2259459937403775/4373538981659057", "correct_value": 0.516620509587106, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 4257923265569541/8760931988387643 or 2140105390002006/4172364584159390?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.026911277928638455, "pure_math": { "question": "Which is larger: 4257923265569541/8760931988387643 or 2140105390002006/4172364584159390?", "options": [ "4257923265569541/8760931988387643", "2140105390002006/4172364584159390", "4942317274958244/8522405866564943", "3637092241011150/8257310160871239" ], "answer": "2140105390002006/4172364584159390", "correct_value": 0.5129238701064219, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 5281050516198242/9601910029451349 or 3648829720203973/6291085724489609?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 5281050516198242/9601910029451349 or 3648829720203973/6291085724489609?", "options": [ "5281050516198242/9601910029451349", "3278877006788082/5067270096273020", "3588548795549949/6389746851508991", "3648829720203973/6291085724489609" ], "answer": "3648829720203973/6291085724489609", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001304", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 3220662534796948/6647440010120226 or 1928230730472980/3769291125634051?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.027066558791819018, "pure_math": { "question": "Which is larger: 3220662534796948/6647440010120226 or 1928230730472980/3769291125634051?", "options": [ "1928230730472980/3769291125634051", "3220662534796948/6647440010120226", "1237112765199482/2391592955849234", "3088640644904944/5470023248328239" ], "answer": "1928230730472980/3769291125634051", "correct_value": 0.511563226666028, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 1963505072338930/4018644540248816 or 1538172129233847/3019625868580428?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.020792782492592865, "pure_math": { "question": "Which is larger: 1963505072338930/4018644540248816 or 1538172129233847/3019625868580428?", "options": [ "1963505072338930/4018644540248816", "2324972554392452/5289483760989002", "1538172129233847/3019625868580428", "2117296715849548/4892148725709180" ], "answer": "1538172129233847/3019625868580428", "correct_value": 0.5093916253793934, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 2575054336946853/4681916976267005 or 694250715235600/1196983991785517?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 2575054336946853/4681916976267005 or 694250715235600/1196983991785517?", "options": [ "4389391886449678/9107898491873354", "6172700772410794/9809068654205697", "694250715235600/1196983991785517", "2575054336946853/4681916976267005" ], "answer": "694250715235600/1196983991785517", "correct_value": 0.5800000000000001, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001305", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 1454204997919891/3040755933152225 or 3445807942413052/6646985639840719?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.040163605219574294, "pure_math": { "question": "Which is larger: 1454204997919891/3040755933152225 or 3445807942413052/6646985639840719?", "options": [ "3129640954618884/5207434698905855", "1454204997919891/3040755933152225", "4094780089717400/9194811767424842", "3445807942413052/6646985639840719" ], "answer": "3445807942413052/6646985639840719", "correct_value": 0.5184015927098683, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 3930189047528952/7916166648814665 or 3560589629168386/6935027885098984?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.016944807837905584, "pure_math": { "question": "Which is larger: 3930189047528952/7916166648814665 or 3560589629168386/6935027885098984?", "options": [ "3930189047528952/7916166648814665", "3237663950679270/7648144013436969", "3560589629168386/6935027885098984", "2841855633889217/5609524112941494" ], "answer": "3560589629168386/6935027885098984", "correct_value": 0.5134210976741539, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 1155139038892897/2100252797987085 or 3874143305783707/6679557423765012?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 1155139038892897/2100252797987085 or 3874143305783707/6679557423765012?", "options": [ "3874143305783707/6679557423765012", "1155139038892897/2100252797987085", "2069188180744777/3390764182335194", "3230498624662145/6424690693927703" ], "answer": "3874143305783707/6679557423765012", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001306", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 3333052801968624/6860639450354117 or 5132723570042286/9855194266382395?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03499157319173252, "pure_math": { "question": "Which is larger: 3333052801968624/6860639450354117 or 5132723570042286/9855194266382395?", "options": [ "3333052801968624/6860639450354117", "5132723570042286/9855194266382395", "2978177391015356/6730991151732789", "3904620046064212/6999986778416726" ], "answer": "5132723570042286/9855194266382395", "correct_value": 0.5208140429611628, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 1842256977819895/3783352512245763 or 3071786111278982/6017767289138701?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.02351508958239512, "pure_math": { "question": "Which is larger: 1842256977819895/3783352512245763 or 3071786111278982/6017767289138701?", "options": [ "3071786111278982/6017767289138701", "638144502256096/1538954996921627", "1842256977819895/3783352512245763", "4263662399372258/7562434907645223" ], "answer": "3071786111278982/6017767289138701", "correct_value": 0.5104527914901532, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 811009880479418/1474563419053486 or 1400907195323288/2415357233316013?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 811009880479418/1474563419053486 or 1400907195323288/2415357233316013?", "options": [ "811009880479418/1474563419053486", "2979281035625189/4671084778567335", "1400907195323288/2415357233316013", "936965484664366/1743303226046354" ], "answer": "1400907195323288/2415357233316013", "correct_value": 0.5800000000000002, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001307", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 1229537758691493/2519410450378818 or 3007403250535453/5758234843786562?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.0342526907308775, "pure_math": { "question": "Which is larger: 1229537758691493/2519410450378818 or 3007403250535453/5758234843786562?", "options": [ "1229537758691493/2519410450378818", "2911071022476600/6470847518888681", "3007403250535453/5758234843786562", "2474542771774016/4846757333950487" ], "answer": "3007403250535453/5758234843786562", "correct_value": 0.5222786725658818, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 3051744962993482/6177937792904941 or 3261333844822568/6335216347540372?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.020819692315944716, "pure_math": { "question": "Which is larger: 3051744962993482/6177937792904941 or 3261333844822568/6335216347540372?", "options": [ "2496008933528442/5541114818369478", "3051744962993482/6177937792904941", "3261333844822568/6335216347540372", "2071934556275850/4951643436497568" ], "answer": "3261333844822568/6335216347540372", "correct_value": 0.514794391526782, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 4041795351853063/7348718821551023 or 1524160051863781/2627862158385829?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 4041795351853063/7348718821551023 or 1524160051863781/2627862158385829?", "options": [ "3056244361502256/5049600719720822", "4041795351853063/7348718821551023", "4601343183321240/9564830446286305", "1524160051863781/2627862158385829" ], "answer": "1524160051863781/2627862158385829", "correct_value": 0.5800000000000001, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001308", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 621972244946824/1321343939535526 or 4986469870644520/9682642702775566?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.04427874323819353, "pure_math": { "question": "Which is larger: 621972244946824/1321343939535526 or 4986469870644520/9682642702775566?", "options": [ "621972244946824/1321343939535526", "2075686528182336/4770939494887476", "4368710044953868/7799564731614446", "4986469870644520/9682642702775566" ], "answer": "4986469870644520/9682642702775566", "correct_value": 0.5149905892133281, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 3290698361029717/6683556800234541 or 1667560642537979/3254087518089096?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.020093716767469794, "pure_math": { "question": "Which is larger: 3290698361029717/6683556800234541 or 1667560642537979/3254087518089096?", "options": [ "3870771005817290/7303656306789064", "3290698361029717/6683556800234541", "1667560642537979/3254087518089096", "850669737178215/1474455459034722" ], "answer": "1667560642537979/3254087518089096", "correct_value": 0.5124510736936859, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 3710668912477579/6746670749959234 or 1855793764419956/3199644421413716?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 3710668912477579/6746670749959234 or 1855793764419956/3199644421413716?", "options": [ "3710668912477579/6746670749959234", "1855793764419956/3199644421413716", "1880510494246346/3725201813978980", "1909594229114240/2840493379938222" ], "answer": "1855793764419956/3199644421413716", "correct_value": 0.5800000000000002, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001309", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 3624172306526417/7483531551441204 or 3014998201968400/5888582815017812?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.02772107135150481, "pure_math": { "question": "Which is larger: 3624172306526417/7483531551441204 or 3014998201968400/5888582815017812?", "options": [ "3624172306526417/7483531551441204", "3180084534395234/5294796712005516", "3014998201968400/5888582815017812", "1679752661713731/2959045653674508" ], "answer": "3014998201968400/5888582815017812", "correct_value": 0.5120074382377995, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 2473348213231356/5084378278631433 or 2941843566423737/5744432217471448?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.025660553828265376, "pure_math": { "question": "Which is larger: 2473348213231356/5084378278631433 or 2941843566423737/5744432217471448?", "options": [ "5150453885676395/9854837634193719", "2473348213231356/5084378278631433", "2941843566423737/5744432217471448", "1022830830242634/2391075853590735" ], "answer": "2941843566423737/5744432217471448", "correct_value": 0.5121208598260145, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 3837831986636319/6977876339338760 or 4615591470243281/7957916328005657?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 3837831986636319/6977876339338760 or 4615591470243281/7957916328005657?", "options": [ "4655253455983604/8549222969853635", "1803559305690937/3277752941741463", "4615591470243281/7957916328005657", "3837831986636319/6977876339338760" ], "answer": "4615591470243281/7957916328005657", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001310", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 4093817194031311/8548352818694374 or 4350635498424729/8368514596708697?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.04098015925861709, "pure_math": { "question": "Which is larger: 4093817194031311/8548352818694374 or 4350635498424729/8368514596708697?", "options": [ "4591492187946326/9003276877623646", "4350635498424729/8368514596708697", "1595663420655602/2745857036705163", "4093817194031311/8548352818694374" ], "answer": "4350635498424729/8368514596708697", "correct_value": 0.5198814494672467, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 4706781351734568/9567613416409113 or 3414157819514092/6784340129631510?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.011291591281408886, "pure_math": { "question": "Which is larger: 4706781351734568/9567613416409113 or 3414157819514092/6784340129631510?", "options": [ "3300191058131380/5805226944840016", "3414157819514092/6784340129631510", "2874179384824560/5973215343387199", "4706781351734568/9567613416409113" ], "answer": "3414157819514092/6784340129631510", "correct_value": 0.5032409570095554, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 1989139181926951/3616616694412638 or 1231469667569955/2123223564775784?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 1989139181926951/3616616694412638 or 1231469667569955/2123223564775784?", "options": [ "1231469667569955/2123223564775784", "1989139181926951/3616616694412638", "5063973183873033/9755263462960607", "5770094070614239/8588805202473470" ], "answer": "1231469667569955/2123223564775784", "correct_value": 0.5800000000000002, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001311", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 3400950152504542/6946881746689933 or 5099402649947071/9987378353437411?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.021019708378544943, "pure_math": { "question": "Which is larger: 3400950152504542/6946881746689933 or 5099402649947071/9987378353437411?", "options": [ "3400950152504542/6946881746689933", "3608246193151117/8778455359131540", "5099402649947071/9987378353437411", "4509407413009303/9974769704465236" ], "answer": "5099402649947071/9987378353437411", "correct_value": 0.5105847069658658, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 3800087307234718/7777935815831479 or 1564605571542898/3105797740800920?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.015196579313861758, "pure_math": { "question": "Which is larger: 3800087307234718/7777935815831479 or 1564605571542898/3105797740800920?", "options": [ "3372565788121730/5945949483890539", "1564605571542898/3105797740800920", "3800087307234718/7777935815831479", "1332634088041347/3136037756952679" ], "answer": "1564605571542898/3105797740800920", "correct_value": 0.5037693057048265, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 714715855461390/1299483373566162 or 3196356830879319/5510960053240205?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 714715855461390/1299483373566162 or 3196356830879319/5510960053240205?", "options": [ "4386475992409966/8443955736178859", "3196356830879319/5510960053240205", "714715855461390/1299483373566162", "1362388578538712/2030383546674213" ], "answer": "3196356830879319/5510960053240205", "correct_value": 0.5800000000000001, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001312", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 1809049684448755/3793677693721527 or 2097520633542843/4100768123955861?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03463548349140205, "pure_math": { "question": "Which is larger: 1809049684448755/3793677693721527 or 2097520633542843/4100768123955861?", "options": [ "1809049684448755/3793677693721527", "2359104552781194/5267865246457108", "2097520633542843/4100768123955861", "2938563784889089/6589269830009136" ], "answer": "2097520633542843/4100768123955861", "correct_value": 0.5114945712949606, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 733723295160259/1484406961123012 or 1922388678223931/3816535393576681?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.00941279172194115, "pure_math": { "question": "Which is larger: 733723295160259/1484406961123012 or 1922388678223931/3816535393576681?", "options": [ "1922388678223931/3816535393576681", "733723295160259/1484406961123012", "4112475697612940/8052288475087787", "2689646493658724/4725255150779860" ], "answer": "1922388678223931/3816535393576681", "correct_value": 0.5036999477220508, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 4850925502184806/8819864549426919 or 3225083018516997/5560487962960339?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 4850925502184806/8819864549426919 or 3225083018516997/5560487962960339?", "options": [ "2088582831520776/3870837424383381", "5295912612133356/8794836921300009", "3225083018516997/5560487962960339", "4850925502184806/8819864549426919" ], "answer": "3225083018516997/5560487962960339", "correct_value": 0.5800000000000001, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001313", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 3832135623781000/7990113075730424 or 1217161281479986/2355849665581661?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03704521738239802, "pure_math": { "question": "Which is larger: 3832135623781000/7990113075730424 or 1217161281479986/2355849665581661?", "options": [ "4746046912840130/8457970966921286", "3316142736988988/5528718857816152", "1217161281479986/2355849665581661", "3832135623781000/7990113075730424" ], "answer": "1217161281479986/2355849665581661", "correct_value": 0.5166549034356435, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 3646445341004400/7328529707232140 or 3264228631497663/6418934773087698?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.010962730743062421, "pure_math": { "question": "Which is larger: 3646445341004400/7328529707232140 or 3264228631497663/6418934773087698?", "options": [ "1242313365642037/2374622536615810", "3264228631497663/6418934773087698", "3646445341004400/7328529707232140", "1619090798040308/2797047622732739" ], "answer": "3264228631497663/6418934773087698", "correct_value": 0.5085312044583173, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 3681699039027596/6693998252777446 or 4813112772027922/8298470296599866?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 3681699039027596/6693998252777446 or 4813112772027922/8298470296599866?", "options": [ "562339164345563/1024278819423814", "4106561054545598/6405105331827948", "4813112772027922/8298470296599866", "3681699039027596/6693998252777446" ], "answer": "4813112772027922/8298470296599866", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001314", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 2328099629426930/4802289850285624 or 1085692948671045/2052630831071951?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.044138040415096835, "pure_math": { "question": "Which is larger: 2328099629426930/4802289850285624 or 1085692948671045/2052630831071951?", "options": [ "2328099629426930/4802289850285624", "3112887959791793/6474412118515400", "4736166046620089/9637331951400662", "1085692948671045/2052630831071951" ], "answer": "1085692948671045/2052630831071951", "correct_value": 0.5289275266824579, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 1651586676297210/3374259162052785 or 909990441952020/1786374564022702?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.0199398347584554, "pure_math": { "question": "Which is larger: 1651586676297210/3374259162052785 or 909990441952020/1786374564022702?", "options": [ "2438843441109301/5066384740074769", "909990441952020/1786374564022702", "1651586676297210/3374259162052785", "1362638275094165/2419192855137853" ], "answer": "909990441952020/1786374564022702", "correct_value": 0.509406291535427, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 3737414519217084/6795299125849243 or 5265740317699619/9078862616723482?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 3737414519217084/6795299125849243 or 5265740317699619/9078862616723482?", "options": [ "3983293953208783/6283020137650891", "2482314485567700/4343196485347070", "3737414519217084/6795299125849243", "5265740317699619/9078862616723482" ], "answer": "5265740317699619/9078862616723482", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001315", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 2571455766889687/5314128421507481 or 1474968005701703/2852085805696115?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03326374088668799, "pure_math": { "question": "Which is larger: 2571455766889687/5314128421507481 or 1474968005701703/2852085805696115?", "options": [ "5003619631207823/8166942172815245", "1958557955976782/3253129134936374", "1474968005701703/2852085805696115", "2571455766889687/5314128421507481" ], "answer": "1474968005701703/2852085805696115", "correct_value": 0.5171541482924299, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 2151123623073786/4403882701826968 or 1763829668391729/3434867733523728?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.0250466046441154, "pure_math": { "question": "Which is larger: 2151123623073786/4403882701826968 or 1763829668391729/3434867733523728?", "options": [ "1763829668391729/3434867733523728", "3927104112885930/7596147399798533", "2466573936728772/4895933282394038", "2151123623073786/4403882701826968" ], "answer": "1763829668391729/3434867733523728", "correct_value": 0.5135073036952922, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 4961589883296470/9021072515084490 or 4064394552799012/7007576815170711?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 4961589883296470/9021072515084490 or 4064394552799012/7007576815170711?", "options": [ "5653773258249729/8688835813771749", "4961589883296470/9021072515084490", "3920541137415851/7537515983576144", "4064394552799012/7007576815170711" ], "answer": "4064394552799012/7007576815170711", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001316", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 4400235113990462/9133296478578571 or 4292907636683648/8228332662690457?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.0399431526564924, "pure_math": { "question": "Which is larger: 4400235113990462/9133296478578571 or 4292907636683648/8228332662690457?", "options": [ "4292907636683648/8228332662690457", "2801801662865984/4676285543733454", "4400235113990462/9133296478578571", "4350315373265544/8401711986879224" ], "answer": "4292907636683648/8228332662690457", "correct_value": 0.5217226639546165, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 713362764052804/1453633670149995 or 1198132274424776/2385567644476762?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.011497479632861074, "pure_math": { "question": "Which is larger: 713362764052804/1453633670149995 or 1198132274424776/2385567644476762?", "options": [ "713362764052804/1453633670149995", "1198132274424776/2385567644476762", "3089087600357570/7153904165941948", "2561250561506201/4374257384172332" ], "answer": "1198132274424776/2385567644476762", "correct_value": 0.5022420039937991, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 2194415693531519/3989846715511852 or 4132098666261770/7124308045278914?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 2194415693531519/3989846715511852 or 4132098666261770/7124308045278914?", "options": [ "4132098666261770/7124308045278914", "4396017839217536/7522750990228412", "5298331555144480/9107148567063869", "2194415693531519/3989846715511852" ], "answer": "4132098666261770/7124308045278914", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001317", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 899353677509414/1876610312189529 or 1953773664469476/3752265578237291?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.04144802814689097, "pure_math": { "question": "Which is larger: 899353677509414/1876610312189529 or 1953773664469476/3752265578237291?", "options": [ "1953773664469476/3752265578237291", "2378704202311726/5560404585212965", "3005285997303963/6282496631690872", "899353677509414/1876610312189529" ], "answer": "1953773664469476/3752265578237291", "correct_value": 0.5206917324299054, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 3475729804195854/7055115658396344 or 4282882197468053/8345242411101316?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.020558599623763885, "pure_math": { "question": "Which is larger: 3475729804195854/7055115658396344 or 4282882197468053/8345242411101316?", "options": [ "3475729804195854/7055115658396344", "1823694826992158/3285178990916632", "3598167828063282/8347552758408573", "4282882197468053/8345242411101316" ], "answer": "4282882197468053/8345242411101316", "correct_value": 0.5132124372769231, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 4161586802510395/7566521459109808 or 3913488263172457/6747393557193891?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 4161586802510395/7566521459109808 or 3913488263172457/6747393557193891?", "options": [ "3913488263172457/6747393557193891", "2994999338814920/5429687432046532", "3062128785755546/5961157524320735", "4161586802510395/7566521459109808" ], "answer": "3913488263172457/6747393557193891", "correct_value": 0.5800000000000001, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001318", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 2544934428522855/5197074050005155 or 2941839518116505/5625160278864343?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03329278547857595, "pure_math": { "question": "Which is larger: 2544934428522855/5197074050005155 or 2941839518116505/5625160278864343?", "options": [ "2941839518116505/5625160278864343", "2681454867017476/6106115904144565", "2544934428522855/5197074050005155", "4864991466173950/8938345843954084" ], "answer": "2941839518116505/5625160278864343", "correct_value": 0.522978790341318, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 2297377069965648/4664001722315931 or 3552671990262190/7071392335349837?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.009824256835011758, "pure_math": { "question": "Which is larger: 2297377069965648/4664001722315931 or 3552671990262190/7071392335349837?", "options": [ "3878071165744446/9227115706937936", "2297377069965648/4664001722315931", "3341703628809730/7190330958350584", "3552671990262190/7071392335349837" ], "answer": "3552671990262190/7071392335349837", "correct_value": 0.5024006336775304, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 5183878942571820/9425234441039672 or 3676087368763872/6338081670282538?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 5183878942571820/9425234441039672 or 3676087368763872/6338081670282538?", "options": [ "3273457772714356/5244026943640966", "5183878942571820/9425234441039672", "5359704138248868/9807528949030234", "3676087368763872/6338081670282538" ], "answer": "3676087368763872/6338081670282538", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001319", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 910646369219713/1871334268902897 or 3089540991586700/6044075948185964?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.024539007810614244, "pure_math": { "question": "Which is larger: 910646369219713/1871334268902897 or 3089540991586700/6044075948185964?", "options": [ "4859218225158536/9258901846692782", "910646369219713/1871334268902897", "2224000349275634/5259140082640715", "3089540991586700/6044075948185964" ], "answer": "3089540991586700/6044075948185964", "correct_value": 0.5111684595085173, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 3355513585936439/6915585738415527 or 572184954916716/1111106539763659?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.02975825568422308, "pure_math": { "question": "Which is larger: 3355513585936439/6915585738415527 or 572184954916716/1111106539763659?", "options": [ "2793163766124851/5586676880153876", "572184954916716/1111106539763659", "3355513585936439/6915585738415527", "4194418494223416/8725643450669567" ], "answer": "572184954916716/1111106539763659", "correct_value": 0.5149685781153122, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 601789337951075/1094162432638318 or 883439868815099/1523172187612239?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 601789337951075/1094162432638318 or 883439868815099/1523172187612239?", "options": [ "883439868815099/1523172187612239", "716686942101947/1241282289313064", "601789337951075/1094162432638318", "3782438002925814/6085027624888039" ], "answer": "883439868815099/1523172187612239", "correct_value": 0.5800000000000003, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001320", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 3950685376183907/8071990796507909 or 4004028028668073/7741107408862846?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.02781092486936687, "pure_math": { "question": "Which is larger: 3950685376183907/8071990796507909 or 4004028028668073/7741107408862846?", "options": [ "4004028028668073/7741107408862846", "644310969319432/1218456905875296", "5060906674020351/8237340062704028", "3950685376183907/8071990796507909" ], "answer": "4004028028668073/7741107408862846", "correct_value": 0.5172422777758947, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 1953786165390229/3984967560087258 or 1642719039332376/3267801120498895?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.012409495526573944, "pure_math": { "question": "Which is larger: 1953786165390229/3984967560087258 or 1642719039332376/3267801120498895?", "options": [ "3250293045313914/7241735290492555", "1953786165390229/3984967560087258", "1642719039332376/3267801120498895", "5865852235968630/9948416552852361" ], "answer": "1642719039332376/3267801120498895", "correct_value": 0.5026985972394741, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 2103244890203102/3824081618551094 or 5424461217143410/9352519339902431?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 2103244890203102/3824081618551094 or 5424461217143410/9352519339902431?", "options": [ "2103244890203102/3824081618551094", "2578300913909242/4669849232015448", "1027935006989502/1607955914991646", "5424461217143410/9352519339902431" ], "answer": "5424461217143410/9352519339902431", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001321", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 3818190347847252/7859509924251032 or 4331070142258588/8443016131360446?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.027171507678161178, "pure_math": { "question": "Which is larger: 3818190347847252/7859509924251032 or 4331070142258588/8443016131360446?", "options": [ "3818190347847252/7859509924251032", "4331070142258588/8443016131360446", "5465112892269918/9951476506088829", "5199941974409009/9946252436873207" ], "answer": "4331070142258588/8443016131360446", "correct_value": 0.51297665133807, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 2452928009910006/5041099278184445 or 2745053558374209/5444258490046201?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.017624796149114608, "pure_math": { "question": "Which is larger: 2452928009910006/5041099278184445 or 2745053558374209/5444258490046201?", "options": [ "4236284112346408/8532523817990687", "2745053558374209/5444258490046201", "838150938621901/1554976177127418", "2452928009910006/5041099278184445" ], "answer": "2745053558374209/5444258490046201", "correct_value": 0.5042107319836157, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 1475079906462510/2681963466295472 or 5761457835990353/9933547993086815?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 1475079906462510/2681963466295472 or 5761457835990353/9933547993086815?", "options": [ "1475079906462510/2681963466295472", "2086013543328417/3831223799933664", "4632794301576571/8917894668890351", "5761457835990353/9933547993086815" ], "answer": "5761457835990353/9933547993086815", "correct_value": 0.5800000000000001, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001322", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 2607013388966683/5321421034883884 or 2811022505423814/5470967110370684?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.023897988804157777, "pure_math": { "question": "Which is larger: 2607013388966683/5321421034883884 or 2811022505423814/5470967110370684?", "options": [ "2607013388966683/5321421034883884", "2227438836053826/5106690397020671", "2811022505423814/5470967110370684", "3345048941385950/5864483503007188" ], "answer": "2811022505423814/5470967110370684", "correct_value": 0.5138072389606001, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 4550154378899129/9280124164396468 or 3656219672547532/7117577908780174?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.02337694675591645, "pure_math": { "question": "Which is larger: 4550154378899129/9280124164396468 or 3656219672547532/7117577908780174?", "options": [ "1089162586293145/2039023606313356", "4550154378899129/9280124164396468", "1395077101973392/2507816574500963", "3656219672547532/7117577908780174" ], "answer": "3656219672547532/7117577908780174", "correct_value": 0.5136887462850608, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 4624885607478622/8408882922688403 or 5640818746280330/9725549562552294?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 4624885607478622/8408882922688403 or 5640818746280330/9725549562552294?", "options": [ "5640818746280330/9725549562552294", "5444070713551233/9769947318420437", "4624885607478622/8408882922688403", "3333529742952483/5330390373562849" ], "answer": "5640818746280330/9725549562552294", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001323", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 4412182227556401/9018022577686047 or 2640276677828445/5043640536194472?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03422357416440208, "pure_math": { "question": "Which is larger: 4412182227556401/9018022577686047 or 2640276677828445/5043640536194472?", "options": [ "2640276677828445/5043640536194472", "3427153018122302/7353643867046593", "2326940021900240/4139572061383055", "4412182227556401/9018022577686047" ], "answer": "2640276677828445/5043640536194472", "correct_value": 0.5234862910790599, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 2019538362979077/4079083105366376 or 1003585043493176/1968628573859549?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.014692769154363461, "pure_math": { "question": "Which is larger: 2019538362979077/4079083105366376 or 1003585043493176/1968628573859549?", "options": [ "590355372323102/1038420137814867", "2864613691644326/6889008812509295", "1003585043493176/1968628573859549", "2019538362979077/4079083105366376" ], "answer": "1003585043493176/1968628573859549", "correct_value": 0.5097889245433539, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 2686119213140158/4883853114800287 or 5047443207912410/8702488289504155?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 2686119213140158/4883853114800287 or 5047443207912410/8702488289504155?", "options": [ "5047443207912410/8702488289504155", "1742620330191822/3274268029688133", "2686119213140158/4883853114800287", "5662467448121169/9078087709099722" ], "answer": "5047443207912410/8702488289504155", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001324", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 3762873463966898/7736193396426065 or 606334248785514/1138338364057463?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.04624994584783848, "pure_math": { "question": "Which is larger: 3762873463966898/7736193396426065 or 606334248785514/1138338364057463?", "options": [ "2965846903386792/5168387757701654", "3192815450819356/6803868512465385", "606334248785514/1138338364057463", "3762873463966898/7736193396426065" ], "answer": "606334248785514/1138338364057463", "correct_value": 0.5326485234357844, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 4821209642193859/9717811369442923 or 893899496711254/1745751479412805?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.01592187919644822, "pure_math": { "question": "Which is larger: 4821209642193859/9717811369442923 or 893899496711254/1745751479412805?", "options": [ "4821209642193859/9717811369442923", "3465142081056995/7818059129193969", "893899496711254/1745751479412805", "645029535122236/1134123340523095" ], "answer": "893899496711254/1745751479412805", "correct_value": 0.5120428120799434, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 3458260023419260/6287745497125927 or 3147890474551938/5427397369917135?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 3458260023419260/6287745497125927 or 3147890474551938/5427397369917135?", "options": [ "1522455454985539/3170811620377464", "3147890474551938/5427397369917135", "3458260023419260/6287745497125927", "1922696657445385/3010293319102051" ], "answer": "3147890474551938/5427397369917135", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001325", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 2334093559660546/4820774190763534 or 4650408078602261/8956359906833583?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03505569364100736, "pure_math": { "question": "Which is larger: 2334093559660546/4820774190763534 or 4650408078602261/8956359906833583?", "options": [ "2456153839037698/5372274207447582", "4650408078602261/8956359906833583", "2334093559660546/4820774190763534", "2252191763471898/4683139030731783" ], "answer": "4650408078602261/8956359906833583", "correct_value": 0.5192297012376715, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 4919930319518848/9937768064140201 or 3886063331178023/7649442921273683?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.012945156001739289, "pure_math": { "question": "Which is larger: 4919930319518848/9937768064140201 or 3886063331178023/7649442921273683?", "options": [ "4257079773957318/9505916388177287", "3886063331178023/7649442921273683", "4919930319518848/9937768064140201", "3084607375181382/7281707838544679" ], "answer": "3886063331178023/7649442921273683", "correct_value": 0.5080191291277676, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 4894498530814632/8899088237844785 or 1750770861012757/3018570450021995?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 4894498530814632/8899088237844785 or 1750770861012757/3018570450021995?", "options": [ "2607026480660291/4745470332838100", "1750770861012757/3018570450021995", "4894498530814632/8899088237844785", "4168885193157192/6357465287205160" ], "answer": "1750770861012757/3018570450021995", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001326", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 534837061794351/1130789586333723 or 2875399962050632/5553475060032459?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.04478924850978466, "pure_math": { "question": "Which is larger: 534837061794351/1130789586333723 or 2875399962050632/5553475060032459?", "options": [ "534837061794351/1130789586333723", "1121067616966639/2058479517514305", "2875399962050632/5553475060032459", "5748208199720642/9585993587552760" ], "answer": "2875399962050632/5553475060032459", "correct_value": 0.5177658909003592, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 4366255093163884/8883576839284705 or 2212322675925314/4392211548616846?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.01219477544067954, "pure_math": { "question": "Which is larger: 4366255093163884/8883576839284705 or 2212322675925314/4392211548616846?", "options": [ "2212322675925314/4392211548616846", "1000963882704460/1870504935150829", "4366255093163884/8883576839284705", "3523346741834662/5873339912067862" ], "answer": "2212322675925314/4392211548616846", "correct_value": 0.5036921950223454, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 2378677727339270/4324868595162308 or 2243102837060376/3867418684586855?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 2378677727339270/4324868595162308 or 2243102837060376/3867418684586855?", "options": [ "1790604976805330/3038728371130342", "6117100713363486/9924275641047440", "2378677727339270/4324868595162308", "2243102837060376/3867418684586855" ], "answer": "2243102837060376/3867418684586855", "correct_value": 0.5800000000000001, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001327", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 2735390981649663/5691084238440991 or 4295109095246166/8351862018618441?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.033624678104418104, "pure_math": { "question": "Which is larger: 2735390981649663/5691084238440991 or 4295109095246166/8351862018618441?", "options": [ "617236428868181/1223862747558723", "2031456334205041/3580982707859069", "2735390981649663/5691084238440991", "4295109095246166/8351862018618441" ], "answer": "4295109095246166/8351862018618441", "correct_value": 0.5142696425864397, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 2290601367143787/4703775571670791 or 5028021769094710/9903621976464959?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.02072444034688009, "pure_math": { "question": "Which is larger: 2290601367143787/4703775571670791 or 5028021769094710/9903621976464959?", "options": [ "2290601367143787/4703775571670791", "5028021769094710/9903621976464959", "5887091354936243/9985533119019792", "948755934326244/2120434006407571" ], "answer": "5028021769094710/9903621976464959", "correct_value": 0.5076952433204073, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 4760153649387807/8654824817068740 or 1006626787530420/1735563426776585?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 4760153649387807/8654824817068740 or 1006626787530420/1735563426776585?", "options": [ "4760153649387807/8654824817068740", "1006626787530420/1735563426776585", "6273355377898368/9687264767622686", "3595322834925330/6092785559050952" ], "answer": "1006626787530420/1735563426776585", "correct_value": 0.5800000000000004, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001328", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 3398067461346230/7100579156150744 or 2867940035574290/5608231587068869?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03281844367140946, "pure_math": { "question": "Which is larger: 3398067461346230/7100579156150744 or 2867940035574290/5608231587068869?", "options": [ "2867940035574290/5608231587068869", "3398067461346230/7100579156150744", "4038835917224014/8130230740243585", "3526847817698832/7330477268960436" ], "answer": "2867940035574290/5608231587068869", "correct_value": 0.5113804576449763, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 4663702514573769/9502206491512501 or 879787191995726/1736207779474908?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.01592714804397516, "pure_math": { "question": "Which is larger: 4663702514573769/9502206491512501 or 879787191995726/1736207779474908?", "options": [ "4663702514573769/9502206491512501", "3701906347095936/6961560148816440", "1569702331953697/2883135099564296", "879787191995726/1736207779474908" ], "answer": "879787191995726/1736207779474908", "correct_value": 0.5067292074119178, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 2185370478211734/3973400869475879 or 4217972244057987/7272365938031013?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 2185370478211734/3973400869475879 or 4217972244057987/7272365938031013?", "options": [ "2185370478211734/3973400869475879", "4217972244057987/7272365938031013", "1366538836238242/2048130517685016", "1027053370563694/1880209476152048" ], "answer": "4217972244057987/7272365938031013", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001329", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 4118043687492473/8543280461730780 or 2492270044633080/4833345504534993?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03361941732807627, "pure_math": { "question": "Which is larger: 4118043687492473/8543280461730780 or 2492270044633080/4833345504534993?", "options": [ "3408642894745590/7075401161197349", "4118043687492473/8543280461730780", "2492270044633080/4833345504534993", "1570119869193244/3050770287516046" ], "answer": "2492270044633080/4833345504534993", "correct_value": 0.5156407797238275, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 2190970081489660/4447369192746225 or 669651770074654/1319716273614575?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.014776939393268351, "pure_math": { "question": "Which is larger: 2190970081489660/4447369192746225 or 669651770074654/1319716273614575?", "options": [ "3888459134820571/6997808022412214", "2190970081489660/4447369192746225", "962367813007527/1755782766088950", "669651770074654/1319716273614575" ], "answer": "669651770074654/1319716273614575", "correct_value": 0.5074210142461475, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 4609971706783611/8381766739606565 or 2005897578690569/3458444101190635?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 4609971706783611/8381766739606565 or 2005897578690569/3458444101190635?", "options": [ "4572201298847648/8871086570862480", "4609971706783611/8381766739606565", "2005897578690569/3458444101190635", "4903830923497090/9207406918926752" ], "answer": "2005897578690569/3458444101190635", "correct_value": 0.5800000000000002, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001330", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 523121356736527/1122113587913851 or 3338924647595120/6440772830084977?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.05221147681703825, "pure_math": { "question": "Which is larger: 523121356736527/1122113587913851 or 3338924647595120/6440772830084977?", "options": [ "2516100279844157/4764622874148550", "3338924647595120/6440772830084977", "523121356736527/1122113587913851", "4231780825652356/7576969195842783" ], "answer": "3338924647595120/6440772830084977", "correct_value": 0.518404349241901, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 1357678798992610/2745623415921136 or 4695003609586796/9146054663174490?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.018848138944960513, "pure_math": { "question": "Which is larger: 1357678798992610/2745623415921136 or 4695003609586796/9146054663174490?", "options": [ "4695003609586796/9146054663174490", "454714514028190/1016516926150135", "923295564680861/1588040509772651", "1357678798992610/2745623415921136" ], "answer": "4695003609586796/9146054663174490", "correct_value": 0.5133364912513233, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 4572946460884796/8314448110699629 or 4864606572552940/8387252711298173?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 4572946460884796/8314448110699629 or 4864606572552940/8387252711298173?", "options": [ "4572946460884796/8314448110699629", "4864606572552940/8387252711298173", "3883979111385546/5724015006102090", "4069740355347359/7648103724929683" ], "answer": "4864606572552940/8387252711298173", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001331", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 2337137697930081/4820992535670831 or 4069499609858078/7841469570042440?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03418803601306808, "pure_math": { "question": "Which is larger: 2337137697930081/4820992535670831 or 4069499609858078/7841469570042440?", "options": [ "2767752600646682/6310717749500977", "4069499609858078/7841469570042440", "4555053149955965/7929667348677205", "2337137697930081/4820992535670831" ], "answer": "4069499609858078/7841469570042440", "correct_value": 0.5189715490837584, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 4151980129238030/8434121218697235 or 3175755140307945/6324186912728718?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.009876559808911778, "pure_math": { "question": "Which is larger: 4151980129238030/8434121218697235 or 3175755140307945/6324186912728718?", "options": [ "1328590436815466/2339383448759758", "3175755140307945/6324186912728718", "4151980129238030/8434121218697235", "3120226302436280/6722619281185166" ], "answer": "3175755140307945/6324186912728718", "correct_value": 0.5021602277307916, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 5082015476290782/9240028138710513 or 3892008974015782/6710360300027210?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 5082015476290782/9240028138710513 or 3892008974015782/6710360300027210?", "options": [ "3892008974015782/6710360300027210", "1571366238504614/2798510633030528", "3697932573277365/7609545816765941", "5082015476290782/9240028138710513" ], "answer": "3892008974015782/6710360300027210", "correct_value": 0.5800000000000001, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001332", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 4686843661616532/9773225767096842 or 3130737125680045/6057647470054147?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.037264381486155684, "pure_math": { "question": "Which is larger: 4686843661616532/9773225767096842 or 3130737125680045/6057647470054147?", "options": [ "3130737125680045/6057647470054147", "4686843661616532/9773225767096842", "3935216554476600/6604596303808471", "3891441734539914/9037461188341525" ], "answer": "3130737125680045/6057647470054147", "correct_value": 0.516823922348862, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 2067200038734377/4173786374762110 or 1402267345056736/2766726222716309?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.011551001856902499, "pure_math": { "question": "Which is larger: 2067200038734377/4173786374762110 or 1402267345056736/2766726222716309?", "options": [ "4288792382854759/9184574065742254", "2067200038734377/4173786374762110", "1402267345056736/2766726222716309", "2411653069979286/4249720249540360" ], "answer": "1402267345056736/2766726222716309", "correct_value": 0.5068327084708879, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 4718238855131213/8578616100238569 or 1693957204831094/2920615870398438?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 4718238855131213/8578616100238569 or 1693957204831094/2920615870398438?", "options": [ "1829589709164500/2727899493299769", "1693957204831094/2920615870398438", "4718238855131213/8578616100238569", "1493079285869078/2694572429954983" ], "answer": "1693957204831094/2920615870398438", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001333", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 1625216608778890/3324879232130241 or 3407637128664700/6552641746983606?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03123549665135572, "pure_math": { "question": "Which is larger: 1625216608778890/3324879232130241 or 3407637128664700/6552641746983606?", "options": [ "4549412076835150/9033955399039167", "3407637128664700/6552641746983606", "1625216608778890/3324879232130241", "5274042311726651/8871265304802566" ], "answer": "3407637128664700/6552641746983606", "correct_value": 0.5200402006157816, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 3735810553504051/7606840451308897 or 566130854510678/1113219859552959?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.017440610262984435, "pure_math": { "question": "Which is larger: 3735810553504051/7606840451308897 or 566130854510678/1113219859552959?", "options": [ "3217103839739998/6320995120518642", "3735810553504051/7606840451308897", "2789267296907942/6190547494016540", "566130854510678/1113219859552959" ], "answer": "566130854510678/1113219859552959", "correct_value": 0.5085526005060867, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 1303975793000345/2370865078182445 or 600568012159500/1035462089930171?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 1303975793000345/2370865078182445 or 600568012159500/1035462089930171?", "options": [ "616500614384905/1209863167333742", "1303975793000345/2370865078182445", "3502853659716746/6331636881884912", "600568012159500/1035462089930171" ], "answer": "600568012159500/1035462089930171", "correct_value": 0.5800000000000007, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001334", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 2019145687252705/4146076980235333 or 4228380328927378/8128192755779531?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03321012607834173, "pure_math": { "question": "Which is larger: 2019145687252705/4146076980235333 or 4228380328927378/8128192755779531?", "options": [ "2019145687252705/4146076980235333", "4228380328927378/8128192755779531", "4127586459268053/7566662875029198", "1672708367847358/3478675003084052" ], "answer": "4228380328927378/8128192755779531", "correct_value": 0.5202116209560604, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 2455303244156940/4961613497846706 or 1169803089163102/2286598516469136?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.016731105597211915, "pure_math": { "question": "Which is larger: 2455303244156940/4961613497846706 or 1169803089163102/2286598516469136?", "options": [ "2981824855060226/5445646880425277", "4663823983708041/9947797342963520", "2455303244156940/4961613497846706", "1169803089163102/2286598516469136" ], "answer": "1169803089163102/2286598516469136", "correct_value": 0.5115909420642238, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 1601841727843736/2912439505170428 or 3222896302702020/5556717763279344?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 1601841727843736/2912439505170428 or 3222896302702020/5556717763279344?", "options": [ "3222896302702020/5556717763279344", "1601841727843736/2912439505170428", "5156941608335202/9403899579418829", "1871193465246943/3878904190860438" ], "answer": "3222896302702020/5556717763279344", "correct_value": 0.5800000000000001, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001335", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 1084059988850201/2273968299967842 or 2286429987581696/4464879691951442?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03536610497019438, "pure_math": { "question": "Which is larger: 1084059988850201/2273968299967842 or 2286429987581696/4464879691951442?", "options": [ "949012624262388/1623787856898585", "2286429987581696/4464879691951442", "2940171283250512/6162002287297619", "1084059988850201/2273968299967842" ], "answer": "2286429987581696/4464879691951442", "correct_value": 0.5120921828427538, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 954540566866734/1967430496322492 or 2205827926865648/4339317640379422?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.023164030428434457, "pure_math": { "question": "Which is larger: 954540566866734/1967430496322492 or 2205827926865648/4339317640379422?", "options": [ "774203620250975/1568367134274264", "1005171808766176/2042160626843561", "954540566866734/1967430496322492", "2205827926865648/4339317640379422" ], "answer": "2205827926865648/4339317640379422", "correct_value": 0.5083352060562164, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 2090560450309943/3801019000563532 or 5226476081713045/9011165658125939?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 2090560450309943/3801019000563532 or 5226476081713045/9011165658125939?", "options": [ "1822688025030948/3097854428266356", "2090560450309943/3801019000563532", "5226476081713045/9011165658125939", "2753562731215419/5613939836078244" ], "answer": "5226476081713045/9011165658125939", "correct_value": 0.5800000000000001, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001336", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 3184319397227633/6537363710836453 or 4735064900814904/9084226096404750?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03414497850706527, "pure_math": { "question": "Which is larger: 3184319397227633/6537363710836453 or 4735064900814904/9084226096404750?", "options": [ "1361303571343678/2583866449998175", "2147287797774732/4919877706598877", "4735064900814904/9084226096404750", "3184319397227633/6537363710836453" ], "answer": "4735064900814904/9084226096404750", "correct_value": 0.521240318169634, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 4280842766102084/8790908275021653 or 1649883149166605/3208234898409106?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.027302560588856217, "pure_math": { "question": "Which is larger: 4280842766102084/8790908275021653 or 1649883149166605/3208234898409106?", "options": [ "907394138304993/1739573698928892", "4280842766102084/8790908275021653", "1474013443098467/3418804474348912", "1649883149166605/3208234898409106" ], "answer": "1649883149166605/3208234898409106", "correct_value": 0.514265071421281, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 858876027344172/1561592776989402 or 976992479307539/1684469791909549?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 858876027344172/1561592776989402 or 976992479307539/1684469791909549?", "options": [ "2656181287220544/4424188846685126", "976992479307539/1684469791909549", "3275163012097004/4832456372172441", "858876027344172/1561592776989402" ], "answer": "976992479307539/1684469791909549", "correct_value": 0.5800000000000003, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001337", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 3357867569362663/6896901737119165 or 3508689182343676/6854721606247319?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.024998489820744763, "pure_math": { "question": "Which is larger: 3357867569362663/6896901737119165 or 3508689182343676/6854721606247319?", "options": [ "3508689182343676/6854721606247319", "3357867569362663/6896901737119165", "3113051997313280/5985360217372915", "3937898031415452/7008676083650705" ], "answer": "3508689182343676/6854721606247319", "correct_value": 0.5118645780079376, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 3918171302961720/8043202777600170 or 1537694615852214/3031369449113607?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.020120024910669054, "pure_math": { "question": "Which is larger: 3918171302961720/8043202777600170 or 1537694615852214/3031369449113607?", "options": [ "1537694615852214/3031369449113607", "4069095967496376/7697547514485231", "4406720150529692/9765137645209172", "3918171302961720/8043202777600170" ], "answer": "1537694615852214/3031369449113607", "correct_value": 0.5072607089518061, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 3965166756125506/7209394102046374 or 3985850659649933/6872156309741264?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 3965166756125506/7209394102046374 or 3985850659649933/6872156309741264?", "options": [ "3985850659649933/6872156309741264", "3965166756125506/7209394102046374", "1098279609546152/2274623490065534", "1236082494589244/2008505140825774" ], "answer": "3985850659649933/6872156309741264", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001338", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 868845075584592/1833935462398168 or 2830049248898475/5519126244147128?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03901135052470528, "pure_math": { "question": "Which is larger: 868845075584592/1833935462398168 or 2830049248898475/5519126244147128?", "options": [ "2457027297834989/5222233173903642", "2163939436660623/3879160627252800", "2830049248898475/5519126244147128", "868845075584592/1833935462398168" ], "answer": "2830049248898475/5519126244147128", "correct_value": 0.5127712474233869, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 3025658439119534/6159427615997753 or 4576296843995681/9047154908565248?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.014603216533017382, "pure_math": { "question": "Which is larger: 3025658439119534/6159427615997753 or 4576296843995681/9047154908565248?", "options": [ "2316060690529954/4035889938519996", "3259067151437114/6550281156545806", "3025658439119534/6159427615997753", "4576296843995681/9047154908565248" ], "answer": "4576296843995681/9047154908565248", "correct_value": 0.505827178847479, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 1304687838849693/2372159706999441 or 3684375518214571/6352371583128570?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 1304687838849693/2372159706999441 or 3684375518214571/6352371583128570?", "options": [ "5217493705867827/9439566958230191", "3684375518214571/6352371583128570", "1161653333803923/1854000496766655", "1304687838849693/2372159706999441" ], "answer": "3684375518214571/6352371583128570", "correct_value": 0.5800000000000001, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001339", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 1764942955006914/3609737891832292 or 1501284997370891/2853901654670184?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03710714156055317, "pure_math": { "question": "Which is larger: 1764942955006914/3609737891832292 or 1501284997370891/2853901654670184?", "options": [ "2512882737604272/5493221901481943", "5869632320999520/9404599401982372", "1764942955006914/3609737891832292", "1501284997370891/2853901654670184" ], "answer": "1501284997370891/2853901654670184", "correct_value": 0.5260465072137847, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 3610604033953450/7294690953850435 or 3482223572809720/6824231230581686?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.015310122834740314, "pure_math": { "question": "Which is larger: 3610604033953450/7294690953850435 or 3482223572809720/6824231230581686?", "options": [ "3482223572809720/6824231230581686", "3515815408424384/5940989860873307", "1889832051803909/3765293520207879", "3610604033953450/7294690953850435" ], "answer": "3482223572809720/6824231230581686", "correct_value": 0.5102733854041609, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 1649492839039788/2999077889163250 or 5738162613041919/9893383815589517?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 1649492839039788/2999077889163250 or 5738162613041919/9893383815589517?", "options": [ "5738162613041919/9893383815589517", "3448778944830642/5244954294335088", "4401721037423114/8187151841345460", "1649492839039788/2999077889163250" ], "answer": "5738162613041919/9893383815589517", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001340", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 4346702105191448/9061446599882290 or 3223360345624141/6244385429075352?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03650952274721364, "pure_math": { "question": "Which is larger: 4346702105191448/9061446599882290 or 3223360345624141/6244385429075352?", "options": [ "3223360345624141/6244385429075352", "3502200693007688/5815011153392818", "3597839034805179/6201464759292738", "4346702105191448/9061446599882290" ], "answer": "3223360345624141/6244385429075352", "correct_value": 0.5162013751770357, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 3477685179767593/7025794002303377 or 2069756507867359/4050048134799183?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.01605670821560251, "pure_math": { "question": "Which is larger: 3477685179767593/7025794002303377 or 2069756507867359/4050048134799183?", "options": [ "1944551476003737/3197434332903640", "3948955896284518/7752868886091889", "2069756507867359/4050048134799183", "3477685179767593/7025794002303377" ], "answer": "2069756507867359/4050048134799183", "correct_value": 0.5110449157587569, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 4857509660182905/8831835745787100 or 4980380569899551/8586863051550951?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 4857509660182905/8831835745787100 or 4980380569899551/8586863051550951?", "options": [ "4857509660182905/8831835745787100", "4980380569899551/8586863051550951", "3461399778938974/6486995465200153", "3716951044735863/6536425135240954" ], "answer": "4980380569899551/8586863051550951", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001341", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 1020076506990537/2122399223857063 or 932765900007092/1757835664258900?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.05000892709898208, "pure_math": { "question": "Which is larger: 1020076506990537/2122399223857063 or 932765900007092/1757835664258900?", "options": [ "5189866683564466/8476172031212809", "2514334797134233/4143770615781618", "932765900007092/1757835664258900", "1020076506990537/2122399223857063" ], "answer": "932765900007092/1757835664258900", "correct_value": 0.5306331638232771, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 4882122712277066/9949157996743252 or 2793273194906070/5468860323376772?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.0200525273221292, "pure_math": { "question": "Which is larger: 4882122712277066/9949157996743252 or 2793273194906070/5468860323376772?", "options": [ "4573935773120604/9851389365326994", "2793273194906070/5468860323376772", "4087262072370951/8283477726814000", "4882122712277066/9949157996743252" ], "answer": "2793273194906070/5468860323376772", "correct_value": 0.5107596518722847, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 3735413737163654/6791661340297551 or 2444359227159502/4214412460619831?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 3735413737163654/6791661340297551 or 2444359227159502/4214412460619831?", "options": [ "3178052751053222/5990155524625409", "2444359227159502/4214412460619831", "3451430251610944/5853556535967252", "3735413737163654/6791661340297551" ], "answer": "2444359227159502/4214412460619831", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001342", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 3842046505581364/8014540785646586 or 2079021598737597/4074142207125622?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.030912282707871108, "pure_math": { "question": "Which is larger: 3842046505581364/8014540785646586 or 2079021598737597/4074142207125622?", "options": [ "2546441418693976/4211496326731502", "2079021598737597/4074142207125622", "2571173230323056/5078702610485445", "3842046505581364/8014540785646586" ], "answer": "2079021598737597/4074142207125622", "correct_value": 0.5102967675260366, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 4566998352606383/9384341596401488 or 1837927661125912/3652609981355893?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.01652045454787282, "pure_math": { "question": "Which is larger: 4566998352606383/9384341596401488 or 1837927661125912/3652609981355893?", "options": [ "1837927661125912/3652609981355893", "4628647679015465/9023928657999394", "1430096137084976/3048819136382965", "4566998352606383/9384341596401488" ], "answer": "1837927661125912/3652609981355893", "correct_value": 0.5031820179289033, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 3433027144391820/6241867535257853 or 5279657858942153/9102858377486473?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 3433027144391820/6241867535257853 or 5279657858942153/9102858377486473?", "options": [ "5279657858942153/9102858377486473", "3433027144391820/6241867535257853", "5410745186934916/9203139415666231", "5820333535559982/9885994348382550" ], "answer": "5279657858942153/9102858377486473", "correct_value": 0.5799999999999998, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001343", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 2699181860479841/5570746350276855 or 2260496818825161/4309799787303831?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.039973715375536956, "pure_math": { "question": "Which is larger: 2699181860479841/5570746350276855 or 2260496818825161/4309799787303831?", "options": [ "2195712951693809/4851865973747759", "2260496818825161/4309799787303831", "2699181860479841/5570746350276855", "2526950718672560/5804439968004486" ], "answer": "2260496818825161/4309799787303831", "correct_value": 0.5245015848495612, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 2411359745650468/4906083988565643 or 892614852989565/1762269871574573?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.015010311186909653, "pure_math": { "question": "Which is larger: 2411359745650468/4906083988565643 or 892614852989565/1762269871574573?", "options": [ "801296250844713/1637655130544094", "892614852989565/1762269871574573", "3548756253224950/6813772661910614", "2411359745650468/4906083988565643" ], "answer": "892614852989565/1762269871574573", "correct_value": 0.506514278764819, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 848228667648923/1542233941179859 or 1471728980427774/2537463759358231?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 848228667648923/1542233941179859 or 1471728980427774/2537463759358231?", "options": [ "1471728980427774/2537463759358231", "912475414456964/1450986385541645", "2873846704156870/4674876294853463", "848228667648923/1542233941179859" ], "answer": "1471728980427774/2537463759358231", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001344", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 3621708825890507/7452671663685288 or 1492794305435217/2890616693142384?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.030466481071862173, "pure_math": { "question": "Which is larger: 3621708825890507/7452671663685288 or 1492794305435217/2890616693142384?", "options": [ "3621708825890507/7452671663685288", "3262680298056836/5922730934314292", "1492794305435217/2890616693142384", "2813884828699646/5983305954626777" ], "answer": "1492794305435217/2890616693142384", "correct_value": 0.5164276221668128, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 2952429997147698/6049155652411894 or 2857179685663812/5625730634265959?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.019804012344002753, "pure_math": { "question": "Which is larger: 2952429997147698/6049155652411894 or 2857179685663812/5625730634265959?", "options": [ "2952429997147698/6049155652411894", "4468144572402968/9328792895390672", "2278259587298962/5314973427394324", "2857179685663812/5625730634265959" ], "answer": "2857179685663812/5625730634265959", "correct_value": 0.5078770868020087, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 4159767551635229/7563213730245870 or 1660393523062978/2862747453556858?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 4159767551635229/7563213730245870 or 1660393523062978/2862747453556858?", "options": [ "1660393523062978/2862747453556858", "3072491440442170/4733018207296292", "5231864442496922/8031614635696683", "4159767551635229/7563213730245870" ], "answer": "1660393523062978/2862747453556858", "correct_value": 0.5800000000000001, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001345", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 4017905752275166/8280230891777500 or 3551818234881982/6954246519204120?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.025500128915934617, "pure_math": { "question": "Which is larger: 4017905752275166/8280230891777500 or 3551818234881982/6954246519204120?", "options": [ "3384677958801852/7237252955759563", "4259549853349834/8597784492212904", "3551818234881982/6954246519204120", "4017905752275166/8280230891777500" ], "answer": "3551818234881982/6954246519204120", "correct_value": 0.510740915651128, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 1702967706089885/3452247204125610 or 4513505898424737/8832293174558281?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.01773059171617042, "pure_math": { "question": "Which is larger: 1702967706089885/3452247204125610 or 4513505898424737/8832293174558281?", "options": [ "4513505898424737/8832293174558281", "4593299210122059/8454688449273046", "2661428196781989/5967407350846953", "1702967706089885/3452247204125610" ], "answer": "4513505898424737/8832293174558281", "correct_value": 0.5110231068219115, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 1006131235611399/1829329519293452 or 5723939706188107/9868861562393289?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 1006131235611399/1829329519293452 or 5723939706188107/9868861562393289?", "options": [ "5723939706188107/9868861562393289", "2188054346681618/3340912773128747", "1006131235611399/1829329519293452", "3544117965402570/7266390895576074" ], "answer": "5723939706188107/9868861562393289", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001346", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 1826661504330556/3750083738054417 or 1889569696386313/3662348912254606?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.0288458931676861, "pure_math": { "question": "Which is larger: 1826661504330556/3750083738054417 or 1889569696386313/3662348912254606?", "options": [ "1826661504330556/3750083738054417", "1889569696386313/3662348912254606", "5499674714660168/9598858562068483", "4992537151528782/9357006574357913" ], "answer": "1889569696386313/3662348912254606", "correct_value": 0.5159447506663315, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 3091795815947612/6319241858847855 or 3286129086726898/6490623644348618?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.017021759928805613, "pure_math": { "question": "Which is larger: 3091795815947612/6319241858847855 or 3286129086726898/6490623644348618?", "options": [ "3286129086726898/6490623644348618", "3564361663965602/7638636160073444", "3181952539084172/6815574434214378", "3091795815947612/6319241858847855" ], "answer": "3286129086726898/6490623644348618", "correct_value": 0.506288650642705, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 682109100591654/1240198364712097 or 2588761817061519/4463382443209516?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 682109100591654/1240198364712097 or 2588761817061519/4463382443209516?", "options": [ "931714019916183/1519752294519979", "4072118219743160/6398357034550668", "682109100591654/1240198364712097", "2588761817061519/4463382443209516" ], "answer": "2588761817061519/4463382443209516", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001347", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 1096250179524652/2296558316495842 or 1784808749420023/3400676061909729?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.0474945858257278, "pure_math": { "question": "Which is larger: 1096250179524652/2296558316495842 or 1784808749420023/3400676061909729?", "options": [ "3036772285432694/5436869902447668", "1096250179524652/2296558316495842", "802898003638481/1851625148795914", "1784808749420023/3400676061909729" ], "answer": "1784808749420023/3400676061909729", "correct_value": 0.5248393898528877, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 699403959181126/1406969445565431 or 4744875408322916/9407134962829650?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.007291519755757092, "pure_math": { "question": "Which is larger: 699403959181126/1406969445565431 or 4744875408322916/9407134962829650?", "options": [ "699403959181126/1406969445565431", "1425867556058326/2406777180737023", "2553390980863545/5211276509125470", "4744875408322916/9407134962829650" ], "answer": "4744875408322916/9407134962829650", "correct_value": 0.5043911272742775, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 3866653637213065/7030279340387390 or 4671024546168615/8053490596842440?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 3866653637213065/7030279340387390 or 4671024546168615/8053490596842440?", "options": [ "1833486152750768/3450457265323523", "4671024546168615/8053490596842440", "3866653637213065/7030279340387390", "3331074583081438/6078826257788332" ], "answer": "4671024546168615/8053490596842440", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001348", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 2298717635817654/4805662446333514 or 5001350293010689/9725754866200039?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.035902522616023225, "pure_math": { "question": "Which is larger: 2298717635817654/4805662446333514 or 5001350293010689/9725754866200039?", "options": [ "5001350293010689/9725754866200039", "5799553835734272/9540859078332400", "3818012474684758/8057882924126315", "2298717635817654/4805662446333514" ], "answer": "5001350293010689/9725754866200039", "correct_value": 0.5142377493382961, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 3347449571050627/6806356823192244 or 2715721504533959/5332752313458164?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.01744102436029371, "pure_math": { "question": "Which is larger: 3347449571050627/6806356823192244 or 2715721504533959/5332752313458164?", "options": [ "2715721504533959/5332752313458164", "3859215684066802/6656396771908862", "1517759020020946/3193750157580736", "3347449571050627/6806356823192244" ], "answer": "2715721504533959/5332752313458164", "correct_value": 0.5092532607749932, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 4883093768562807/8878352306477831 or 3905760067025311/6734069081078122?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 4883093768562807/8878352306477831 or 3905760067025311/6734069081078122?", "options": [ "4883093768562807/8878352306477831", "3709512976487030/5843888615522402", "5237938060975568/9998715313347575", "3905760067025311/6734069081078122" ], "answer": "3905760067025311/6734069081078122", "correct_value": 0.5800000000000001, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001349", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 1027571917473488/2152901084379104 or 3734325433337088/7289257027522810?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.035008967445614936, "pure_math": { "question": "Which is larger: 1027571917473488/2152901084379104 or 3734325433337088/7289257027522810?", "options": [ "1027571917473488/2152901084379104", "4529271578469340/9331157566977177", "1392319536516942/2719151490458338", "3734325433337088/7289257027522810" ], "answer": "3734325433337088/7289257027522810", "correct_value": 0.5123053583152584, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 557411526323285/1130709848730534 or 3546689171745906/6899542062418101?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.02107218229689406, "pure_math": { "question": "Which is larger: 557411526323285/1130709848730534 or 3546689171745906/6899542062418101?", "options": [ "557411526323285/1130709848730534", "3731970706143090/6113008614288064", "1621094735634453/2672551080042966", "3546689171745906/6899542062418101" ], "answer": "3546689171745906/6899542062418101", "correct_value": 0.5140470395948117, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 5342268111289720/9713214747799490 or 2618287552669848/4514288883913531?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 5342268111289720/9713214747799490 or 2618287552669848/4514288883913531?", "options": [ "1925161218557222/3125537690952804", "2618287552669848/4514288883913531", "5342268111289720/9713214747799490", "4722548331758751/9353302870040205" ], "answer": "2618287552669848/4514288883913531", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001350", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 3448731705750732/7182974452955876 or 3291552320887094/6360664293114701?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.03735977369852833, "pure_math": { "question": "Which is larger: 3448731705750732/7182974452955876 or 3291552320887094/6360664293114701?", "options": [ "3448731705750732/7182974452955876", "986241500587478/2176322891567777", "3291552320887094/6360664293114701", "5302167276349741/9154611338433432" ], "answer": "3291552320887094/6360664293114701", "correct_value": 0.5174856224451488, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 4424126540464797/9053339842894726 or 4674456197221998/9140524968174877?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.022725675403263257, "pure_math": { "question": "Which is larger: 4424126540464797/9053339842894726 or 4674456197221998/9140524968174877?", "options": [ "3341409808678908/6179155380632538", "4674456197221998/9140524968174877", "4614244916286202/9993913919507941", "4424126540464797/9053339842894726" ], "answer": "4674456197221998/9140524968174877", "correct_value": 0.5113990950735693, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 860250341689244/1564091530344079 or 5469455236399968/9430095235172361?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 860250341689244/1564091530344079 or 5469455236399968/9430095235172361?", "options": [ "860250341689244/1564091530344079", "5806163335720849/9450590292868555", "5469455236399968/9430095235172361", "6170480191789436/9488027472527746" ], "answer": "5469455236399968/9430095235172361", "correct_value": 0.5799999999999998, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "relative_distance_001351", "category": "relative_distance", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "Which is larger: 4080460046716636/8330135989229760 or 1615192380814155/3090076401254617?", "ns_shortcut": "One is below 1/2, the other above; no computation needed.", "shortcut_strength": 0.032859907584392556, "pure_math": { "question": "Which is larger: 4080460046716636/8330135989229760 or 1615192380814155/3090076401254617?", "options": [ "2716179641619403/4597548158142664", "4080460046716636/8330135989229760", "3086518073099383/6853099662633411", "1615192380814155/3090076401254617" ], "answer": "1615192380814155/3090076401254617", "correct_value": 0.5227030568429838, "explanation": "Both straddle 1/2; compare to benchmark." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "Which is larger: 1696212746935578/3487303918873317 or 2140037932990052/4206847130692648?", "ns_shortcut": "Compare to 1/2, but margin is tight.", "shortcut_strength": 0.022306923920217547, "pure_math": { "question": "Which is larger: 1696212746935578/3487303918873317 or 2140037932990052/4206847130692648?", "options": [ "3224031964394206/5515766130666183", "1696212746935578/3487303918873317", "2140037932990052/4206847130692648", "4048408288742299/7238870431254089" ], "answer": "2140037932990052/4206847130692648", "correct_value": 0.5087035175052105, "explanation": "One is very close to 1/2, shortcut harder to apply." } }, "control": { "task_type": "control", "math_expression": "Which is larger: 3053434746819050/5551699539670999 or 3586256826101075/6183201424312199?", "ns_shortcut": "Both on same side; must cross-multiply.", "shortcut_strength": 0.0, "pure_math": { "question": "Which is larger: 3053434746819050/5551699539670999 or 3586256826101075/6183201424312199?", "options": [ "4741776921689394/7992634183251650", "2164937090226504/3626730178380152", "3586256826101075/6183201424312199", "3053434746819050/5551699539670999" ], "answer": "3586256826101075/6183201424312199", "correct_value": 0.58, "explanation": "Both above 1/2; benchmark comparison won't help." } } }, { "family_id": "cancellation_identity_001352", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9353983493802460 + 5630721528224774 - 5630721528224775", "ns_shortcut": "Near-cancellation: 5630721528224774 - 5630721528224775 = -1, so answer ≈ 9353983493802460 + -1.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "Evaluate: 9353983493802460 + 5630721528224774 - 5630721528224775", "options": [ "9353983493802469", "9353983493802459", "9353983493802449", "9353983493802479" ], "answer": "9353983493802459", "correct_value": 9353983493802459, "explanation": "B-C = -1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6932464695698539 + 8723964766500443 - 8380847847347378", "ns_shortcut": "Near-cancellation: 8723964766500443 - 8380847847347378 = 343116919153065, offset is manageable.", "shortcut_strength": 0.9606696120013443, "pure_math": { "question": "Evaluate: 6932464695698539 + 8723964766500443 - 8380847847347378", "options": [ "7275581614851604", "7275581614851624", "7275581614851614", "7275581614851594" ], "answer": "7275581614851604", "correct_value": 7275581614851604, "explanation": "B-C = 343116919153065; still a moderate shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "9353983493802460 + 5630721528224774 - 8551746009954387", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 9353983493802460 + 5630721528224774 - 8551746009954387", "options": [ "6432959012072847", "6432959012072867", "6432959012072837", "6432959012072857" ], "answer": "6432959012072847", "correct_value": 6432959012072847, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001353", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "4910714678897393 + 4527555332700942 - 4527555332700941", "ns_shortcut": "Near-cancellation: 4527555332700942 - 4527555332700941 = 1, so answer ≈ 4910714678897393 + 1.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "Evaluate: 4910714678897393 + 4527555332700942 - 4527555332700941", "options": [ "4910714678897404", "4910714678897384", "4910714678897394", "4910714678897414" ], "answer": "4910714678897394", "correct_value": 4910714678897394, "explanation": "B-C = 1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6117261882326184 + 6172180626665470 - 6449815536146337", "ns_shortcut": "Near-cancellation: 6172180626665470 - 6449815536146337 = -277634909480867, offset is manageable.", "shortcut_strength": 0.9550183433904365, "pure_math": { "question": "Evaluate: 6117261882326184 + 6172180626665470 - 6449815536146337", "options": [ "5839626972845317", "5839626972845337", "5839626972845327", "5839626972845307" ], "answer": "5839626972845317", "correct_value": 5839626972845317, "explanation": "B-C = -277634909480867; still a moderate shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4910714678897393 + 4527555332700942 - 2108691550966365", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 4910714678897393 + 4527555332700942 - 2108691550966365", "options": [ "7329578460631980", "7329578460631960", "7329578460631970", "7329578460631990" ], "answer": "7329578460631970", "correct_value": 7329578460631970, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001354", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8960759563483501 + 4679561405662685 - 4679561405662686", "ns_shortcut": "Near-cancellation: 4679561405662685 - 4679561405662686 = -1, so answer ≈ 8960759563483501 + -1.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "Evaluate: 8960759563483501 + 4679561405662685 - 4679561405662686", "options": [ "8960759563483490", "8960759563483500", "8960759563483510", "8960759563483520" ], "answer": "8960759563483500", "correct_value": 8960759563483500, "explanation": "B-C = -1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1131468034302009 + 6652232431651857 - 6927018998142107", "ns_shortcut": "Near-cancellation: 6652232431651857 - 6927018998142107 = -274786566490250, offset is manageable.", "shortcut_strength": 0.9586925788728017, "pure_math": { "question": "Evaluate: 1131468034302009 + 6652232431651857 - 6927018998142107", "options": [ "856681467811779", "856681467811769", "856681467811749", "856681467811759" ], "answer": "856681467811759", "correct_value": 856681467811759, "explanation": "B-C = -274786566490250; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "8960759563483501 + 4679561405662685 - 6136976713782523", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 8960759563483501 + 4679561405662685 - 6136976713782523", "options": [ "7503344255363653", "7503344255363673", "7503344255363663", "7503344255363683" ], "answer": "7503344255363663", "correct_value": 7503344255363663, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001355", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7491770029949762 + 6367214608309250 - 6367214608309251", "ns_shortcut": "Near-cancellation: 6367214608309250 - 6367214608309251 = -1, so answer ≈ 7491770029949762 + -1.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "Evaluate: 7491770029949762 + 6367214608309250 - 6367214608309251", "options": [ "7491770029949761", "7491770029949751", "7491770029949771", "7491770029949781" ], "answer": "7491770029949761", "correct_value": 7491770029949761, "explanation": "B-C = -1; result ≈ A + offset." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2435945251662927 + 4700585321318475 - 4876028297576304", "ns_shortcut": "Near-cancellation: 4700585321318475 - 4876028297576304 = -175442976257829, offset is manageable.", "shortcut_strength": 0.9626763553334208, "pure_math": { "question": "Evaluate: 2435945251662927 + 4700585321318475 - 4876028297576304", "options": [ "2260502275405088", "2260502275405108", "2260502275405118", "2260502275405098" ], "answer": "2260502275405098", "correct_value": 2260502275405098, "explanation": "B-C = -175442976257829; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "7491770029949762 + 6367214608309250 - 4069256424506097", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 7491770029949762 + 6367214608309250 - 4069256424506097", "options": [ "9789728213752905", "9789728213752925", "9789728213752935", "9789728213752915" ], "answer": "9789728213752915", "correct_value": 9789728213752915, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "cancellation_identity_001356", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3047125066342880 + 7650052116765667 - 7650052116765665", "ns_shortcut": "Near-cancellation: 7650052116765667 - 7650052116765665 = 2, so answer ≈ 3047125066342880 + 2.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "Evaluate: 3047125066342880 + 7650052116765667 - 7650052116765665", "options": [ "3047125066342882", "3047125066342872", "3047125066342902", "3047125066342892" ], "answer": "3047125066342882", "correct_value": 3047125066342882, "explanation": "B-C = 2; result ≈ A + offset." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4097188869600102 + 3822104829843905 - 3715016059969892", "ns_shortcut": "Near-cancellation: 3822104829843905 - 3715016059969892 = 107088769874013, offset is manageable.", "shortcut_strength": 0.971981728748558, "pure_math": { "question": "Evaluate: 4097188869600102 + 3822104829843905 - 3715016059969892", "options": [ "4204277639474105", "4204277639474135", "4204277639474115", "4204277639474125" ], "answer": "4204277639474115", "correct_value": 4204277639474115, "explanation": "B-C = 107088769874013; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "3047125066342880 + 7650052116765667 - 2822989013939863", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 3047125066342880 + 7650052116765667 - 2822989013939863", "options": [ "7874188169168704", "7874188169168674", "7874188169168694", "7874188169168684" ], "answer": "7874188169168684", "correct_value": 7874188169168684, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "cancellation_identity_001357", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "6035876363532134 + 4163007159408050 - 4163007159408052", "ns_shortcut": "Near-cancellation: 4163007159408050 - 4163007159408052 = -2, so answer ≈ 6035876363532134 + -2.", "shortcut_strength": 0.9999999999999996, "pure_math": { "question": "Evaluate: 6035876363532134 + 4163007159408050 - 4163007159408052", "options": [ "6035876363532142", "6035876363532122", "6035876363532132", "6035876363532152" ], "answer": "6035876363532132", "correct_value": 6035876363532132, "explanation": "B-C = -2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9062968696989280 + 4468478095973296 - 4570133501018174", "ns_shortcut": "Near-cancellation: 4468478095973296 - 4570133501018174 = -101655405044878, offset is manageable.", "shortcut_strength": 0.9772505531275887, "pure_math": { "question": "Evaluate: 9062968696989280 + 4468478095973296 - 4570133501018174", "options": [ "8961313291944402", "8961313291944392", "8961313291944412", "8961313291944422" ], "answer": "8961313291944402", "correct_value": 8961313291944402, "explanation": "B-C = -101655405044878; still a moderate shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6035876363532134 + 4163007159408050 - 1647854337727897", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 6035876363532134 + 4163007159408050 - 1647854337727897", "options": [ "8551029185212277", "8551029185212297", "8551029185212287", "8551029185212307" ], "answer": "8551029185212287", "correct_value": 8551029185212287, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001358", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8968831756244665 + 2254422786222179 - 2254422786222177", "ns_shortcut": "Near-cancellation: 2254422786222179 - 2254422786222177 = 2, so answer ≈ 8968831756244665 + 2.", "shortcut_strength": 0.9999999999999991, "pure_math": { "question": "Evaluate: 8968831756244665 + 2254422786222179 - 2254422786222177", "options": [ "8968831756244677", "8968831756244667", "8968831756244657", "8968831756244687" ], "answer": "8968831756244667", "correct_value": 8968831756244667, "explanation": "B-C = 2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7118494637116223 + 7566864667992923 - 7917281360912575", "ns_shortcut": "Near-cancellation: 7566864667992923 - 7917281360912575 = -350416692919652, offset is manageable.", "shortcut_strength": 0.9536906356470363, "pure_math": { "question": "Evaluate: 7118494637116223 + 7566864667992923 - 7917281360912575", "options": [ "6768077944196561", "6768077944196591", "6768077944196571", "6768077944196581" ], "answer": "6768077944196571", "correct_value": 6768077944196571, "explanation": "B-C = -350416692919652; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "8968831756244665 + 2254422786222179 - 1452360350818663", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 8968831756244665 + 2254422786222179 - 1452360350818663", "options": [ "9770894191648181", "9770894191648171", "9770894191648201", "9770894191648191" ], "answer": "9770894191648181", "correct_value": 9770894191648181, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001359", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1262944667178215 + 9950983607937084 - 9950983607937085", "ns_shortcut": "Near-cancellation: 9950983607937084 - 9950983607937085 = -1, so answer ≈ 1262944667178215 + -1.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "Evaluate: 1262944667178215 + 9950983607937084 - 9950983607937085", "options": [ "1262944667178234", "1262944667178214", "1262944667178224", "1262944667178204" ], "answer": "1262944667178214", "correct_value": 1262944667178214, "explanation": "B-C = -1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1825069733510673 + 3337740315817758 - 3462818393281205", "ns_shortcut": "Near-cancellation: 3337740315817758 - 3462818393281205 = -125078077463447, offset is manageable.", "shortcut_strength": 0.96252612077977, "pure_math": { "question": "Evaluate: 1825069733510673 + 3337740315817758 - 3462818393281205", "options": [ "1699991656047226", "1699991656047246", "1699991656047216", "1699991656047236" ], "answer": "1699991656047226", "correct_value": 1699991656047226, "explanation": "B-C = -125078077463447; still a moderate shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1262944667178215 + 9950983607937084 - 15979112010600517", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 1262944667178215 + 9950983607937084 - 15979112010600517", "options": [ "-4765183735485218", "-4765183735485232", "-4765183735485202", "-4765183735485212" ], "answer": "-4765183735485218", "correct_value": -4765183735485218, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001360", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7232083922624080 + 5715921555175029 - 5715921555175028", "ns_shortcut": "Near-cancellation: 5715921555175029 - 5715921555175028 = 1, so answer ≈ 7232083922624080 + 1.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "Evaluate: 7232083922624080 + 5715921555175029 - 5715921555175028", "options": [ "7232083922624091", "7232083922624101", "7232083922624081", "7232083922624071" ], "answer": "7232083922624081", "correct_value": 7232083922624081, "explanation": "B-C = 1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1049152041646239 + 5267674693859319 - 5426348742776347", "ns_shortcut": "Near-cancellation: 5267674693859319 - 5426348742776347 = -158674048917028, offset is manageable.", "shortcut_strength": 0.969877781347812, "pure_math": { "question": "Evaluate: 1049152041646239 + 5267674693859319 - 5426348742776347", "options": [ "890477992729211", "890477992729231", "890477992729221", "890477992729201" ], "answer": "890477992729211", "correct_value": 890477992729211, "explanation": "B-C = -158674048917028; still a moderate shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "7232083922624080 + 5715921555175029 - 8853656161363979", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 7232083922624080 + 5715921555175029 - 8853656161363979", "options": [ "4094349316435140", "4094349316435120", "4094349316435150", "4094349316435130" ], "answer": "4094349316435130", "correct_value": 4094349316435130, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "cancellation_identity_001361", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3091024518766016 + 1844576646325284 - 1844576646325286", "ns_shortcut": "Near-cancellation: 1844576646325284 - 1844576646325286 = -2, so answer ≈ 3091024518766016 + -2.", "shortcut_strength": 0.9999999999999989, "pure_math": { "question": "Evaluate: 3091024518766016 + 1844576646325284 - 1844576646325286", "options": [ "3091024518766014", "3091024518766024", "3091024518766034", "3091024518766004" ], "answer": "3091024518766014", "correct_value": 3091024518766014, "explanation": "B-C = -2; result ≈ A + offset." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7180261019784450 + 2148455851236618 - 2206230806703453", "ns_shortcut": "Near-cancellation: 2148455851236618 - 2206230806703453 = -57774955466835, offset is manageable.", "shortcut_strength": 0.9731086140617781, "pure_math": { "question": "Evaluate: 7180261019784450 + 2148455851236618 - 2206230806703453", "options": [ "7122486064317605", "7122486064317625", "7122486064317615", "7122486064317635" ], "answer": "7122486064317615", "correct_value": 7122486064317615, "explanation": "B-C = -57774955466835; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "3091024518766016 + 1844576646325284 - 2908834662791459", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 3091024518766016 + 1844576646325284 - 2908834662791459", "options": [ "2026766502299861", "2026766502299851", "2026766502299841", "2026766502299831" ], "answer": "2026766502299841", "correct_value": 2026766502299841, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001362", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5898260678145156 + 1969656921692371 - 1969656921692369", "ns_shortcut": "Near-cancellation: 1969656921692371 - 1969656921692369 = 2, so answer ≈ 5898260678145156 + 2.", "shortcut_strength": 0.999999999999999, "pure_math": { "question": "Evaluate: 5898260678145156 + 1969656921692371 - 1969656921692369", "options": [ "5898260678145168", "5898260678145178", "5898260678145158", "5898260678145148" ], "answer": "5898260678145158", "correct_value": 5898260678145158, "explanation": "B-C = 2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1258754971128922 + 8791475070802598 - 8606538127542485", "ns_shortcut": "Near-cancellation: 8791475070802598 - 8606538127542485 = 184936943260113, offset is manageable.", "shortcut_strength": 0.9789640598681434, "pure_math": { "question": "Evaluate: 1258754971128922 + 8791475070802598 - 8606538127542485", "options": [ "1443691914389035", "1443691914389045", "1443691914389055", "1443691914389025" ], "answer": "1443691914389035", "correct_value": 1443691914389035, "explanation": "B-C = 184936943260113; still a moderate shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5898260678145156 + 1969656921692371 - 2658362626388779", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 5898260678145156 + 1969656921692371 - 2658362626388779", "options": [ "5209554973448748", "5209554973448768", "5209554973448758", "5209554973448738" ], "answer": "5209554973448748", "correct_value": 5209554973448748, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001363", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7502420366077232 + 8831652800854053 - 8831652800854054", "ns_shortcut": "Near-cancellation: 8831652800854053 - 8831652800854054 = -1, so answer ≈ 7502420366077232 + -1.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "Evaluate: 7502420366077232 + 8831652800854053 - 8831652800854054", "options": [ "7502420366077251", "7502420366077231", "7502420366077241", "7502420366077221" ], "answer": "7502420366077231", "correct_value": 7502420366077231, "explanation": "B-C = -1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4406749603386068 + 3433358178924696 - 3303039207991090", "ns_shortcut": "Near-cancellation: 3433358178924696 - 3303039207991090 = 130318970933606, offset is manageable.", "shortcut_strength": 0.962043292851426, "pure_math": { "question": "Evaluate: 4406749603386068 + 3433358178924696 - 3303039207991090", "options": [ "4537068574319664", "4537068574319684", "4537068574319694", "4537068574319674" ], "answer": "4537068574319674", "correct_value": 4537068574319674, "explanation": "B-C = 130318970933606; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "7502420366077232 + 8831652800854053 - 12014950909439814", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 7502420366077232 + 8831652800854053 - 12014950909439814", "options": [ "4319122257491481", "4319122257491491", "4319122257491471", "4319122257491461" ], "answer": "4319122257491471", "correct_value": 4319122257491471, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001364", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "6597401479046184 + 1533322980861231 - 1533322980861229", "ns_shortcut": "Near-cancellation: 1533322980861231 - 1533322980861229 = 2, so answer ≈ 6597401479046184 + 2.", "shortcut_strength": 0.9999999999999987, "pure_math": { "question": "Evaluate: 6597401479046184 + 1533322980861231 - 1533322980861229", "options": [ "6597401479046206", "6597401479046176", "6597401479046186", "6597401479046196" ], "answer": "6597401479046186", "correct_value": 6597401479046186, "explanation": "B-C = 2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "8350627460268120 + 1272733644786462 - 1241234753080512", "ns_shortcut": "Near-cancellation: 1272733644786462 - 1241234753080512 = 31498891705950, offset is manageable.", "shortcut_strength": 0.9752509947112816, "pure_math": { "question": "Evaluate: 8350627460268120 + 1272733644786462 - 1241234753080512", "options": [ "8382126351974060", "8382126351974090", "8382126351974070", "8382126351974080" ], "answer": "8382126351974070", "correct_value": 8382126351974070, "explanation": "B-C = 31498891705950; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6597401479046184 + 1533322980861231 - 2034843403417515", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 6597401479046184 + 1533322980861231 - 2034843403417515", "options": [ "6095881056489920", "6095881056489910", "6095881056489900", "6095881056489890" ], "answer": "6095881056489900", "correct_value": 6095881056489900, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001365", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1292608973276385 + 3971746765883373 - 3971746765883374", "ns_shortcut": "Near-cancellation: 3971746765883373 - 3971746765883374 = -1, so answer ≈ 1292608973276385 + -1.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "Evaluate: 1292608973276385 + 3971746765883373 - 3971746765883374", "options": [ "1292608973276384", "1292608973276374", "1292608973276404", "1292608973276394" ], "answer": "1292608973276384", "correct_value": 1292608973276384, "explanation": "B-C = -1; result ≈ A + offset." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9250212932742928 + 1094878449760473 - 1071046220038691", "ns_shortcut": "Near-cancellation: 1094878449760473 - 1071046220038691 = 23832229721782, offset is manageable.", "shortcut_strength": 0.9782329904045551, "pure_math": { "question": "Evaluate: 9250212932742928 + 1094878449760473 - 1071046220038691", "options": [ "9274045162464720", "9274045162464730", "9274045162464710", "9274045162464700" ], "answer": "9274045162464710", "correct_value": 9274045162464710, "explanation": "B-C = 23832229721782; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1292608973276385 + 3971746765883373 - 1584479035761489", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 1292608973276385 + 3971746765883373 - 1584479035761489", "options": [ "3679876703398269", "3679876703398279", "3679876703398259", "3679876703398289" ], "answer": "3679876703398269", "correct_value": 3679876703398269, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001366", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9428576322309430 + 6922867663852684 - 6922867663852683", "ns_shortcut": "Near-cancellation: 6922867663852684 - 6922867663852683 = 1, so answer ≈ 9428576322309430 + 1.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "Evaluate: 9428576322309430 + 6922867663852684 - 6922867663852683", "options": [ "9428576322309421", "9428576322309451", "9428576322309431", "9428576322309441" ], "answer": "9428576322309431", "correct_value": 9428576322309431, "explanation": "B-C = 1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1144964561939826 + 7485357783684748 - 7701007639517961", "ns_shortcut": "Near-cancellation: 7485357783684748 - 7701007639517961 = -215649855833213, offset is manageable.", "shortcut_strength": 0.971190441116489, "pure_math": { "question": "Evaluate: 1144964561939826 + 7485357783684748 - 7701007639517961", "options": [ "929314706106633", "929314706106623", "929314706106613", "929314706106603" ], "answer": "929314706106613", "correct_value": 929314706106613, "explanation": "B-C = -215649855833213; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "9428576322309430 + 6922867663852684 - 4637948985835889", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 9428576322309430 + 6922867663852684 - 4637948985835889", "options": [ "11713495000326215", "11713495000326225", "11713495000326245", "11713495000326235" ], "answer": "11713495000326225", "correct_value": 11713495000326225, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001367", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3218621651172795 + 2193585393187522 - 2193585393187520", "ns_shortcut": "Near-cancellation: 2193585393187522 - 2193585393187520 = 2, so answer ≈ 3218621651172795 + 2.", "shortcut_strength": 0.9999999999999991, "pure_math": { "question": "Evaluate: 3218621651172795 + 2193585393187522 - 2193585393187520", "options": [ "3218621651172797", "3218621651172787", "3218621651172807", "3218621651172817" ], "answer": "3218621651172797", "correct_value": 3218621651172797, "explanation": "B-C = 2; result ≈ A + offset." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1803572945425034 + 7256356429152883 - 7088111589818837", "ns_shortcut": "Near-cancellation: 7256356429152883 - 7088111589818837 = 168244839334046, offset is manageable.", "shortcut_strength": 0.9768141434373163, "pure_math": { "question": "Evaluate: 1803572945425034 + 7256356429152883 - 7088111589818837", "options": [ "1971817784759100", "1971817784759070", "1971817784759090", "1971817784759080" ], "answer": "1971817784759080", "correct_value": 1971817784759080, "explanation": "B-C = 168244839334046; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "3218621651172795 + 2193585393187522 - 1205843842814193", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 3218621651172795 + 2193585393187522 - 1205843842814193", "options": [ "4206363201546124", "4206363201546134", "4206363201546114", "4206363201546144" ], "answer": "4206363201546124", "correct_value": 4206363201546124, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001368", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9064145529201681 + 5695786894384243 - 5695786894384244", "ns_shortcut": "Near-cancellation: 5695786894384243 - 5695786894384244 = -1, so answer ≈ 9064145529201681 + -1.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "Evaluate: 9064145529201681 + 5695786894384243 - 5695786894384244", "options": [ "9064145529201670", "9064145529201690", "9064145529201680", "9064145529201700" ], "answer": "9064145529201680", "correct_value": 9064145529201680, "explanation": "B-C = -1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1841587699957485 + 5394487508839473 - 5175655578119812", "ns_shortcut": "Near-cancellation: 5394487508839473 - 5175655578119812 = 218831930719661, offset is manageable.", "shortcut_strength": 0.9594341574874202, "pure_math": { "question": "Evaluate: 1841587699957485 + 5394487508839473 - 5175655578119812", "options": [ "2060419630677136", "2060419630677156", "2060419630677146", "2060419630677166" ], "answer": "2060419630677146", "correct_value": 2060419630677146, "explanation": "B-C = 218831930719661; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "9064145529201681 + 5695786894384243 - 8493965405820755", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 9064145529201681 + 5695786894384243 - 8493965405820755", "options": [ "6265967017765179", "6265967017765159", "6265967017765189", "6265967017765169" ], "answer": "6265967017765169", "correct_value": 6265967017765169, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "cancellation_identity_001369", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1142310823752700 + 2324754350027802 - 2324754350027804", "ns_shortcut": "Near-cancellation: 2324754350027802 - 2324754350027804 = -2, so answer ≈ 1142310823752700 + -2.", "shortcut_strength": 0.9999999999999991, "pure_math": { "question": "Evaluate: 1142310823752700 + 2324754350027802 - 2324754350027804", "options": [ "1142310823752708", "1142310823752698", "1142310823752718", "1142310823752688" ], "answer": "1142310823752698", "correct_value": 1142310823752698, "explanation": "B-C = -2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9803016209186119 + 8949311547580064 - 9145349842732507", "ns_shortcut": "Near-cancellation: 8949311547580064 - 9145349842732507 = -196038295152443, offset is manageable.", "shortcut_strength": 0.9780945948623888, "pure_math": { "question": "Evaluate: 9803016209186119 + 8949311547580064 - 9145349842732507", "options": [ "9606977914033666", "9606977914033696", "9606977914033676", "9606977914033686" ], "answer": "9606977914033676", "correct_value": 9606977914033676, "explanation": "B-C = -196038295152443; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1142310823752700 + 2324754350027802 - 1445243519684742", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 1142310823752700 + 2324754350027802 - 1445243519684742", "options": [ "2021821654095770", "2021821654095780", "2021821654095760", "2021821654095750" ], "answer": "2021821654095760", "correct_value": 2021821654095760, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001370", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1482564244448419 + 5534401978133828 - 5534401978133826", "ns_shortcut": "Near-cancellation: 5534401978133828 - 5534401978133826 = 2, so answer ≈ 1482564244448419 + 2.", "shortcut_strength": 0.9999999999999997, "pure_math": { "question": "Evaluate: 1482564244448419 + 5534401978133828 - 5534401978133826", "options": [ "1482564244448411", "1482564244448421", "1482564244448441", "1482564244448431" ], "answer": "1482564244448421", "correct_value": 1482564244448421, "explanation": "B-C = 2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5211999502552818 + 8497077976829595 - 8680881856976673", "ns_shortcut": "Near-cancellation: 8497077976829595 - 8680881856976673 = -183803880147078, offset is manageable.", "shortcut_strength": 0.9783685779219295, "pure_math": { "question": "Evaluate: 5211999502552818 + 8497077976829595 - 8680881856976673", "options": [ "5028195622405730", "5028195622405750", "5028195622405760", "5028195622405740" ], "answer": "5028195622405740", "correct_value": 5028195622405740, "explanation": "B-C = -183803880147078; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "1482564244448419 + 5534401978133828 - 7920395035450401", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 1482564244448419 + 5534401978133828 - 7920395035450401", "options": [ "-903428812868154", "-903428812868166", "-903428812868136", "-903428812868146" ], "answer": "-903428812868154", "correct_value": -903428812868154, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001371", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8007726343681213 + 2025996751533766 - 2025996751533768", "ns_shortcut": "Near-cancellation: 2025996751533766 - 2025996751533768 = -2, so answer ≈ 8007726343681213 + -2.", "shortcut_strength": 0.999999999999999, "pure_math": { "question": "Evaluate: 8007726343681213 + 2025996751533766 - 2025996751533768", "options": [ "8007726343681201", "8007726343681221", "8007726343681211", "8007726343681231" ], "answer": "8007726343681211", "correct_value": 8007726343681211, "explanation": "B-C = -2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7557425138598869 + 2527862048526318 - 2629100737794445", "ns_shortcut": "Near-cancellation: 2527862048526318 - 2629100737794445 = -101238689268127, offset is manageable.", "shortcut_strength": 0.9599508646735107, "pure_math": { "question": "Evaluate: 7557425138598869 + 2527862048526318 - 2629100737794445", "options": [ "7456186449330732", "7456186449330762", "7456186449330752", "7456186449330742" ], "answer": "7456186449330742", "correct_value": 7456186449330742, "explanation": "B-C = -101238689268127; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "8007726343681213 + 2025996751533766 - 2724968178798653", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 8007726343681213 + 2025996751533766 - 2724968178798653", "options": [ "7308754916416346", "7308754916416316", "7308754916416326", "7308754916416336" ], "answer": "7308754916416326", "correct_value": 7308754916416326, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001372", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3862450851421550 + 1093243959548208 - 1093243959548209", "ns_shortcut": "Near-cancellation: 1093243959548208 - 1093243959548209 = -1, so answer ≈ 3862450851421550 + -1.", "shortcut_strength": 0.9999999999999991, "pure_math": { "question": "Evaluate: 3862450851421550 + 1093243959548208 - 1093243959548209", "options": [ "3862450851421569", "3862450851421559", "3862450851421539", "3862450851421549" ], "answer": "3862450851421549", "correct_value": 3862450851421549, "explanation": "B-C = -1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "8965705948669061 + 3141782249628854 - 3025476866879805", "ns_shortcut": "Near-cancellation: 3141782249628854 - 3025476866879805 = 116305382749049, offset is manageable.", "shortcut_strength": 0.9629810809572216, "pure_math": { "question": "Evaluate: 8965705948669061 + 3141782249628854 - 3025476866879805", "options": [ "9082011331418120", "9082011331418110", "9082011331418100", "9082011331418130" ], "answer": "9082011331418110", "correct_value": 9082011331418110, "explanation": "B-C = 116305382749049; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "3862450851421550 + 1093243959548208 - 1541153624055630", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 3862450851421550 + 1093243959548208 - 1541153624055630", "options": [ "3414541186914138", "3414541186914118", "3414541186914128", "3414541186914148" ], "answer": "3414541186914128", "correct_value": 3414541186914128, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001373", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3848154280890826 + 4600765302376846 - 4600765302376844", "ns_shortcut": "Near-cancellation: 4600765302376846 - 4600765302376844 = 2, so answer ≈ 3848154280890826 + 2.", "shortcut_strength": 0.9999999999999996, "pure_math": { "question": "Evaluate: 3848154280890826 + 4600765302376846 - 4600765302376844", "options": [ "3848154280890838", "3848154280890848", "3848154280890828", "3848154280890818" ], "answer": "3848154280890828", "correct_value": 3848154280890828, "explanation": "B-C = 2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4280455298074716 + 6410053414153669 - 6635532181248744", "ns_shortcut": "Near-cancellation: 6410053414153669 - 6635532181248744 = -225478767095075, offset is manageable.", "shortcut_strength": 0.9648241984072694, "pure_math": { "question": "Evaluate: 4280455298074716 + 6410053414153669 - 6635532181248744", "options": [ "4054976530979641", "4054976530979631", "4054976530979661", "4054976530979651" ], "answer": "4054976530979641", "correct_value": 4054976530979641, "explanation": "B-C = -225478767095075; still a moderate shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "3848154280890826 + 4600765302376846 - 6899925794043708", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 3848154280890826 + 4600765302376846 - 6899925794043708", "options": [ "1548993789223954", "1548993789223964", "1548993789223974", "1548993789223984" ], "answer": "1548993789223964", "correct_value": 1548993789223964, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001374", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7403970341146625 + 2316814310994107 - 2316814310994108", "ns_shortcut": "Near-cancellation: 2316814310994107 - 2316814310994108 = -1, so answer ≈ 7403970341146625 + -1.", "shortcut_strength": 0.9999999999999996, "pure_math": { "question": "Evaluate: 7403970341146625 + 2316814310994107 - 2316814310994108", "options": [ "7403970341146634", "7403970341146624", "7403970341146644", "7403970341146614" ], "answer": "7403970341146624", "correct_value": 7403970341146624, "explanation": "B-C = -1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1782830431497243 + 8519139115344043 - 8137311997997429", "ns_shortcut": "Near-cancellation: 8519139115344043 - 8137311997997429 = 381827117346614, offset is manageable.", "shortcut_strength": 0.9551800819100494, "pure_math": { "question": "Evaluate: 1782830431497243 + 8519139115344043 - 8137311997997429", "options": [ "2164657548843867", "2164657548843857", "2164657548843847", "2164657548843877" ], "answer": "2164657548843857", "correct_value": 2164657548843857, "explanation": "B-C = 381827117346614; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "7403970341146625 + 2316814310994107 - 1427528204429881", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 7403970341146625 + 2316814310994107 - 1427528204429881", "options": [ "8293256447710871", "8293256447710841", "8293256447710861", "8293256447710851" ], "answer": "8293256447710851", "correct_value": 8293256447710851, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "cancellation_identity_001375", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3202864002931054 + 4380372396287022 - 4380372396287024", "ns_shortcut": "Near-cancellation: 4380372396287022 - 4380372396287024 = -2, so answer ≈ 3202864002931054 + -2.", "shortcut_strength": 0.9999999999999996, "pure_math": { "question": "Evaluate: 3202864002931054 + 4380372396287022 - 4380372396287024", "options": [ "3202864002931042", "3202864002931062", "3202864002931072", "3202864002931052" ], "answer": "3202864002931052", "correct_value": 3202864002931052, "explanation": "B-C = -2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2473971916805469 + 4584160127129365 - 4778623548761421", "ns_shortcut": "Near-cancellation: 4584160127129365 - 4778623548761421 = -194463421632056, offset is manageable.", "shortcut_strength": 0.9575792694323201, "pure_math": { "question": "Evaluate: 2473971916805469 + 4584160127129365 - 4778623548761421", "options": [ "2279508495173433", "2279508495173423", "2279508495173403", "2279508495173413" ], "answer": "2279508495173413", "correct_value": 2279508495173413, "explanation": "B-C = -194463421632056; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "3202864002931054 + 4380372396287022 - 6075170890392789", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 3202864002931054 + 4380372396287022 - 6075170890392789", "options": [ "1508065508825277", "1508065508825307", "1508065508825287", "1508065508825297" ], "answer": "1508065508825287", "correct_value": 1508065508825287, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001376", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5955519397453081 + 2774301413890383 - 2774301413890385", "ns_shortcut": "Near-cancellation: 2774301413890383 - 2774301413890385 = -2, so answer ≈ 5955519397453081 + -2.", "shortcut_strength": 0.9999999999999993, "pure_math": { "question": "Evaluate: 5955519397453081 + 2774301413890383 - 2774301413890385", "options": [ "5955519397453089", "5955519397453079", "5955519397453099", "5955519397453069" ], "answer": "5955519397453079", "correct_value": 5955519397453079, "explanation": "B-C = -2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7691829430599997 + 2506943551435156 - 2408011304108513", "ns_shortcut": "Near-cancellation: 2506943551435156 - 2408011304108513 = 98932247326643, offset is manageable.", "shortcut_strength": 0.9605367072306008, "pure_math": { "question": "Evaluate: 7691829430599997 + 2506943551435156 - 2408011304108513", "options": [ "7790761677926640", "7790761677926630", "7790761677926650", "7790761677926660" ], "answer": "7790761677926640", "correct_value": 7790761677926640, "explanation": "B-C = 98932247326643; still a moderate shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5955519397453081 + 2774301413890383 - 1305494528035585", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 5955519397453081 + 2774301413890383 - 1305494528035585", "options": [ "7424326283307899", "7424326283307889", "7424326283307879", "7424326283307869" ], "answer": "7424326283307879", "correct_value": 7424326283307879, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001377", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1113763376748804 + 7747167989588306 - 7747167989588305", "ns_shortcut": "Near-cancellation: 7747167989588306 - 7747167989588305 = 1, so answer ≈ 1113763376748804 + 1.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "Evaluate: 1113763376748804 + 7747167989588306 - 7747167989588305", "options": [ "1113763376748805", "1113763376748825", "1113763376748815", "1113763376748795" ], "answer": "1113763376748805", "correct_value": 1113763376748805, "explanation": "B-C = 1; result ≈ A + offset." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5054839549442162 + 2991950401165783 - 2867772101155726", "ns_shortcut": "Near-cancellation: 2991950401165783 - 2867772101155726 = 124178300010057, offset is manageable.", "shortcut_strength": 0.9584958694630525, "pure_math": { "question": "Evaluate: 5054839549442162 + 2991950401165783 - 2867772101155726", "options": [ "5179017849452219", "5179017849452239", "5179017849452229", "5179017849452209" ], "answer": "5179017849452219", "correct_value": 5179017849452219, "explanation": "B-C = 124178300010057; still a moderate shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1113763376748804 + 7747167989588306 - 10435175686655470", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 1113763376748804 + 7747167989588306 - 10435175686655470", "options": [ "-1574244320318340", "-1574244320318350", "-1574244320318370", "-1574244320318360" ], "answer": "-1574244320318360", "correct_value": -1574244320318360, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "cancellation_identity_001378", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1079581076760174 + 2028079693443884 - 2028079693443882", "ns_shortcut": "Near-cancellation: 2028079693443884 - 2028079693443882 = 2, so answer ≈ 1079581076760174 + 2.", "shortcut_strength": 0.999999999999999, "pure_math": { "question": "Evaluate: 1079581076760174 + 2028079693443884 - 2028079693443882", "options": [ "1079581076760176", "1079581076760196", "1079581076760166", "1079581076760186" ], "answer": "1079581076760176", "correct_value": 1079581076760176, "explanation": "B-C = 2; result ≈ A + offset." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1780464232782456 + 4141059571836620 - 4000708423812371", "ns_shortcut": "Near-cancellation: 4141059571836620 - 4000708423812371 = 140351148024249, offset is manageable.", "shortcut_strength": 0.9661074308182431, "pure_math": { "question": "Evaluate: 1780464232782456 + 4141059571836620 - 4000708423812371", "options": [ "1920815380806725", "1920815380806715", "1920815380806705", "1920815380806695" ], "answer": "1920815380806705", "correct_value": 1920815380806705, "explanation": "B-C = 140351148024249; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1079581076760174 + 2028079693443884 - 3153038774013756", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 1079581076760174 + 2028079693443884 - 3153038774013756", "options": [ "-45378003809692", "-45378003809712", "-45378003809698", "-45378003809682" ], "answer": "-45378003809698", "correct_value": -45378003809698, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001379", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8035349198224981 + 7786699376607413 - 7786699376607415", "ns_shortcut": "Near-cancellation: 7786699376607413 - 7786699376607415 = -2, so answer ≈ 8035349198224981 + -2.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "Evaluate: 8035349198224981 + 7786699376607413 - 7786699376607415", "options": [ "8035349198224979", "8035349198224989", "8035349198224969", "8035349198224999" ], "answer": "8035349198224979", "correct_value": 8035349198224979, "explanation": "B-C = -2; result ≈ A + offset." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7278997814478573 + 6520945870093188 - 6223631879683651", "ns_shortcut": "Near-cancellation: 6520945870093188 - 6223631879683651 = 297313990409537, offset is manageable.", "shortcut_strength": 0.9544063090949583, "pure_math": { "question": "Evaluate: 7278997814478573 + 6520945870093188 - 6223631879683651", "options": [ "7576311804888120", "7576311804888100", "7576311804888130", "7576311804888110" ], "answer": "7576311804888110", "correct_value": 7576311804888110, "explanation": "B-C = 297313990409537; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "8035349198224981 + 7786699376607413 - 11672472527289751", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 8035349198224981 + 7786699376607413 - 11672472527289751", "options": [ "4149576047542653", "4149576047542663", "4149576047542643", "4149576047542633" ], "answer": "4149576047542643", "correct_value": 4149576047542643, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001380", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "4853154548046597 + 1704789094515621 - 1704789094515622", "ns_shortcut": "Near-cancellation: 1704789094515621 - 1704789094515622 = -1, so answer ≈ 4853154548046597 + -1.", "shortcut_strength": 0.9999999999999994, "pure_math": { "question": "Evaluate: 4853154548046597 + 1704789094515621 - 1704789094515622", "options": [ "4853154548046616", "4853154548046596", "4853154548046606", "4853154548046586" ], "answer": "4853154548046596", "correct_value": 4853154548046596, "explanation": "B-C = -1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6055747953863791 + 2969118782577616 - 3034035219801003", "ns_shortcut": "Near-cancellation: 2969118782577616 - 3034035219801003 = -64916437223387, offset is manageable.", "shortcut_strength": 0.9781361265826387, "pure_math": { "question": "Evaluate: 6055747953863791 + 2969118782577616 - 3034035219801003", "options": [ "5990831516640424", "5990831516640404", "5990831516640394", "5990831516640414" ], "answer": "5990831516640404", "correct_value": 5990831516640404, "explanation": "B-C = -64916437223387; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4853154548046597 + 1704789094515621 - 1073387913227803", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 4853154548046597 + 1704789094515621 - 1073387913227803", "options": [ "5484555729334415", "5484555729334435", "5484555729334425", "5484555729334405" ], "answer": "5484555729334415", "correct_value": 5484555729334415, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001381", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2970404482430650 + 1227517044682432 - 1227517044682431", "ns_shortcut": "Near-cancellation: 1227517044682432 - 1227517044682431 = 1, so answer ≈ 2970404482430650 + 1.", "shortcut_strength": 0.9999999999999992, "pure_math": { "question": "Evaluate: 2970404482430650 + 1227517044682432 - 1227517044682431", "options": [ "2970404482430641", "2970404482430651", "2970404482430661", "2970404482430671" ], "answer": "2970404482430651", "correct_value": 2970404482430651, "explanation": "B-C = 1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2140840669508292 + 2039676315182189 - 2082861053129637", "ns_shortcut": "Near-cancellation: 2039676315182189 - 2082861053129637 = -43184737947448, offset is manageable.", "shortcut_strength": 0.9788276514140968, "pure_math": { "question": "Evaluate: 2140840669508292 + 2039676315182189 - 2082861053129637", "options": [ "2097655931560834", "2097655931560844", "2097655931560864", "2097655931560854" ], "answer": "2097655931560844", "correct_value": 2097655931560844, "explanation": "B-C = -43184737947448; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2970404482430650 + 1227517044682432 - 1757055685248408", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 2970404482430650 + 1227517044682432 - 1757055685248408", "options": [ "2440865841864664", "2440865841864674", "2440865841864694", "2440865841864684" ], "answer": "2440865841864674", "correct_value": 2440865841864674, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001382", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7191462578948404 + 9355341997717434 - 9355341997717433", "ns_shortcut": "Near-cancellation: 9355341997717434 - 9355341997717433 = 1, so answer ≈ 7191462578948404 + 1.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "Evaluate: 7191462578948404 + 9355341997717434 - 9355341997717433", "options": [ "7191462578948425", "7191462578948415", "7191462578948405", "7191462578948395" ], "answer": "7191462578948405", "correct_value": 7191462578948405, "explanation": "B-C = 1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9560508097372620 + 2409537388782123 - 2357324478019387", "ns_shortcut": "Near-cancellation: 2409537388782123 - 2357324478019387 = 52212910762736, offset is manageable.", "shortcut_strength": 0.9783307322783954, "pure_math": { "question": "Evaluate: 9560508097372620 + 2409537388782123 - 2357324478019387", "options": [ "9612721008135356", "9612721008135346", "9612721008135366", "9612721008135376" ], "answer": "9612721008135356", "correct_value": 9612721008135356, "explanation": "B-C = 52212910762736; still a moderate shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "7191462578948404 + 9355341997717434 - 5626432004096479", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 7191462578948404 + 9355341997717434 - 5626432004096479", "options": [ "10920372572569379", "10920372572569349", "10920372572569369", "10920372572569359" ], "answer": "10920372572569359", "correct_value": 10920372572569359, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "cancellation_identity_001383", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8255621704717552 + 4747859145104852 - 4747859145104854", "ns_shortcut": "Near-cancellation: 4747859145104852 - 4747859145104854 = -2, so answer ≈ 8255621704717552 + -2.", "shortcut_strength": 0.9999999999999996, "pure_math": { "question": "Evaluate: 8255621704717552 + 4747859145104852 - 4747859145104854", "options": [ "8255621704717560", "8255621704717540", "8255621704717550", "8255621704717570" ], "answer": "8255621704717550", "correct_value": 8255621704717550, "explanation": "B-C = -2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1141528761137667 + 4712169457298319 - 4595582492692089", "ns_shortcut": "Near-cancellation: 4712169457298319 - 4595582492692089 = 116586964606230, offset is manageable.", "shortcut_strength": 0.9752583251381893, "pure_math": { "question": "Evaluate: 1141528761137667 + 4712169457298319 - 4595582492692089", "options": [ "1258115725743917", "1258115725743897", "1258115725743907", "1258115725743887" ], "answer": "1258115725743897", "correct_value": 1258115725743897, "explanation": "B-C = 116586964606230; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "8255621704717552 + 4747859145104852 - 7525467078345214", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 8255621704717552 + 4747859145104852 - 7525467078345214", "options": [ "5478013771477210", "5478013771477200", "5478013771477190", "5478013771477180" ], "answer": "5478013771477190", "correct_value": 5478013771477190, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001384", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5316228509104056 + 8581527927386372 - 8581527927386371", "ns_shortcut": "Near-cancellation: 8581527927386372 - 8581527927386371 = 1, so answer ≈ 5316228509104056 + 1.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "Evaluate: 5316228509104056 + 8581527927386372 - 8581527927386371", "options": [ "5316228509104057", "5316228509104077", "5316228509104067", "5316228509104047" ], "answer": "5316228509104057", "correct_value": 5316228509104057, "explanation": "B-C = 1; result ≈ A + offset." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9893192993666332 + 5441588940057179 - 5212679305221599", "ns_shortcut": "Near-cancellation: 5441588940057179 - 5212679305221599 = 228909634835580, offset is manageable.", "shortcut_strength": 0.957933310039186, "pure_math": { "question": "Evaluate: 9893192993666332 + 5441588940057179 - 5212679305221599", "options": [ "10122102628501902", "10122102628501922", "10122102628501912", "10122102628501932" ], "answer": "10122102628501912", "correct_value": 10122102628501912, "explanation": "B-C = 228909634835580; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5316228509104056 + 8581527927386372 - 3524658435521293", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 5316228509104056 + 8581527927386372 - 3524658435521293", "options": [ "10373098000969155", "10373098000969145", "10373098000969125", "10373098000969135" ], "answer": "10373098000969135", "correct_value": 10373098000969135, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "cancellation_identity_001385", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2980587838640542 + 7151121106508808 - 7151121106508809", "ns_shortcut": "Near-cancellation: 7151121106508808 - 7151121106508809 = -1, so answer ≈ 2980587838640542 + -1.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "Evaluate: 2980587838640542 + 7151121106508808 - 7151121106508809", "options": [ "2980587838640551", "2980587838640541", "2980587838640561", "2980587838640531" ], "answer": "2980587838640541", "correct_value": 2980587838640541, "explanation": "B-C = -1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4453139839135248 + 1669509554860676 - 1747305902488522", "ns_shortcut": "Near-cancellation: 1669509554860676 - 1747305902488522 = -77796347627846, offset is manageable.", "shortcut_strength": 0.953401675718868, "pure_math": { "question": "Evaluate: 4453139839135248 + 1669509554860676 - 1747305902488522", "options": [ "4375343491507392", "4375343491507422", "4375343491507402", "4375343491507412" ], "answer": "4375343491507402", "correct_value": 4375343491507402, "explanation": "B-C = -77796347627846; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2980587838640542 + 7151121106508808 - 10051562699049249", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 2980587838640542 + 7151121106508808 - 10051562699049249", "options": [ "80146246100091", "80146246100121", "80146246100111", "80146246100101" ], "answer": "80146246100101", "correct_value": 80146246100101, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "cancellation_identity_001386", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3009093781510667 + 1580186860456278 - 1580186860456280", "ns_shortcut": "Near-cancellation: 1580186860456278 - 1580186860456280 = -2, so answer ≈ 3009093781510667 + -2.", "shortcut_strength": 0.9999999999999988, "pure_math": { "question": "Evaluate: 3009093781510667 + 1580186860456278 - 1580186860456280", "options": [ "3009093781510655", "3009093781510675", "3009093781510665", "3009093781510685" ], "answer": "3009093781510665", "correct_value": 3009093781510665, "explanation": "B-C = -2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "3449114411267930 + 9631175605839490 - 9999999999999999", "ns_shortcut": "Near-cancellation: 9631175605839490 - 9999999999999999 = -368824394160509, offset is manageable.", "shortcut_strength": 0.960515520160038, "pure_math": { "question": "Evaluate: 3449114411267930 + 9631175605839490 - 9999999999999999", "options": [ "3080290017107441", "3080290017107431", "3080290017107421", "3080290017107411" ], "answer": "3080290017107421", "correct_value": 3080290017107421, "explanation": "B-C = -368824394160509; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "3009093781510667 + 1580186860456278 - 1033914490604551", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 3009093781510667 + 1580186860456278 - 1033914490604551", "options": [ "3555366151362384", "3555366151362394", "3555366151362404", "3555366151362414" ], "answer": "3555366151362394", "correct_value": 3555366151362394, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001387", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2734602079244710 + 1788173831169177 - 1788173831169179", "ns_shortcut": "Near-cancellation: 1788173831169177 - 1788173831169179 = -2, so answer ≈ 2734602079244710 + -2.", "shortcut_strength": 0.9999999999999989, "pure_math": { "question": "Evaluate: 2734602079244710 + 1788173831169177 - 1788173831169179", "options": [ "2734602079244708", "2734602079244718", "2734602079244728", "2734602079244698" ], "answer": "2734602079244708", "correct_value": 2734602079244708, "explanation": "B-C = -2; result ≈ A + offset." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5410018196170384 + 8340354497092603 - 8694468992565301", "ns_shortcut": "Near-cancellation: 8340354497092603 - 8694468992565301 = -354114495472698, offset is manageable.", "shortcut_strength": 0.9575420330639255, "pure_math": { "question": "Evaluate: 5410018196170384 + 8340354497092603 - 8694468992565301", "options": [ "5055903700697706", "5055903700697686", "5055903700697696", "5055903700697676" ], "answer": "5055903700697686", "correct_value": 5055903700697686, "explanation": "B-C = -354114495472698; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2734602079244710 + 1788173831169177 - 2861752771858599", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 2734602079244710 + 1788173831169177 - 2861752771858599", "options": [ "1661023138555298", "1661023138555288", "1661023138555308", "1661023138555278" ], "answer": "1661023138555288", "correct_value": 1661023138555288, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001388", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2452228583048936 + 8820490899666081 - 8820490899666082", "ns_shortcut": "Near-cancellation: 8820490899666081 - 8820490899666082 = -1, so answer ≈ 2452228583048936 + -1.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "Evaluate: 2452228583048936 + 8820490899666081 - 8820490899666082", "options": [ "2452228583048955", "2452228583048925", "2452228583048945", "2452228583048935" ], "answer": "2452228583048935", "correct_value": 2452228583048935, "explanation": "B-C = -1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7391138933240773 + 6988743124604126 - 6651533549824121", "ns_shortcut": "Near-cancellation: 6988743124604126 - 6651533549824121 = 337209574780005, offset is manageable.", "shortcut_strength": 0.9517496109432257, "pure_math": { "question": "Evaluate: 7391138933240773 + 6988743124604126 - 6651533549824121", "options": [ "7728348508020788", "7728348508020798", "7728348508020768", "7728348508020778" ], "answer": "7728348508020778", "correct_value": 7728348508020778, "explanation": "B-C = 337209574780005; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "2452228583048936 + 8820490899666081 - 3846955157861624", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 2452228583048936 + 8820490899666081 - 3846955157861624", "options": [ "7425764324853413", "7425764324853403", "7425764324853383", "7425764324853393" ], "answer": "7425764324853393", "correct_value": 7425764324853393, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "cancellation_identity_001389", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5436186151273914 + 6691128927295273 - 6691128927295272", "ns_shortcut": "Near-cancellation: 6691128927295273 - 6691128927295272 = 1, so answer ≈ 5436186151273914 + 1.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "Evaluate: 5436186151273914 + 6691128927295273 - 6691128927295272", "options": [ "5436186151273925", "5436186151273935", "5436186151273915", "5436186151273905" ], "answer": "5436186151273915", "correct_value": 5436186151273915, "explanation": "B-C = 1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2076167037959536 + 9846573003649629 - 9475495315338918", "ns_shortcut": "Near-cancellation: 9846573003649629 - 9475495315338918 = 371077688310711, offset is manageable.", "shortcut_strength": 0.9623140265985769, "pure_math": { "question": "Evaluate: 2076167037959536 + 9846573003649629 - 9475495315338918", "options": [ "2447244726270237", "2447244726270247", "2447244726270267", "2447244726270257" ], "answer": "2447244726270247", "correct_value": 2447244726270247, "explanation": "B-C = 371077688310711; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5436186151273914 + 6691128927295273 - 10849129447978382", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 5436186151273914 + 6691128927295273 - 10849129447978382", "options": [ "1278185630590795", "1278185630590805", "1278185630590815", "1278185630590825" ], "answer": "1278185630590805", "correct_value": 1278185630590805, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001390", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8806849288913202 + 4732742132459529 - 4732742132459531", "ns_shortcut": "Near-cancellation: 4732742132459529 - 4732742132459531 = -2, so answer ≈ 8806849288913202 + -2.", "shortcut_strength": 0.9999999999999996, "pure_math": { "question": "Evaluate: 8806849288913202 + 4732742132459529 - 4732742132459531", "options": [ "8806849288913200", "8806849288913210", "8806849288913220", "8806849288913190" ], "answer": "8806849288913200", "correct_value": 8806849288913200, "explanation": "B-C = -2; result ≈ A + offset." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4670567744021902 + 2802608386909537 - 2906782871150999", "ns_shortcut": "Near-cancellation: 2802608386909537 - 2906782871150999 = -104174484241462, offset is manageable.", "shortcut_strength": 0.9628294539015719, "pure_math": { "question": "Evaluate: 4670567744021902 + 2802608386909537 - 2906782871150999", "options": [ "4566393259780430", "4566393259780450", "4566393259780460", "4566393259780440" ], "answer": "4566393259780440", "correct_value": 4566393259780440, "explanation": "B-C = -104174484241462; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "8806849288913202 + 4732742132459529 - 2020481848099117", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 8806849288913202 + 4732742132459529 - 2020481848099117", "options": [ "11519109573273634", "11519109573273604", "11519109573273624", "11519109573273614" ], "answer": "11519109573273614", "correct_value": 11519109573273614, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "cancellation_identity_001391", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1426592033336676 + 7877982591541203 - 7877982591541204", "ns_shortcut": "Near-cancellation: 7877982591541203 - 7877982591541204 = -1, so answer ≈ 1426592033336676 + -1.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "Evaluate: 1426592033336676 + 7877982591541203 - 7877982591541204", "options": [ "1426592033336695", "1426592033336665", "1426592033336685", "1426592033336675" ], "answer": "1426592033336675", "correct_value": 1426592033336675, "explanation": "B-C = -1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "3684355428677358 + 2619070755002426 - 2531731586987423", "ns_shortcut": "Near-cancellation: 2619070755002426 - 2531731586987423 = 87339168015003, offset is manageable.", "shortcut_strength": 0.9666526122487584, "pure_math": { "question": "Evaluate: 3684355428677358 + 2619070755002426 - 2531731586987423", "options": [ "3771694596692381", "3771694596692371", "3771694596692361", "3771694596692351" ], "answer": "3771694596692361", "correct_value": 3771694596692361, "explanation": "B-C = 87339168015003; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1426592033336676 + 7877982591541203 - 3762610030923778", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 1426592033336676 + 7877982591541203 - 3762610030923778", "options": [ "5541964593954121", "5541964593954101", "5541964593954091", "5541964593954111" ], "answer": "5541964593954101", "correct_value": 5541964593954101, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001392", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8544238055458542 + 1692487206781134 - 1692487206781132", "ns_shortcut": "Near-cancellation: 1692487206781134 - 1692487206781132 = 2, so answer ≈ 8544238055458542 + 2.", "shortcut_strength": 0.9999999999999988, "pure_math": { "question": "Evaluate: 8544238055458542 + 1692487206781134 - 1692487206781132", "options": [ "8544238055458534", "8544238055458544", "8544238055458554", "8544238055458564" ], "answer": "8544238055458544", "correct_value": 8544238055458544, "explanation": "B-C = 2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5710521770186968 + 5931135809373146 - 6179126848634044", "ns_shortcut": "Near-cancellation: 5931135809373146 - 6179126848634044 = -247991039260898, offset is manageable.", "shortcut_strength": 0.9581882716512762, "pure_math": { "question": "Evaluate: 5710521770186968 + 5931135809373146 - 6179126848634044", "options": [ "5462530730926060", "5462530730926090", "5462530730926080", "5462530730926070" ], "answer": "5462530730926070", "correct_value": 5462530730926070, "explanation": "B-C = -247991039260898; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "8544238055458542 + 1692487206781134 - 2384142462066265", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 8544238055458542 + 1692487206781134 - 2384142462066265", "options": [ "7852582800173411", "7852582800173401", "7852582800173421", "7852582800173431" ], "answer": "7852582800173411", "correct_value": 7852582800173411, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001393", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3025787912279006 + 7258602041716913 - 7258602041716911", "ns_shortcut": "Near-cancellation: 7258602041716913 - 7258602041716911 = 2, so answer ≈ 3025787912279006 + 2.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "Evaluate: 3025787912279006 + 7258602041716913 - 7258602041716911", "options": [ "3025787912278998", "3025787912279008", "3025787912279028", "3025787912279018" ], "answer": "3025787912279008", "correct_value": 3025787912279008, "explanation": "B-C = 2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4015822108784906 + 2084277139593633 - 2015595330536300", "ns_shortcut": "Near-cancellation: 2084277139593633 - 2015595330536300 = 68681809057333, offset is manageable.", "shortcut_strength": 0.9670476599523978, "pure_math": { "question": "Evaluate: 4015822108784906 + 2084277139593633 - 2015595330536300", "options": [ "4084503917842239", "4084503917842259", "4084503917842249", "4084503917842229" ], "answer": "4084503917842239", "correct_value": 4084503917842239, "explanation": "B-C = 68681809057333; still a moderate shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "3025787912279006 + 7258602041716913 - 10091453591651899", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 3025787912279006 + 7258602041716913 - 10091453591651899", "options": [ "192936362344010", "192936362344030", "192936362344020", "192936362344040" ], "answer": "192936362344020", "correct_value": 192936362344020, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001394", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3452923720689472 + 8508642515024003 - 8508642515024004", "ns_shortcut": "Near-cancellation: 8508642515024003 - 8508642515024004 = -1, so answer ≈ 3452923720689472 + -1.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "Evaluate: 3452923720689472 + 8508642515024003 - 8508642515024004", "options": [ "3452923720689481", "3452923720689471", "3452923720689461", "3452923720689491" ], "answer": "3452923720689471", "correct_value": 3452923720689471, "explanation": "B-C = -1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "3192378564063252 + 9840946500686358 - 9512174475010171", "ns_shortcut": "Near-cancellation: 9840946500686358 - 9512174475010171 = 328772025676187, offset is manageable.", "shortcut_strength": 0.9665914223135694, "pure_math": { "question": "Evaluate: 3192378564063252 + 9840946500686358 - 9512174475010171", "options": [ "3521150589739429", "3521150589739439", "3521150589739459", "3521150589739449" ], "answer": "3521150589739439", "correct_value": 3521150589739439, "explanation": "B-C = 328772025676187; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "3452923720689472 + 8508642515024003 - 5135504885239440", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 3452923720689472 + 8508642515024003 - 5135504885239440", "options": [ "6826061350474045", "6826061350474055", "6826061350474035", "6826061350474025" ], "answer": "6826061350474035", "correct_value": 6826061350474035, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001395", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5881903337016386 + 3101116878293009 - 3101116878293008", "ns_shortcut": "Near-cancellation: 3101116878293009 - 3101116878293008 = 1, so answer ≈ 5881903337016386 + 1.", "shortcut_strength": 0.9999999999999997, "pure_math": { "question": "Evaluate: 5881903337016386 + 3101116878293009 - 3101116878293008", "options": [ "5881903337016377", "5881903337016387", "5881903337016397", "5881903337016407" ], "answer": "5881903337016387", "correct_value": 5881903337016387, "explanation": "B-C = 1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1801255933184228 + 5260298652573074 - 5070102609629308", "ns_shortcut": "Near-cancellation: 5260298652573074 - 5070102609629308 = 190196042943766, offset is manageable.", "shortcut_strength": 0.9638431093925187, "pure_math": { "question": "Evaluate: 1801255933184228 + 5260298652573074 - 5070102609629308", "options": [ "1991451976128014", "1991451976127984", "1991451976127994", "1991451976128004" ], "answer": "1991451976127994", "correct_value": 1991451976127994, "explanation": "B-C = 190196042943766; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5881903337016386 + 3101116878293009 - 4384928153178447", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 5881903337016386 + 3101116878293009 - 4384928153178447", "options": [ "4598092062130938", "4598092062130968", "4598092062130948", "4598092062130958" ], "answer": "4598092062130948", "correct_value": 4598092062130948, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001396", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9059244175206189 + 8214586299514995 - 8214586299514993", "ns_shortcut": "Near-cancellation: 8214586299514995 - 8214586299514993 = 2, so answer ≈ 9059244175206189 + 2.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "Evaluate: 9059244175206189 + 8214586299514995 - 8214586299514993", "options": [ "9059244175206191", "9059244175206201", "9059244175206211", "9059244175206181" ], "answer": "9059244175206191", "correct_value": 9059244175206191, "explanation": "B-C = 2; result ≈ A + offset." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7135798708326236 + 1827220320163975 - 1887770499010697", "ns_shortcut": "Near-cancellation: 1827220320163975 - 1887770499010697 = -60550178846722, offset is manageable.", "shortcut_strength": 0.9668621357925309, "pure_math": { "question": "Evaluate: 7135798708326236 + 1827220320163975 - 1887770499010697", "options": [ "7075248529479534", "7075248529479504", "7075248529479514", "7075248529479524" ], "answer": "7075248529479514", "correct_value": 7075248529479514, "explanation": "B-C = -60550178846722; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "9059244175206189 + 8214586299514995 - 13287874739398554", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 9059244175206189 + 8214586299514995 - 13287874739398554", "options": [ "3985955735322630", "3985955735322620", "3985955735322650", "3985955735322640" ], "answer": "3985955735322630", "correct_value": 3985955735322630, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001397", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7569671803769991 + 6065677840344046 - 6065677840344047", "ns_shortcut": "Near-cancellation: 6065677840344046 - 6065677840344047 = -1, so answer ≈ 7569671803769991 + -1.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "Evaluate: 7569671803769991 + 6065677840344046 - 6065677840344047", "options": [ "7569671803769980", "7569671803770010", "7569671803770000", "7569671803769990" ], "answer": "7569671803769990", "correct_value": 7569671803769990, "explanation": "B-C = -1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5235300124491457 + 1838748042025660 - 1924907235642080", "ns_shortcut": "Near-cancellation: 1838748042025660 - 1924907235642080 = -86159193616420, offset is manageable.", "shortcut_strength": 0.9531424688716446, "pure_math": { "question": "Evaluate: 5235300124491457 + 1838748042025660 - 1924907235642080", "options": [ "5149140930875057", "5149140930875027", "5149140930875047", "5149140930875037" ], "answer": "5149140930875037", "correct_value": 5149140930875037, "explanation": "B-C = -86159193616420; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "7569671803769991 + 6065677840344046 - 4101021454170077", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 7569671803769991 + 6065677840344046 - 4101021454170077", "options": [ "9534328189943980", "9534328189943950", "9534328189943960", "9534328189943970" ], "answer": "9534328189943960", "correct_value": 9534328189943960, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001398", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9415473845540360 + 5078653109786016 - 5078653109786018", "ns_shortcut": "Near-cancellation: 5078653109786016 - 5078653109786018 = -2, so answer ≈ 9415473845540360 + -2.", "shortcut_strength": 0.9999999999999996, "pure_math": { "question": "Evaluate: 9415473845540360 + 5078653109786016 - 5078653109786018", "options": [ "9415473845540368", "9415473845540358", "9415473845540378", "9415473845540348" ], "answer": "9415473845540358", "correct_value": 9415473845540358, "explanation": "B-C = -2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7115339003600441 + 1668580966784889 - 1634801212842571", "ns_shortcut": "Near-cancellation: 1668580966784889 - 1634801212842571 = 33779753942318, offset is manageable.", "shortcut_strength": 0.9797554001785082, "pure_math": { "question": "Evaluate: 7115339003600441 + 1668580966784889 - 1634801212842571", "options": [ "7149118757542759", "7149118757542769", "7149118757542749", "7149118757542779" ], "answer": "7149118757542759", "correct_value": 7149118757542759, "explanation": "B-C = 33779753942318; still a moderate shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "9415473845540360 + 5078653109786016 - 3529824352988296", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 9415473845540360 + 5078653109786016 - 3529824352988296", "options": [ "10964302602338090", "10964302602338070", "10964302602338080", "10964302602338100" ], "answer": "10964302602338080", "correct_value": 10964302602338080, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001399", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2537854471044720 + 8664982181762846 - 8664982181762845", "ns_shortcut": "Near-cancellation: 8664982181762846 - 8664982181762845 = 1, so answer ≈ 2537854471044720 + 1.", "shortcut_strength": 0.9999999999999999, "pure_math": { "question": "Evaluate: 2537854471044720 + 8664982181762846 - 8664982181762845", "options": [ "2537854471044731", "2537854471044711", "2537854471044741", "2537854471044721" ], "answer": "2537854471044721", "correct_value": 2537854471044721, "explanation": "B-C = 1; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7489377885647361 + 5312625832658185 - 5148436741115562", "ns_shortcut": "Near-cancellation: 5312625832658185 - 5148436741115562 = 164189091542623, offset is manageable.", "shortcut_strength": 0.9690945500936078, "pure_math": { "question": "Evaluate: 7489377885647361 + 5312625832658185 - 5148436741115562", "options": [ "7653566977189994", "7653566977189974", "7653566977190004", "7653566977189984" ], "answer": "7653566977189984", "correct_value": 7653566977189984, "explanation": "B-C = 164189091542623; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "2537854471044720 + 8664982181762846 - 12672622382489264", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 2537854471044720 + 8664982181762846 - 12672622382489264", "options": [ "-1469785729681692", "-1469785729681682", "-1469785729681712", "-1469785729681698" ], "answer": "-1469785729681698", "correct_value": -1469785729681698, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "cancellation_identity_001400", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8123820227372139 + 3613793873750583 - 3613793873750582", "ns_shortcut": "Near-cancellation: 3613793873750583 - 3613793873750582 = 1, so answer ≈ 8123820227372139 + 1.", "shortcut_strength": 0.9999999999999998, "pure_math": { "question": "Evaluate: 8123820227372139 + 3613793873750583 - 3613793873750582", "options": [ "8123820227372140", "8123820227372160", "8123820227372130", "8123820227372150" ], "answer": "8123820227372140", "correct_value": 8123820227372140, "explanation": "B-C = 1; result ≈ A + offset." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9850062723915098 + 5958186192025091 - 6101045492791627", "ns_shortcut": "Near-cancellation: 5958186192025091 - 6101045492791627 = -142859300766536, offset is manageable.", "shortcut_strength": 0.9760230217448138, "pure_math": { "question": "Evaluate: 9850062723915098 + 5958186192025091 - 6101045492791627", "options": [ "9707203423148552", "9707203423148582", "9707203423148572", "9707203423148562" ], "answer": "9707203423148562", "correct_value": 9707203423148562, "explanation": "B-C = -142859300766536; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "8123820227372139 + 3613793873750583 - 4714421995253350", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 8123820227372139 + 3613793873750583 - 4714421995253350", "options": [ "7023192105869372", "7023192105869392", "7023192105869382", "7023192105869362" ], "answer": "7023192105869372", "correct_value": 7023192105869372, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "cancellation_identity_001401", "category": "cancellation_identity", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9056451127371004 + 1818410046845831 - 1818410046845833", "ns_shortcut": "Near-cancellation: 1818410046845831 - 1818410046845833 = -2, so answer ≈ 9056451127371004 + -2.", "shortcut_strength": 0.9999999999999989, "pure_math": { "question": "Evaluate: 9056451127371004 + 1818410046845831 - 1818410046845833", "options": [ "9056451127371012", "9056451127371002", "9056451127371022", "9056451127370992" ], "answer": "9056451127371002", "correct_value": 9056451127371002, "explanation": "B-C = -2; result ≈ A + offset." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1759057509254454 + 6080357899437250 - 5946796702449351", "ns_shortcut": "Near-cancellation: 6080357899437250 - 5946796702449351 = 133561196987899, offset is manageable.", "shortcut_strength": 0.9780339908938156, "pure_math": { "question": "Evaluate: 1759057509254454 + 6080357899437250 - 5946796702449351", "options": [ "1892618706242373", "1892618706242353", "1892618706242343", "1892618706242363" ], "answer": "1892618706242353", "correct_value": 1892618706242353, "explanation": "B-C = 133561196987899; still a moderate shortcut." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "9056451127371004 + 1818410046845831 - 2711752840028592", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Evaluate: 9056451127371004 + 1818410046845831 - 2711752840028592", "options": [ "8163108334188253", "8163108334188233", "8163108334188243", "8163108334188263" ], "answer": "8163108334188243", "correct_value": 8163108334188243, "explanation": "Large gap between B and C; no cancellation shortcut." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001402", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5004 * 21", "ns_shortcut": "Round to 5000×20 for easy mental computation.", "shortcut_strength": 0.9746, "pure_math": { "question": "Estimate 5004 × 21.", "options": [ "105084", "105104", "105074", "105094" ], "answer": "105084", "correct_value": 105084, "explanation": "Round to compatible pair 5000×20 = 100000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4913 * 14", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8412999999999999, "pure_math": { "question": "Estimate 4913 × 14.", "options": [ "68802", "68782", "68772", "68792" ], "answer": "68782", "correct_value": 68782, "explanation": "One number rounds to 5000, other is farther from 20." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6183 * 26", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 6183 × 26.", "options": [ "160768", "160748", "160778", "160758" ], "answer": "160758", "correct_value": 160758, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "compatible_numbers_001403", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1260 * 82", "ns_shortcut": "Round to 1250×80 for easy mental computation.", "shortcut_strength": 0.9835, "pure_math": { "question": "Estimate 1260 × 82.", "options": [ "103310", "103340", "103320", "103330" ], "answer": "103320", "correct_value": 103320, "explanation": "Round to compatible pair 1250×80 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1235 * 102", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8565, "pure_math": { "question": "Estimate 1235 × 102.", "options": [ "125970", "125980", "125990", "125960" ], "answer": "125970", "correct_value": 125970, "explanation": "One number rounds to 1250, other is farther from 80." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1528 * 108", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 1528 × 108.", "options": [ "165024", "165014", "165044", "165034" ], "answer": "165024", "correct_value": 165024, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001404", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "4932 * 20", "ns_shortcut": "Round to 5000×20 for easy mental computation.", "shortcut_strength": 0.9932, "pure_math": { "question": "Estimate 4932 × 20.", "options": [ "98650", "98630", "98640", "98660" ], "answer": "98640", "correct_value": 98640, "explanation": "Round to compatible pair 5000×20 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4923 * 14", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8423, "pure_math": { "question": "Estimate 4923 × 14.", "options": [ "68942", "68922", "68912", "68932" ], "answer": "68922", "correct_value": 68922, "explanation": "One number rounds to 5000, other is farther from 20." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5787 * 27", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 5787 × 27.", "options": [ "156239", "156259", "156249", "156269" ], "answer": "156249", "correct_value": 156249, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001405", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "25075 * 3813", "ns_shortcut": "Round to 25000×4000 for easy mental computation.", "shortcut_strength": 0.975125, "pure_math": { "question": "Estimate 25075 × 3813.", "options": [ "95610975", "95610995", "95610965", "95610985" ], "answer": "95610975", "correct_value": 95610975, "explanation": "Round to compatible pair 25000×4000 = 100000000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "24585 * 5236", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8371999999999999, "pure_math": { "question": "Estimate 24585 × 5236.", "options": [ "128727060", "128727080", "128727050", "128727070" ], "answer": "128727060", "correct_value": 128727060, "explanation": "One number rounds to 25000, other is farther from 4000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "8099664881692863 * 8848130205665746", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 8099664881692863 × 8848130205665746.", "options": [ "71666889495476692139554411770788", "71666889495476692139554411770818", "71666889495476692139554411770798", "71666889495476692139554411770808" ], "answer": "71666889495476692139554411770798", "correct_value": 71666889495476692139554411770798, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001406", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2549 * 42", "ns_shortcut": "Round to 2500×40 for easy mental computation.", "shortcut_strength": 0.9652000000000001, "pure_math": { "question": "Estimate 2549 × 42.", "options": [ "107078", "107058", "107048", "107068" ], "answer": "107058", "correct_value": 107058, "explanation": "Round to compatible pair 2500×40 = 100000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2476 * 28", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8452, "pure_math": { "question": "Estimate 2476 × 28.", "options": [ "69318", "69328", "69338", "69348" ], "answer": "69328", "correct_value": 69328, "explanation": "One number rounds to 2500, other is farther from 40." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2980 * 57", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 2980 × 57.", "options": [ "169880", "169860", "169850", "169870" ], "answer": "169860", "correct_value": 169860, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001407", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1226 * 84", "ns_shortcut": "Round to 1250×80 for easy mental computation.", "shortcut_strength": 0.9654, "pure_math": { "question": "Estimate 1226 × 84.", "options": [ "102994", "102984", "103004", "102974" ], "answer": "102984", "correct_value": 102984, "explanation": "Round to compatible pair 1250×80 = 100000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1232 * 105", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.83655, "pure_math": { "question": "Estimate 1232 × 105.", "options": [ "129360", "129380", "129370", "129350" ], "answer": "129360", "correct_value": 129360, "explanation": "One number rounds to 1250, other is farther from 80." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1428 * 111", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 1428 × 111.", "options": [ "158498", "158528", "158518", "158508" ], "answer": "158508", "correct_value": 158508, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "compatible_numbers_001408", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1242 * 81", "ns_shortcut": "Round to 1250×80 for easy mental computation.", "shortcut_strength": 0.99055, "pure_math": { "question": "Estimate 1242 × 81.", "options": [ "100592", "100622", "100602", "100612" ], "answer": "100602", "correct_value": 100602, "explanation": "Round to compatible pair 1250×80 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1258 * 56", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8468, "pure_math": { "question": "Estimate 1258 × 56.", "options": [ "70468", "70458", "70448", "70438" ], "answer": "70448", "correct_value": 70448, "explanation": "One number rounds to 1250, other is farther from 80." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1447 * 117", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 1447 × 117.", "options": [ "169299", "169309", "169319", "169289" ], "answer": "169299", "correct_value": 169299, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001409", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "25172 * 3789", "ns_shortcut": "Round to 25000×4000 for easy mental computation.", "shortcut_strength": 0.970185, "pure_math": { "question": "Estimate 25172 × 3789.", "options": [ "95376728", "95376698", "95376708", "95376718" ], "answer": "95376708", "correct_value": 95376708, "explanation": "Round to compatible pair 25000×4000 = 100000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "25076 * 5304", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.83548, "pure_math": { "question": "Estimate 25076 × 5304.", "options": [ "133003104", "133003124", "133003094", "133003114" ], "answer": "133003104", "correct_value": 133003104, "explanation": "One number rounds to 25000, other is farther from 4000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6472790500899456 * 7451613880107531", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 6472790500899456 × 7451613880107531.", "options": [ "48232735539530564449401499403156", "48232735539530564449401499403136", "48232735539530564449401499403126", "48232735539530564449401499403146" ], "answer": "48232735539530564449401499403136", "correct_value": 48232735539530564449401499403136, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001410", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5091 * 20", "ns_shortcut": "Round to 5000×20 for easy mental computation.", "shortcut_strength": 0.9909, "pure_math": { "question": "Estimate 5091 × 20.", "options": [ "101820", "101810", "101830", "101840" ], "answer": "101820", "correct_value": 101820, "explanation": "Round to compatible pair 5000×20 = 100000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5053 * 26", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8447, "pure_math": { "question": "Estimate 5053 × 26.", "options": [ "131368", "131388", "131378", "131398" ], "answer": "131378", "correct_value": 131378, "explanation": "One number rounds to 5000, other is farther from 20." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5578 * 30", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 5578 × 30.", "options": [ "167350", "167360", "167340", "167330" ], "answer": "167340", "correct_value": 167340, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001411", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "25194 * 4220", "ns_shortcut": "Round to 25000×4000 for easy mental computation.", "shortcut_strength": 0.96862, "pure_math": { "question": "Estimate 25194 × 4220.", "options": [ "106318690", "106318700", "106318670", "106318680" ], "answer": "106318680", "correct_value": 106318680, "explanation": "Round to compatible pair 25000×4000 = 100000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "25164 * 5315", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.832345, "pure_math": { "question": "Estimate 25164 × 5315.", "options": [ "133746660", "133746670", "133746650", "133746680" ], "answer": "133746660", "correct_value": 133746660, "explanation": "One number rounds to 25000, other is farther from 4000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "3773823270149475 * 2856702546260156", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 3773823270149475 × 2856702546260156.", "options": [ "10780690544971833799732356818120", "10780690544971833799732356818100", "10780690544971833799732356818110", "10780690544971833799732356818090" ], "answer": "10780690544971833799732356818100", "correct_value": 10780690544971833799732356818100, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001412", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5043 * 20", "ns_shortcut": "Round to 5000×20 for easy mental computation.", "shortcut_strength": 0.9957, "pure_math": { "question": "Estimate 5043 × 20.", "options": [ "100860", "100850", "100880", "100870" ], "answer": "100860", "correct_value": 100860, "explanation": "Round to compatible pair 5000×20 = 100000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4929 * 24", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8929, "pure_math": { "question": "Estimate 4929 × 24.", "options": [ "118306", "118296", "118316", "118286" ], "answer": "118296", "correct_value": 118296, "explanation": "One number rounds to 5000, other is farther from 20." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5839 * 29", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 5839 × 29.", "options": [ "169351", "169321", "169341", "169331" ], "answer": "169331", "correct_value": 169331, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "compatible_numbers_001413", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5007 * 18", "ns_shortcut": "Round to 5000×20 for easy mental computation.", "shortcut_strength": 0.9493, "pure_math": { "question": "Estimate 5007 × 18.", "options": [ "90126", "90136", "90116", "90146" ], "answer": "90126", "correct_value": 90126, "explanation": "Round to compatible pair 5000×20 = 100000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5095 * 15", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8654999999999999, "pure_math": { "question": "Estimate 5095 × 15.", "options": [ "76435", "76425", "76415", "76445" ], "answer": "76425", "correct_value": 76425, "explanation": "One number rounds to 5000, other is farther from 20." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6144 * 29", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 6144 × 29.", "options": [ "178176", "178166", "178196", "178186" ], "answer": "178176", "correct_value": 178176, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001414", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5071 * 18", "ns_shortcut": "Round to 5000×20 for easy mental computation.", "shortcut_strength": 0.9429, "pure_math": { "question": "Estimate 5071 × 18.", "options": [ "91298", "91288", "91278", "91268" ], "answer": "91278", "correct_value": 91278, "explanation": "Round to compatible pair 5000×20 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5028 * 14", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8472, "pure_math": { "question": "Estimate 5028 × 14.", "options": [ "70402", "70412", "70382", "70392" ], "answer": "70392", "correct_value": 70392, "explanation": "One number rounds to 5000, other is farther from 20." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "5644 * 29", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 5644 × 29.", "options": [ "163676", "163686", "163696", "163666" ], "answer": "163676", "correct_value": 163676, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001415", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2500 * 44", "ns_shortcut": "Round to 2500×40 for easy mental computation.", "shortcut_strength": 0.95, "pure_math": { "question": "Estimate 2500 × 44.", "options": [ "109990", "110010", "110000", "110020" ], "answer": "110000", "correct_value": 110000, "explanation": "Round to compatible pair 2500×40 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2522 * 51", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8581, "pure_math": { "question": "Estimate 2522 × 51.", "options": [ "128642", "128632", "128612", "128622" ], "answer": "128622", "correct_value": 128622, "explanation": "One number rounds to 2500, other is farther from 40." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "2917 * 59", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 2917 × 59.", "options": [ "172093", "172123", "172103", "172113" ], "answer": "172103", "correct_value": 172103, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001416", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "12320 * 8126", "ns_shortcut": "Round to 12500×8000 for easy mental computation.", "shortcut_strength": 0.984925, "pure_math": { "question": "Estimate 12320 × 8126.", "options": [ "100112330", "100112320", "100112340", "100112310" ], "answer": "100112320", "correct_value": 100112320, "explanation": "Round to compatible pair 12500×8000 = 100000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "12451 * 10594", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.835915, "pure_math": { "question": "Estimate 12451 × 10594.", "options": [ "131905914", "131905894", "131905904", "131905884" ], "answer": "131905894", "correct_value": 131905894, "explanation": "One number rounds to 12500, other is farther from 8000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "9659902900757571 * 8347009167623358", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 9659902900757571 × 8347009167623358.", "options": [ "80631298070974914134063794943428", "80631298070974914134063794943418", "80631298070974914134063794943408", "80631298070974914134063794943438" ], "answer": "80631298070974914134063794943418", "correct_value": 80631298070974914134063794943418, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001417", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "12306 * 8178", "ns_shortcut": "Round to 12500×8000 for easy mental computation.", "shortcut_strength": 0.981115, "pure_math": { "question": "Estimate 12306 × 8178.", "options": [ "100638488", "100638468", "100638458", "100638478" ], "answer": "100638468", "correct_value": 100638468, "explanation": "Round to compatible pair 12500×8000 = 100000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "12319 * 6129", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8758225, "pure_math": { "question": "Estimate 12319 × 6129.", "options": [ "75503151", "75503171", "75503161", "75503141" ], "answer": "75503151", "correct_value": 75503151, "explanation": "One number rounds to 12500, other is farther from 8000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1174774348535227 * 3143675754891268", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 1174774348535227 × 3143675754891268.", "options": [ "3693109636958377318856452697836", "3693109636958377318856452697856", "3693109636958377318856452697846", "3693109636958377318856452697826" ], "answer": "3693109636958377318856452697836", "correct_value": 3693109636958377318856452697836, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001418", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "12367 * 7555", "ns_shortcut": "Round to 12500×8000 for easy mental computation.", "shortcut_strength": 0.9668675, "pure_math": { "question": "Estimate 12367 × 7555.", "options": [ "93432685", "93432675", "93432695", "93432705" ], "answer": "93432685", "correct_value": 93432685, "explanation": "Round to compatible pair 12500×8000 = 100000000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "12624 * 6211", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8832275, "pure_math": { "question": "Estimate 12624 × 6211.", "options": [ "78407664", "78407684", "78407674", "78407654" ], "answer": "78407664", "correct_value": 78407664, "explanation": "One number rounds to 12500, other is farther from 8000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5466174183547032 * 1855605852007037", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 5466174183547032 × 1855605852007037.", "options": [ "10143064803079660164160884464204", "10143064803079660164160884464184", "10143064803079660164160884464194", "10143064803079660164160884464174" ], "answer": "10143064803079660164160884464184", "correct_value": 10143064803079660164160884464184, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001419", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5095 * 21", "ns_shortcut": "Round to 5000×20 for easy mental computation.", "shortcut_strength": 0.9655, "pure_math": { "question": "Estimate 5095 × 21.", "options": [ "107015", "107005", "106985", "106995" ], "answer": "106995", "correct_value": 106995, "explanation": "Round to compatible pair 5000×20 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5059 * 26", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8441000000000001, "pure_math": { "question": "Estimate 5059 × 26.", "options": [ "131554", "131534", "131544", "131524" ], "answer": "131534", "correct_value": 131534, "explanation": "One number rounds to 5000, other is farther from 20." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5771 * 27", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 5771 × 27.", "options": [ "155807", "155837", "155827", "155817" ], "answer": "155817", "correct_value": 155817, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "compatible_numbers_001420", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "12278 * 7286", "ns_shortcut": "Round to 12500×8000 for easy mental computation.", "shortcut_strength": 0.946495, "pure_math": { "question": "Estimate 12278 × 7286.", "options": [ "89457528", "89457498", "89457508", "89457518" ], "answer": "89457508", "correct_value": 89457508, "explanation": "Round to compatible pair 12500×8000 = 100000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "12383 * 10057", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8667575000000001, "pure_math": { "question": "Estimate 12383 × 10057.", "options": [ "124535831", "124535851", "124535841", "124535821" ], "answer": "124535831", "correct_value": 124535831, "explanation": "One number rounds to 12500, other is farther from 8000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "8919221252986174 * 4752771007203461", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 8919221252986174 × 4752771007203461.", "options": [ "42391016178025613634411037948224", "42391016178025613634411037948234", "42391016178025613634411037948214", "42391016178025613634411037948204" ], "answer": "42391016178025613634411037948214", "correct_value": 42391016178025613634411037948214, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001421", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1225 * 85", "ns_shortcut": "Round to 1250×80 for easy mental computation.", "shortcut_strength": 0.95875, "pure_math": { "question": "Estimate 1225 × 85.", "options": [ "104145", "104125", "104135", "104115" ], "answer": "104125", "correct_value": 104125, "explanation": "Round to compatible pair 1250×80 = 100000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1227 * 100", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8658, "pure_math": { "question": "Estimate 1227 × 100.", "options": [ "122690", "122710", "122700", "122720" ], "answer": "122700", "correct_value": 122700, "explanation": "One number rounds to 1250, other is farther from 80." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1542 * 116", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 1542 × 116.", "options": [ "178892", "178872", "178882", "178862" ], "answer": "178872", "correct_value": 178872, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001422", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1244 * 74", "ns_shortcut": "Round to 1250×80 for easy mental computation.", "shortcut_strength": 0.9601, "pure_math": { "question": "Estimate 1244 × 74.", "options": [ "92066", "92076", "92056", "92046" ], "answer": "92056", "correct_value": 92056, "explanation": "Round to compatible pair 1250×80 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1242 * 96", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8968, "pure_math": { "question": "Estimate 1242 × 96.", "options": [ "119242", "119252", "119222", "119232" ], "answer": "119232", "correct_value": 119232, "explanation": "One number rounds to 1250, other is farther from 80." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "1437 * 119", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 1437 × 119.", "options": [ "171003", "171013", "170993", "171023" ], "answer": "171003", "correct_value": 171003, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001423", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "12539 * 8265", "ns_shortcut": "Round to 12500×8000 for easy mental computation.", "shortcut_strength": 0.9818775, "pure_math": { "question": "Estimate 12539 × 8265.", "options": [ "103634825", "103634845", "103634835", "103634855" ], "answer": "103634835", "correct_value": 103634835, "explanation": "Round to compatible pair 12500×8000 = 100000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "12454 * 9933", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8773475, "pure_math": { "question": "Estimate 12454 × 9933.", "options": [ "123705572", "123705592", "123705582", "123705602" ], "answer": "123705582", "correct_value": 123705582, "explanation": "One number rounds to 12500, other is farther from 8000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "3378032915225661 * 1485563249512574", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 3378032915225661 × 1485563249512574.", "options": [ "5018281554503066366821466961424", "5018281554503066366821466961404", "5018281554503066366821466961414", "5018281554503066366821466961434" ], "answer": "5018281554503066366821466961414", "correct_value": 5018281554503066366821466961414, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001424", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "12265 * 8114", "ns_shortcut": "Round to 12500×8000 for easy mental computation.", "shortcut_strength": 0.983475, "pure_math": { "question": "Estimate 12265 × 8114.", "options": [ "99518220", "99518200", "99518230", "99518210" ], "answer": "99518210", "correct_value": 99518210, "explanation": "Round to compatible pair 12500×8000 = 100000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "12281 * 6128", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.87424, "pure_math": { "question": "Estimate 12281 × 6128.", "options": [ "75257958", "75257978", "75257968", "75257988" ], "answer": "75257968", "correct_value": 75257968, "explanation": "One number rounds to 12500, other is farther from 8000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1499948280203443 * 9484714131858546", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 1499948280203443 × 9484714131858546.", "options": [ "14226580650302517973128578173888", "14226580650302517973128578173878", "14226580650302517973128578173868", "14226580650302517973128578173898" ], "answer": "14226580650302517973128578173878", "correct_value": 14226580650302517973128578173878, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001425", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1234 * 76", "ns_shortcut": "Round to 1250×80 for easy mental computation.", "shortcut_strength": 0.9686, "pure_math": { "question": "Estimate 1234 × 76.", "options": [ "93774", "93784", "93804", "93794" ], "answer": "93784", "correct_value": 93784, "explanation": "Round to compatible pair 1250×80 = 100000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1250 * 55", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.84375, "pure_math": { "question": "Estimate 1250 × 55.", "options": [ "68750", "68770", "68740", "68760" ], "answer": "68750", "correct_value": 68750, "explanation": "One number rounds to 1250, other is farther from 80." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1448 * 119", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 1448 × 119.", "options": [ "172302", "172332", "172322", "172312" ], "answer": "172312", "correct_value": 172312, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "compatible_numbers_001426", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "12631 * 7301", "ns_shortcut": "Round to 12500×8000 for easy mental computation.", "shortcut_strength": 0.9510725, "pure_math": { "question": "Estimate 12631 × 7301.", "options": [ "92218921", "92218931", "92218941", "92218951" ], "answer": "92218931", "correct_value": 92218931, "explanation": "Round to compatible pair 12500×8000 = 100000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "12678 * 5849", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8584425, "pure_math": { "question": "Estimate 12678 × 5849.", "options": [ "74153612", "74153622", "74153642", "74153632" ], "answer": "74153622", "correct_value": 74153622, "explanation": "One number rounds to 12500, other is farther from 8000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5231253370538667 * 9052636122378665", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 5231253370538667 × 9052636122378665.", "options": [ "47356633227453480039484398339565", "47356633227453480039484398339545", "47356633227453480039484398339555", "47356633227453480039484398339575" ], "answer": "47356633227453480039484398339555", "correct_value": 47356633227453480039484398339555, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001427", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "12384 * 7642", "ns_shortcut": "Round to 12500×8000 for easy mental computation.", "shortcut_strength": 0.972985, "pure_math": { "question": "Estimate 12384 × 7642.", "options": [ "94638528", "94638538", "94638518", "94638548" ], "answer": "94638528", "correct_value": 94638528, "explanation": "Round to compatible pair 12500×8000 = 100000000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "12307 * 5480", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.83478, "pure_math": { "question": "Estimate 12307 × 5480.", "options": [ "67442370", "67442350", "67442360", "67442380" ], "answer": "67442360", "correct_value": 67442360, "explanation": "One number rounds to 12500, other is farther from 8000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "3671576375942932 * 6648265023707863", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 3671576375942932 × 6648265023707863.", "options": [ "24409612802053466527871527674326", "24409612802053466527871527674316", "24409612802053466527871527674336", "24409612802053466527871527674306" ], "answer": "24409612802053466527871527674316", "correct_value": 24409612802053466527871527674316, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001428", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2454 * 36", "ns_shortcut": "Round to 2500×40 for easy mental computation.", "shortcut_strength": 0.9408, "pure_math": { "question": "Estimate 2454 × 36.", "options": [ "88354", "88344", "88364", "88334" ], "answer": "88344", "correct_value": 88344, "explanation": "Round to compatible pair 2500×40 = 100000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2505 * 51", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8614999999999999, "pure_math": { "question": "Estimate 2505 × 51.", "options": [ "127755", "127745", "127765", "127775" ], "answer": "127755", "correct_value": 127755, "explanation": "One number rounds to 2500, other is farther from 40." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2904 * 54", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 2904 × 54.", "options": [ "156806", "156836", "156816", "156826" ], "answer": "156816", "correct_value": 156816, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001429", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1234 * 80", "ns_shortcut": "Round to 1250×80 for easy mental computation.", "shortcut_strength": 0.9936, "pure_math": { "question": "Estimate 1234 × 80.", "options": [ "98720", "98740", "98710", "98730" ], "answer": "98720", "correct_value": 98720, "explanation": "Round to compatible pair 1250×80 = 100000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1242 * 101", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.86555, "pure_math": { "question": "Estimate 1242 × 101.", "options": [ "125462", "125432", "125452", "125442" ], "answer": "125442", "correct_value": 125442, "explanation": "One number rounds to 1250, other is farther from 80." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "1534 * 120", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 1534 × 120.", "options": [ "184090", "184080", "184070", "184100" ], "answer": "184080", "correct_value": 184080, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001430", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2535 * 42", "ns_shortcut": "Round to 2500×40 for easy mental computation.", "shortcut_strength": 0.968, "pure_math": { "question": "Estimate 2535 × 42.", "options": [ "106460", "106480", "106490", "106470" ], "answer": "106470", "correct_value": 106470, "explanation": "Round to compatible pair 2500×40 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2490 * 50", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.873, "pure_math": { "question": "Estimate 2490 × 50.", "options": [ "124510", "124500", "124490", "124520" ], "answer": "124500", "correct_value": 124500, "explanation": "One number rounds to 2500, other is farther from 40." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2942 * 60", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 2942 × 60.", "options": [ "176530", "176520", "176510", "176540" ], "answer": "176520", "correct_value": 176520, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001431", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "4988 * 19", "ns_shortcut": "Round to 5000×20 for easy mental computation.", "shortcut_strength": 0.9738, "pure_math": { "question": "Estimate 4988 × 19.", "options": [ "94782", "94762", "94792", "94772" ], "answer": "94772", "correct_value": 94772, "explanation": "Round to compatible pair 5000×20 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4975 * 15", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8725, "pure_math": { "question": "Estimate 4975 × 15.", "options": [ "74645", "74635", "74625", "74615" ], "answer": "74625", "correct_value": 74625, "explanation": "One number rounds to 5000, other is farther from 20." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5630 * 30", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 5630 × 30.", "options": [ "168920", "168900", "168890", "168910" ], "answer": "168900", "correct_value": 168900, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001432", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2536 * 41", "ns_shortcut": "Round to 2500×40 for easy mental computation.", "shortcut_strength": 0.9803, "pure_math": { "question": "Estimate 2536 × 41.", "options": [ "103986", "103966", "103996", "103976" ], "answer": "103976", "correct_value": 103976, "explanation": "Round to compatible pair 2500×40 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2496 * 49", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8867, "pure_math": { "question": "Estimate 2496 × 49.", "options": [ "122324", "122314", "122304", "122294" ], "answer": "122304", "correct_value": 122304, "explanation": "One number rounds to 2500, other is farther from 40." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2796 * 54", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 2796 × 54.", "options": [ "150984", "150994", "150974", "151004" ], "answer": "150984", "correct_value": 150984, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001433", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2485 * 38", "ns_shortcut": "Round to 2500×40 for easy mental computation.", "shortcut_strength": 0.972, "pure_math": { "question": "Estimate 2485 × 38.", "options": [ "94420", "94430", "94440", "94450" ], "answer": "94430", "correct_value": 94430, "explanation": "Round to compatible pair 2500×40 = 100000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2496 * 49", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8867, "pure_math": { "question": "Estimate 2496 × 49.", "options": [ "122314", "122324", "122294", "122304" ], "answer": "122304", "correct_value": 122304, "explanation": "One number rounds to 2500, other is farther from 40." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "2907 * 57", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 2907 × 57.", "options": [ "165689", "165719", "165709", "165699" ], "answer": "165699", "correct_value": 165699, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "compatible_numbers_001434", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2497 * 42", "ns_shortcut": "Round to 2500×40 for easy mental computation.", "shortcut_strength": 0.9744, "pure_math": { "question": "Estimate 2497 × 42.", "options": [ "104874", "104884", "104864", "104894" ], "answer": "104874", "correct_value": 104874, "explanation": "Round to compatible pair 2500×40 = 100000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2529 * 53", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8317, "pure_math": { "question": "Estimate 2529 × 53.", "options": [ "134057", "134047", "134037", "134027" ], "answer": "134037", "correct_value": 134037, "explanation": "One number rounds to 2500, other is farther from 40." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2916 * 57", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 2916 × 57.", "options": [ "166222", "166202", "166232", "166212" ], "answer": "166212", "correct_value": 166212, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "compatible_numbers_001435", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2474 * 42", "ns_shortcut": "Round to 2500×40 for easy mental computation.", "shortcut_strength": 0.9698, "pure_math": { "question": "Estimate 2474 × 42.", "options": [ "103928", "103898", "103908", "103918" ], "answer": "103908", "correct_value": 103908, "explanation": "Round to compatible pair 2500×40 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2543 * 28", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8414, "pure_math": { "question": "Estimate 2543 × 28.", "options": [ "71224", "71214", "71194", "71204" ], "answer": "71204", "correct_value": 71204, "explanation": "One number rounds to 2500, other is farther from 40." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "2803 * 59", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 2803 × 59.", "options": [ "165387", "165377", "165397", "165367" ], "answer": "165377", "correct_value": 165377, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001436", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2452 * 39", "ns_shortcut": "Round to 2500×40 for easy mental computation.", "shortcut_strength": 0.9779, "pure_math": { "question": "Estimate 2452 × 39.", "options": [ "95638", "95618", "95628", "95648" ], "answer": "95628", "correct_value": 95628, "explanation": "Round to compatible pair 2500×40 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2534 * 29", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8557, "pure_math": { "question": "Estimate 2534 × 29.", "options": [ "73476", "73506", "73486", "73496" ], "answer": "73486", "correct_value": 73486, "explanation": "One number rounds to 2500, other is farther from 40." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "3074 * 60", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 3074 × 60.", "options": [ "184440", "184450", "184430", "184460" ], "answer": "184440", "correct_value": 184440, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001437", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5079 * 20", "ns_shortcut": "Round to 5000×20 for easy mental computation.", "shortcut_strength": 0.9921, "pure_math": { "question": "Estimate 5079 × 20.", "options": [ "101590", "101570", "101600", "101580" ], "answer": "101580", "correct_value": 101580, "explanation": "Round to compatible pair 5000×20 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5063 * 14", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8437, "pure_math": { "question": "Estimate 5063 × 14.", "options": [ "70902", "70892", "70872", "70882" ], "answer": "70882", "correct_value": 70882, "explanation": "One number rounds to 5000, other is farther from 20." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6082 * 27", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 6082 × 27.", "options": [ "164214", "164204", "164234", "164224" ], "answer": "164214", "correct_value": 164214, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001438", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "25115 * 3718", "ns_shortcut": "Round to 25000×4000 for easy mental computation.", "shortcut_strength": 0.96245, "pure_math": { "question": "Estimate 25115 × 3718.", "options": [ "93377560", "93377570", "93377580", "93377590" ], "answer": "93377570", "correct_value": 93377570, "explanation": "Round to compatible pair 25000×4000 = 100000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "25083 * 2897", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.860465, "pure_math": { "question": "Estimate 25083 × 2897.", "options": [ "72665461", "72665451", "72665441", "72665471" ], "answer": "72665451", "correct_value": 72665451, "explanation": "One number rounds to 25000, other is farther from 4000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "9629849444815274 * 9393404118088864", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 9629849444815274 × 9393404118088864.", "options": [ "90457067431503555481962396508756", "90457067431503555481962396508736", "90457067431503555481962396508726", "90457067431503555481962396508746" ], "answer": "90457067431503555481962396508736", "correct_value": 90457067431503555481962396508736, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001439", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "12580 * 8354", "ns_shortcut": "Round to 12500×8000 for easy mental computation.", "shortcut_strength": 0.974675, "pure_math": { "question": "Estimate 12580 × 8354.", "options": [ "105093340", "105093320", "105093310", "105093330" ], "answer": "105093320", "correct_value": 105093320, "explanation": "Round to compatible pair 12500×8000 = 100000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "12630 * 6310", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.889175, "pure_math": { "question": "Estimate 12630 × 6310.", "options": [ "79695290", "79695300", "79695320", "79695310" ], "answer": "79695300", "correct_value": 79695300, "explanation": "One number rounds to 12500, other is farther from 8000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6887467545148327 * 6152846126004433", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 6887467545148327 × 6152846126004433.", "options": [ "42377528003147146020958844533601", "42377528003147146020958844533591", "42377528003147146020958844533611", "42377528003147146020958844533581" ], "answer": "42377528003147146020958844533591", "correct_value": 42377528003147146020958844533591, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001440", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1237 * 84", "ns_shortcut": "Round to 1250×80 for easy mental computation.", "shortcut_strength": 0.9698, "pure_math": { "question": "Estimate 1237 × 84.", "options": [ "103908", "103928", "103918", "103898" ], "answer": "103908", "correct_value": 103908, "explanation": "Round to compatible pair 1250×80 = 100000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1272 * 96", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8912, "pure_math": { "question": "Estimate 1272 × 96.", "options": [ "122132", "122102", "122122", "122112" ], "answer": "122112", "correct_value": 122112, "explanation": "One number rounds to 1250, other is farther from 80." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "1537 * 114", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 1537 × 114.", "options": [ "175218", "175208", "175228", "175238" ], "answer": "175218", "correct_value": 175218, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001441", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1249 * 74", "ns_shortcut": "Round to 1250×80 for easy mental computation.", "shortcut_strength": 0.9621, "pure_math": { "question": "Estimate 1249 × 74.", "options": [ "92426", "92436", "92416", "92446" ], "answer": "92426", "correct_value": 92426, "explanation": "Round to compatible pair 1250×80 = 100000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1275 * 61", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.87125, "pure_math": { "question": "Estimate 1275 × 61.", "options": [ "77775", "77785", "77795", "77765" ], "answer": "77775", "correct_value": 77775, "explanation": "One number rounds to 1250, other is farther from 80." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1561 * 120", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 1561 × 120.", "options": [ "187320", "187340", "187330", "187310" ], "answer": "187320", "correct_value": 187320, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001442", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "4974 * 22", "ns_shortcut": "Round to 5000×20 for easy mental computation.", "shortcut_strength": 0.9474, "pure_math": { "question": "Estimate 4974 × 22.", "options": [ "109448", "109418", "109438", "109428" ], "answer": "109428", "correct_value": 109428, "explanation": "Round to compatible pair 5000×20 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4925 * 16", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8925, "pure_math": { "question": "Estimate 4925 × 16.", "options": [ "78810", "78790", "78820", "78800" ], "answer": "78800", "correct_value": 78800, "explanation": "One number rounds to 5000, other is farther from 20." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6069 * 29", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 6069 × 29.", "options": [ "175991", "176021", "176011", "176001" ], "answer": "176001", "correct_value": 176001, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "compatible_numbers_001443", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1252 * 77", "ns_shortcut": "Round to 1250×80 for easy mental computation.", "shortcut_strength": 0.98045, "pure_math": { "question": "Estimate 1252 × 77.", "options": [ "96424", "96414", "96394", "96404" ], "answer": "96404", "correct_value": 96404, "explanation": "Round to compatible pair 1250×80 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1226 * 104", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8404, "pure_math": { "question": "Estimate 1226 × 104.", "options": [ "127514", "127504", "127494", "127524" ], "answer": "127504", "correct_value": 127504, "explanation": "One number rounds to 1250, other is farther from 80." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1518 * 108", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 1518 × 108.", "options": [ "163934", "163944", "163954", "163964" ], "answer": "163944", "correct_value": 163944, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001444", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2529 * 38", "ns_shortcut": "Round to 2500×40 for easy mental computation.", "shortcut_strength": 0.9692, "pure_math": { "question": "Estimate 2529 × 38.", "options": [ "96102", "96122", "96092", "96112" ], "answer": "96102", "correct_value": 96102, "explanation": "Round to compatible pair 2500×40 = 100000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2465 * 28", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.843, "pure_math": { "question": "Estimate 2465 × 28.", "options": [ "69010", "69030", "69020", "69040" ], "answer": "69020", "correct_value": 69020, "explanation": "One number rounds to 2500, other is farther from 40." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2992 * 56", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 2992 × 56.", "options": [ "167542", "167562", "167572", "167552" ], "answer": "167552", "correct_value": 167552, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "compatible_numbers_001445", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "4916 * 21", "ns_shortcut": "Round to 5000×20 for easy mental computation.", "shortcut_strength": 0.9666, "pure_math": { "question": "Estimate 4916 × 21.", "options": [ "103226", "103246", "103256", "103236" ], "answer": "103236", "correct_value": 103236, "explanation": "Round to compatible pair 5000×20 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5080 * 25", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.867, "pure_math": { "question": "Estimate 5080 × 25.", "options": [ "127010", "127000", "127020", "126990" ], "answer": "127000", "correct_value": 127000, "explanation": "One number rounds to 5000, other is farther from 20." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6107 * 27", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 6107 × 27.", "options": [ "164899", "164889", "164879", "164909" ], "answer": "164889", "correct_value": 164889, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001446", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "4909 * 20", "ns_shortcut": "Round to 5000×20 for easy mental computation.", "shortcut_strength": 0.9909, "pure_math": { "question": "Estimate 4909 × 20.", "options": [ "98170", "98200", "98180", "98190" ], "answer": "98180", "correct_value": 98180, "explanation": "Round to compatible pair 5000×20 = 100000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5026 * 15", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8724000000000001, "pure_math": { "question": "Estimate 5026 × 15.", "options": [ "75390", "75380", "75400", "75410" ], "answer": "75390", "correct_value": 75390, "explanation": "One number rounds to 5000, other is farther from 20." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6197 * 29", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 6197 × 29.", "options": [ "179713", "179733", "179723", "179703" ], "answer": "179713", "correct_value": 179713, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001447", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2517 * 44", "ns_shortcut": "Round to 2500×40 for easy mental computation.", "shortcut_strength": 0.9466, "pure_math": { "question": "Estimate 2517 × 44.", "options": [ "110748", "110768", "110738", "110758" ], "answer": "110748", "correct_value": 110748, "explanation": "Round to compatible pair 2500×40 = 100000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2536 * 52", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8428, "pure_math": { "question": "Estimate 2536 × 52.", "options": [ "131892", "131872", "131882", "131862" ], "answer": "131872", "correct_value": 131872, "explanation": "One number rounds to 2500, other is farther from 40." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2790 * 56", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 2790 × 56.", "options": [ "156250", "156240", "156230", "156260" ], "answer": "156240", "correct_value": 156240, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001448", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2520 * 42", "ns_shortcut": "Round to 2500×40 for easy mental computation.", "shortcut_strength": 0.971, "pure_math": { "question": "Estimate 2520 × 42.", "options": [ "105840", "105830", "105860", "105850" ], "answer": "105840", "correct_value": 105840, "explanation": "Round to compatible pair 2500×40 = 100000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2534 * 50", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8682, "pure_math": { "question": "Estimate 2534 × 50.", "options": [ "126710", "126690", "126720", "126700" ], "answer": "126700", "correct_value": 126700, "explanation": "One number rounds to 2500, other is farther from 40." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "3111 * 57", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 3111 × 57.", "options": [ "177337", "177327", "177347", "177317" ], "answer": "177327", "correct_value": 177327, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001449", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "12604 * 8724", "ns_shortcut": "Round to 12500×8000 for easy mental computation.", "shortcut_strength": 0.95059, "pure_math": { "question": "Estimate 12604 × 8724.", "options": [ "109957286", "109957296", "109957306", "109957316" ], "answer": "109957296", "correct_value": 109957296, "explanation": "Round to compatible pair 12500×8000 = 100000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "12707 * 5559", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.8391575, "pure_math": { "question": "Estimate 12707 × 5559.", "options": [ "70638213", "70638203", "70638233", "70638223" ], "answer": "70638213", "correct_value": 70638213, "explanation": "One number rounds to 12500, other is farther from 8000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "3183236220829731 * 9057157894685061", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 3183236220829731 × 9057157894685061.", "options": [ "28831073068135436345224350348581", "28831073068135436345224350348591", "28831073068135436345224350348611", "28831073068135436345224350348601" ], "answer": "28831073068135436345224350348591", "correct_value": 28831073068135436345224350348591, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001450", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "50726 * 2178", "ns_shortcut": "Round to 50000×2000 for easy mental computation.", "shortcut_strength": 0.94824, "pure_math": { "question": "Estimate 50726 × 2178.", "options": [ "110481238", "110481218", "110481228", "110481248" ], "answer": "110481228", "correct_value": 110481228, "explanation": "Round to compatible pair 50000×2000 = 100000000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "49978 * 2500", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.87478, "pure_math": { "question": "Estimate 49978 × 2500.", "options": [ "124945000", "124945010", "124945020", "124944990" ], "answer": "124945000", "correct_value": 124945000, "explanation": "One number rounds to 50000, other is farther from 2000." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1720962281842074 * 1674691518836428", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 1720962281842074 × 1674691518836428.", "options": [ "2882080937638307782804934271662", "2882080937638307782804934271692", "2882080937638307782804934271672", "2882080937638307782804934271682" ], "answer": "2882080937638307782804934271672", "correct_value": 2882080937638307782804934271672, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "compatible_numbers_001451", "category": "compatible_numbers", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "12731 * 8051", "ns_shortcut": "Round to 12500×8000 for easy mental computation.", "shortcut_strength": 0.9875725, "pure_math": { "question": "Estimate 12731 × 8051.", "options": [ "102497301", "102497281", "102497271", "102497291" ], "answer": "102497281", "correct_value": 102497281, "explanation": "Round to compatible pair 12500×8000 = 100000000." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "12677 * 10324", "ns_shortcut": "Compatible pair shortcut works but less precise.", "shortcut_strength": 0.84767, "pure_math": { "question": "Estimate 12677 × 10324.", "options": [ "130877358", "130877338", "130877368", "130877348" ], "answer": "130877348", "correct_value": 130877348, "explanation": "One number rounds to 12500, other is farther from 8000." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "3630342636561734 * 7839339991393261", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Estimate 3630342636561734 × 7839339991393261.", "options": [ "28459490213258452262101298074564", "28459490213258452262101298074584", "28459490213258452262101298074574", "28459490213258452262101298074594" ], "answer": "28459490213258452262101298074574", "correct_value": 28459490213258452262101298074574, "explanation": "Not near any compatible pair; full multiplication needed." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "landmark_comparison_001452", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "50% of 17459110716362590 vs 8050826308646471", "ns_shortcut": "Use 50% as landmark: 50% of 17459110716362590 = 8729555358181295.", "shortcut_strength": 1.0, "pure_math": { "question": "Is 50% of 17459110716362590 greater than 8050826308646471?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 8729555358181295.0, "explanation": "50% ≈ 50%. 50% of 17459110716362590 = 8729555358181295. Compare to 8050826308646471." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "46% of 17459110716362590 vs 8050826308646471", "ns_shortcut": "Use 50% as approximate anchor, then adjust.", "shortcut_strength": 0.96, "pure_math": { "question": "Is 46% of 17459110716362590 greater than 8050826308646471?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 8031190929526791.0, "explanation": "46% is near 50% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "67% of 17459110716362590 vs 8050826308646471", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 67% of 17459110716362590 greater than 8050826308646471?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.1697604179962936e+16, "explanation": "67% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001453", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "52% of 10451081240165040 vs 3592814225867691", "ns_shortcut": "Use 50% as landmark: 50% of 10451081240165040 = 5225540620082520.", "shortcut_strength": 0.98, "pure_math": { "question": "Is 52% of 10451081240165040 greater than 3592814225867691?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 5434562244885821.0, "explanation": "52% ≈ 50%. 50% of 10451081240165040 = 5225540620082520. Compare to 3592814225867691." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "45% of 10451081240165040 vs 3592814225867691", "ns_shortcut": "Use 50% as approximate anchor, then adjust.", "shortcut_strength": 0.95, "pure_math": { "question": "Is 45% of 10451081240165040 greater than 3592814225867691?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 4702986558074268.0, "explanation": "45% is near 50% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "83% of 10451081240165040 vs 3592814225867691", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 83% of 10451081240165040 greater than 3592814225867691?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 8674397429336983.0, "explanation": "83% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001454", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "24% of 46050855933712470 vs 22664602861215469", "ns_shortcut": "Use 25% as landmark: 25% of 46050855933712470 = 11512713983428117.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 24% of 46050855933712470 greater than 22664602861215469?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.1052205424090992e+16, "explanation": "24% ≈ 25%. 25% of 46050855933712470 = 11512713983428117. Compare to 22664602861215469." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "30% of 46050855933712470 vs 22664602861215469", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.95, "pure_math": { "question": "Is 30% of 46050855933712470 greater than 22664602861215469?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.381525678011374e+16, "explanation": "30% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "67% of 46050855933712470 vs 22664602861215469", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 67% of 46050855933712470 greater than 22664602861215469?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 3.0854073475587356e+16, "explanation": "67% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001455", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "49% of 89331027387217860 vs 30297949728752351", "ns_shortcut": "Use 50% as landmark: 50% of 89331027387217860 = 44665513693608930.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 49% of 89331027387217860 greater than 30297949728752351?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 4.377220341973675e+16, "explanation": "49% ≈ 50%. 50% of 89331027387217860 = 44665513693608930. Compare to 30297949728752351." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "43% of 89331027387217860 vs 30297949728752351", "ns_shortcut": "Use 50% as approximate anchor, then adjust.", "shortcut_strength": 0.9299999999999999, "pure_math": { "question": "Is 43% of 89331027387217860 greater than 30297949728752351?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 3.841234177650368e+16, "explanation": "43% is near 50% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "73% of 89331027387217860 vs 30297949728752351", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 73% of 89331027387217860 greater than 30297949728752351?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 6.521164999266904e+16, "explanation": "73% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001456", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9% of 87082915564319140 vs 31932610460138434", "ns_shortcut": "Use 10% as landmark: 10% of 87082915564319140 = 8708291556431914.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 9% of 87082915564319140 greater than 31932610460138434?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 7837462400788723.0, "explanation": "9% ≈ 10%. 10% of 87082915564319140 = 8708291556431914. Compare to 31932610460138434." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7% of 87082915564319140 vs 31932610460138434", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.97, "pure_math": { "question": "Is 7% of 87082915564319140 greater than 31932610460138434?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 6095804089502340.0, "explanation": "7% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "67% of 87082915564319140 vs 31932610460138434", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 67% of 87082915564319140 greater than 31932610460138434?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 5.834555342809382e+16, "explanation": "67% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001457", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "12% of 28573790056161910 vs 14048027054512305", "ns_shortcut": "Use 10% as landmark: 10% of 28573790056161910 = 2857379005616191.", "shortcut_strength": 0.98, "pure_math": { "question": "Is 12% of 28573790056161910 greater than 14048027054512305?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 3428854806739429.0, "explanation": "12% ≈ 10%. 10% of 28573790056161910 = 2857379005616191. Compare to 14048027054512305." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7% of 28573790056161910 vs 14048027054512305", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.97, "pure_math": { "question": "Is 7% of 28573790056161910 greater than 14048027054512305?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 2000165303931333.8, "explanation": "7% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "83% of 28573790056161910 vs 14048027054512305", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 83% of 28573790056161910 greater than 14048027054512305?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.3716245746614384e+16, "explanation": "83% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001458", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "26% of 23795163979851260 vs 9539295793704942", "ns_shortcut": "Use 25% as landmark: 25% of 23795163979851260 = 5948790994962815.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 26% of 23795163979851260 greater than 9539295793704942?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 6186742634761328.0, "explanation": "26% ≈ 25%. 25% of 23795163979851260 = 5948790994962815. Compare to 9539295793704942." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "18% of 23795163979851260 vs 9539295793704942", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.9299999999999999, "pure_math": { "question": "Is 18% of 23795163979851260 greater than 9539295793704942?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 4283129516373227.0, "explanation": "18% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "67% of 23795163979851260 vs 9539295793704942", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 67% of 23795163979851260 greater than 9539295793704942?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.5942759866500344e+16, "explanation": "67% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001459", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "11% of 27011507141015730 vs 10554357249151569", "ns_shortcut": "Use 10% as landmark: 10% of 27011507141015730 = 2701150714101573.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 11% of 27011507141015730 greater than 10554357249151569?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 2971265785511730.5, "explanation": "11% ≈ 10%. 10% of 27011507141015730 = 2701150714101573. Compare to 10554357249151569." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6% of 27011507141015730 vs 10554357249151569", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.96, "pure_math": { "question": "Is 6% of 27011507141015730 greater than 10554357249151569?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1620690428460943.8, "explanation": "6% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "87% of 27011507141015730 vs 10554357249151569", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 87% of 27011507141015730 greater than 10554357249151569?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.3500011212683684e+16, "explanation": "87% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001460", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10% of 38429370135837590 vs 11686612358710279", "ns_shortcut": "Use 10% as landmark: 10% of 38429370135837590 = 3842937013583759.", "shortcut_strength": 1.0, "pure_math": { "question": "Is 10% of 38429370135837590 greater than 11686612358710279?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 3842937013583759.0, "explanation": "10% ≈ 10%. 10% of 38429370135837590 = 3842937013583759. Compare to 11686612358710279." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6% of 38429370135837590 vs 11686612358710279", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.96, "pure_math": { "question": "Is 6% of 38429370135837590 greater than 11686612358710279?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 2305762208150255.5, "explanation": "6% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "43% of 38429370135837590 vs 11686612358710279", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 43% of 38429370135837590 greater than 11686612358710279?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.6524629158410164e+16, "explanation": "43% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001461", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "51% of 26428481246050850 vs 7625961294369026", "ns_shortcut": "Use 50% as landmark: 50% of 26428481246050850 = 13214240623025425.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 51% of 26428481246050850 greater than 7625961294369026?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.3478525435485934e+16, "explanation": "51% ≈ 50%. 50% of 26428481246050850 = 13214240623025425. Compare to 7625961294369026." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "57% of 26428481246050850 vs 7625961294369026", "ns_shortcut": "Use 50% as approximate anchor, then adjust.", "shortcut_strength": 0.9299999999999999, "pure_math": { "question": "Is 57% of 26428481246050850 greater than 7625961294369026?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.5064234310248984e+16, "explanation": "57% is near 50% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "67% of 26428481246050850 vs 7625961294369026", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 67% of 26428481246050850 greater than 7625961294369026?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.770708243485407e+16, "explanation": "67% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001462", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "24% of 77408836909998320 vs 15572835271909931", "ns_shortcut": "Use 25% as landmark: 25% of 77408836909998320 = 19352209227499580.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 24% of 77408836909998320 greater than 15572835271909931?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.8578120858399596e+16, "explanation": "24% ≈ 25%. 25% of 77408836909998320 = 19352209227499580. Compare to 15572835271909931." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "28% of 77408836909998320 vs 15572835271909931", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.97, "pure_math": { "question": "Is 28% of 77408836909998320 greater than 15572835271909931?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.167447433479953e+16, "explanation": "28% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "83% of 77408836909998320 vs 15572835271909931", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 83% of 77408836909998320 greater than 15572835271909931?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 6.424933463529861e+16, "explanation": "83% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001463", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9% of 61297616079390110 vs 15864383491083749", "ns_shortcut": "Use 10% as landmark: 10% of 61297616079390110 = 6129761607939011.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 9% of 61297616079390110 greater than 15864383491083749?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 5516785447145110.0, "explanation": "9% ≈ 10%. 10% of 61297616079390110 = 6129761607939011. Compare to 15864383491083749." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "15% of 61297616079390110 vs 15864383491083749", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.95, "pure_math": { "question": "Is 15% of 61297616079390110 greater than 15864383491083749?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 9194642411908516.0, "explanation": "15% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "67% of 61297616079390110 vs 15864383491083749", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 67% of 61297616079390110 greater than 15864383491083749?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 4.106940277319138e+16, "explanation": "67% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001464", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "11% of 23796898186616470 vs 10862534919147394", "ns_shortcut": "Use 10% as landmark: 10% of 23796898186616470 = 2379689818661647.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 11% of 23796898186616470 greater than 10862534919147394?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 2617658800527811.5, "explanation": "11% ≈ 10%. 10% of 23796898186616470 = 2379689818661647. Compare to 10862534919147394." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5% of 23796898186616470 vs 10862534919147394", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.95, "pure_math": { "question": "Is 5% of 23796898186616470 greater than 10862534919147394?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1189844909330823.5, "explanation": "5% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "83% of 23796898186616470 vs 10862534919147394", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 83% of 23796898186616470 greater than 10862534919147394?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.975142549489167e+16, "explanation": "83% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001465", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9% of 24880935572536320 vs 6831747586587914", "ns_shortcut": "Use 10% as landmark: 10% of 24880935572536320 = 2488093557253632.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 9% of 24880935572536320 greater than 6831747586587914?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 2239284201528268.8, "explanation": "9% ≈ 10%. 10% of 24880935572536320 = 2488093557253632. Compare to 6831747586587914." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "17% of 24880935572536320 vs 6831747586587914", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.9299999999999999, "pure_math": { "question": "Is 17% of 24880935572536320 greater than 6831747586587914?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 4229759047331174.5, "explanation": "17% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "83% of 24880935572536320 vs 6831747586587914", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 83% of 24880935572536320 greater than 6831747586587914?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.0651176525205144e+16, "explanation": "83% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001466", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "52% of 56838644635808480 vs 23490767007714413", "ns_shortcut": "Use 50% as landmark: 50% of 56838644635808480 = 28419322317904240.", "shortcut_strength": 0.98, "pure_math": { "question": "Is 52% of 56838644635808480 greater than 23490767007714413?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.955609521062041e+16, "explanation": "52% ≈ 50%. 50% of 56838644635808480 = 28419322317904240. Compare to 23490767007714413." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "55% of 56838644635808480 vs 23490767007714413", "ns_shortcut": "Use 50% as approximate anchor, then adjust.", "shortcut_strength": 0.95, "pure_math": { "question": "Is 55% of 56838644635808480 greater than 23490767007714413?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 3.1261254549694664e+16, "explanation": "55% is near 50% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "73% of 56838644635808480 vs 23490767007714413", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 73% of 56838644635808480 greater than 23490767007714413?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 4.149221058414019e+16, "explanation": "73% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001467", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "24% of 49124851894302840 vs 16656667124369784", "ns_shortcut": "Use 25% as landmark: 25% of 49124851894302840 = 12281212973575710.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 24% of 49124851894302840 greater than 16656667124369784?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.1789964454632682e+16, "explanation": "24% ≈ 25%. 25% of 49124851894302840 = 12281212973575710. Compare to 16656667124369784." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "20% of 49124851894302840 vs 16656667124369784", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.95, "pure_math": { "question": "Is 20% of 49124851894302840 greater than 16656667124369784?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 9824970378860568.0, "explanation": "20% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "67% of 49124851894302840 vs 16656667124369784", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 67% of 49124851894302840 greater than 16656667124369784?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 3.2913650769182904e+16, "explanation": "67% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001468", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8% of 75327797025445290 vs 24552805491308206", "ns_shortcut": "Use 10% as landmark: 10% of 75327797025445290 = 7532779702544529.", "shortcut_strength": 0.98, "pure_math": { "question": "Is 8% of 75327797025445290 greater than 24552805491308206?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 6026223762035623.0, "explanation": "8% ≈ 10%. 10% of 75327797025445290 = 7532779702544529. Compare to 24552805491308206." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "14% of 75327797025445290 vs 24552805491308206", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.96, "pure_math": { "question": "Is 14% of 75327797025445290 greater than 24552805491308206?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.054589158356234e+16, "explanation": "14% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "73% of 75327797025445290 vs 24552805491308206", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 73% of 75327797025445290 greater than 24552805491308206?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 5.4989291828575064e+16, "explanation": "73% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001469", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "51% of 52401443450506650 vs 19354707558755145", "ns_shortcut": "Use 50% as landmark: 50% of 52401443450506650 = 26200721725253325.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 51% of 52401443450506650 greater than 19354707558755145?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.672473615975839e+16, "explanation": "51% ≈ 50%. 50% of 52401443450506650 = 26200721725253325. Compare to 19354707558755145." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "46% of 52401443450506650 vs 19354707558755145", "ns_shortcut": "Use 50% as approximate anchor, then adjust.", "shortcut_strength": 0.96, "pure_math": { "question": "Is 46% of 52401443450506650 greater than 19354707558755145?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.410466398723306e+16, "explanation": "46% is near 50% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "37% of 52401443450506650 vs 19354707558755145", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 37% of 52401443450506650 greater than 19354707558755145?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.938853407668746e+16, "explanation": "37% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001470", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "27% of 62445580654912800 vs 14426322035310753", "ns_shortcut": "Use 25% as landmark: 25% of 62445580654912800 = 15611395163728200.", "shortcut_strength": 0.98, "pure_math": { "question": "Is 27% of 62445580654912800 greater than 14426322035310753?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.6860306776826456e+16, "explanation": "27% ≈ 25%. 25% of 62445580654912800 = 15611395163728200. Compare to 14426322035310753." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "20% of 62445580654912800 vs 14426322035310753", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.95, "pure_math": { "question": "Is 20% of 62445580654912800 greater than 14426322035310753?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.248911613098256e+16, "explanation": "20% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "37% of 62445580654912800 vs 14426322035310753", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 37% of 62445580654912800 greater than 14426322035310753?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.3104864842317736e+16, "explanation": "37% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001471", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "25% of 28960964022867870 vs 11140145893571161", "ns_shortcut": "Use 25% as landmark: 25% of 28960964022867870 = 7240241005716967.", "shortcut_strength": 1.0, "pure_math": { "question": "Is 25% of 28960964022867870 greater than 11140145893571161?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 7240241005716968.0, "explanation": "25% ≈ 25%. 25% of 28960964022867870 = 7240241005716967. Compare to 11140145893571161." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "22% of 28960964022867870 vs 11140145893571161", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.97, "pure_math": { "question": "Is 22% of 28960964022867870 greater than 11140145893571161?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 6371412085030931.0, "explanation": "22% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "37% of 28960964022867870 vs 11140145893571161", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 37% of 28960964022867870 greater than 11140145893571161?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.0715556688461112e+16, "explanation": "37% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001472", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "26% of 95545574749616180 vs 42291807950807293", "ns_shortcut": "Use 25% as landmark: 25% of 95545574749616180 = 23886393687404045.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 26% of 95545574749616180 greater than 42291807950807293?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 2.484184943490021e+16, "explanation": "26% ≈ 25%. 25% of 95545574749616180 = 23886393687404045. Compare to 42291807950807293." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "20% of 95545574749616180 vs 42291807950807293", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.95, "pure_math": { "question": "Is 20% of 95545574749616180 greater than 42291807950807293?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.9109114949923236e+16, "explanation": "20% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "83% of 95545574749616180 vs 42291807950807293", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 83% of 95545574749616180 greater than 42291807950807293?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 7.930282704218142e+16, "explanation": "83% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001473", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "49% of 33673686259299930 vs 15449322689230523", "ns_shortcut": "Use 50% as landmark: 50% of 33673686259299930 = 16836843129649965.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 49% of 33673686259299930 greater than 15449322689230523?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.6500106267056966e+16, "explanation": "49% ≈ 50%. 50% of 33673686259299930 = 16836843129649965. Compare to 15449322689230523." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "56% of 33673686259299930 vs 15449322689230523", "ns_shortcut": "Use 50% as approximate anchor, then adjust.", "shortcut_strength": 0.94, "pure_math": { "question": "Is 56% of 33673686259299930 greater than 15449322689230523?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.885726430520796e+16, "explanation": "56% is near 50% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "83% of 33673686259299930 vs 15449322689230523", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 83% of 33673686259299930 greater than 15449322689230523?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.794915959521894e+16, "explanation": "83% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001474", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "51% of 46145041887937950 vs 11100026074081148", "ns_shortcut": "Use 50% as landmark: 50% of 46145041887937950 = 23072520943968975.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 51% of 46145041887937950 greater than 11100026074081148?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.3533971362848356e+16, "explanation": "51% ≈ 50%. 50% of 46145041887937950 = 23072520943968975. Compare to 11100026074081148." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "55% of 46145041887937950 vs 11100026074081148", "ns_shortcut": "Use 50% as approximate anchor, then adjust.", "shortcut_strength": 0.95, "pure_math": { "question": "Is 55% of 46145041887937950 greater than 11100026074081148?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.537977303836587e+16, "explanation": "55% is near 50% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "37% of 46145041887937950 vs 11100026074081148", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 37% of 46145041887937950 greater than 11100026074081148?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.7073665498537042e+16, "explanation": "37% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001475", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "11% of 58095490178373490 vs 27391357525101115", "ns_shortcut": "Use 10% as landmark: 10% of 58095490178373490 = 5809549017837349.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 11% of 58095490178373490 greater than 27391357525101115?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 6390503919621084.0, "explanation": "11% ≈ 10%. 10% of 58095490178373490 = 5809549017837349. Compare to 27391357525101115." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "3% of 58095490178373490 vs 27391357525101115", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.9299999999999999, "pure_math": { "question": "Is 3% of 58095490178373490 greater than 27391357525101115?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1742864705351204.8, "explanation": "3% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "67% of 58095490178373490 vs 27391357525101115", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 67% of 58095490178373490 greater than 27391357525101115?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 3.892397841951024e+16, "explanation": "67% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001476", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "11% of 31417977626260880 vs 8887891296580495", "ns_shortcut": "Use 10% as landmark: 10% of 31417977626260880 = 3141797762626088.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 11% of 31417977626260880 greater than 8887891296580495?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 3455977538888697.0, "explanation": "11% ≈ 10%. 10% of 31417977626260880 = 3141797762626088. Compare to 8887891296580495." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "17% of 31417977626260880 vs 8887891296580495", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.9299999999999999, "pure_math": { "question": "Is 17% of 31417977626260880 greater than 8887891296580495?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 5341056196464350.0, "explanation": "17% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "37% of 31417977626260880 vs 8887891296580495", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 37% of 31417977626260880 greater than 8887891296580495?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.1624651721716526e+16, "explanation": "37% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001477", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "11% of 79828043614688640 vs 33475040815007993", "ns_shortcut": "Use 10% as landmark: 10% of 79828043614688640 = 7982804361468864.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 11% of 79828043614688640 greater than 33475040815007993?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 8781084797615750.0, "explanation": "11% ≈ 10%. 10% of 79828043614688640 = 7982804361468864. Compare to 33475040815007993." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "17% of 79828043614688640 vs 33475040815007993", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.9299999999999999, "pure_math": { "question": "Is 17% of 79828043614688640 greater than 33475040815007993?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.3570767414497068e+16, "explanation": "17% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "37% of 79828043614688640 vs 33475040815007993", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 37% of 79828043614688640 greater than 33475040815007993?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 2.9536376137434796e+16, "explanation": "37% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001478", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "52% of 46820009607307570 vs 10162244796777552", "ns_shortcut": "Use 50% as landmark: 50% of 46820009607307570 = 23410004803653785.", "shortcut_strength": 0.98, "pure_math": { "question": "Is 52% of 46820009607307570 greater than 10162244796777552?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.4346404995799936e+16, "explanation": "52% ≈ 50%. 50% of 46820009607307570 = 23410004803653785. Compare to 10162244796777552." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "46% of 46820009607307570 vs 10162244796777552", "ns_shortcut": "Use 50% as approximate anchor, then adjust.", "shortcut_strength": 0.96, "pure_math": { "question": "Is 46% of 46820009607307570 greater than 10162244796777552?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.1537204419361484e+16, "explanation": "46% is near 50% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "73% of 46820009607307570 vs 10162244796777552", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 73% of 46820009607307570 greater than 10162244796777552?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 3.417860701333453e+16, "explanation": "73% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001479", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "24% of 31467413386844740 vs 11732962714053500", "ns_shortcut": "Use 25% as landmark: 25% of 31467413386844740 = 7866853346711185.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 24% of 31467413386844740 greater than 11732962714053500?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 7552179212842738.0, "explanation": "24% ≈ 25%. 25% of 31467413386844740 = 7866853346711185. Compare to 11732962714053500." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "19% of 31467413386844740 vs 11732962714053500", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.94, "pure_math": { "question": "Is 19% of 31467413386844740 greater than 11732962714053500?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 5978808543500501.0, "explanation": "19% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "37% of 31467413386844740 vs 11732962714053500", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 37% of 31467413386844740 greater than 11732962714053500?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.1642942953132554e+16, "explanation": "37% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001480", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "24% of 46069060259573650 vs 19894604613570500", "ns_shortcut": "Use 25% as landmark: 25% of 46069060259573650 = 11517265064893412.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 24% of 46069060259573650 greater than 19894604613570500?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.1056574462297676e+16, "explanation": "24% ≈ 25%. 25% of 46069060259573650 = 11517265064893412. Compare to 19894604613570500." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "32% of 46069060259573650 vs 19894604613570500", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.9299999999999999, "pure_math": { "question": "Is 32% of 46069060259573650 greater than 19894604613570500?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.4742099283063568e+16, "explanation": "32% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "73% of 46069060259573650 vs 19894604613570500", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 73% of 46069060259573650 greater than 19894604613570500?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 3.3630413989488764e+16, "explanation": "73% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001481", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "23% of 82900443589700560 vs 25427208930332104", "ns_shortcut": "Use 25% as landmark: 25% of 82900443589700560 = 20725110897425140.", "shortcut_strength": 0.98, "pure_math": { "question": "Is 23% of 82900443589700560 greater than 25427208930332104?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.906710202563113e+16, "explanation": "23% ≈ 25%. 25% of 82900443589700560 = 20725110897425140. Compare to 25427208930332104." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "30% of 82900443589700560 vs 25427208930332104", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.95, "pure_math": { "question": "Is 30% of 82900443589700560 greater than 25427208930332104?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 2.487013307691017e+16, "explanation": "30% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "83% of 82900443589700560 vs 25427208930332104", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 83% of 82900443589700560 greater than 25427208930332104?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 6.8807368179451464e+16, "explanation": "83% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001482", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9% of 99594063191560240 vs 38588988408656215", "ns_shortcut": "Use 10% as landmark: 10% of 99594063191560240 = 9959406319156024.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 9% of 99594063191560240 greater than 38588988408656215?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 8963465687240422.0, "explanation": "9% ≈ 10%. 10% of 99594063191560240 = 9959406319156024. Compare to 38588988408656215." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "15% of 99594063191560240 vs 38588988408656215", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.95, "pure_math": { "question": "Is 15% of 99594063191560240 greater than 38588988408656215?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.4939109478734036e+16, "explanation": "15% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "73% of 99594063191560240 vs 38588988408656215", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 73% of 99594063191560240 greater than 38588988408656215?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 7.270366612983898e+16, "explanation": "73% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001483", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "50% of 21048332515114690 vs 4438435623116392", "ns_shortcut": "Use 50% as landmark: 50% of 21048332515114690 = 10524166257557345.", "shortcut_strength": 1.0, "pure_math": { "question": "Is 50% of 21048332515114690 greater than 4438435623116392?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.0524166257557344e+16, "explanation": "50% ≈ 50%. 50% of 21048332515114690 = 10524166257557345. Compare to 4438435623116392." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "47% of 21048332515114690 vs 4438435623116392", "ns_shortcut": "Use 50% as approximate anchor, then adjust.", "shortcut_strength": 0.97, "pure_math": { "question": "Is 47% of 21048332515114690 greater than 4438435623116392?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 9892716282103904.0, "explanation": "47% is near 50% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "73% of 21048332515114690 vs 4438435623116392", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 73% of 21048332515114690 greater than 4438435623116392?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.5365282736033724e+16, "explanation": "73% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001484", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10% of 65464062963051550 vs 14218346043413672", "ns_shortcut": "Use 10% as landmark: 10% of 65464062963051550 = 6546406296305155.", "shortcut_strength": 1.0, "pure_math": { "question": "Is 10% of 65464062963051550 greater than 14218346043413672?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 6546406296305155.0, "explanation": "10% ≈ 10%. 10% of 65464062963051550 = 6546406296305155. Compare to 14218346043413672." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "13% of 65464062963051550 vs 14218346043413672", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.97, "pure_math": { "question": "Is 13% of 65464062963051550 greater than 14218346043413672?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 8510328185196702.0, "explanation": "13% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "73% of 65464062963051550 vs 14218346043413672", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 73% of 65464062963051550 greater than 14218346043413672?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 4.778876596302763e+16, "explanation": "73% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001485", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "12% of 61591295765932360 vs 29932057616674953", "ns_shortcut": "Use 10% as landmark: 10% of 61591295765932360 = 6159129576593236.", "shortcut_strength": 0.98, "pure_math": { "question": "Is 12% of 61591295765932360 greater than 29932057616674953?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 7390955491911883.0, "explanation": "12% ≈ 10%. 10% of 61591295765932360 = 6159129576593236. Compare to 29932057616674953." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "3% of 61591295765932360 vs 29932057616674953", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.9299999999999999, "pure_math": { "question": "Is 3% of 61591295765932360 greater than 29932057616674953?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1847738872977970.8, "explanation": "3% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "67% of 61591295765932360 vs 29932057616674953", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 67% of 61591295765932360 greater than 29932057616674953?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 4.126616816317468e+16, "explanation": "67% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001486", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "50% of 14938538847356380 vs 3181972412748525", "ns_shortcut": "Use 50% as landmark: 50% of 14938538847356380 = 7469269423678190.", "shortcut_strength": 1.0, "pure_math": { "question": "Is 50% of 14938538847356380 greater than 3181972412748525?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 7469269423678190.0, "explanation": "50% ≈ 50%. 50% of 14938538847356380 = 7469269423678190. Compare to 3181972412748525." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "44% of 14938538847356380 vs 3181972412748525", "ns_shortcut": "Use 50% as approximate anchor, then adjust.", "shortcut_strength": 0.94, "pure_math": { "question": "Is 44% of 14938538847356380 greater than 3181972412748525?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 6572957092836807.0, "explanation": "44% is near 50% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "63% of 14938538847356380 vs 3181972412748525", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 63% of 14938538847356380 greater than 3181972412748525?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 9411279473834520.0, "explanation": "63% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001487", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "11% of 59510521746277510 vs 26159801354059813", "ns_shortcut": "Use 10% as landmark: 10% of 59510521746277510 = 5951052174627751.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 11% of 59510521746277510 greater than 26159801354059813?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 6546157392090526.0, "explanation": "11% ≈ 10%. 10% of 59510521746277510 = 5951052174627751. Compare to 26159801354059813." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "14% of 59510521746277510 vs 26159801354059813", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.96, "pure_math": { "question": "Is 14% of 59510521746277510 greater than 26159801354059813?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 8331473044478851.0, "explanation": "14% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "67% of 59510521746277510 vs 26159801354059813", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 67% of 59510521746277510 greater than 26159801354059813?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 3.987204957000593e+16, "explanation": "67% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001488", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "24% of 27829053414098080 vs 9424825013950520", "ns_shortcut": "Use 25% as landmark: 25% of 27829053414098080 = 6957263353524520.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 24% of 27829053414098080 greater than 9424825013950520?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 6678972819383539.0, "explanation": "24% ≈ 25%. 25% of 27829053414098080 = 6957263353524520. Compare to 9424825013950520." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "32% of 27829053414098080 vs 9424825013950520", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.9299999999999999, "pure_math": { "question": "Is 32% of 27829053414098080 greater than 9424825013950520?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 8905297092511386.0, "explanation": "32% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "83% of 27829053414098080 vs 9424825013950520", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 83% of 27829053414098080 greater than 9424825013950520?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.309811433370141e+16, "explanation": "83% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001489", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "25% of 14096041913667020 vs 3079485501323762", "ns_shortcut": "Use 25% as landmark: 25% of 14096041913667020 = 3524010478416755.", "shortcut_strength": 1.0, "pure_math": { "question": "Is 25% of 14096041913667020 greater than 3079485501323762?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 3524010478416755.0, "explanation": "25% ≈ 25%. 25% of 14096041913667020 = 3524010478416755. Compare to 3079485501323762." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "20% of 14096041913667020 vs 3079485501323762", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.95, "pure_math": { "question": "Is 20% of 14096041913667020 greater than 3079485501323762?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 2819208382733404.0, "explanation": "20% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "37% of 14096041913667020 vs 3079485501323762", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 37% of 14096041913667020 greater than 3079485501323762?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 5215535508056797.0, "explanation": "37% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001490", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "25% of 29242225164514270 vs 13061817826652028", "ns_shortcut": "Use 25% as landmark: 25% of 29242225164514270 = 7310556291128567.", "shortcut_strength": 1.0, "pure_math": { "question": "Is 25% of 29242225164514270 greater than 13061817826652028?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 7310556291128568.0, "explanation": "25% ≈ 25%. 25% of 29242225164514270 = 7310556291128567. Compare to 13061817826652028." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "22% of 29242225164514270 vs 13061817826652028", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.97, "pure_math": { "question": "Is 22% of 29242225164514270 greater than 13061817826652028?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 6433289536193139.0, "explanation": "22% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "67% of 29242225164514270 vs 13061817826652028", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 67% of 29242225164514270 greater than 13061817826652028?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.959229086022456e+16, "explanation": "67% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001491", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8% of 78340976882500670 vs 23153379741504276", "ns_shortcut": "Use 10% as landmark: 10% of 78340976882500670 = 7834097688250067.", "shortcut_strength": 0.98, "pure_math": { "question": "Is 8% of 78340976882500670 greater than 23153379741504276?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 6267278150600054.0, "explanation": "8% ≈ 10%. 10% of 78340976882500670 = 7834097688250067. Compare to 23153379741504276." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7% of 78340976882500670 vs 23153379741504276", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.97, "pure_math": { "question": "Is 7% of 78340976882500670 greater than 23153379741504276?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 5483868381775047.0, "explanation": "7% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "43% of 78340976882500670 vs 23153379741504276", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 43% of 78340976882500670 greater than 23153379741504276?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 3.368662005947529e+16, "explanation": "43% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001492", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "51% of 46496159563794750 vs 21411610278461941", "ns_shortcut": "Use 50% as landmark: 50% of 46496159563794750 = 23248079781897375.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 51% of 46496159563794750 greater than 21411610278461941?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.3713041377535324e+16, "explanation": "51% ≈ 50%. 50% of 46496159563794750 = 23248079781897375. Compare to 21411610278461941." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "56% of 46496159563794750 vs 21411610278461941", "ns_shortcut": "Use 50% as approximate anchor, then adjust.", "shortcut_strength": 0.94, "pure_math": { "question": "Is 56% of 46496159563794750 greater than 21411610278461941?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.603784935572506e+16, "explanation": "56% is near 50% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "37% of 46496159563794750 vs 21411610278461941", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 37% of 46496159563794750 greater than 21411610278461941?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.7203579038604058e+16, "explanation": "37% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001493", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "48% of 72555380504779420 vs 33251814860914235", "ns_shortcut": "Use 50% as landmark: 50% of 72555380504779420 = 36277690252389710.", "shortcut_strength": 0.98, "pure_math": { "question": "Is 48% of 72555380504779420 greater than 33251814860914235?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 3.482658264229412e+16, "explanation": "48% ≈ 50%. 50% of 72555380504779420 = 36277690252389710. Compare to 33251814860914235." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "47% of 72555380504779420 vs 33251814860914235", "ns_shortcut": "Use 50% as approximate anchor, then adjust.", "shortcut_strength": 0.97, "pure_math": { "question": "Is 47% of 72555380504779420 greater than 33251814860914235?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 3.410102883724633e+16, "explanation": "47% is near 50% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "73% of 72555380504779420 vs 33251814860914235", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 73% of 72555380504779420 greater than 33251814860914235?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 5.296542776848898e+16, "explanation": "73% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001494", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "24% of 85617785986819640 vs 39210145274455251", "ns_shortcut": "Use 25% as landmark: 25% of 85617785986819640 = 21404446496704910.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 24% of 85617785986819640 greater than 39210145274455251?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 2.054826863683671e+16, "explanation": "24% ≈ 25%. 25% of 85617785986819640 = 21404446496704910. Compare to 39210145274455251." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "28% of 85617785986819640 vs 39210145274455251", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.97, "pure_math": { "question": "Is 28% of 85617785986819640 greater than 39210145274455251?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 2.39729800763095e+16, "explanation": "28% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "37% of 85617785986819640 vs 39210145274455251", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 37% of 85617785986819640 greater than 39210145274455251?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 3.1678580815123268e+16, "explanation": "37% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001495", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10% of 86543273958337710 vs 23846970877621619", "ns_shortcut": "Use 10% as landmark: 10% of 86543273958337710 = 8654327395833771.", "shortcut_strength": 1.0, "pure_math": { "question": "Is 10% of 86543273958337710 greater than 23846970877621619?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 8654327395833771.0, "explanation": "10% ≈ 10%. 10% of 86543273958337710 = 8654327395833771. Compare to 23846970877621619." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "17% of 86543273958337710 vs 23846970877621619", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.9299999999999999, "pure_math": { "question": "Is 17% of 86543273958337710 greater than 23846970877621619?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.471235657291741e+16, "explanation": "17% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "67% of 86543273958337710 vs 23846970877621619", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 67% of 86543273958337710 greater than 23846970877621619?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 5.7983993552086264e+16, "explanation": "67% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001496", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "27% of 25414068387281590 vs 6636124686799575", "ns_shortcut": "Use 25% as landmark: 25% of 25414068387281590 = 6353517096820397.", "shortcut_strength": 0.98, "pure_math": { "question": "Is 27% of 25414068387281590 greater than 6636124686799575?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 6861798464566029.0, "explanation": "27% ≈ 25%. 25% of 25414068387281590 = 6353517096820397. Compare to 6636124686799575." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "32% of 25414068387281590 vs 6636124686799575", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.9299999999999999, "pure_math": { "question": "Is 32% of 25414068387281590 greater than 6636124686799575?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 8132501883930109.0, "explanation": "32% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "43% of 25414068387281590 vs 6636124686799575", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 43% of 25414068387281590 greater than 6636124686799575?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 1.0928049406531084e+16, "explanation": "43% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001497", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "25% of 99113393492527500 vs 27131593840766456", "ns_shortcut": "Use 25% as landmark: 25% of 99113393492527500 = 24778348373131875.", "shortcut_strength": 1.0, "pure_math": { "question": "Is 25% of 99113393492527500 greater than 27131593840766456?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 2.4778348373131876e+16, "explanation": "25% ≈ 25%. 25% of 99113393492527500 = 24778348373131875. Compare to 27131593840766456." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "22% of 99113393492527500 vs 27131593840766456", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.97, "pure_math": { "question": "Is 22% of 99113393492527500 greater than 27131593840766456?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 2.180494656835605e+16, "explanation": "22% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "43% of 99113393492527500 vs 27131593840766456", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 43% of 99113393492527500 greater than 27131593840766456?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 4.2618759201786824e+16, "explanation": "43% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001498", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "24% of 44012181243688880 vs 19351323069051348", "ns_shortcut": "Use 25% as landmark: 25% of 44012181243688880 = 11003045310922220.", "shortcut_strength": 0.99, "pure_math": { "question": "Is 24% of 44012181243688880 greater than 19351323069051348?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.0562923498485332e+16, "explanation": "24% ≈ 25%. 25% of 44012181243688880 = 11003045310922220. Compare to 19351323069051348." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "31% of 44012181243688880 vs 19351323069051348", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.94, "pure_math": { "question": "Is 31% of 44012181243688880 greater than 19351323069051348?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.3643776185543552e+16, "explanation": "31% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "67% of 44012181243688880 vs 19351323069051348", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 67% of 44012181243688880 greater than 19351323069051348?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.9488161433271548e+16, "explanation": "67% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001499", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "50% of 99093244084081020 vs 35340469217099591", "ns_shortcut": "Use 50% as landmark: 50% of 99093244084081020 = 49546622042040510.", "shortcut_strength": 1.0, "pure_math": { "question": "Is 50% of 99093244084081020 greater than 35340469217099591?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 4.954662204204051e+16, "explanation": "50% ≈ 50%. 50% of 99093244084081020 = 49546622042040510. Compare to 35340469217099591." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "57% of 99093244084081020 vs 35340469217099591", "ns_shortcut": "Use 50% as approximate anchor, then adjust.", "shortcut_strength": 0.9299999999999999, "pure_math": { "question": "Is 57% of 99093244084081020 greater than 35340469217099591?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 5.6483149127926184e+16, "explanation": "57% is near 50% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "37% of 99093244084081020 vs 35340469217099591", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 37% of 99093244084081020 greater than 35340469217099591?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 3.6664500311109976e+16, "explanation": "37% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001500", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "25% of 71829076799906750 vs 25787867789323824", "ns_shortcut": "Use 25% as landmark: 25% of 71829076799906750 = 17957269199976687.", "shortcut_strength": 1.0, "pure_math": { "question": "Is 25% of 71829076799906750 greater than 25787867789323824?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.7957269199976688e+16, "explanation": "25% ≈ 25%. 25% of 71829076799906750 = 17957269199976687. Compare to 25787867789323824." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "18% of 71829076799906750 vs 25787867789323824", "ns_shortcut": "Use 25% as approximate anchor, then adjust.", "shortcut_strength": 0.9299999999999999, "pure_math": { "question": "Is 18% of 71829076799906750 greater than 25787867789323824?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 1.2929233823983216e+16, "explanation": "18% is near 25% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "37% of 71829076799906750 vs 25787867789323824", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 37% of 71829076799906750 greater than 25787867789323824?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 2.6576758415965496e+16, "explanation": "37% is not near any standard landmark; compute directly." } } }, { "family_id": "landmark_comparison_001501", "category": "landmark_comparison", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "10% of 79577516596178380 vs 35094476574023310", "ns_shortcut": "Use 10% as landmark: 10% of 79577516596178380 = 7957751659617838.", "shortcut_strength": 1.0, "pure_math": { "question": "Is 10% of 79577516596178380 greater than 35094476574023310?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 7957751659617838.0, "explanation": "10% ≈ 10%. 10% of 79577516596178380 = 7957751659617838. Compare to 35094476574023310." } }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4% of 79577516596178380 vs 35094476574023310", "ns_shortcut": "Use 10% as approximate anchor, then adjust.", "shortcut_strength": 0.94, "pure_math": { "question": "Is 4% of 79577516596178380 greater than 35094476574023310?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "No", "correct_value": 3183100663847135.0, "explanation": "4% is near 10% but requires adjustment." } }, "control": { "task_type": "control", "math_expression": "73% of 79577516596178380 vs 35094476574023310", "ns_shortcut": "No clean shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "Is 73% of 79577516596178380 greater than 35094476574023310?", "options": [ "Yes", "No", "Cannot be determined", "Need more information" ], "answer": "Yes", "correct_value": 5.8091587115210216e+16, "explanation": "73% is not near any standard landmark; compute directly." } } }, { "family_id": "equation_reasoning_001502", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9439839984087080 + 5680357738248731 = ___ + 6721737474255938", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 9439839984087080 + 5680357738248731 = ___ + 6721737474255938", "options": [ "8398460248079893", "8398460248079873", "8398460248079883", "8398460248079863" ], "answer": "8398460248079873", "correct_value": 8398460248079873, "explanation": "Blank = A + B − C = 9439839984087080 + 5680357738248731 − 6721737474255938 = 8398460248079873." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4647263882519947 + 5531351252492803 + ___ = 5531351252492803 + 5770025176328175 + 9003434594886877", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 4647263882519947 + 5531351252492803 + ___ = 5531351252492803 + 5770025176328175 + 9003434594886877", "options": [ "10126195888695105", "10126195888695095", "10126195888695115", "10126195888695125" ], "answer": "10126195888695105", "correct_value": 10126195888695105, "explanation": "B cancels; blank = C + D − A = 5770025176328175 + 9003434594886877 − 4647263882519947 = 10126195888695105." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "9809306984370131 + 5392916904357432 + ___ = 1949456289674931 + 2946559008092851", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 9809306984370131 + 5392916904357432 + ___ = 1949456289674931 + 2946559008092851", "options": [ "-10306208590959781", "-10306208590959769", "-10306208590959779", "-10306208590959799" ], "answer": "-10306208590959781", "correct_value": -10306208590959781, "explanation": "No matching terms; blank = 1949456289674931 + 2946559008092851 − 9809306984370131 − 5392916904357432 = -10306208590959781." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001503", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7909921596135286 × 6968059258509573 × 5076712796416482 = ___ × 5076712796416482 × 7909921596135286", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 7909921596135286 × 6968059258509573 × 5076712796416482 = ___ × 5076712796416482 × 7909921596135286", "options": [ "6968059258509573", "6968059258509593", "6968059258509563", "6968059258509583" ], "answer": "6968059258509573", "correct_value": 6968059258509573, "explanation": "A and C appear on both sides; blank = B = 6968059258509573." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2731919332502536 + 6344046269483549 + ___ = 6344046269483549 + 7824767275565529 + 6959515523447174", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 2731919332502536 + 6344046269483549 + ___ = 6344046269483549 + 7824767275565529 + 6959515523447174", "options": [ "12052363466510167", "12052363466510157", "12052363466510177", "12052363466510187" ], "answer": "12052363466510167", "correct_value": 12052363466510167, "explanation": "B cancels; blank = C + D − A = 7824767275565529 + 6959515523447174 − 2731919332502536 = 12052363466510167." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "7977782355592264 + 3269479168870098 + ___ = 9559212115258965 + 3074702339819025", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 7977782355592264 + 3269479168870098 + ___ = 9559212115258965 + 3074702339819025", "options": [ "1386652930615648", "1386652930615618", "1386652930615628", "1386652930615638" ], "answer": "1386652930615628", "correct_value": 1386652930615628, "explanation": "No matching terms; blank = 9559212115258965 + 3074702339819025 − 7977782355592264 − 3269479168870098 = 1386652930615628." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001504", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1224754964184304 + 2704385503800682 = ___ + 6654494801878172", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 1224754964184304 + 2704385503800682 = ___ + 6654494801878172", "options": [ "-2725354333893186", "-2725354333893204", "-2725354333893174", "-2725354333893184" ], "answer": "-2725354333893186", "correct_value": -2725354333893186, "explanation": "Blank = A + B − C = 1224754964184304 + 2704385503800682 − 6654494801878172 = -2725354333893186." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7229121268105070 + 8747354521770206 + ___ = 8747354521770206 + 7277272901012433 + 7373730950440976", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 7229121268105070 + 8747354521770206 + ___ = 8747354521770206 + 7277272901012433 + 7373730950440976", "options": [ "7421882583348349", "7421882583348339", "7421882583348329", "7421882583348359" ], "answer": "7421882583348339", "correct_value": 7421882583348339, "explanation": "B cancels; blank = C + D − A = 7277272901012433 + 7373730950440976 − 7229121268105070 = 7421882583348339." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "8657611627518062 + 6051155890340580 + ___ = 6990492710298970 + 4633377579646802", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 8657611627518062 + 6051155890340580 + ___ = 6990492710298970 + 4633377579646802", "options": [ "-3084897227912870", "-3084897227912850", "-3084897227912880", "-3084897227912860" ], "answer": "-3084897227912870", "correct_value": -3084897227912870, "explanation": "No matching terms; blank = 6990492710298970 + 4633377579646802 − 8657611627518062 − 6051155890340580 = -3084897227912870." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001505", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8612658183754669 × (9293786595441622 + 8136416571009627) = 8612658183754669 × ___ + 8612658183754669 × 8136416571009627", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 8612658183754669 × (9293786595441622 + 8136416571009627) = 8612658183754669 × ___ + 8612658183754669 × 8136416571009627", "options": [ "9293786595441622", "9293786595441642", "9293786595441632", "9293786595441612" ], "answer": "9293786595441622", "correct_value": 9293786595441622, "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 9293786595441622." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5775209737122446 + 2846262099739843 + ___ = 2846262099739843 + 8350638490566745 + 9140569559694007", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 5775209737122446 + 2846262099739843 + ___ = 2846262099739843 + 8350638490566745 + 9140569559694007", "options": [ "11715998313138296", "11715998313138326", "11715998313138316", "11715998313138306" ], "answer": "11715998313138306", "correct_value": 11715998313138306, "explanation": "B cancels; blank = C + D − A = 8350638490566745 + 9140569559694007 − 5775209737122446 = 11715998313138306." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "3549641158719565 + 7314660177521054 + ___ = 1747721233648055 + 2828101486056783", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 3549641158719565 + 7314660177521054 + ___ = 1747721233648055 + 2828101486056783", "options": [ "-6288478616535769", "-6288478616535781", "-6288478616535799", "-6288478616535779" ], "answer": "-6288478616535781", "correct_value": -6288478616535781, "explanation": "No matching terms; blank = 1747721233648055 + 2828101486056783 − 3549641158719565 − 7314660177521054 = -6288478616535781." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001506", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5974521391895561 + 8205556615168986 = ___ + 3502022703234047", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 5974521391895561 + 8205556615168986 = ___ + 3502022703234047", "options": [ "10678055303830500", "10678055303830510", "10678055303830520", "10678055303830490" ], "answer": "10678055303830500", "correct_value": 10678055303830500, "explanation": "Blank = A + B − C = 5974521391895561 + 8205556615168986 − 3502022703234047 = 10678055303830500." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6265103165085395 + 9684307404438930 + ___ = 9684307404438930 + 1631162435031096 + 2280586721861644", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 6265103165085395 + 9684307404438930 + ___ = 9684307404438930 + 1631162435031096 + 2280586721861644", "options": [ "-2353354008192635", "-2353354008192645", "-2353354008192665", "-2353354008192655" ], "answer": "-2353354008192655", "correct_value": -2353354008192655, "explanation": "B cancels; blank = C + D − A = 1631162435031096 + 2280586721861644 − 6265103165085395 = -2353354008192655." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "5811932851708041 + 6247120149805131 + ___ = 3149219295718641 + 1842384905083632", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 5811932851708041 + 6247120149805131 + ___ = 3149219295718641 + 1842384905083632", "options": [ "-7067448800710891", "-7067448800710911", "-7067448800710899", "-7067448800710881" ], "answer": "-7067448800710899", "correct_value": -7067448800710899, "explanation": "No matching terms; blank = 3149219295718641 + 1842384905083632 − 5811932851708041 − 6247120149805131 = -7067448800710899." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001507", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5179102363628298 × 2466728830248527 × 5907368591580582 = ___ × 5907368591580582 × 5179102363628298", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 5179102363628298 × 2466728830248527 × 5907368591580582 = ___ × 5907368591580582 × 5179102363628298", "options": [ "2466728830248527", "2466728830248537", "2466728830248517", "2466728830248547" ], "answer": "2466728830248527", "correct_value": 2466728830248527, "explanation": "A and C appear on both sides; blank = B = 2466728830248527." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "8007220611479638 + 7442815449765966 + ___ = 7442815449765966 + 6837654827511075 + 7185271377355697", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 8007220611479638 + 7442815449765966 + ___ = 7442815449765966 + 6837654827511075 + 7185271377355697", "options": [ "6015705593387154", "6015705593387124", "6015705593387134", "6015705593387144" ], "answer": "6015705593387134", "correct_value": 6015705593387134, "explanation": "B cancels; blank = C + D − A = 6837654827511075 + 7185271377355697 − 8007220611479638 = 6015705593387134." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4781287458112570 + 1343359791635911 + ___ = 6117269957183821 + 8360353002955122", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 4781287458112570 + 1343359791635911 + ___ = 6117269957183821 + 8360353002955122", "options": [ "8352975710390452", "8352975710390472", "8352975710390482", "8352975710390462" ], "answer": "8352975710390462", "correct_value": 8352975710390462, "explanation": "No matching terms; blank = 6117269957183821 + 8360353002955122 − 4781287458112570 − 1343359791635911 = 8352975710390462." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "equation_reasoning_001508", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3503444304661885 × 2217833446729142 × 4487150853184415 = ___ × 4487150853184415 × 3503444304661885", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 3503444304661885 × 2217833446729142 × 4487150853184415 = ___ × 4487150853184415 × 3503444304661885", "options": [ "2217833446729132", "2217833446729142", "2217833446729152", "2217833446729162" ], "answer": "2217833446729142", "correct_value": 2217833446729142, "explanation": "A and C appear on both sides; blank = B = 2217833446729142." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1249360224389294 + 4492262323170989 + ___ = 4492262323170989 + 2197302396419487 + 1301488027998843", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 1249360224389294 + 4492262323170989 + ___ = 4492262323170989 + 2197302396419487 + 1301488027998843", "options": [ "2249430200029056", "2249430200029026", "2249430200029036", "2249430200029046" ], "answer": "2249430200029036", "correct_value": 2249430200029036, "explanation": "B cancels; blank = C + D − A = 2197302396419487 + 1301488027998843 − 1249360224389294 = 2249430200029036." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2141995407863297 + 6004066561484669 + ___ = 8888578569874782 + 9870768075143887", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 2141995407863297 + 6004066561484669 + ___ = 8888578569874782 + 9870768075143887", "options": [ "10613284675670693", "10613284675670713", "10613284675670703", "10613284675670723" ], "answer": "10613284675670703", "correct_value": 10613284675670703, "explanation": "No matching terms; blank = 8888578569874782 + 9870768075143887 − 2141995407863297 − 6004066561484669 = 10613284675670703." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001509", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "6167470022133045 + 1945830225624289 = ___ + 6671291864107536", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 6167470022133045 + 1945830225624289 = ___ + 6671291864107536", "options": [ "1442008383649818", "1442008383649798", "1442008383649808", "1442008383649788" ], "answer": "1442008383649798", "correct_value": 1442008383649798, "explanation": "Blank = A + B − C = 6167470022133045 + 1945830225624289 − 6671291864107536 = 1442008383649798." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6690536995196885 + 8697590516018011 + ___ = 8697590516018011 + 5608649569405207 + 1352768725591595", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 6690536995196885 + 8697590516018011 + ___ = 8697590516018011 + 5608649569405207 + 1352768725591595", "options": [ "270881299799937", "270881299799927", "270881299799917", "270881299799907" ], "answer": "270881299799917", "correct_value": 270881299799917, "explanation": "B cancels; blank = C + D − A = 5608649569405207 + 1352768725591595 − 6690536995196885 = 270881299799917." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5760739749337140 + 4484505648916694 + ___ = 7079121924670356 + 7680979884042997", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 5760739749337140 + 4484505648916694 + ___ = 7079121924670356 + 7680979884042997", "options": [ "4514856410459529", "4514856410459539", "4514856410459519", "4514856410459509" ], "answer": "4514856410459519", "correct_value": 4514856410459519, "explanation": "No matching terms; blank = 7079121924670356 + 7680979884042997 − 5760739749337140 − 4484505648916694 = 4514856410459519." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001510", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2154647614665123 × 2297658492912723 × 9528783858458853 = ___ × 9528783858458853 × 2154647614665123", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 2154647614665123 × 2297658492912723 × 9528783858458853 = ___ × 9528783858458853 × 2154647614665123", "options": [ "2297658492912713", "2297658492912723", "2297658492912743", "2297658492912733" ], "answer": "2297658492912723", "correct_value": 2297658492912723, "explanation": "A and C appear on both sides; blank = B = 2297658492912723." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7000811481908131 + 5856310734043744 + ___ = 5856310734043744 + 5922648152861091 + 8509379635428475", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 7000811481908131 + 5856310734043744 + ___ = 5856310734043744 + 5922648152861091 + 8509379635428475", "options": [ "7431216306381455", "7431216306381445", "7431216306381425", "7431216306381435" ], "answer": "7431216306381435", "correct_value": 7431216306381435, "explanation": "B cancels; blank = C + D − A = 5922648152861091 + 8509379635428475 − 7000811481908131 = 7431216306381435." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "7857392149871659 + 2875405980562751 + ___ = 9924427766561797 + 1201689030062336", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 7857392149871659 + 2875405980562751 + ___ = 9924427766561797 + 1201689030062336", "options": [ "393318666189713", "393318666189723", "393318666189733", "393318666189743" ], "answer": "393318666189723", "correct_value": 393318666189723, "explanation": "No matching terms; blank = 9924427766561797 + 1201689030062336 − 7857392149871659 − 2875405980562751 = 393318666189723." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001511", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1352932065089713 + 7922599369077142 = ___ + 3918108682507707", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 1352932065089713 + 7922599369077142 = ___ + 3918108682507707", "options": [ "5357422751659138", "5357422751659148", "5357422751659168", "5357422751659158" ], "answer": "5357422751659148", "correct_value": 5357422751659148, "explanation": "Blank = A + B − C = 1352932065089713 + 7922599369077142 − 3918108682507707 = 5357422751659148." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "8904700791745449 + 9936763267108657 + ___ = 9936763267108657 + 3968370350681155 + 9291612054874840", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 8904700791745449 + 9936763267108657 + ___ = 9936763267108657 + 3968370350681155 + 9291612054874840", "options": [ "4355281613810556", "4355281613810566", "4355281613810546", "4355281613810536" ], "answer": "4355281613810546", "correct_value": 4355281613810546, "explanation": "B cancels; blank = C + D − A = 3968370350681155 + 9291612054874840 − 8904700791745449 = 4355281613810546." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4366283419240008 + 6165962382764854 + ___ = 4266458290910921 + 9816298188983972", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 4366283419240008 + 6165962382764854 + ___ = 4266458290910921 + 9816298188983972", "options": [ "3550510677890031", "3550510677890051", "3550510677890021", "3550510677890041" ], "answer": "3550510677890031", "correct_value": 3550510677890031, "explanation": "No matching terms; blank = 4266458290910921 + 9816298188983972 − 4366283419240008 − 6165962382764854 = 3550510677890031." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001512", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1289538925669141 + 2685669920518577 + ___ = 2685669920518577 + 9455848728341831 + 1289538925669141", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 1289538925669141 + 2685669920518577 + ___ = 2685669920518577 + 9455848728341831 + 1289538925669141", "options": [ "9455848728341831", "9455848728341841", "9455848728341851", "9455848728341821" ], "answer": "9455848728341831", "correct_value": 9455848728341831, "explanation": "A and B appear on both sides and cancel; blank = C = 9455848728341831." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9700712202697092 + 7711748503208479 + ___ = 7711748503208479 + 6318697878099063 + 6425313924595461", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 9700712202697092 + 7711748503208479 + ___ = 7711748503208479 + 6318697878099063 + 6425313924595461", "options": [ "3043299599997452", "3043299599997442", "3043299599997432", "3043299599997422" ], "answer": "3043299599997432", "correct_value": 3043299599997432, "explanation": "B cancels; blank = C + D − A = 6318697878099063 + 6425313924595461 − 9700712202697092 = 3043299599997432." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2484252941877543 + 6953708637481571 + ___ = 3563259901808394 + 6238864316542788", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 2484252941877543 + 6953708637481571 + ___ = 3563259901808394 + 6238864316542788", "options": [ "364162638992068", "364162638992078", "364162638992088", "364162638992058" ], "answer": "364162638992068", "correct_value": 364162638992068, "explanation": "No matching terms; blank = 3563259901808394 + 6238864316542788 − 2484252941877543 − 6953708637481571 = 364162638992068." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001513", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8040019777565046 + 9342123878230022 + ___ = 9342123878230022 + 3066258182699380 + 8040019777565046", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 8040019777565046 + 9342123878230022 + ___ = 9342123878230022 + 3066258182699380 + 8040019777565046", "options": [ "3066258182699390", "3066258182699380", "3066258182699370", "3066258182699400" ], "answer": "3066258182699380", "correct_value": 3066258182699380, "explanation": "A and B appear on both sides and cancel; blank = C = 3066258182699380." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1666645147493674 + 7524344790140717 + ___ = 7524344790140717 + 7591128516309375 + 5119185510908635", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 1666645147493674 + 7524344790140717 + ___ = 7524344790140717 + 7591128516309375 + 5119185510908635", "options": [ "11043668879724326", "11043668879724336", "11043668879724356", "11043668879724346" ], "answer": "11043668879724336", "correct_value": 11043668879724336, "explanation": "B cancels; blank = C + D − A = 7591128516309375 + 5119185510908635 − 1666645147493674 = 11043668879724336." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1857939131804874 + 3209797707082725 + ___ = 8412858250339722 + 9379659681320141", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 1857939131804874 + 3209797707082725 + ___ = 8412858250339722 + 9379659681320141", "options": [ "12724781092772274", "12724781092772284", "12724781092772264", "12724781092772254" ], "answer": "12724781092772264", "correct_value": 12724781092772264, "explanation": "No matching terms; blank = 8412858250339722 + 9379659681320141 − 1857939131804874 − 3209797707082725 = 12724781092772264." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001514", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5109907754692016 + 2594650108515777 + ___ = 2594650108515777 + 5777013124315858 + 5109907754692016", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 5109907754692016 + 2594650108515777 + ___ = 2594650108515777 + 5777013124315858 + 5109907754692016", "options": [ "5777013124315848", "5777013124315878", "5777013124315868", "5777013124315858" ], "answer": "5777013124315858", "correct_value": 5777013124315858, "explanation": "A and B appear on both sides and cancel; blank = C = 5777013124315858." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "3626867301443028 + 8790300678105402 + ___ = 8790300678105402 + 2412902667882084 + 4111331848260626", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 3626867301443028 + 8790300678105402 + ___ = 8790300678105402 + 2412902667882084 + 4111331848260626", "options": [ "2897367214699682", "2897367214699702", "2897367214699672", "2897367214699692" ], "answer": "2897367214699682", "correct_value": 2897367214699682, "explanation": "B cancels; blank = C + D − A = 2412902667882084 + 4111331848260626 − 3626867301443028 = 2897367214699682." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2904924619645161 + 2473102519052394 + ___ = 2978298170996849 + 5476957687659638", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 2904924619645161 + 2473102519052394 + ___ = 2978298170996849 + 5476957687659638", "options": [ "3077228719958922", "3077228719958942", "3077228719958952", "3077228719958932" ], "answer": "3077228719958932", "correct_value": 3077228719958932, "explanation": "No matching terms; blank = 2978298170996849 + 5476957687659638 − 2904924619645161 − 2473102519052394 = 3077228719958932." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "equation_reasoning_001515", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2985677485819603 × (5646076198799709 + 3130145396661229) = 2985677485819603 × ___ + 2985677485819603 × 3130145396661229", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 2985677485819603 × (5646076198799709 + 3130145396661229) = 2985677485819603 × ___ + 2985677485819603 × 3130145396661229", "options": [ "5646076198799699", "5646076198799709", "5646076198799719", "5646076198799729" ], "answer": "5646076198799709", "correct_value": 5646076198799709, "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 5646076198799709." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4502429267327816 + 8972237822316753 + ___ = 8972237822316753 + 8318961080764666 + 4826110909076862", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 4502429267327816 + 8972237822316753 + ___ = 8972237822316753 + 8318961080764666 + 4826110909076862", "options": [ "8642642722513722", "8642642722513702", "8642642722513732", "8642642722513712" ], "answer": "8642642722513712", "correct_value": 8642642722513712, "explanation": "B cancels; blank = C + D − A = 8318961080764666 + 4826110909076862 − 4502429267327816 = 8642642722513712." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "5375864411024493 + 9224696592961774 + ___ = 2383676791322264 + 8303919470119732", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 5375864411024493 + 9224696592961774 + ___ = 2383676791322264 + 8303919470119732", "options": [ "-3912964742544259", "-3912964742544289", "-3912964742544269", "-3912964742544271" ], "answer": "-3912964742544271", "correct_value": -3912964742544271, "explanation": "No matching terms; blank = 2383676791322264 + 8303919470119732 − 5375864411024493 − 9224696592961774 = -3912964742544271." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "equation_reasoning_001516", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8150906495039786 × 1923367957727910 × 3835687087073904 = ___ × 3835687087073904 × 8150906495039786", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 8150906495039786 × 1923367957727910 × 3835687087073904 = ___ × 3835687087073904 × 8150906495039786", "options": [ "1923367957727920", "1923367957727930", "1923367957727900", "1923367957727910" ], "answer": "1923367957727910", "correct_value": 1923367957727910, "explanation": "A and C appear on both sides; blank = B = 1923367957727910." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "8900636316856695 + 6558344635305247 + ___ = 6558344635305247 + 7339461356284370 + 5399672464654445", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 8900636316856695 + 6558344635305247 + ___ = 6558344635305247 + 7339461356284370 + 5399672464654445", "options": [ "3838497504082140", "3838497504082110", "3838497504082130", "3838497504082120" ], "answer": "3838497504082120", "correct_value": 3838497504082120, "explanation": "B cancels; blank = C + D − A = 7339461356284370 + 5399672464654445 − 8900636316856695 = 3838497504082120." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "8421641457724341 + 2167394213787652 + ___ = 9896436579820751 + 4081880727477021", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 8421641457724341 + 2167394213787652 + ___ = 9896436579820751 + 4081880727477021", "options": [ "3389281635785769", "3389281635785789", "3389281635785799", "3389281635785779" ], "answer": "3389281635785779", "correct_value": 3389281635785779, "explanation": "No matching terms; blank = 9896436579820751 + 4081880727477021 − 8421641457724341 − 2167394213787652 = 3389281635785779." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "equation_reasoning_001517", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1177407098922862 + 7018385328557726 = ___ + 8650949767640421", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 1177407098922862 + 7018385328557726 = ___ + 8650949767640421", "options": [ "-455157340159847", "-455157340159817", "-455157340159833", "-455157340159827" ], "answer": "-455157340159833", "correct_value": -455157340159833, "explanation": "Blank = A + B − C = 1177407098922862 + 7018385328557726 − 8650949767640421 = -455157340159833." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6346614295363390 + 1572340138898159 + ___ = 1572340138898159 + 7827761145344733 + 1793631559764931", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 6346614295363390 + 1572340138898159 + ___ = 1572340138898159 + 7827761145344733 + 1793631559764931", "options": [ "3274778409746284", "3274778409746294", "3274778409746274", "3274778409746264" ], "answer": "3274778409746274", "correct_value": 3274778409746274, "explanation": "B cancels; blank = C + D − A = 7827761145344733 + 1793631559764931 − 6346614295363390 = 3274778409746274." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1600553480410154 + 1050171560575380 + ___ = 5110200140154331 + 4806303018954805", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 1600553480410154 + 1050171560575380 + ___ = 5110200140154331 + 4806303018954805", "options": [ "7265778118123602", "7265778118123612", "7265778118123622", "7265778118123592" ], "answer": "7265778118123602", "correct_value": 7265778118123602, "explanation": "No matching terms; blank = 5110200140154331 + 4806303018954805 − 1600553480410154 − 1050171560575380 = 7265778118123602." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001518", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2112773581227864 + 1414579604105222 + ___ = 1414579604105222 + 6678690112241287 + 2112773581227864", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 2112773581227864 + 1414579604105222 + ___ = 1414579604105222 + 6678690112241287 + 2112773581227864", "options": [ "6678690112241307", "6678690112241297", "6678690112241287", "6678690112241277" ], "answer": "6678690112241287", "correct_value": 6678690112241287, "explanation": "A and B appear on both sides and cancel; blank = C = 6678690112241287." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4803860560914905 + 3018065474221164 + ___ = 3018065474221164 + 7303677807736335 + 2127195074626803", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 4803860560914905 + 3018065474221164 + ___ = 3018065474221164 + 7303677807736335 + 2127195074626803", "options": [ "4627012321448233", "4627012321448243", "4627012321448223", "4627012321448253" ], "answer": "4627012321448233", "correct_value": 4627012321448233, "explanation": "B cancels; blank = C + D − A = 7303677807736335 + 2127195074626803 − 4803860560914905 = 4627012321448233." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "8242404637826093 + 4366852062762029 + ___ = 8143899447524841 + 2151552225614427", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 8242404637826093 + 4366852062762029 + ___ = 8143899447524841 + 2151552225614427", "options": [ "-2313805027448836", "-2313805027448846", "-2313805027448866", "-2313805027448854" ], "answer": "-2313805027448854", "correct_value": -2313805027448854, "explanation": "No matching terms; blank = 8143899447524841 + 2151552225614427 − 8242404637826093 − 4366852062762029 = -2313805027448854." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "equation_reasoning_001519", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "6623478785750054 + 8014713585438972 + ___ = 8014713585438972 + 7046952963218514 + 6623478785750054", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 6623478785750054 + 8014713585438972 + ___ = 8014713585438972 + 7046952963218514 + 6623478785750054", "options": [ "7046952963218524", "7046952963218534", "7046952963218504", "7046952963218514" ], "answer": "7046952963218514", "correct_value": 7046952963218514, "explanation": "A and B appear on both sides and cancel; blank = C = 7046952963218514." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5490537961674600 + 6828358005395124 + ___ = 6828358005395124 + 5774022162915461 + 4894615404225851", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 5490537961674600 + 6828358005395124 + ___ = 6828358005395124 + 5774022162915461 + 4894615404225851", "options": [ "5178099605466712", "5178099605466732", "5178099605466702", "5178099605466722" ], "answer": "5178099605466712", "correct_value": 5178099605466712, "explanation": "B cancels; blank = C + D − A = 5774022162915461 + 4894615404225851 − 5490537961674600 = 5178099605466712." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "6009023038881706 + 7293337910089244 + ___ = 6600795267894023 + 7624449218669616", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 6009023038881706 + 7293337910089244 + ___ = 6600795267894023 + 7624449218669616", "options": [ "922883537592699", "922883537592689", "922883537592709", "922883537592679" ], "answer": "922883537592689", "correct_value": 922883537592689, "explanation": "No matching terms; blank = 6600795267894023 + 7624449218669616 − 6009023038881706 − 7293337910089244 = 922883537592689." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001520", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3783232999236154 + 5540993599874260 = ___ + 8578080502857062", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 3783232999236154 + 5540993599874260 = ___ + 8578080502857062", "options": [ "746146096253342", "746146096253362", "746146096253352", "746146096253372" ], "answer": "746146096253352", "correct_value": 746146096253352, "explanation": "Blank = A + B − C = 3783232999236154 + 5540993599874260 − 8578080502857062 = 746146096253352." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4496489777806444 + 8708018234559428 + ___ = 8708018234559428 + 2185876600708184 + 1726288414321106", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 4496489777806444 + 8708018234559428 + ___ = 8708018234559428 + 2185876600708184 + 1726288414321106", "options": [ "-584324762777146", "-584324762777154", "-584324762777166", "-584324762777136" ], "answer": "-584324762777154", "correct_value": -584324762777154, "explanation": "B cancels; blank = C + D − A = 2185876600708184 + 1726288414321106 − 4496489777806444 = -584324762777154." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4400656012946353 + 6322755340699458 + ___ = 2618864962276194 + 6043779327385039", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 4400656012946353 + 6322755340699458 + ___ = 2618864962276194 + 6043779327385039", "options": [ "-2060767063984562", "-2060767063984578", "-2060767063984572", "-2060767063984592" ], "answer": "-2060767063984578", "correct_value": -2060767063984578, "explanation": "No matching terms; blank = 2618864962276194 + 6043779327385039 − 4400656012946353 − 6322755340699458 = -2060767063984578." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001521", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9799822745904316 × (4955072442514443 + 2727017033639288) = 9799822745904316 × ___ + 9799822745904316 × 2727017033639288", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 9799822745904316 × (4955072442514443 + 2727017033639288) = 9799822745904316 × ___ + 9799822745904316 × 2727017033639288", "options": [ "4955072442514433", "4955072442514443", "4955072442514453", "4955072442514463" ], "answer": "4955072442514443", "correct_value": 4955072442514443, "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 4955072442514443." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "3744478940441406 + 9426503303185783 + ___ = 9426503303185783 + 9030784457070724 + 5147544767118164", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 3744478940441406 + 9426503303185783 + ___ = 9426503303185783 + 9030784457070724 + 5147544767118164", "options": [ "10433850283747492", "10433850283747502", "10433850283747482", "10433850283747472" ], "answer": "10433850283747482", "correct_value": 10433850283747482, "explanation": "B cancels; blank = C + D − A = 9030784457070724 + 5147544767118164 − 3744478940441406 = 10433850283747482." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "3218081288377565 + 1522931031786912 + ___ = 6228184581566130 + 1262176782801083", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 3218081288377565 + 1522931031786912 + ___ = 6228184581566130 + 1262176782801083", "options": [ "2749349044202726", "2749349044202756", "2749349044202736", "2749349044202746" ], "answer": "2749349044202736", "correct_value": 2749349044202736, "explanation": "No matching terms; blank = 6228184581566130 + 1262176782801083 − 3218081288377565 − 1522931031786912 = 2749349044202736." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001522", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "6177521085584930 × (8637858249115732 + 2500401195567394) = 6177521085584930 × ___ + 6177521085584930 × 2500401195567394", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 6177521085584930 × (8637858249115732 + 2500401195567394) = 6177521085584930 × ___ + 6177521085584930 × 2500401195567394", "options": [ "8637858249115732", "8637858249115742", "8637858249115752", "8637858249115722" ], "answer": "8637858249115732", "correct_value": 8637858249115732, "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 8637858249115732." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7678157765547604 + 2257814850270630 + ___ = 2257814850270630 + 3552847933856027 + 8548343318885714", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 7678157765547604 + 2257814850270630 + ___ = 2257814850270630 + 3552847933856027 + 8548343318885714", "options": [ "4423033487194127", "4423033487194147", "4423033487194157", "4423033487194137" ], "answer": "4423033487194137", "correct_value": 4423033487194137, "explanation": "B cancels; blank = C + D − A = 3552847933856027 + 8548343318885714 − 7678157765547604 = 4423033487194137." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6361739528366893 + 2311966976517363 + ___ = 1370107699456199 + 8438935617172475", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 6361739528366893 + 2311966976517363 + ___ = 1370107699456199 + 8438935617172475", "options": [ "1135336811744438", "1135336811744408", "1135336811744418", "1135336811744428" ], "answer": "1135336811744418", "correct_value": 1135336811744418, "explanation": "No matching terms; blank = 1370107699456199 + 8438935617172475 − 6361739528366893 − 2311966976517363 = 1135336811744418." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001523", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5279848701229619 × 8067058568554400 × 6848080896658693 = ___ × 6848080896658693 × 5279848701229619", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 5279848701229619 × 8067058568554400 × 6848080896658693 = ___ × 6848080896658693 × 5279848701229619", "options": [ "8067058568554410", "8067058568554390", "8067058568554420", "8067058568554400" ], "answer": "8067058568554400", "correct_value": 8067058568554400, "explanation": "A and C appear on both sides; blank = B = 8067058568554400." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6409571735225285 + 8819001476294742 + ___ = 8819001476294742 + 4509081856907372 + 2364434605004501", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 6409571735225285 + 8819001476294742 + ___ = 8819001476294742 + 4509081856907372 + 2364434605004501", "options": [ "463944726686578", "463944726686598", "463944726686588", "463944726686608" ], "answer": "463944726686588", "correct_value": 463944726686588, "explanation": "B cancels; blank = C + D − A = 4509081856907372 + 2364434605004501 − 6409571735225285 = 463944726686588." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "9665244905236245 + 5027905169318434 + ___ = 3376800092758929 + 8965168085357801", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 9665244905236245 + 5027905169318434 + ___ = 3376800092758929 + 8965168085357801", "options": [ "-2351181896437941", "-2351181896437949", "-2351181896437961", "-2351181896437931" ], "answer": "-2351181896437949", "correct_value": -2351181896437949, "explanation": "No matching terms; blank = 3376800092758929 + 8965168085357801 − 9665244905236245 − 5027905169318434 = -2351181896437949." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001524", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3641883926565584 + 4987457468571662 = ___ + 2678808084509157", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 3641883926565584 + 4987457468571662 = ___ + 2678808084509157", "options": [ "5950533310628109", "5950533310628079", "5950533310628099", "5950533310628089" ], "answer": "5950533310628089", "correct_value": 5950533310628089, "explanation": "Blank = A + B − C = 3641883926565584 + 4987457468571662 − 2678808084509157 = 5950533310628089." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4651905840594284 + 5022947496227180 + ___ = 5022947496227180 + 7641311254811318 + 8448416746792996", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 4651905840594284 + 5022947496227180 + ___ = 5022947496227180 + 7641311254811318 + 8448416746792996", "options": [ "11437822161010040", "11437822161010020", "11437822161010050", "11437822161010030" ], "answer": "11437822161010030", "correct_value": 11437822161010030, "explanation": "B cancels; blank = C + D − A = 7641311254811318 + 8448416746792996 − 4651905840594284 = 11437822161010030." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "8555603530971725 + 5728804618040901 + ___ = 6695559803388096 + 5238189805176004", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 8555603530971725 + 5728804618040901 + ___ = 6695559803388096 + 5238189805176004", "options": [ "-2350658540448526", "-2350658540448524", "-2350658540448544", "-2350658540448514" ], "answer": "-2350658540448526", "correct_value": -2350658540448526, "explanation": "No matching terms; blank = 6695559803388096 + 5238189805176004 − 8555603530971725 − 5728804618040901 = -2350658540448526." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001525", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8220675246136888 × 5716956132596674 × 7882204116131020 = ___ × 7882204116131020 × 8220675246136888", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 8220675246136888 × 5716956132596674 × 7882204116131020 = ___ × 7882204116131020 × 8220675246136888", "options": [ "5716956132596684", "5716956132596664", "5716956132596694", "5716956132596674" ], "answer": "5716956132596674", "correct_value": 5716956132596674, "explanation": "A and C appear on both sides; blank = B = 5716956132596674." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4238520678086216 + 5704118530477137 + ___ = 5704118530477137 + 6808587733194744 + 3065104510693303", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 4238520678086216 + 5704118530477137 + ___ = 5704118530477137 + 6808587733194744 + 3065104510693303", "options": [ "5635171565801851", "5635171565801821", "5635171565801841", "5635171565801831" ], "answer": "5635171565801831", "correct_value": 5635171565801831, "explanation": "B cancels; blank = C + D − A = 6808587733194744 + 3065104510693303 − 4238520678086216 = 5635171565801831." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "2477609906096385 + 9729057415790933 + ___ = 4792895096745303 + 6194809155064944", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 2477609906096385 + 9729057415790933 + ___ = 4792895096745303 + 6194809155064944", "options": [ "-1218963070077069", "-1218963070077071", "-1218963070077059", "-1218963070077089" ], "answer": "-1218963070077071", "correct_value": -1218963070077071, "explanation": "No matching terms; blank = 4792895096745303 + 6194809155064944 − 2477609906096385 − 9729057415790933 = -1218963070077071." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001526", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1635940226141318 + 4528076738061711 + ___ = 4528076738061711 + 3025322630736057 + 1635940226141318", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 1635940226141318 + 4528076738061711 + ___ = 4528076738061711 + 3025322630736057 + 1635940226141318", "options": [ "3025322630736077", "3025322630736067", "3025322630736047", "3025322630736057" ], "answer": "3025322630736057", "correct_value": 3025322630736057, "explanation": "A and B appear on both sides and cancel; blank = C = 3025322630736057." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7422552463665516 + 1702979937337549 + ___ = 1702979937337549 + 4556207391042188 + 1937014635407686", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 7422552463665516 + 1702979937337549 + ___ = 1702979937337549 + 4556207391042188 + 1937014635407686", "options": [ "-929330437215638", "-929330437215628", "-929330437215658", "-929330437215642" ], "answer": "-929330437215642", "correct_value": -929330437215642, "explanation": "B cancels; blank = C + D − A = 4556207391042188 + 1937014635407686 − 7422552463665516 = -929330437215642." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "8244252261533432 + 6345410395154468 + ___ = 1045173059623523 + 5500859413577684", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 8244252261533432 + 6345410395154468 + ___ = 1045173059623523 + 5500859413577684", "options": [ "-8043630183486707", "-8043630183486677", "-8043630183486687", "-8043630183486693" ], "answer": "-8043630183486693", "correct_value": -8043630183486693, "explanation": "No matching terms; blank = 1045173059623523 + 5500859413577684 − 8244252261533432 − 6345410395154468 = -8043630183486693." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "equation_reasoning_001527", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2331554701290316 + 2750223032444389 = ___ + 2943991846229781", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 2331554701290316 + 2750223032444389 = ___ + 2943991846229781", "options": [ "2137785887504924", "2137785887504934", "2137785887504944", "2137785887504914" ], "answer": "2137785887504924", "correct_value": 2137785887504924, "explanation": "Blank = A + B − C = 2331554701290316 + 2750223032444389 − 2943991846229781 = 2137785887504924." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2969667866174914 + 2835208904672195 + ___ = 2835208904672195 + 9411621780563791 + 8787938837962200", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 2969667866174914 + 2835208904672195 + ___ = 2835208904672195 + 9411621780563791 + 8787938837962200", "options": [ "15229892752351087", "15229892752351097", "15229892752351077", "15229892752351067" ], "answer": "15229892752351077", "correct_value": 15229892752351077, "explanation": "B cancels; blank = C + D − A = 9411621780563791 + 8787938837962200 − 2969667866174914 = 15229892752351077." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1975576113516337 + 3160842760844015 + ___ = 1359145942987413 + 8641094422753294", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 1975576113516337 + 3160842760844015 + ___ = 1359145942987413 + 8641094422753294", "options": [ "4863821491380355", "4863821491380375", "4863821491380345", "4863821491380365" ], "answer": "4863821491380355", "correct_value": 4863821491380355, "explanation": "No matching terms; blank = 1359145942987413 + 8641094422753294 − 1975576113516337 − 3160842760844015 = 4863821491380355." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001528", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1053178554533564 × 7227177900025980 × 4634679686962713 = ___ × 4634679686962713 × 1053178554533564", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 1053178554533564 × 7227177900025980 × 4634679686962713 = ___ × 4634679686962713 × 1053178554533564", "options": [ "7227177900025970", "7227177900025980", "7227177900025990", "7227177900026000" ], "answer": "7227177900025980", "correct_value": 7227177900025980, "explanation": "A and C appear on both sides; blank = B = 7227177900025980." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1269926382503418 + 9150809689568890 + ___ = 9150809689568890 + 4479553786449994 + 2047706625547910", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 1269926382503418 + 9150809689568890 + ___ = 9150809689568890 + 4479553786449994 + 2047706625547910", "options": [ "5257334029494476", "5257334029494486", "5257334029494496", "5257334029494506" ], "answer": "5257334029494486", "correct_value": 5257334029494486, "explanation": "B cancels; blank = C + D − A = 4479553786449994 + 2047706625547910 − 1269926382503418 = 5257334029494486." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4383358308500639 + 2043346310459568 + ___ = 2252358307425676 + 8426730381445619", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 4383358308500639 + 2043346310459568 + ___ = 2252358307425676 + 8426730381445619", "options": [ "4252384069911078", "4252384069911088", "4252384069911098", "4252384069911108" ], "answer": "4252384069911088", "correct_value": 4252384069911088, "explanation": "No matching terms; blank = 2252358307425676 + 8426730381445619 − 4383358308500639 − 2043346310459568 = 4252384069911088." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001529", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3804984043419640 × (5109926858958150 + 1399339712696575) = 3804984043419640 × ___ + 3804984043419640 × 1399339712696575", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 3804984043419640 × (5109926858958150 + 1399339712696575) = 3804984043419640 × ___ + 3804984043419640 × 1399339712696575", "options": [ "5109926858958170", "5109926858958160", "5109926858958150", "5109926858958140" ], "answer": "5109926858958150", "correct_value": 5109926858958150, "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 5109926858958150." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "8993585900192683 + 3063606481066844 + ___ = 3063606481066844 + 5472040642910330 + 7807134294419628", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 8993585900192683 + 3063606481066844 + ___ = 3063606481066844 + 5472040642910330 + 7807134294419628", "options": [ "4285589037137265", "4285589037137295", "4285589037137285", "4285589037137275" ], "answer": "4285589037137275", "correct_value": 4285589037137275, "explanation": "B cancels; blank = C + D − A = 5472040642910330 + 7807134294419628 − 8993585900192683 = 4285589037137275." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6989359793732632 + 6168817558749705 + ___ = 2993279725679857 + 7070223707602262", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 6989359793732632 + 6168817558749705 + ___ = 2993279725679857 + 7070223707602262", "options": [ "-3094673919200202", "-3094673919200212", "-3094673919200218", "-3094673919200232" ], "answer": "-3094673919200218", "correct_value": -3094673919200218, "explanation": "No matching terms; blank = 2993279725679857 + 7070223707602262 − 6989359793732632 − 6168817558749705 = -3094673919200218." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001530", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1558248099715812 + 2493640720624898 + ___ = 2493640720624898 + 8297637911266294 + 1558248099715812", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 1558248099715812 + 2493640720624898 + ___ = 2493640720624898 + 8297637911266294 + 1558248099715812", "options": [ "8297637911266314", "8297637911266304", "8297637911266294", "8297637911266284" ], "answer": "8297637911266294", "correct_value": 8297637911266294, "explanation": "A and B appear on both sides and cancel; blank = C = 8297637911266294." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1672408278892294 + 3681912780186421 + ___ = 3681912780186421 + 5630047012415380 + 1346210079936730", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 1672408278892294 + 3681912780186421 + ___ = 3681912780186421 + 5630047012415380 + 1346210079936730", "options": [ "5303848813459836", "5303848813459806", "5303848813459826", "5303848813459816" ], "answer": "5303848813459816", "correct_value": 5303848813459816, "explanation": "B cancels; blank = C + D − A = 5630047012415380 + 1346210079936730 − 1672408278892294 = 5303848813459816." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "5554919492266349 + 9230798831324212 + ___ = 1834549347427670 + 9764518853299343", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 5554919492266349 + 9230798831324212 + ___ = 1834549347427670 + 9764518853299343", "options": [ "-3186650122863542", "-3186650122863562", "-3186650122863548", "-3186650122863532" ], "answer": "-3186650122863548", "correct_value": -3186650122863548, "explanation": "No matching terms; blank = 1834549347427670 + 9764518853299343 − 5554919492266349 − 9230798831324212 = -3186650122863548." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001531", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "6527972947479958 + 5331973890368190 = ___ + 1213658118102539", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 6527972947479958 + 5331973890368190 = ___ + 1213658118102539", "options": [ "10646288719745599", "10646288719745629", "10646288719745609", "10646288719745619" ], "answer": "10646288719745609", "correct_value": 10646288719745609, "explanation": "Blank = A + B − C = 6527972947479958 + 5331973890368190 − 1213658118102539 = 10646288719745609." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "3957337784694626 + 7220197236470882 + ___ = 7220197236470882 + 1228007244388564 + 3058940027635768", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 3957337784694626 + 7220197236470882 + ___ = 7220197236470882 + 1228007244388564 + 3058940027635768", "options": [ "329609487329726", "329609487329716", "329609487329696", "329609487329706" ], "answer": "329609487329706", "correct_value": 329609487329706, "explanation": "B cancels; blank = C + D − A = 1228007244388564 + 3058940027635768 − 3957337784694626 = 329609487329706." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "7482360483237204 + 4884662767184543 + ___ = 5799027556998043 + 6863182341505315", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 7482360483237204 + 4884662767184543 + ___ = 5799027556998043 + 6863182341505315", "options": [ "295186648081631", "295186648081621", "295186648081601", "295186648081611" ], "answer": "295186648081611", "correct_value": 295186648081611, "explanation": "No matching terms; blank = 5799027556998043 + 6863182341505315 − 7482360483237204 − 4884662767184543 = 295186648081611." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "equation_reasoning_001532", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5210909806817207 + 8687748902934265 + ___ = 8687748902934265 + 4766604526544077 + 5210909806817207", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 5210909806817207 + 8687748902934265 + ___ = 8687748902934265 + 4766604526544077 + 5210909806817207", "options": [ "4766604526544067", "4766604526544097", "4766604526544087", "4766604526544077" ], "answer": "4766604526544077", "correct_value": 4766604526544077, "explanation": "A and B appear on both sides and cancel; blank = C = 4766604526544077." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2608821142028725 + 6551140325300192 + ___ = 6551140325300192 + 8441602010359219 + 1180593286726693", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 2608821142028725 + 6551140325300192 + ___ = 6551140325300192 + 8441602010359219 + 1180593286726693", "options": [ "7013374155057197", "7013374155057207", "7013374155057187", "7013374155057177" ], "answer": "7013374155057187", "correct_value": 7013374155057187, "explanation": "B cancels; blank = C + D − A = 8441602010359219 + 1180593286726693 − 2608821142028725 = 7013374155057187." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "9648695889988034 + 1811024757233040 + ___ = 4643537247595849 + 2744594432365468", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 9648695889988034 + 1811024757233040 + ___ = 4643537247595849 + 2744594432365468", "options": [ "-4071588967259773", "-4071588967259757", "-4071588967259743", "-4071588967259753" ], "answer": "-4071588967259757", "correct_value": -4071588967259757, "explanation": "No matching terms; blank = 4643537247595849 + 2744594432365468 − 9648695889988034 − 1811024757233040 = -4071588967259757." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001533", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7362474948979208 × (3096403967180910 + 2868708106811189) = 7362474948979208 × ___ + 7362474948979208 × 2868708106811189", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 7362474948979208 × (3096403967180910 + 2868708106811189) = 7362474948979208 × ___ + 7362474948979208 × 2868708106811189", "options": [ "3096403967180900", "3096403967180920", "3096403967180930", "3096403967180910" ], "answer": "3096403967180910", "correct_value": 3096403967180910, "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 3096403967180910." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9314607380742707 + 7751766541961534 + ___ = 7751766541961534 + 9269139053443230 + 1003982980841327", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 9314607380742707 + 7751766541961534 + ___ = 7751766541961534 + 9269139053443230 + 1003982980841327", "options": [ "958514653541850", "958514653541840", "958514653541870", "958514653541860" ], "answer": "958514653541850", "correct_value": 958514653541850, "explanation": "B cancels; blank = C + D − A = 9269139053443230 + 1003982980841327 − 9314607380742707 = 958514653541850." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "7372970343022899 + 1932788624818773 + ___ = 7647218364977824 + 3372934120545147", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 7372970343022899 + 1932788624818773 + ___ = 7647218364977824 + 3372934120545147", "options": [ "1714393517681319", "1714393517681309", "1714393517681299", "1714393517681289" ], "answer": "1714393517681299", "correct_value": 1714393517681299, "explanation": "No matching terms; blank = 7647218364977824 + 3372934120545147 − 7372970343022899 − 1932788624818773 = 1714393517681299." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001534", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8602606603691833 × (1258085147733797 + 9069041956716424) = 8602606603691833 × ___ + 8602606603691833 × 9069041956716424", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 8602606603691833 × (1258085147733797 + 9069041956716424) = 8602606603691833 × ___ + 8602606603691833 × 9069041956716424", "options": [ "1258085147733797", "1258085147733817", "1258085147733807", "1258085147733787" ], "answer": "1258085147733797", "correct_value": 1258085147733797, "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 1258085147733797." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2591932087263427 + 7708309892032582 + ___ = 7708309892032582 + 6997721657690295 + 3363939889587527", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 2591932087263427 + 7708309892032582 + ___ = 7708309892032582 + 6997721657690295 + 3363939889587527", "options": [ "7769729460014395", "7769729460014385", "7769729460014415", "7769729460014405" ], "answer": "7769729460014395", "correct_value": 7769729460014395, "explanation": "B cancels; blank = C + D − A = 6997721657690295 + 3363939889587527 − 2591932087263427 = 7769729460014395." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2034089414628812 + 8964343016150275 + ___ = 1415296126326191 + 5804444176093503", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 2034089414628812 + 8964343016150275 + ___ = 1415296126326191 + 5804444176093503", "options": [ "-3778692128359393", "-3778692128359407", "-3778692128359377", "-3778692128359387" ], "answer": "-3778692128359393", "correct_value": -3778692128359393, "explanation": "No matching terms; blank = 1415296126326191 + 5804444176093503 − 2034089414628812 − 8964343016150275 = -3778692128359393." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001535", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "4121120999123190 + 2324915079786940 + ___ = 2324915079786940 + 4851750228294284 + 4121120999123190", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 4121120999123190 + 2324915079786940 + ___ = 2324915079786940 + 4851750228294284 + 4121120999123190", "options": [ "4851750228294294", "4851750228294304", "4851750228294284", "4851750228294274" ], "answer": "4851750228294284", "correct_value": 4851750228294284, "explanation": "A and B appear on both sides and cancel; blank = C = 4851750228294284." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6363324370274417 + 9801247958144421 + ___ = 9801247958144421 + 7773067894115681 + 1396790104218304", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 6363324370274417 + 9801247958144421 + ___ = 9801247958144421 + 7773067894115681 + 1396790104218304", "options": [ "2806533628059588", "2806533628059578", "2806533628059558", "2806533628059568" ], "answer": "2806533628059568", "correct_value": 2806533628059568, "explanation": "B cancels; blank = C + D − A = 7773067894115681 + 1396790104218304 − 6363324370274417 = 2806533628059568." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "4473744137432268 + 8770445041402556 + ___ = 3866549158744246 + 2894603128571256", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 4473744137432268 + 8770445041402556 + ___ = 3866549158744246 + 2894603128571256", "options": [ "-6483036891519338", "-6483036891519322", "-6483036891519308", "-6483036891519318" ], "answer": "-6483036891519322", "correct_value": -6483036891519322, "explanation": "No matching terms; blank = 3866549158744246 + 2894603128571256 − 4473744137432268 − 8770445041402556 = -6483036891519322." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001536", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7092693933349702 + 6549579805424133 = ___ + 5140582229494738", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 7092693933349702 + 6549579805424133 = ___ + 5140582229494738", "options": [ "8501691509279087", "8501691509279097", "8501691509279117", "8501691509279107" ], "answer": "8501691509279097", "correct_value": 8501691509279097, "explanation": "Blank = A + B − C = 7092693933349702 + 6549579805424133 − 5140582229494738 = 8501691509279097." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5966648575683013 + 6773720296764147 + ___ = 6773720296764147 + 7052806545995782 + 7273360208528615", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 5966648575683013 + 6773720296764147 + ___ = 6773720296764147 + 7052806545995782 + 7273360208528615", "options": [ "8359518178841384", "8359518178841404", "8359518178841394", "8359518178841374" ], "answer": "8359518178841384", "correct_value": 8359518178841384, "explanation": "B cancels; blank = C + D − A = 7052806545995782 + 7273360208528615 − 5966648575683013 = 8359518178841384." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4248551597586604 + 7819374017845253 + ___ = 7581676545272329 + 6307386984137290", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 4248551597586604 + 7819374017845253 + ___ = 7581676545272329 + 6307386984137290", "options": [ "1821137913977752", "1821137913977762", "1821137913977772", "1821137913977782" ], "answer": "1821137913977762", "correct_value": 1821137913977762, "explanation": "No matching terms; blank = 7581676545272329 + 6307386984137290 − 4248551597586604 − 7819374017845253 = 1821137913977762." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001537", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9705526572272258 × (9682265322061810 + 8826504239402538) = 9705526572272258 × ___ + 9705526572272258 × 8826504239402538", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 9705526572272258 × (9682265322061810 + 8826504239402538) = 9705526572272258 × ___ + 9705526572272258 × 8826504239402538", "options": [ "9682265322061820", "9682265322061800", "9682265322061810", "9682265322061830" ], "answer": "9682265322061810", "correct_value": 9682265322061810, "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 9682265322061810." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4486659640098485 + 3168176662677883 + ___ = 3168176662677883 + 3034901893688522 + 1239614113100124", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 4486659640098485 + 3168176662677883 + ___ = 3168176662677883 + 3034901893688522 + 1239614113100124", "options": [ "-212143633309839", "-212143633309831", "-212143633309851", "-212143633309821" ], "answer": "-212143633309839", "correct_value": -212143633309839, "explanation": "B cancels; blank = C + D − A = 3034901893688522 + 1239614113100124 − 4486659640098485 = -212143633309839." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1660415726549050 + 8287736090551328 + ___ = 5548294621840376 + 6096120602703683", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 1660415726549050 + 8287736090551328 + ___ = 5548294621840376 + 6096120602703683", "options": [ "1696263407443701", "1696263407443671", "1696263407443691", "1696263407443681" ], "answer": "1696263407443681", "correct_value": 1696263407443681, "explanation": "No matching terms; blank = 5548294621840376 + 6096120602703683 − 1660415726549050 − 8287736090551328 = 1696263407443681." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "equation_reasoning_001538", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8567110385018472 + 1284928631894407 = ___ + 7368772355517850", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 8567110385018472 + 1284928631894407 = ___ + 7368772355517850", "options": [ "2483266661395049", "2483266661395029", "2483266661395039", "2483266661395019" ], "answer": "2483266661395029", "correct_value": 2483266661395029, "explanation": "Blank = A + B − C = 8567110385018472 + 1284928631894407 − 7368772355517850 = 2483266661395029." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4142055866604761 + 6961119553855947 + ___ = 6961119553855947 + 6198989687222077 + 4251662158579775", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 4142055866604761 + 6961119553855947 + ___ = 6961119553855947 + 6198989687222077 + 4251662158579775", "options": [ "6308595979197081", "6308595979197101", "6308595979197111", "6308595979197091" ], "answer": "6308595979197091", "correct_value": 6308595979197091, "explanation": "B cancels; blank = C + D − A = 6198989687222077 + 4251662158579775 − 4142055866604761 = 6308595979197091." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "2366439511432283 + 6736053632823127 + ___ = 6632272192440828 + 7197749211610348", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 2366439511432283 + 6736053632823127 + ___ = 6632272192440828 + 7197749211610348", "options": [ "4727528259795786", "4727528259795776", "4727528259795756", "4727528259795766" ], "answer": "4727528259795766", "correct_value": 4727528259795766, "explanation": "No matching terms; blank = 6632272192440828 + 7197749211610348 − 2366439511432283 − 6736053632823127 = 4727528259795766." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "equation_reasoning_001539", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5546435633866435 + 1789414082090709 + ___ = 1789414082090709 + 8744302183355268 + 5546435633866435", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 5546435633866435 + 1789414082090709 + ___ = 1789414082090709 + 8744302183355268 + 5546435633866435", "options": [ "8744302183355288", "8744302183355278", "8744302183355258", "8744302183355268" ], "answer": "8744302183355268", "correct_value": 8744302183355268, "explanation": "A and B appear on both sides and cancel; blank = C = 8744302183355268." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9177031969368272 + 3756324981116268 + ___ = 3756324981116268 + 5619686992164732 + 9841527275780593", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 9177031969368272 + 3756324981116268 + ___ = 3756324981116268 + 5619686992164732 + 9841527275780593", "options": [ "6284182298577063", "6284182298577053", "6284182298577073", "6284182298577043" ], "answer": "6284182298577053", "correct_value": 6284182298577053, "explanation": "B cancels; blank = C + D − A = 5619686992164732 + 9841527275780593 − 9177031969368272 = 6284182298577053." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2623290847445194 + 9627740616700838 + ___ = 5473770064427568 + 3279583191885382", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 2623290847445194 + 9627740616700838 + ___ = 5473770064427568 + 3279583191885382", "options": [ "-3497678207833068", "-3497678207833082", "-3497678207833098", "-3497678207833078" ], "answer": "-3497678207833082", "correct_value": -3497678207833082, "explanation": "No matching terms; blank = 5473770064427568 + 3279583191885382 − 2623290847445194 − 9627740616700838 = -3497678207833082." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001540", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5779586281922916 + 9649000890147332 + ___ = 9649000890147332 + 5445616426713288 + 5779586281922916", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 5779586281922916 + 9649000890147332 + ___ = 9649000890147332 + 5445616426713288 + 5779586281922916", "options": [ "5445616426713288", "5445616426713298", "5445616426713278", "5445616426713308" ], "answer": "5445616426713288", "correct_value": 5445616426713288, "explanation": "A and B appear on both sides and cancel; blank = C = 5445616426713288." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9279154286019750 + 9376151788091536 + ___ = 9376151788091536 + 5892203765188854 + 5416298213014737", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 9279154286019750 + 9376151788091536 + ___ = 9376151788091536 + 5892203765188854 + 5416298213014737", "options": [ "2029347692183841", "2029347692183851", "2029347692183861", "2029347692183831" ], "answer": "2029347692183841", "correct_value": 2029347692183841, "explanation": "B cancels; blank = C + D − A = 5892203765188854 + 5416298213014737 − 9279154286019750 = 2029347692183841." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "9161078002344448 + 8072009496056966 + ___ = 5149091563954446 + 8229613880656996", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 9161078002344448 + 8072009496056966 + ___ = 5149091563954446 + 8229613880656996", "options": [ "-3854382053789968", "-3854382053789972", "-3854382053789988", "-3854382053789958" ], "answer": "-3854382053789972", "correct_value": -3854382053789972, "explanation": "No matching terms; blank = 5149091563954446 + 8229613880656996 − 9161078002344448 − 8072009496056966 = -3854382053789972." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001541", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7930357640608270 × (2686770542842925 + 8828674763399836) = 7930357640608270 × ___ + 7930357640608270 × 8828674763399836", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 7930357640608270 × (2686770542842925 + 8828674763399836) = 7930357640608270 × ___ + 7930357640608270 × 8828674763399836", "options": [ "2686770542842945", "2686770542842915", "2686770542842935", "2686770542842925" ], "answer": "2686770542842925", "correct_value": 2686770542842925, "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 2686770542842925." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6420697250042561 + 8566801919529005 + ___ = 8566801919529005 + 6738470214947001 + 2773095636631891", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 6420697250042561 + 8566801919529005 + ___ = 8566801919529005 + 6738470214947001 + 2773095636631891", "options": [ "3090868601536321", "3090868601536341", "3090868601536331", "3090868601536351" ], "answer": "3090868601536331", "correct_value": 3090868601536331, "explanation": "B cancels; blank = C + D − A = 6738470214947001 + 2773095636631891 − 6420697250042561 = 3090868601536331." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "4162435170458615 + 5144009478775509 + ___ = 3688980578438333 + 4256043172532636", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 4162435170458615 + 5144009478775509 + ___ = 3688980578438333 + 4256043172532636", "options": [ "-1361420898263135", "-1361420898263165", "-1361420898263145", "-1361420898263155" ], "answer": "-1361420898263155", "correct_value": -1361420898263155, "explanation": "No matching terms; blank = 3688980578438333 + 4256043172532636 − 4162435170458615 − 5144009478775509 = -1361420898263155." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "equation_reasoning_001542", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "4040350569059189 × (8101456112862280 + 7951044226508976) = 4040350569059189 × ___ + 4040350569059189 × 7951044226508976", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 4040350569059189 × (8101456112862280 + 7951044226508976) = 4040350569059189 × ___ + 4040350569059189 × 7951044226508976", "options": [ "8101456112862280", "8101456112862300", "8101456112862270", "8101456112862290" ], "answer": "8101456112862280", "correct_value": 8101456112862280, "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 8101456112862280." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9381869575195913 + 3181148814224181 + ___ = 3181148814224181 + 9957683638004322 + 5326675778838333", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 9381869575195913 + 3181148814224181 + ___ = 3181148814224181 + 9957683638004322 + 5326675778838333", "options": [ "5902489841646762", "5902489841646752", "5902489841646742", "5902489841646732" ], "answer": "5902489841646742", "correct_value": 5902489841646742, "explanation": "B cancels; blank = C + D − A = 9957683638004322 + 5326675778838333 − 9381869575195913 = 5902489841646742." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2764481616711720 + 2932945505346258 + ___ = 6100028888533813 + 3643025948340725", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 2764481616711720 + 2932945505346258 + ___ = 6100028888533813 + 3643025948340725", "options": [ "4045627714816560", "4045627714816570", "4045627714816550", "4045627714816580" ], "answer": "4045627714816560", "correct_value": 4045627714816560, "explanation": "No matching terms; blank = 6100028888533813 + 3643025948340725 − 2764481616711720 − 2932945505346258 = 4045627714816560." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001543", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "6434842961228522 + 2406174307339384 + ___ = 2406174307339384 + 7967513799762804 + 6434842961228522", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 6434842961228522 + 2406174307339384 + ___ = 2406174307339384 + 7967513799762804 + 6434842961228522", "options": [ "7967513799762824", "7967513799762804", "7967513799762814", "7967513799762794" ], "answer": "7967513799762804", "correct_value": 7967513799762804, "explanation": "A and B appear on both sides and cancel; blank = C = 7967513799762804." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1759428135988573 + 2885624518936929 + ___ = 2885624518936929 + 4674916132093709 + 1443614856555700", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 1759428135988573 + 2885624518936929 + ___ = 2885624518936929 + 4674916132093709 + 1443614856555700", "options": [ "4359102852660826", "4359102852660856", "4359102852660836", "4359102852660846" ], "answer": "4359102852660836", "correct_value": 4359102852660836, "explanation": "B cancels; blank = C + D − A = 4674916132093709 + 1443614856555700 − 1759428135988573 = 4359102852660836." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "7684146107471134 + 3479922420634625 + ___ = 4710462794975655 + 1378621435304972", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 7684146107471134 + 3479922420634625 + ___ = 4710462794975655 + 1378621435304972", "options": [ "-5074984297825148", "-5074984297825128", "-5074984297825132", "-5074984297825118" ], "answer": "-5074984297825132", "correct_value": -5074984297825132, "explanation": "No matching terms; blank = 4710462794975655 + 1378621435304972 − 7684146107471134 − 3479922420634625 = -5074984297825132." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001544", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "4808451641380022 + 7159593992100287 + ___ = 7159593992100287 + 7413149497672321 + 4808451641380022", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 4808451641380022 + 7159593992100287 + ___ = 7159593992100287 + 7413149497672321 + 4808451641380022", "options": [ "7413149497672331", "7413149497672311", "7413149497672341", "7413149497672321" ], "answer": "7413149497672321", "correct_value": 7413149497672321, "explanation": "A and B appear on both sides and cancel; blank = C = 7413149497672321." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4999612740312860 + 4339578254099992 + ___ = 4339578254099992 + 2083299081016873 + 7187169998390544", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 4999612740312860 + 4339578254099992 + ___ = 4339578254099992 + 2083299081016873 + 7187169998390544", "options": [ "4270856339094547", "4270856339094577", "4270856339094557", "4270856339094567" ], "answer": "4270856339094557", "correct_value": 4270856339094557, "explanation": "B cancels; blank = C + D − A = 2083299081016873 + 7187169998390544 − 4999612740312860 = 4270856339094557." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2583488210897906 + 8223587992672437 + ___ = 3668419096589427 + 8281620012018478", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 2583488210897906 + 8223587992672437 + ___ = 3668419096589427 + 8281620012018478", "options": [ "1142962905037572", "1142962905037562", "1142962905037552", "1142962905037582" ], "answer": "1142962905037562", "correct_value": 1142962905037562, "explanation": "No matching terms; blank = 3668419096589427 + 8281620012018478 − 2583488210897906 − 8223587992672437 = 1142962905037562." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001545", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7588589430236939 × (2786356109818763 + 3888250562473902) = 7588589430236939 × ___ + 7588589430236939 × 3888250562473902", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 7588589430236939 × (2786356109818763 + 3888250562473902) = 7588589430236939 × ___ + 7588589430236939 × 3888250562473902", "options": [ "2786356109818773", "2786356109818753", "2786356109818783", "2786356109818763" ], "answer": "2786356109818763", "correct_value": 2786356109818763, "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 2786356109818763." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9592244723381687 + 1276780178492779 + ___ = 1276780178492779 + 1905660082081232 + 8622618628948106", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 9592244723381687 + 1276780178492779 + ___ = 1276780178492779 + 1905660082081232 + 8622618628948106", "options": [ "936033987647661", "936033987647671", "936033987647651", "936033987647641" ], "answer": "936033987647651", "correct_value": 936033987647651, "explanation": "B cancels; blank = C + D − A = 1905660082081232 + 8622618628948106 − 9592244723381687 = 936033987647651." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2633186068596676 + 1276205797427340 + ___ = 5250573011182617 + 4538143596295017", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 2633186068596676 + 1276205797427340 + ___ = 5250573011182617 + 4538143596295017", "options": [ "5879324741453608", "5879324741453638", "5879324741453628", "5879324741453618" ], "answer": "5879324741453618", "correct_value": 5879324741453618, "explanation": "No matching terms; blank = 5250573011182617 + 4538143596295017 − 2633186068596676 − 1276205797427340 = 5879324741453618." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "equation_reasoning_001546", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2179499626991260 + 4090609559610474 + ___ = 4090609559610474 + 7540453928018498 + 2179499626991260", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 2179499626991260 + 4090609559610474 + ___ = 4090609559610474 + 7540453928018498 + 2179499626991260", "options": [ "7540453928018518", "7540453928018488", "7540453928018508", "7540453928018498" ], "answer": "7540453928018498", "correct_value": 7540453928018498, "explanation": "A and B appear on both sides and cancel; blank = C = 7540453928018498." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6974378193549744 + 8259878256643015 + ___ = 8259878256643015 + 3781521404367227 + 7421030228673248", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 6974378193549744 + 8259878256643015 + ___ = 8259878256643015 + 3781521404367227 + 7421030228673248", "options": [ "4228173439490721", "4228173439490741", "4228173439490751", "4228173439490731" ], "answer": "4228173439490731", "correct_value": 4228173439490731, "explanation": "B cancels; blank = C + D − A = 3781521404367227 + 7421030228673248 − 6974378193549744 = 4228173439490731." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "4904884027639950 + 8581618191050997 + ___ = 2588523606788874 + 1994917565483293", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 4904884027639950 + 8581618191050997 + ___ = 2588523606788874 + 1994917565483293", "options": [ "-8903061046418780", "-8903061046418760", "-8903061046418770", "-8903061046418790" ], "answer": "-8903061046418780", "correct_value": -8903061046418780, "explanation": "No matching terms; blank = 2588523606788874 + 1994917565483293 − 4904884027639950 − 8581618191050997 = -8903061046418780." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001547", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8271184018436236 × 1961806206927511 × 6660562264930959 = ___ × 6660562264930959 × 8271184018436236", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 8271184018436236 × 1961806206927511 × 6660562264930959 = ___ × 6660562264930959 × 8271184018436236", "options": [ "1961806206927531", "1961806206927511", "1961806206927501", "1961806206927521" ], "answer": "1961806206927511", "correct_value": 1961806206927511, "explanation": "A and C appear on both sides; blank = B = 1961806206927511." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4302516795408079 + 6059218749366943 + ___ = 6059218749366943 + 5310895210652928 + 5971568305995865", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 4302516795408079 + 6059218749366943 + ___ = 6059218749366943 + 5310895210652928 + 5971568305995865", "options": [ "6979946721240724", "6979946721240714", "6979946721240734", "6979946721240704" ], "answer": "6979946721240714", "correct_value": 6979946721240714, "explanation": "B cancels; blank = C + D − A = 5310895210652928 + 5971568305995865 − 4302516795408079 = 6979946721240714." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "3574711165269986 + 2070305504398164 + ___ = 7970841844637652 + 2075064048686238", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 3574711165269986 + 2070305504398164 + ___ = 7970841844637652 + 2075064048686238", "options": [ "4400889223655730", "4400889223655740", "4400889223655760", "4400889223655750" ], "answer": "4400889223655740", "correct_value": 4400889223655740, "explanation": "No matching terms; blank = 7970841844637652 + 2075064048686238 − 3574711165269986 − 2070305504398164 = 4400889223655740." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001548", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2450248408144842 × 6844204389213416 × 4189558827380758 = ___ × 4189558827380758 × 2450248408144842", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 2450248408144842 × 6844204389213416 × 4189558827380758 = ___ × 4189558827380758 × 2450248408144842", "options": [ "6844204389213416", "6844204389213436", "6844204389213426", "6844204389213406" ], "answer": "6844204389213416", "correct_value": 6844204389213416, "explanation": "A and C appear on both sides; blank = B = 6844204389213416." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5675011729469962 + 4212894898055065 + ___ = 4212894898055065 + 3150277309359643 + 8373655772025434", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 5675011729469962 + 4212894898055065 + ___ = 4212894898055065 + 3150277309359643 + 8373655772025434", "options": [ "5848921351915115", "5848921351915135", "5848921351915105", "5848921351915125" ], "answer": "5848921351915115", "correct_value": 5848921351915115, "explanation": "B cancels; blank = C + D − A = 3150277309359643 + 8373655772025434 − 5675011729469962 = 5848921351915115." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "2227768180762713 + 4802803633413909 + ___ = 4065402958608648 + 4315068719599908", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 2227768180762713 + 4802803633413909 + ___ = 4065402958608648 + 4315068719599908", "options": [ "1349899864031954", "1349899864031944", "1349899864031924", "1349899864031934" ], "answer": "1349899864031934", "correct_value": 1349899864031934, "explanation": "No matching terms; blank = 4065402958608648 + 4315068719599908 − 2227768180762713 − 4802803633413909 = 1349899864031934." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "equation_reasoning_001549", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1075520320366280 + 2258145426800559 + ___ = 2258145426800559 + 5625074371984145 + 1075520320366280", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 1075520320366280 + 2258145426800559 + ___ = 2258145426800559 + 5625074371984145 + 1075520320366280", "options": [ "5625074371984155", "5625074371984165", "5625074371984145", "5625074371984135" ], "answer": "5625074371984145", "correct_value": 5625074371984145, "explanation": "A and B appear on both sides and cancel; blank = C = 5625074371984145." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1097727547347667 + 2238939376301181 + ___ = 2238939376301181 + 5699300420409497 + 4166752520351224", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 1097727547347667 + 2238939376301181 + ___ = 2238939376301181 + 5699300420409497 + 4166752520351224", "options": [ "8768325393413044", "8768325393413054", "8768325393413064", "8768325393413074" ], "answer": "8768325393413054", "correct_value": 8768325393413054, "explanation": "B cancels; blank = C + D − A = 5699300420409497 + 4166752520351224 − 1097727547347667 = 8768325393413054." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "5243751945272956 + 8825149157449464 + ___ = 2450480957709452 + 5479400679294102", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 5243751945272956 + 8825149157449464 + ___ = 2450480957709452 + 5479400679294102", "options": [ "-6139019465718866", "-6139019465718884", "-6139019465718854", "-6139019465718864" ], "answer": "-6139019465718866", "correct_value": -6139019465718866, "explanation": "No matching terms; blank = 2450480957709452 + 5479400679294102 − 5243751945272956 − 8825149157449464 = -6139019465718866." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001550", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9701365724720143 + 9872498440745459 + ___ = 9872498440745459 + 1944242811036027 + 9701365724720143", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 9701365724720143 + 9872498440745459 + ___ = 9872498440745459 + 1944242811036027 + 9701365724720143", "options": [ "1944242811036047", "1944242811036027", "1944242811036037", "1944242811036017" ], "answer": "1944242811036027", "correct_value": 1944242811036027, "explanation": "A and B appear on both sides and cancel; blank = C = 1944242811036027." }, "distractor_types": [ "nearby_same_digit", "correct", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6456116102479139 + 1514001708300421 + ___ = 1514001708300421 + 8265966107208020 + 1697186552362223", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 6456116102479139 + 1514001708300421 + ___ = 1514001708300421 + 8265966107208020 + 1697186552362223", "options": [ "3507036557091104", "3507036557091124", "3507036557091114", "3507036557091094" ], "answer": "3507036557091104", "correct_value": 3507036557091104, "explanation": "B cancels; blank = C + D − A = 8265966107208020 + 1697186552362223 − 6456116102479139 = 3507036557091104." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "control": { "task_type": "control", "math_expression": "1335604668120552 + 7938165956294774 + ___ = 9158241693554797 + 6297339728436959", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 1335604668120552 + 7938165956294774 + ___ = 9158241693554797 + 6297339728436959", "options": [ "6181810797576440", "6181810797576450", "6181810797576430", "6181810797576420" ], "answer": "6181810797576430", "correct_value": 6181810797576430, "explanation": "No matching terms; blank = 9158241693554797 + 6297339728436959 − 1335604668120552 − 7938165956294774 = 6181810797576430." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "correct", "nearby_same_digit" ] } }, { "family_id": "equation_reasoning_001551", "category": "equation_reasoning", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3067540763036053 × 8656456881613739 × 1982939910026357 = ___ × 1982939910026357 × 3067540763036053", "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", "shortcut_strength": 1.0, "pure_math": { "question": "What value fills the blank? 3067540763036053 × 8656456881613739 × 1982939910026357 = ___ × 1982939910026357 × 3067540763036053", "options": [ "8656456881613739", "8656456881613749", "8656456881613759", "8656456881613729" ], "answer": "8656456881613739", "correct_value": 8656456881613739, "explanation": "A and C appear on both sides; blank = B = 8656456881613739." }, "distractor_types": [ "correct", "nearby_same_digit", "nearby_same_digit", "nearby_same_digit" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7889855830438321 + 1218209599272062 + ___ = 1218209599272062 + 4748615990751375 + 1394874416459976", "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", "shortcut_strength": 0.5, "pure_math": { "question": "What value fills the blank? 7889855830438321 + 1218209599272062 + ___ = 1218209599272062 + 4748615990751375 + 1394874416459976", "options": [ "-1746365423226950", "-1746365423226980", "-1746365423226960", "-1746365423226970" ], "answer": "-1746365423226970", "correct_value": -1746365423226970, "explanation": "B cancels; blank = C + D − A = 4748615990751375 + 1394874416459976 − 7889855830438321 = -1746365423226970." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] }, "control": { "task_type": "control", "math_expression": "6944630308971588 + 4770037120131624 + ___ = 8726397552817575 + 9816875847622960", "ns_shortcut": "No algebraic shortcut; compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What value fills the blank? 6944630308971588 + 4770037120131624 + ___ = 8726397552817575 + 9816875847622960", "options": [ "6828605971337313", "6828605971337343", "6828605971337333", "6828605971337323" ], "answer": "6828605971337323", "correct_value": 6828605971337323, "explanation": "No matching terms; blank = 8726397552817575 + 9816875847622960 − 6944630308971588 − 4770037120131624 = 6828605971337323." }, "distractor_types": [ "nearby_same_digit", "nearby_same_digit", "nearby_same_digit", "correct" ] } }, { "family_id": "option_elimination_001552", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9701770548774270 × 3786889370837892", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 9701770548774270 × 3786889370837892?", "options": [ "36739531769461385521163470638840", "3673953176946138552116347063884076", "367395317694613855211634706388475", "3673953176946138552116347063884" ], "answer": "36739531769461385521163470638840", "correct_value": 36739531769461385521163470638840, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5796549157068825 × 4495584738272522", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 5796549157068825 × 4495584738272522?", "options": [ "26058877925165061654975560326650", "26058877925165061654975560326651", "26058877925165061654975560326649", "26058877925165061654975560326640" ], "answer": "26058877925165061654975560326650", "correct_value": 26058877925165061654975560326650, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "1548919619890571 × 9564396928689351", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 1548919619890571 × 9564396928689351?", "options": [ "17142751318182894257457073009421", "14814482055268054257457073009421", "15556831448477464257457073009421", "15593714292370214257457073009421" ], "answer": "14814482055268054257457073009421", "correct_value": 14814482055268054257457073009421, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001553", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9701770548774270 × 3786889370837892", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 9701770548774270 × 3786889370837892?", "options": [ "36739531769461385521163470638840", "3673953176946138552116347063884076", "367395317694613855211634706388475", "3673953176946138552116347063884" ], "answer": "36739531769461385521163470638840", "correct_value": 36739531769461385521163470638840, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5796549157068825 × 4495584738272522", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 5796549157068825 × 4495584738272522?", "options": [ "26058877925165061654975560326650", "26058877925165061654975560326651", "26058877925165061654975560326649", "26058877925165061654975560326640" ], "answer": "26058877925165061654975560326650", "correct_value": 26058877925165061654975560326650, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "1548919619890571 × 9564396928689351", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 1548919619890571 × 9564396928689351?", "options": [ "17142751318182894257457073009421", "14814482055268054257457073009421", "15556831448477464257457073009421", "15593714292370214257457073009421" ], "answer": "14814482055268054257457073009421", "correct_value": 14814482055268054257457073009421, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001554", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9894722443785397 × 1999149853124014", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 9894722443785397 × 1999149853124014?", "options": [ "19781032920196461285240243223558", "1978103292019646128524024322355814", "1978103292019646128524024322354", "3956206584039292257048048644710" ], "answer": "19781032920196461285240243223558", "correct_value": 19781032920196461285240243223558, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2305857800177520 × 8222437279440358", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 2305857800177520 × 8222437279440358?", "options": [ "18959771137267976194938652352160", "18959771137267976194938652352159", "18959771137267976194938652352161", "18959771137267976194938652352140" ], "answer": "18959771137267976194938652352160", "correct_value": 18959771137267976194938652352160, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "2634734013801253 × 2714068522933927", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 2634734013801253 × 2714068522933927?", "options": [ "7150848653161343564765428810531", "8459075884082677564765428810531", "6258097218066191564765428810531", "5979526457831248564765428810531" ], "answer": "7150848653161343564765428810531", "correct_value": 7150848653161343564765428810531, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001555", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "6678355599032622 × 6328880242283253", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 6678355599032622 × 6328880242283253?", "options": [ "422665128016592999477473112793701", "422665128016592999477473112793724", "42266512801659299947747311279366", "4226651280165929994774731127933" ], "answer": "42266512801659299947747311279366", "correct_value": 42266512801659299947747311279366, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "8879760874513620 × 1460092885435774", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 8879760874513620 × 1460092885435774?", "options": [ "12965275677248283312923998241881", "12965275677248283312923998241860", "12965275677248283312923998241880", "12965275677248283312923998241879" ], "answer": "12965275677248283312923998241880", "correct_value": 12965275677248283312923998241880, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "control": { "task_type": "control", "math_expression": "9731543961643649 × 8367365962109257", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 9731543961643649 × 8367365962109257?", "options": [ "67551866688432111514692638158793", "81427389703426941514692638158793", "85635906098207471514692638158793", "87127750795425281514692638158793" ], "answer": "81427389703426941514692638158793", "correct_value": 81427389703426941514692638158793, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001556", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5332517376285870 × 3819109221392685", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 5332517376285870 × 3819109221392685?", "options": [ "20365466285010092434914086860951", "20365466285010092434914086860952", "20365466285010092434914086860949", "20365466285010092434914086860950" ], "answer": "20365466285010092434914086860950", "correct_value": 20365466285010092434914086860950, "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5281644763579870 × 5699237064456672", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 5281644763579870 × 5699237064456672?", "options": [ "30101345597887891705775226392640", "30101345597887891705775226392641", "30101345597887891705775226392639", "30101345597887891705775226392660" ], "answer": "30101345597887891705775226392640", "correct_value": 30101345597887891705775226392640, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "8889817708746851 × 4861702720971151", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 8889817708746851 × 4861702720971151?", "options": [ "43219650943552088655801933095501", "34913488002996478655801933095501", "42213904505954208655801933095501", "35198387935107068655801933095501" ], "answer": "43219650943552088655801933095501", "correct_value": 43219650943552088655801933095501, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001557", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9894722443785397 × 1999149853124014", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 9894722443785397 × 1999149853124014?", "options": [ "19781032920196461285240243223558", "1978103292019646128524024322355814", "1978103292019646128524024322354", "3956206584039292257048048644710" ], "answer": "19781032920196461285240243223558", "correct_value": 19781032920196461285240243223558, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2305857800177520 × 8222437279440358", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 2305857800177520 × 8222437279440358?", "options": [ "18959771137267976194938652352160", "18959771137267976194938652352159", "18959771137267976194938652352161", "18959771137267976194938652352140" ], "answer": "18959771137267976194938652352160", "correct_value": 18959771137267976194938652352160, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "2634734013801253 × 2714068522933927", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 2634734013801253 × 2714068522933927?", "options": [ "7150848653161343564765428810531", "8459075884082677564765428810531", "6258097218066191564765428810531", "5979526457831248564765428810531" ], "answer": "7150848653161343564765428810531", "correct_value": 7150848653161343564765428810531, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001558", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5591935194458535 × 7063918300464727", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 5591935194458535 × 7063918300464727?", "options": [ "39500973355148427244805631594944", "39500973355148427244805631594948", "39500973355148427244805631594945", "39500973355148427244805631594946" ], "answer": "39500973355148427244805631594945", "correct_value": 39500973355148427244805631594945, "explanation": "Product odd (odd × odd); only one option matches." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2171165790781490 × 2005770742435570", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 2171165790781490 × 2005770742435570?", "options": [ "4354860820126500640656273599320", "4354860820126500640656273599300", "4354860820126500640656273599299", "4354860820126500640656273599301" ], "answer": "4354860820126500640656273599300", "correct_value": 4354860820126500640656273599300, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "4904278846389561 × 5225199829061953", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4904278846389561 × 5225199829061953?", "options": [ "25200284730975606191955441472633", "26648977144190646191955441472633", "25625836989826886191955441472633", "25178230381341876191955441472633" ], "answer": "25625836989826886191955441472633", "correct_value": 25625836989826886191955441472633, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] } }, { "family_id": "option_elimination_001559", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "6678355599032622 × 6328880242283253", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 6678355599032622 × 6328880242283253?", "options": [ "422665128016592999477473112793701", "422665128016592999477473112793724", "42266512801659299947747311279366", "4226651280165929994774731127933" ], "answer": "42266512801659299947747311279366", "correct_value": 42266512801659299947747311279366, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "8879760874513620 × 1460092885435774", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 8879760874513620 × 1460092885435774?", "options": [ "12965275677248283312923998241881", "12965275677248283312923998241860", "12965275677248283312923998241880", "12965275677248283312923998241879" ], "answer": "12965275677248283312923998241880", "correct_value": 12965275677248283312923998241880, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "control": { "task_type": "control", "math_expression": "9731543961643649 × 8367365962109257", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 9731543961643649 × 8367365962109257?", "options": [ "67551866688432111514692638158793", "81427389703426941514692638158793", "85635906098207471514692638158793", "87127750795425281514692638158793" ], "answer": "81427389703426941514692638158793", "correct_value": 81427389703426941514692638158793, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001560", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5332517376285870 × 3819109221392685", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 5332517376285870 × 3819109221392685?", "options": [ "20365466285010092434914086860951", "20365466285010092434914086860952", "20365466285010092434914086860949", "20365466285010092434914086860950" ], "answer": "20365466285010092434914086860950", "correct_value": 20365466285010092434914086860950, "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5281644763579870 × 5699237064456672", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 5281644763579870 × 5699237064456672?", "options": [ "30101345597887891705775226392640", "30101345597887891705775226392641", "30101345597887891705775226392639", "30101345597887891705775226392660" ], "answer": "30101345597887891705775226392640", "correct_value": 30101345597887891705775226392640, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "8889817708746851 × 4861702720971151", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 8889817708746851 × 4861702720971151?", "options": [ "43219650943552088655801933095501", "34913488002996478655801933095501", "42213904505954208655801933095501", "35198387935107068655801933095501" ], "answer": "43219650943552088655801933095501", "correct_value": 43219650943552088655801933095501, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001561", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5963386418730146 × 1941811390038431", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 5963386418730146 × 1941811390038431?", "options": [ "1157977167109068574262115824089", "11579771671090685742621158240926", "2315954334218137148524231648182", "1157977167109068574262115824092671" ], "answer": "11579771671090685742621158240926", "correct_value": 11579771671090685742621158240926, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7104144620250465 × 1801487161267680", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 7104144620250465 × 1801487161267680?", "options": [ "12798025325170070733728509471199", "12798025325170070733728509471190", "12798025325170070733728509471201", "12798025325170070733728509471200" ], "answer": "12798025325170070733728509471200", "correct_value": 12798025325170070733728509471200, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "control": { "task_type": "control", "math_expression": "2438730543144625 × 5046821684534037", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 2438730543144625 × 5046821684534037?", "options": [ "13387671963083243341117826101125", "14523299660449533341117826101125", "12307838187877763341117826101125", "14373529610625103341117826101125" ], "answer": "12307838187877763341117826101125", "correct_value": 12307838187877763341117826101125, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] } }, { "family_id": "option_elimination_001562", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7213731379752666 × 3087612137773096", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 7213731379752666 × 3087612137773096?", "options": [ "4454640913351798894893701814784", "2227320456675899447446850907390", "22273204566758994474468509073936", "222732045667589944744685090739402" ], "answer": "22273204566758994474468509073936", "correct_value": 22273204566758994474468509073936, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "8239672695843940 × 5633478364354926", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 8239672695843940 × 5633478364354926?", "options": [ "46418017861402862781759266248439", "46418017861402862781759266248441", "46418017861402862781759266248420", "46418017861402862781759266248440" ], "answer": "46418017861402862781759266248440", "correct_value": 46418017861402862781759266248440, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "control": { "task_type": "control", "math_expression": "9153190170892871 × 8012294981653827", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 9153190170892871 × 8012294981653827?", "options": [ "82214400917405455471844824167317", "72759159757192865471844824167317", "73338059672368085471844824167317", "81720260669730595471844824167317" ], "answer": "73338059672368085471844824167317", "correct_value": 73338059672368085471844824167317, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] } }, { "family_id": "option_elimination_001563", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "6014195717486709 × 7956601730014575", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 6014195717486709 × 7956601730014575?", "options": [ "47852560050400996983988938783674", "47852560050400996983988938783675", "47852560050400996983988938783676", "47852560050400996983988938783677" ], "answer": "47852560050400996983988938783675", "correct_value": 47852560050400996983988938783675, "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "3050834015351900 × 2874530445678702", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 3050834015351900 × 2874530445678702?", "options": [ "8769715261841241086482865233800", "8769715261841241086482865233799", "8769715261841241086482865233810", "8769715261841241086482865233801" ], "answer": "8769715261841241086482865233800", "correct_value": 8769715261841241086482865233800, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "5314545849152773 × 9870476216872057", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5314545849152773 × 9870476216872057?", "options": [ "43767791207246466556521587764061", "62644484343349236556521587764061", "46356535312419606556521587764061", "52457098407538556556521587764061" ], "answer": "52457098407538556556521587764061", "correct_value": 52457098407538556556521587764061, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] } }, { "family_id": "option_elimination_001564", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5591935194458535 × 7063918300464727", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 5591935194458535 × 7063918300464727?", "options": [ "39500973355148427244805631594944", "39500973355148427244805631594948", "39500973355148427244805631594945", "39500973355148427244805631594946" ], "answer": "39500973355148427244805631594945", "correct_value": 39500973355148427244805631594945, "explanation": "Product odd (odd × odd); only one option matches." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2171165790781490 × 2005770742435570", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 2171165790781490 × 2005770742435570?", "options": [ "4354860820126500640656273599320", "4354860820126500640656273599300", "4354860820126500640656273599299", "4354860820126500640656273599301" ], "answer": "4354860820126500640656273599300", "correct_value": 4354860820126500640656273599300, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "4904278846389561 × 5225199829061953", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4904278846389561 × 5225199829061953?", "options": [ "25200284730975606191955441472633", "26648977144190646191955441472633", "25625836989826886191955441472633", "25178230381341876191955441472633" ], "answer": "25625836989826886191955441472633", "correct_value": 25625836989826886191955441472633, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] } }, { "family_id": "option_elimination_001565", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2409644947927275 × 9114446616307930", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 2409644947927275 × 9114446616307930?", "options": [ "21962580242139249806666645790750", "21962580242139249806666645790753", "21962580242139249806666645790751", "21962580242139249806666645790749" ], "answer": "21962580242139249806666645790750", "correct_value": 21962580242139249806666645790750, "explanation": "Product ends in 0 (5 × even); only one option matches." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "8136276330904985 × 8324273751164176", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 8136276330904985 × 8324273751164176?", "options": [ "67728591493570738013451391817361", "67728591493570738013451391817350", "67728591493570738013451391817359", "67728591493570738013451391817360" ], "answer": "67728591493570738013451391817360", "correct_value": 67728591493570738013451391817360, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "control": { "task_type": "control", "math_expression": "6621053849287665 × 3193913882299449", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6621053849287665 × 3193913882299449?", "options": [ "20960412925696817008980871996585", "21147075804692077008980871996585", "17281837588610317008980871996585", "19847006240986717008980871996585" ], "answer": "21147075804692077008980871996585", "correct_value": 21147075804692077008980871996585, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001566", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5963386418730146 × 1941811390038431", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 5963386418730146 × 1941811390038431?", "options": [ "1157977167109068574262115824089", "11579771671090685742621158240926", "2315954334218137148524231648182", "1157977167109068574262115824092671" ], "answer": "11579771671090685742621158240926", "correct_value": 11579771671090685742621158240926, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7104144620250465 × 1801487161267680", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 7104144620250465 × 1801487161267680?", "options": [ "12798025325170070733728509471199", "12798025325170070733728509471190", "12798025325170070733728509471201", "12798025325170070733728509471200" ], "answer": "12798025325170070733728509471200", "correct_value": 12798025325170070733728509471200, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "control": { "task_type": "control", "math_expression": "2438730543144625 × 5046821684534037", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 2438730543144625 × 5046821684534037?", "options": [ "13387671963083243341117826101125", "14523299660449533341117826101125", "12307838187877763341117826101125", "14373529610625103341117826101125" ], "answer": "12307838187877763341117826101125", "correct_value": 12307838187877763341117826101125, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] } }, { "family_id": "option_elimination_001567", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9008249277766075 × 4719231914903010", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 9008249277766075 × 4719231914903010?", "options": [ "42512017489035650946833093385753", "42512017489035650946833093385751", "42512017489035650946833093385750", "42512017489035650946833093385749" ], "answer": "42512017489035650946833093385750", "correct_value": 42512017489035650946833093385750, "explanation": "Product ends in 0 (5 × even); only one option matches." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "3961231072948790 × 6920300038169768", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 3961231072948790 × 6920300038169768?", "options": [ "27412907545326782485846390180720", "27412907545326782485846390180740", "27412907545326782485846390180721", "27412907545326782485846390180719" ], "answer": "27412907545326782485846390180720", "correct_value": 27412907545326782485846390180720, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "7594940461319255 × 3569544830163475", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 7594940461319255 × 3569544830163475?", "options": [ "24189195360583884556615815211125", "27110480459101544556615815211125", "22829190544194544556615815211125", "23865571289585134556615815211125" ], "answer": "27110480459101544556615815211125", "correct_value": 27110480459101544556615815211125, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001568", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2947716861205530 × 4360790708480534", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 2947716861205530 × 4360790708480534?", "options": [ "12854376299576479076397778153021", "12854376299576479076397778153023", "12854376299576479076397778153019", "12854376299576479076397778153020" ], "answer": "12854376299576479076397778153020", "correct_value": 12854376299576479076397778153020, "explanation": "Product ends in 0 (5 × even); only one option matches." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9710334381904345 × 1448691498429008", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 9710334381904345 × 1448691498429008?", "options": [ "14067278865967720783270829239760", "14067278865967720783270829239761", "14067278865967720783270829239770", "14067278865967720783270829239759" ], "answer": "14067278865967720783270829239760", "correct_value": 14067278865967720783270829239760, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "7589618085519567 × 2413382906301361", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 7589618085519567 × 2413382906301361?", "options": [ "18069646613584524022191964230687", "18316654552948584022191964230687", "21900940213488564022191964230687", "17813527205003504022191964230687" ], "answer": "18316654552948584022191964230687", "correct_value": 18316654552948584022191964230687, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001569", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7257074709127995 × 4309839363906321", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 7257074709127995 × 4309839363906321?", "options": [ "31276826248208847463718778556394", "31276826248208847463718778556395", "31276826248208847463718778556396", "31276826248208847463718778556398" ], "answer": "31276826248208847463718778556395", "correct_value": 31276826248208847463718778556395, "explanation": "Product odd (odd × odd); only one option matches." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "8392765216632970 × 4481587829716726", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 8392765216632970 × 4481587829716726?", "options": [ "37612914452532179754778612056219", "37612914452532179754778612056230", "37612914452532179754778612056220", "37612914452532179754778612056221" ], "answer": "37612914452532179754778612056220", "correct_value": 37612914452532179754778612056220, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "control": { "task_type": "control", "math_expression": "4752218979054061 × 7428595475812331", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4752218979054061 × 7428595475812331?", "options": [ "39404170481385712120447939426191", "35302312407870492120447939426191", "41161341525070742120447939426191", "36253695873953652120447939426191" ], "answer": "35302312407870492120447939426191", "correct_value": 35302312407870492120447939426191, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001570", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1186576085242495 × 1380148338252802", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 1186576085242495 × 1380148338252802?", "options": [ "1637651012257944608724783220990", "1637651012257944608724783220991", "1637651012257944608724783220993", "1637651012257944608724783220989" ], "answer": "1637651012257944608724783220990", "correct_value": 1637651012257944608724783220990, "explanation": "Product ends in 0 (5 × even); only one option matches." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9125956543514820 × 3923959478246032", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 9125956543514820 × 3923959478246032?", "options": [ "35809883676986374712777998194239", "35809883676986374712777998194241", "35809883676986374712777998194240", "35809883676986374712777998194230" ], "answer": "35809883676986374712777998194240", "correct_value": 35809883676986374712777998194240, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "control": { "task_type": "control", "math_expression": "5840568221869651 × 9272421890100167", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5840568221869651 × 9272421890100167?", "options": [ "54156212631087560856140407331717", "48081020052301500856140407331717", "58389275508354000856140407331717", "49054825372375680856140407331717" ], "answer": "54156212631087560856140407331717", "correct_value": 54156212631087560856140407331717, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001571", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "1550089299659688 × 2382079852886920", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 1550089299659688 × 2382079852886920?", "options": [ "3692436490894938443060346480960", "36924364908949384430603464809664", "369243649089493844306034648096", "36924364908949384430603464809648" ], "answer": "3692436490894938443060346480960", "correct_value": 3692436490894938443060346480960, "explanation": "Product has 31 digits; only one option matches." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "7217036772319915 × 6729809887789582", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 7217036772319915 × 6729809887789582?", "options": [ "48569285430899574222761508125510", "48569285430899574222761508125529", "48569285430899574222761508125530", "48569285430899574222761508125531" ], "answer": "48569285430899574222761508125530", "correct_value": 48569285430899574222761508125530, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "control": { "task_type": "control", "math_expression": "5108123157119671 × 7965649255735447", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5108123157119671 × 7965649255735447?", "options": [ "40689517424715309098529295677937", "43564598793290839098529295677937", "37020156783056139098529295677937", "43709917511051019098529295677937" ], "answer": "40689517424715309098529295677937", "correct_value": 40689517424715309098529295677937, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001572", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7598032523781675 × 6423468101874976", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 7598032523781675 × 6423468101874976?", "options": [ "48805719553520209356377569864800", "48805719553520209356377569864799", "48805719553520209356377569864803", "48805719553520209356377569864801" ], "answer": "48805719553520209356377569864800", "correct_value": 48805719553520209356377569864800, "explanation": "Product ends in 0 (5 × even); only one option matches." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2093862161165225 × 8790662106873714", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 2093862161165225 × 8790662106873714?", "options": [ "18406434757171844896744163395650", "18406434757171844896744163395649", "18406434757171844896744163395640", "18406434757171844896744163395651" ], "answer": "18406434757171844896744163395650", "correct_value": 18406434757171844896744163395650, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "2016239973104857 × 4904643861346173", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 2016239973104857 × 4904643861346173?", "options": [ "8383117178187791834542504662261", "9445789168474064834542504662261", "9888939007089509834542504662261", "9133602747545082834542504662261" ], "answer": "9888939007089509834542504662261", "correct_value": 9888939007089509834542504662261, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] } }, { "family_id": "option_elimination_001573", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "6734630132140245 × 9161805020842608", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 6734630132140245 × 9161805020842608?", "options": [ "61701368158160413211412327558960", "61701368158160413211412327558959", "61701368158160413211412327558963", "61701368158160413211412327558961" ], "answer": "61701368158160413211412327558960", "correct_value": 61701368158160413211412327558960, "explanation": "Product ends in 0 (5 × even); only one option matches." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6276742937807470 × 2637771402884234", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 6276742937807470 × 2637771402884234?", "options": [ "16556613024604118462842190427980", "16556613024604118462842190427981", "16556613024604118462842190427960", "16556613024604118462842190427979" ], "answer": "16556613024604118462842190427980", "correct_value": 16556613024604118462842190427980, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "5018426210631055 × 1371214757075541", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5018426210631055 × 1371214757075541?", "options": [ "7911612293585113832855915525755", "6881340077311989832855915525755", "6115207432099197832855915525755", "6051905532906997832855915525755" ], "answer": "6881340077311989832855915525755", "correct_value": 6881340077311989832855915525755, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001574", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "6536595329094101 × 2212744390773225", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 6536595329094101 × 2212744390773225?", "options": [ "14463814649207434693182176245724", "14463814649207434693182176245726", "14463814649207434693182176245725", "14463814649207434693182176245728" ], "answer": "14463814649207434693182176245725", "correct_value": 14463814649207434693182176245725, "explanation": "Product odd (odd × odd); only one option matches." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1916934649007425 × 3573014363382228", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 1916934649007425 × 3573014363382228?", "options": [ "6849235034568599315666085042901", "6849235034568599315666085042910", "6849235034568599315666085042899", "6849235034568599315666085042900" ], "answer": "6849235034568599315666085042900", "correct_value": 6849235034568599315666085042900, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "control": { "task_type": "control", "math_expression": "6602980250671767 × 6957653357974253", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6602980250671767 × 6957653357974253?", "options": [ "50550949703013174490829540015051", "54663245437716554490829540015051", "43143119127407944490829540015051", "45941247713724094490829540015051" ], "answer": "45941247713724094490829540015051", "correct_value": 45941247713724094490829540015051, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] } }, { "family_id": "option_elimination_001575", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "4210958519718385 × 9675937841090658", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 4210958519718385 × 9675937841090658?", "options": [ "40744972888206223162387414347333", "40744972888206223162387414347329", "40744972888206223162387414347331", "40744972888206223162387414347330" ], "answer": "40744972888206223162387414347330", "correct_value": 40744972888206223162387414347330, "explanation": "Product ends in 0 (5 × even); only one option matches." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6862483568417665 × 8565433400238616", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 6862483568417665 × 8565433400238616?", "options": [ "58780145965513351320172549551639", "58780145965513351320172549551640", "58780145965513351320172549551650", "58780145965513351320172549551641" ], "answer": "58780145965513351320172549551640", "correct_value": 58780145965513351320172549551640, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "4940459504155873 × 9166926327531731", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4940459504155873 × 9166926327531731?", "options": [ "40398223522224633587972777506163", "45288828298750833587972777506163", "53887091955096243587972777506163", "53877558375831943587972777506163" ], "answer": "45288828298750833587972777506163", "correct_value": 45288828298750833587972777506163, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001576", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7213731379752666 × 3087612137773096", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 7213731379752666 × 3087612137773096?", "options": [ "4454640913351798894893701814784", "2227320456675899447446850907390", "22273204566758994474468509073936", "222732045667589944744685090739402" ], "answer": "22273204566758994474468509073936", "correct_value": 22273204566758994474468509073936, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "8239672695843940 × 5633478364354926", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 8239672695843940 × 5633478364354926?", "options": [ "46418017861402862781759266248439", "46418017861402862781759266248441", "46418017861402862781759266248420", "46418017861402862781759266248440" ], "answer": "46418017861402862781759266248440", "correct_value": 46418017861402862781759266248440, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "control": { "task_type": "control", "math_expression": "9153190170892871 × 8012294981653827", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 9153190170892871 × 8012294981653827?", "options": [ "82214400917405455471844824167317", "72759159757192865471844824167317", "73338059672368085471844824167317", "81720260669730595471844824167317" ], "answer": "73338059672368085471844824167317", "correct_value": 73338059672368085471844824167317, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] } }, { "family_id": "option_elimination_001577", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7542925189441836 × 6889952272681644", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 7542925189441836 × 6889952272681644?", "options": [ "51970394551662398057883282858385", "51970394551662398057883282858386", "51970394551662398057883282858384", "51970394551662398057883282858383" ], "answer": "51970394551662398057883282858384", "correct_value": 51970394551662398057883282858384, "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "9210232229508215 × 5365136469613626", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 9210232229508215 × 5365136469613626?", "options": [ "49414152828145340193657042937590", "49414152828145340193657042937589", "49414152828145340193657042937591", "49414152828145340193657042937600" ], "answer": "49414152828145340193657042937590", "correct_value": 49414152828145340193657042937590, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "3172711683938625 × 3534762873314635", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 3172711683938625 × 3534762873314635?", "options": [ "10077434871648968201345654276875", "13440725944866368201345654276875", "11539368232928098201345654276875", "11214783468117808201345654276875" ], "answer": "11214783468117808201345654276875", "correct_value": 11214783468117808201345654276875, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] } }, { "family_id": "option_elimination_001578", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "6014195717486709 × 7956601730014575", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 6014195717486709 × 7956601730014575?", "options": [ "47852560050400996983988938783674", "47852560050400996983988938783675", "47852560050400996983988938783676", "47852560050400996983988938783677" ], "answer": "47852560050400996983988938783675", "correct_value": 47852560050400996983988938783675, "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "3050834015351900 × 2874530445678702", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 3050834015351900 × 2874530445678702?", "options": [ "8769715261841241086482865233800", "8769715261841241086482865233799", "8769715261841241086482865233810", "8769715261841241086482865233801" ], "answer": "8769715261841241086482865233800", "correct_value": 8769715261841241086482865233800, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "5314545849152773 × 9870476216872057", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5314545849152773 × 9870476216872057?", "options": [ "43767791207246466556521587764061", "62644484343349236556521587764061", "46356535312419606556521587764061", "52457098407538556556521587764061" ], "answer": "52457098407538556556521587764061", "correct_value": 52457098407538556556521587764061, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] } }, { "family_id": "option_elimination_001579", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3818572167003824 × 9061807225483345", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 3818572167003824 × 9061807225483345?", "options": [ "3460316485398484668988886331128", "346031648539848466898888633112850", "346031648539848466898888633112819", "34603164853984846689888863311280" ], "answer": "34603164853984846689888863311280", "correct_value": 34603164853984846689888863311280, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2673641747012725 × 4152329224454090", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 2673641747012725 × 4152329224454090?", "options": [ "11101840761841426698276408295250", "11101840761841426698276408295249", "11101840761841426698276408295270", "11101840761841426698276408295251" ], "answer": "11101840761841426698276408295250", "correct_value": 11101840761841426698276408295250, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "4706119946682963 × 8530440955304553", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4706119946682963 × 8530440955304553?", "options": [ "40145278333760026924071801430539", "37541484802808856924071801430539", "33487148554433976924071801430539", "47938211279771816924071801430539" ], "answer": "40145278333760026924071801430539", "correct_value": 40145278333760026924071801430539, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001580", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "4046203707448456 × 3085079634328149", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 4046203707448456 × 3085079634328149?", "options": [ "2496572130838456682148121477584", "1248286065419228341074060738794491", "12482860654192283410740607387944", "1248286065419228341074060738794406" ], "answer": "12482860654192283410740607387944", "correct_value": 12482860654192283410740607387944, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4332547192037445 × 8680380667498968", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 4332547192037445 × 8680380667498968?", "options": [ "37608158886788776325392354856761", "37608158886788776325392354856740", "37608158886788776325392354856759", "37608158886788776325392354856760" ], "answer": "37608158886788776325392354856760", "correct_value": 37608158886788776325392354856760, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "control": { "task_type": "control", "math_expression": "6370520203591549 × 4500299432176029", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6370520203591549 × 4500299432176029?", "options": [ "28495948632628028625618184778921", "28669248454888968625618184778921", "31215493599097218625618184778921", "32552224609684938625618184778921" ], "answer": "28669248454888968625618184778921", "correct_value": 28669248454888968625618184778921, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001581", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3513412892349729 × 6833798894328363", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 3513412892349729 × 6833798894328363?", "options": [ "24009957139058593898970360063626", "24009957139058593898970360063628", "24009957139058593898970360063627", "24009957139058593898970360063629" ], "answer": "24009957139058593898970360063627", "correct_value": 24009957139058593898970360063627, "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4427988671788410 × 1107538359129044", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 4427988671788410 × 1107538359129044?", "options": [ "4904167307794530576781453580039", "4904167307794530576781453580030", "4904167307794530576781453580041", "4904167307794530576781453580040" ], "answer": "4904167307794530576781453580040", "correct_value": 4904167307794530576781453580040, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "control": { "task_type": "control", "math_expression": "5647493434129129 × 9322004850093027", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5647493434129129 × 9322004850093027?", "options": [ "56440578684648225435972380483483", "52645961183820265435972380483483", "52656314680372005435972380483483", "45308846106272905435972380483483" ], "answer": "52645961183820265435972380483483", "correct_value": 52645961183820265435972380483483, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001582", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9592419845424676 × 8870522514249578", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 9592419845424676 × 8870522514249578?", "options": [ "170179552409948090618673727573448", "178688530030445495149607413952120", "153161597168953281556806354816104", "85089776204974045309336863786728" ], "answer": "85089776204974045309336863786728", "correct_value": 85089776204974045309336863786728, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4486065545000635 × 4422630625996590", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 4486065545000635 × 4422630625996590?", "options": [ "19840210869547892056939057834650", "19840210869547892056939057834660", "19840210869547892056939057834649", "19840210869547892056939057834651" ], "answer": "19840210869547892056939057834650", "correct_value": 19840210869547892056939057834650, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "1424388053159637 × 4010367837730863", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 1424388053159637 × 4010367837730863?", "options": [ "5712320036839486977091580776731", "6180263020519376977091580776731", "6176532293478200977091580776731", "5480936340458304977091580776731" ], "answer": "5712320036839486977091580776731", "correct_value": 5712320036839486977091580776731, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001583", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3158730931756257 × 7196143162153134", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 3158730931756257 × 7196143162153134?", "options": [ "22730679995639386563768196659438", "22730679995639386563768196659440", "22730679995639386563768196659437", "22730679995639386563768196659439" ], "answer": "22730679995639386563768196659438", "correct_value": 22730679995639386563768196659438, "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "3208924632123675 × 5539427187449428", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 3208924632123675 × 5539427187449428?", "options": [ "17775604349662039420918304007910", "17775604349662039420918304007900", "17775604349662039420918304007901", "17775604349662039420918304007899" ], "answer": "17775604349662039420918304007900", "correct_value": 17775604349662039420918304007900, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "8063666274288853 × 7268169544787743", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 8063666274288853 × 7268169544787743?", "options": [ "60665662560300588295200155928779", "65851364106591698295200155928779", "69350865676344998295200155928779", "58608093634118288295200155928779" ], "answer": "58608093634118288295200155928779", "correct_value": 58608093634118288295200155928779, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] } }, { "family_id": "option_elimination_001584", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5654104903419940 × 8700877718348512", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 5654104903419940 × 8700877718348512?", "options": [ "49195675371371621350997610129283", "49195675371371621350997610129281", "49195675371371621350997610129280", "49195675371371621350997610129279" ], "answer": "49195675371371621350997610129280", "correct_value": 49195675371371621350997610129280, "explanation": "Product ends in 0 (5 × even); only one option matches." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "4905119398787375 × 5265820313663736", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 4905119398787375 × 5265820313663736?", "options": [ "25829477371080611172221912133000", "25829477371080611172221912132980", "25829477371080611172221912132999", "25829477371080611172221912133001" ], "answer": "25829477371080611172221912133000", "correct_value": 25829477371080611172221912133000, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "2692400653145773 × 2472176115730175", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 2692400653145773 × 2472176115730175?", "options": [ "6494516266841127270722609800275", "5504667816994923270722609800275", "7895237308481934270722609800275", "6656088588683303270722609800275" ], "answer": "6656088588683303270722609800275", "correct_value": 6656088588683303270722609800275, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] } }, { "family_id": "option_elimination_001585", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3081219179671170 × 6765911169612164", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 3081219179671170 × 6765911169612164?", "options": [ "20847255263760398308201952111880", "20847255263760398308201952111883", "20847255263760398308201952111879", "20847255263760398308201952111881" ], "answer": "20847255263760398308201952111880", "correct_value": 20847255263760398308201952111880, "explanation": "Product ends in 0 (5 × even); only one option matches." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6599370524333905 × 6041163997091612", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 6599370524333905 × 6041163997091612?", "options": [ "39867879615073580824893562704861", "39867879615073580824893562704860", "39867879615073580824893562704840", "39867879615073580824893562704859" ], "answer": "39867879615073580824893562704860", "correct_value": 39867879615073580824893562704860, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "6857477010799669 × 8809265397860525", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6857477010799669 × 8809265397860525?", "options": [ "60409334947861549825471978166225", "51112069759274839825471978166225", "69054792109891869825471978166225", "49460947518466639825471978166225" ], "answer": "60409334947861549825471978166225", "correct_value": 60409334947861549825471978166225, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001586", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2093060111840670 × 8462091475368140", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 2093060111840670 × 8462091475368140?", "options": [ "17711666129840019314861274253800", "17711666129840019314861274253803", "17711666129840019314861274253801", "17711666129840019314861274253799" ], "answer": "17711666129840019314861274253800", "correct_value": 17711666129840019314861274253800, "explanation": "Product ends in 0 (5 × even); only one option matches." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5950628423508205 × 5575117212768094", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 5950628423508205 × 5575117212768094?", "options": [ "33175450950687661107050571211269", "33175450950687661107050571211260", "33175450950687661107050571211271", "33175450950687661107050571211270" ], "answer": "33175450950687661107050571211270", "correct_value": 33175450950687661107050571211270, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "control": { "task_type": "control", "math_expression": "5351806069188143 × 2214994378330827", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5351806069188143 × 2214994378330827?", "options": [ "11854220357168537715711359784261", "12965801481961057715711359784261", "13758693394640697715711359784261", "10919298566312827715711359784261" ], "answer": "11854220357168537715711359784261", "correct_value": 11854220357168537715711359784261, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001587", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3472488829255530 × 1565961625968832", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 3472488829255530 × 1565961625968832?", "options": [ "5437784253219595596461543640960", "5437784253219595596461543640963", "5437784253219595596461543640959", "5437784253219595596461543640961" ], "answer": "5437784253219595596461543640960", "correct_value": 5437784253219595596461543640960, "explanation": "Product ends in 0 (5 × even); only one option matches." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "8038867654527440 × 7368580173693902", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 8038867654527440 × 7368580173693902?", "options": [ "59235040818100094411659019670860", "59235040818100094411659019670881", "59235040818100094411659019670879", "59235040818100094411659019670880" ], "answer": "59235040818100094411659019670880", "correct_value": 59235040818100094411659019670880, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "control": { "task_type": "control", "math_expression": "4491516580243833 × 9541993687427473", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4491516580243833 × 9541993687427473?", "options": [ "42858022855662485473788843024009", "45965550121476975473788843024009", "46259919548416575473788843024009", "44729011486175795473788843024009" ], "answer": "42858022855662485473788843024009", "correct_value": 42858022855662485473788843024009, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001588", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "4940674088413356 × 2487942053436903", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 4940674088413356 × 2487942053436903?", "options": [ "12292110836889623770509928476467", "12292110836889623770509928476468", "12292110836889623770509928476469", "12292110836889623770509928476470" ], "answer": "12292110836889623770509928476468", "correct_value": 12292110836889623770509928476468, "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2522750164650235 × 2987719761061568", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 2522750164650235 × 2987719761061568?", "options": [ "7537270519146831444931020668480", "7537270519146831444931020668481", "7537270519146831444931020668479", "7537270519146831444931020668460" ], "answer": "7537270519146831444931020668480", "correct_value": 7537270519146831444931020668480, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "1010350976013247 × 9780388433150063", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 1010350976013247 × 9780388433150063?", "options": [ "9881624999221837712085426884561", "8319912307394945712085426884561", "9453748496225351712085426884561", "8306060953259991712085426884561" ], "answer": "9881624999221837712085426884561", "correct_value": 9881624999221837712085426884561, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001589", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "6318317909183010 × 6478399989537806", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 6318317909183010 × 6478399989537806?", "options": [ "8186518135349548852890593575212", "409325906767477442644529678760643", "4093259067674774426445296787606", "40932590676747744264452967876060" ], "answer": "40932590676747744264452967876060", "correct_value": 40932590676747744264452967876060, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6050979803273290 × 3260366717753688", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 6050979803273290 × 3260366717753688?", "options": [ "19728413160391993237058369393540", "19728413160391993237058369393521", "19728413160391993237058369393520", "19728413160391993237058369393519" ], "answer": "19728413160391993237058369393520", "correct_value": 19728413160391993237058369393520, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "control": { "task_type": "control", "math_expression": "1577805754726375 × 1952636988418475", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 1577805754726375 × 1952636988418475?", "options": [ "3541789761293581907367619778125", "3501883157567717907367619778125", "3201716167643728907367619778125", "3080881877218247907367619778125" ], "answer": "3080881877218247907367619778125", "correct_value": 3080881877218247907367619778125, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] } }, { "family_id": "option_elimination_001590", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "4867467700520370 × 8481282108163580", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 4867467700520370 × 8481282108163580?", "options": [ "41282366720487536736991082124599", "41282366720487536736991082124601", "41282366720487536736991082124603", "41282366720487536736991082124600" ], "answer": "41282366720487536736991082124600", "correct_value": 41282366720487536736991082124600, "explanation": "Product ends in 0 (5 × even); only one option matches." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "2291961403253545 × 6027333872420382", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 2291961403253545 × 6027333872420382?", "options": [ "13814416600110242101198771754180", "13814416600110242101198771754190", "13814416600110242101198771754189", "13814416600110242101198771754191" ], "answer": "13814416600110242101198771754190", "correct_value": 13814416600110242101198771754190, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "7230482141277227 × 5832457397683871", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 7230482141277227 × 5832457397683871?", "options": [ "45071839796679328696235517505717", "44854116159560578696235517505717", "42171479053713478696235517505717", "48710698922281768696235517505717" ], "answer": "42171479053713478696235517505717", "correct_value": 42171479053713478696235517505717, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] } }, { "family_id": "option_elimination_001591", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "4163832105432513 × 5881874901286401", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 4163832105432513 × 5881874901286401?", "options": [ "24491139554114009642884190155714", "24491139554114009642884190155713", "24491139554114009642884190155712", "24491139554114009642884190155716" ], "answer": "24491139554114009642884190155713", "correct_value": 24491139554114009642884190155713, "explanation": "Product odd (odd × odd); only one option matches." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5786743253304555 × 2385886041704744", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 5786743253304555 × 2385886041704744?", "options": [ "13806509954988437483523620308940", "13806509954988437483523620308921", "13806509954988437483523620308920", "13806509954988437483523620308919" ], "answer": "13806509954988437483523620308920", "correct_value": 13806509954988437483523620308920, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "control": { "task_type": "control", "math_expression": "1752368282729443 × 2377934162326559", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 1752368282729443 × 2377934162326559?", "options": [ "4390889937035171746971610176637", "4558470843516006746971610176637", "4167016404479868746971610176637", "4434979362846124746971610176637" ], "answer": "4167016404479868746971610176637", "correct_value": 4167016404479868746971610176637, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] } }, { "family_id": "option_elimination_001592", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "7288523537004620 × 9450332716510620", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 7288523537004620 × 9450332716510620?", "options": [ "6887897243681246291761321906440", "137757944873624925835226438128800", "144645842117306172126987760035240", "68878972436812462917613219064400" ], "answer": "68878972436812462917613219064400", "correct_value": 68878972436812462917613219064400, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "3569997895960310 × 1497389614887062", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 3569997895960310 × 1497389614887062?", "options": [ "5345677774579630223806684509221", "5345677774579630223806684509220", "5345677774579630223806684509210", "5345677774579630223806684509219" ], "answer": "5345677774579630223806684509220", "correct_value": 5345677774579630223806684509220, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "8651630699585929 × 9862165857536867", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 8651630699585929 × 9862165857536867?", "options": [ "94840481676710358040220751944443", "85323816897474148040220751944443", "71171274968004378040220751944443", "77600304805015808040220751944443" ], "answer": "85323816897474148040220751944443", "correct_value": 85323816897474148040220751944443, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001593", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "9819596545092505 × 8066982446959455", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 9819596545092505 × 8066982446959455?", "options": [ "79214512965484946284338959384778", "79214512965484946284338959384774", "79214512965484946284338959384776", "79214512965484946284338959384775" ], "answer": "79214512965484946284338959384775", "correct_value": 79214512965484946284338959384775, "explanation": "Product odd (odd × odd); only one option matches." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "1693037065815600 × 3585193583416618", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 1693037065815600 × 3585193583416618?", "options": [ "6069865624848587497580763640800", "6069865624848587497580763640780", "6069865624848587497580763640801", "6069865624848587497580763640799" ], "answer": "6069865624848587497580763640800", "correct_value": 6069865624848587497580763640800, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "3828154302928653 × 2028008895077469", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 3828154302928653 × 2028008895077469?", "options": [ "7763530978068396120062014819257", "7972882821006243120062014819257", "7648408341006217120062014819257", "8320836777979039120062014819257" ], "answer": "7763530978068396120062014819257", "correct_value": 7763530978068396120062014819257, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001594", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2010477542044524 × 4542838048666181", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 2010477542044524 × 4542838048666181?", "options": [ "10137934000127485057121365697552", "18266547747977450553371830085684", "19179875135376323081040421589968", "9133273873988725276685915042844" ], "answer": "9133273873988725276685915042844", "correct_value": 9133273873988725276685915042844, "explanation": "Product has 31 digits; only one option matches." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5787539708367700 × 1635481249144116", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 5787539708367700 × 1635481249144116?", "options": [ "9465412671712378819868419453199", "9465412671712378819868419453201", "9465412671712378819868419453200", "9465412671712378819868419453210" ], "answer": "9465412671712378819868419453200", "correct_value": 9465412671712378819868419453200, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "control": { "task_type": "control", "math_expression": "3104034420574535 × 6003579712097427", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 3104034420574535 × 6003579712097427?", "options": [ "16922672258666820471327235221445", "18635318073013370471327235221445", "16358528682367520471327235221445", "15226349316690090471327235221445" ], "answer": "18635318073013370471327235221445", "correct_value": 18635318073013370471327235221445, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001595", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8786150048903330 × 7943979796163156", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 8786150048903330 × 7943979796163156?", "options": [ "69796998474545978574499551709479", "69796998474545978574499551709480", "69796998474545978574499551709481", "69796998474545978574499551709483" ], "answer": "69796998474545978574499551709480", "correct_value": 69796998474545978574499551709480, "explanation": "Product ends in 0 (5 × even); only one option matches." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "3983807888961460 × 1115986206578610", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 3983807888961460 × 1115986206578610?", "options": [ "4445874653740040108252750370610", "4445874653740040108252750370601", "4445874653740040108252750370599", "4445874653740040108252750370600" ], "answer": "4445874653740040108252750370600", "correct_value": 4445874653740040108252750370600, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "control": { "task_type": "control", "math_expression": "3283180606240433 × 3181297353322457", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 3283180606240433 × 3181297353322457?", "options": [ "11247361891508039353220320303881", "10444773773112309353220320303881", "10315012892916029353220320303881", "11651367799884419353220320303881" ], "answer": "10444773773112309353220320303881", "correct_value": 10444773773112309353220320303881, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001596", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5190766652348470 × 7440968699690902", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 5190766652348470 × 7440968699690902?", "options": [ "38624332187524291172181392619940", "38624332187524291172181392619943", "38624332187524291172181392619941", "38624332187524291172181392619939" ], "answer": "38624332187524291172181392619940", "correct_value": 38624332187524291172181392619940, "explanation": "Product ends in 0 (5 × even); only one option matches." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5062586882880255 × 1439710909431754", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 5062586882880255 × 1439710909431754?", "options": [ "7288661565228800601232876617269", "7288661565228800601232876617271", "7288661565228800601232876617280", "7288661565228800601232876617270" ], "answer": "7288661565228800601232876617270", "correct_value": 7288661565228800601232876617270, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "control": { "task_type": "control", "math_expression": "2125794470535237 × 6763442068799137", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 2125794470535237 × 6763442068799137?", "options": [ "13505177930659719417950233690469", "11910741032003219417950233690469", "14377687751638609417950233690469", "16745019515681829417950233690469" ], "answer": "14377687751638609417950233690469", "correct_value": 14377687751638609417950233690469, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] } }, { "family_id": "option_elimination_001597", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "5709251040734865 × 1567481107898032", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 5709251040734865 × 1567481107898032?", "options": [ "17898286293198156828617534571360", "894914314659907841430876728568", "8949143146599078414308767285680", "19688114922517972511479288028496" ], "answer": "8949143146599078414308767285680", "correct_value": 8949143146599078414308767285680, "explanation": "Product has 31 digits; only one option matches." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5641156343740710 × 4646356529142848", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 5641156343740710 × 4646356529142848?", "options": [ "26210823609655244092986262942100", "26210823609655244092986262942081", "26210823609655244092986262942080", "26210823609655244092986262942079" ], "answer": "26210823609655244092986262942080", "correct_value": 26210823609655244092986262942080, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "control": { "task_type": "control", "math_expression": "4676073520580933 × 6726916458916965", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4676073520580933 × 6726916458916965?", "options": [ "36400464051454975674494809228345", "31455555928701675674494809228345", "27238457052799345674494809228345", "31085427057357575674494809228345" ], "answer": "31455555928701675674494809228345", "correct_value": 31455555928701675674494809228345, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001598", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "8143624831666060 × 9930551964356277", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 8143624831666060 × 9930551964356277?", "options": [ "177915517051980285048445323488964", "80870689569081947749293328858620", "145567241224347505948727991945516", "8087068956908194774929332885862" ], "answer": "80870689569081947749293328858620", "correct_value": 80870689569081947749293328858620, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "8868286539897005 × 2371388150621778", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 8868286539897005 × 2371388150621778?", "options": [ "21030149617030365345694829974889", "21030149617030365345694829974891", "21030149617030365345694829974870", "21030149617030365345694829974890" ], "answer": "21030149617030365345694829974890", "correct_value": 21030149617030365345694829974890, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "control": { "task_type": "control", "math_expression": "6804183951133473 × 3655748730770145", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 6804183951133473 × 3655748730770145?", "options": [ "27242142628002574229284978563585", "23020721595821604229284978563585", "24874386843282784229284978563585", "26246684950080544229284978563585" ], "answer": "24874386843282784229284978563585", "correct_value": 24874386843282784229284978563585, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] } }, { "family_id": "option_elimination_001599", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "3469029693142585 × 6143139281443207", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 3469029693142585 × 6143139281443207?", "options": [ "21310732576437088490590030670098", "21310732576437088490590030670094", "21310732576437088490590030670096", "21310732576437088490590030670095" ], "answer": "21310732576437088490590030670095", "correct_value": 21310732576437088490590030670095, "explanation": "Product odd (odd × odd); only one option matches." }, "distractor_types": [ "distractor", "distractor", "distractor", "correct" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "8230061953763220 × 9646648567167258", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 8230061953763220 × 9646648567167258?", "options": [ "79392515353967730172568268650759", "79392515353967730172568268650750", "79392515353967730172568268650760", "79392515353967730172568268650761" ], "answer": "79392515353967730172568268650760", "correct_value": 79392515353967730172568268650760, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "control": { "task_type": "control", "math_expression": "4999642993506255 × 4334160961894347", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 4999642993506255 × 4334160961894347?", "options": [ "21467379488442782642624393640485", "21669257485863402642624393640485", "22333236401719532642624393640485", "22781047019545072642624393640485" ], "answer": "21669257485863402642624393640485", "correct_value": 21669257485863402642624393640485, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] } }, { "family_id": "option_elimination_001600", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2319400042090665 × 1754247088243002", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 2319400042090665 × 1754247088243002?", "options": [ "4068800770308245357261635776330", "4068800770308245357261635776329", "4068800770308245357261635776331", "4068800770308245357261635776332" ], "answer": "4068800770308245357261635776330", "correct_value": 4068800770308245357261635776330, "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "6774265252377280 × 8616372167914848", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 6774265252377280 × 8616372167914848?", "options": [ "58369590578656248992772609853450", "58369590578656248992772609853439", "58369590578656248992772609853440", "58369590578656248992772609853441" ], "answer": "58369590578656248992772609853440", "correct_value": 58369590578656248992772609853440, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] }, "control": { "task_type": "control", "math_expression": "3046691762033043 × 6916284114945829", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 3046691762033043 × 6916284114945829?", "options": [ "23671041938071683066570853027647", "22977119835209293066570853027647", "21071785836885453066570853027647", "19789342930086383066570853027647" ], "answer": "21071785836885453066570853027647", "correct_value": 21071785836885453066570853027647, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] } }, { "family_id": "option_elimination_001601", "category": "option_elimination", "strong_shortcut": { "task_type": "strong_shortcut", "math_expression": "2843786192974500 × 6848318830608338", "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", "shortcut_strength": 1.0, "pure_math": { "question": "What is 2843786192974500 × 6848318830608338?", "options": [ "194751545355712652648967213810018", "19475154535571265264896721381000", "1947515453557126526489672138100", "1947515453557126526489672138100012" ], "answer": "19475154535571265264896721381000", "correct_value": 19475154535571265264896721381000, "explanation": "Product has 32 digits; only one option matches." }, "distractor_types": [ "distractor", "correct", "distractor", "distractor" ] }, "weak_shortcut": { "task_type": "weak_shortcut", "math_expression": "5167599468238550 × 6802723170459728", "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", "shortcut_strength": 0.5, "pure_math": { "question": "What is 5167599468238550 × 6802723170459728?", "options": [ "35153748638241753340537872114400", "35153748638241753340537872114401", "35153748638241753340537872114420", "35153748638241753340537872114399" ], "answer": "35153748638241753340537872114400", "correct_value": 35153748638241753340537872114400, "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." }, "distractor_types": [ "correct", "distractor", "distractor", "distractor" ] }, "control": { "task_type": "control", "math_expression": "5317971905409333 × 8283401077171239", "ns_shortcut": "All options have same structural features; must compute precisely.", "shortcut_strength": 0.0, "pure_math": { "question": "What is 5317971905409333 × 8283401077171239?", "options": [ "49083108893223245289162029773587", "39473099415937845289162029773587", "44050894209634055289162029773587", "44854781254400605289162029773587" ], "answer": "44050894209634055289162029773587", "correct_value": 44050894209634055289162029773587, "explanation": "All options have same structural features; must compute precisely." }, "distractor_types": [ "distractor", "distractor", "correct", "distractor" ] } } ]