diff --git "a/data/sensemath_v2_d8.json" "b/data/sensemath_v2_d8.json" new file mode 100644--- /dev/null +++ "b/data/sensemath_v2_d8.json" @@ -0,0 +1,28602 @@ +[ + { + "family_id": "magnitude_estimation_000802", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98000000 * 98000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 98000000?", + "options": [ + "9604000000000020", + "9604000000000000", + "9604000000000010", + "9603999999999990" + ], + "answer": "9604000000000000", + "correct_value": 9604000000000000, + "explanation": "Round to 100000000: 98000000≈100000000, 98000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101000000 * 51150189", + "ns_shortcut": "Round 101000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 51150189?", + "options": [ + "5166169089000000", + "5166169089000010", + "5166169089000020", + "5166169088999990" + ], + "answer": "5166169089000000", + "correct_value": 5166169089000000, + "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": "44487259 * 66251365", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 44487259 * 66251365?", + "options": [ + "2947341633858545", + "2947341633858555", + "2947341633858535", + "2947341633858525" + ], + "answer": "2947341633858535", + "correct_value": 2947341633858535, + "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_000803", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102000000 * 99000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 99000000?", + "options": [ + "10097999999999990", + "10098000000000010", + "10098000000000000", + "10098000000000020" + ], + "answer": "10098000000000000", + "correct_value": 10098000000000000, + "explanation": "Round to 100000000: 102000000≈100000000, 99000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101000000 * 50527433", + "ns_shortcut": "Round 101000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 50527433?", + "options": [ + "5103270733000020", + "5103270733000010", + "5103270733000000", + "5103270732999990" + ], + "answer": "5103270733000000", + "correct_value": 5103270733000000, + "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": "49221069 * 69525738", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 49221069 * 69525738?", + "options": [ + "3422131147373942", + "3422131147373932", + "3422131147373922", + "3422131147373912" + ], + "answer": "3422131147373922", + "correct_value": 3422131147373922, + "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_000804", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101000000 * 99000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 99000000?", + "options": [ + "9999000000000020", + "9999000000000000", + "9998999999999990", + "9999000000000010" + ], + "answer": "9999000000000000", + "correct_value": 9999000000000000, + "explanation": "Round to 100000000: 101000000≈100000000, 99000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98000000 * 51171709", + "ns_shortcut": "Round 98000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 51171709?", + "options": [ + "5014827482000020", + "5014827482000000", + "5014827482000010", + "5014827481999990" + ], + "answer": "5014827482000000", + "correct_value": 5014827482000000, + "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": "63388868 * 58811666", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 63388868 * 58811666?", + "options": [ + "3728004932934098", + "3728004932934078", + "3728004932934108", + "3728004932934088" + ], + "answer": "3728004932934088", + "correct_value": 3728004932934088, + "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_000805", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99000000 * 99000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 99000000?", + "options": [ + "9801000000000000", + "9801000000000020", + "9801000000000010", + "9800999999999990" + ], + "answer": "9801000000000000", + "correct_value": 9801000000000000, + "explanation": "Round to 100000000: 99000000≈100000000, 99000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99000000 * 49678412", + "ns_shortcut": "Round 99000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 49678412?", + "options": [ + "4918162788000020", + "4918162787999990", + "4918162788000010", + "4918162788000000" + ], + "answer": "4918162788000000", + "correct_value": 4918162788000000, + "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": "67427559 * 51434243", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 67427559 * 51434243?", + "options": [ + "3468085454502857", + "3468085454502827", + "3468085454502837", + "3468085454502847" + ], + "answer": "3468085454502837", + "correct_value": 3468085454502837, + "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_000806", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99000000 * 102000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 102000000?", + "options": [ + "10098000000000000", + "10097999999999990", + "10098000000000020", + "10098000000000010" + ], + "answer": "10098000000000000", + "correct_value": 10098000000000000, + "explanation": "Round to 100000000: 99000000≈100000000, 102000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 50410683", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 50410683?", + "options": [ + "5141889666000000", + "5141889666000020", + "5141889666000010", + "5141889665999990" + ], + "answer": "5141889666000000", + "correct_value": 5141889666000000, + "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": "64988447 * 52940733", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 64988447 * 52940733?", + "options": [ + "3440536020711661", + "3440536020711671", + "3440536020711641", + "3440536020711651" + ], + "answer": "3440536020711651", + "correct_value": 3440536020711651, + "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_000807", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99000000 * 98000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 98000000?", + "options": [ + "9702000000000000", + "9702000000000020", + "9702000000000010", + "9701999999999990" + ], + "answer": "9702000000000000", + "correct_value": 9702000000000000, + "explanation": "Round to 100000000: 99000000≈100000000, 98000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99000000 * 50347195", + "ns_shortcut": "Round 99000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 50347195?", + "options": [ + "4984372305000010", + "4984372304999990", + "4984372305000020", + "4984372305000000" + ], + "answer": "4984372305000000", + "correct_value": 4984372305000000, + "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": "40757534 * 52635166", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 40757534 * 52635166?", + "options": [ + "2145279567840664", + "2145279567840654", + "2145279567840644", + "2145279567840634" + ], + "answer": "2145279567840644", + "correct_value": 2145279567840644, + "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_000808", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99000000 * 99000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 99000000?", + "options": [ + "9801000000000010", + "9801000000000000", + "9800999999999990", + "9801000000000020" + ], + "answer": "9801000000000000", + "correct_value": 9801000000000000, + "explanation": "Round to 100000000: 99000000≈100000000, 99000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 48957473", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 48957473?", + "options": [ + "4993662245999990", + "4993662246000010", + "4993662246000020", + "4993662246000000" + ], + "answer": "4993662246000000", + "correct_value": 4993662246000000, + "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": "50793445 * 67617141", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 50793445 * 67617141?", + "options": [ + "3434507532440735", + "3434507532440745", + "3434507532440755", + "3434507532440765" + ], + "answer": "3434507532440745", + "correct_value": 3434507532440745, + "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_000809", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102000000 * 99000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 99000000?", + "options": [ + "10098000000000010", + "10098000000000020", + "10097999999999990", + "10098000000000000" + ], + "answer": "10098000000000000", + "correct_value": 10098000000000000, + "explanation": "Round to 100000000: 102000000≈100000000, 99000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101000000 * 49505651", + "ns_shortcut": "Round 101000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 49505651?", + "options": [ + "5000070751000020", + "5000070750999990", + "5000070751000010", + "5000070751000000" + ], + "answer": "5000070751000000", + "correct_value": 5000070751000000, + "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": "57555861 * 49523335", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 57555861 * 49523335?", + "options": [ + "2850358185516425", + "2850358185516455", + "2850358185516445", + "2850358185516435" + ], + "answer": "2850358185516435", + "correct_value": 2850358185516435, + "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_000810", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101000000 * 102000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 102000000?", + "options": [ + "10302000000000020", + "10301999999999990", + "10302000000000000", + "10302000000000010" + ], + "answer": "10302000000000000", + "correct_value": 10302000000000000, + "explanation": "Round to 100000000: 101000000≈100000000, 102000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101000000 * 48641837", + "ns_shortcut": "Round 101000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 48641837?", + "options": [ + "4912825536999990", + "4912825537000000", + "4912825537000020", + "4912825537000010" + ], + "answer": "4912825537000000", + "correct_value": 4912825537000000, + "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": "42990472 * 42027573", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 42990472 * 42027573?", + "options": [ + "1806785200284466", + "1806785200284446", + "1806785200284476", + "1806785200284456" + ], + "answer": "1806785200284456", + "correct_value": 1806785200284456, + "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_000811", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98000000 * 98000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 98000000?", + "options": [ + "9604000000000010", + "9603999999999990", + "9604000000000020", + "9604000000000000" + ], + "answer": "9604000000000000", + "correct_value": 9604000000000000, + "explanation": "Round to 100000000: 98000000≈100000000, 98000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 50567853", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 50567853?", + "options": [ + "5157921006000020", + "5157921006000010", + "5157921006000000", + "5157921005999990" + ], + "answer": "5157921006000000", + "correct_value": 5157921006000000, + "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": "66797932 * 58037044", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 66797932 * 58037044?", + "options": [ + "3876754518593018", + "3876754518592998", + "3876754518593028", + "3876754518593008" + ], + "answer": "3876754518593008", + "correct_value": 3876754518593008, + "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_000812", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102000000 * 101000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 101000000?", + "options": [ + "10301999999999990", + "10302000000000000", + "10302000000000010", + "10302000000000020" + ], + "answer": "10302000000000000", + "correct_value": 10302000000000000, + "explanation": "Round to 100000000: 102000000≈100000000, 101000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101000000 * 48541165", + "ns_shortcut": "Round 101000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 48541165?", + "options": [ + "4902657664999990", + "4902657665000020", + "4902657665000010", + "4902657665000000" + ], + "answer": "4902657665000000", + "correct_value": 4902657665000000, + "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": "46339547 * 45109558", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 46339547 * 45109558?", + "options": [ + "2090356483090226", + "2090356483090236", + "2090356483090216", + "2090356483090246" + ], + "answer": "2090356483090226", + "correct_value": 2090356483090226, + "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_000813", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101000000 * 98000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 98000000?", + "options": [ + "9898000000000000", + "9898000000000020", + "9897999999999990", + "9898000000000010" + ], + "answer": "9898000000000000", + "correct_value": 9898000000000000, + "explanation": "Round to 100000000: 101000000≈100000000, 98000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 50435493", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 50435493?", + "options": [ + "5144420285999990", + "5144420286000020", + "5144420286000000", + "5144420286000010" + ], + "answer": "5144420286000000", + "correct_value": 5144420286000000, + "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": "67723128 * 60477232", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 67723128 * 60477232?", + "options": [ + "4095707323821686", + "4095707323821696", + "4095707323821706", + "4095707323821716" + ], + "answer": "4095707323821696", + "correct_value": 4095707323821696, + "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_000814", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101000000 * 101000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 101000000?", + "options": [ + "10201000000000010", + "10201000000000020", + "10201000000000000", + "10200999999999990" + ], + "answer": "10201000000000000", + "correct_value": 10201000000000000, + "explanation": "Round to 100000000: 101000000≈100000000, 101000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98000000 * 48379272", + "ns_shortcut": "Round 98000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 48379272?", + "options": [ + "4741168656000010", + "4741168655999990", + "4741168656000020", + "4741168656000000" + ], + "answer": "4741168656000000", + "correct_value": 4741168656000000, + "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": "41251531 * 66421039", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 41251531 * 66421039?", + "options": [ + "2739969549360729", + "2739969549360709", + "2739969549360699", + "2739969549360719" + ], + "answer": "2739969549360709", + "correct_value": 2739969549360709, + "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_000815", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99000000 * 101000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 101000000?", + "options": [ + "9998999999999990", + "9999000000000000", + "9999000000000020", + "9999000000000010" + ], + "answer": "9999000000000000", + "correct_value": 9999000000000000, + "explanation": "Round to 100000000: 99000000≈100000000, 101000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98000000 * 49469976", + "ns_shortcut": "Round 98000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 49469976?", + "options": [ + "4848057647999990", + "4848057648000000", + "4848057648000010", + "4848057648000020" + ], + "answer": "4848057648000000", + "correct_value": 4848057648000000, + "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": "59058448 * 61504675", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 59058448 * 61504675?", + "options": [ + "3632370650244390", + "3632370650244410", + "3632370650244420", + "3632370650244400" + ], + "answer": "3632370650244400", + "correct_value": 3632370650244400, + "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_000816", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101000000 * 102000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 102000000?", + "options": [ + "10302000000000000", + "10302000000000010", + "10301999999999990", + "10302000000000020" + ], + "answer": "10302000000000000", + "correct_value": 10302000000000000, + "explanation": "Round to 100000000: 101000000≈100000000, 102000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99000000 * 50217648", + "ns_shortcut": "Round 99000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 50217648?", + "options": [ + "4971547152000020", + "4971547152000000", + "4971547152000010", + "4971547151999990" + ], + "answer": "4971547152000000", + "correct_value": 4971547152000000, + "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": "45369157 * 63864567", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 45369157 * 63864567?", + "options": [ + "2897481566960019", + "2897481566960029", + "2897481566960009", + "2897481566960039" + ], + "answer": "2897481566960019", + "correct_value": 2897481566960019, + "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_000817", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101000000 * 98000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 98000000?", + "options": [ + "9898000000000020", + "9898000000000000", + "9897999999999990", + "9898000000000010" + ], + "answer": "9898000000000000", + "correct_value": 9898000000000000, + "explanation": "Round to 100000000: 101000000≈100000000, 98000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98000000 * 51100248", + "ns_shortcut": "Round 98000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 51100248?", + "options": [ + "5007824303999990", + "5007824304000020", + "5007824304000010", + "5007824304000000" + ], + "answer": "5007824304000000", + "correct_value": 5007824304000000, + "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": "47949739 * 59794147", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 47949739 * 59794147?", + "options": [ + "2867113742377653", + "2867113742377633", + "2867113742377643", + "2867113742377623" + ], + "answer": "2867113742377633", + "correct_value": 2867113742377633, + "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_000818", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98000000 * 98000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 98000000?", + "options": [ + "9604000000000000", + "9604000000000010", + "9604000000000020", + "9603999999999990" + ], + "answer": "9604000000000000", + "correct_value": 9604000000000000, + "explanation": "Round to 100000000: 98000000≈100000000, 98000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99000000 * 48619145", + "ns_shortcut": "Round 99000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 48619145?", + "options": [ + "4813295354999990", + "4813295355000000", + "4813295355000010", + "4813295355000020" + ], + "answer": "4813295355000000", + "correct_value": 4813295355000000, + "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": "45433348 * 45954075", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 45433348 * 45954075?", + "options": [ + "2087847481493100", + "2087847481493120", + "2087847481493090", + "2087847481493110" + ], + "answer": "2087847481493100", + "correct_value": 2087847481493100, + "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_000819", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101000000 * 102000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 102000000?", + "options": [ + "10302000000000010", + "10301999999999990", + "10302000000000000", + "10302000000000020" + ], + "answer": "10302000000000000", + "correct_value": 10302000000000000, + "explanation": "Round to 100000000: 101000000≈100000000, 102000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101000000 * 50472487", + "ns_shortcut": "Round 101000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 50472487?", + "options": [ + "5097721187000020", + "5097721187000010", + "5097721186999990", + "5097721187000000" + ], + "answer": "5097721187000000", + "correct_value": 5097721187000000, + "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": "46715027 * 46076673", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 46715027 * 46076673?", + "options": [ + "2152473023265161", + "2152473023265181", + "2152473023265171", + "2152473023265191" + ], + "answer": "2152473023265171", + "correct_value": 2152473023265171, + "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_000820", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102000000 * 102000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 102000000?", + "options": [ + "10404000000000000", + "10404000000000010", + "10404000000000020", + "10403999999999990" + ], + "answer": "10404000000000000", + "correct_value": 10404000000000000, + "explanation": "Round to 100000000: 102000000≈100000000, 102000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 49440956", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 49440956?", + "options": [ + "5042977512000020", + "5042977512000010", + "5042977512000000", + "5042977511999990" + ], + "answer": "5042977512000000", + "correct_value": 5042977512000000, + "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": "59011434 * 42152262", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 59011434 * 42152262?", + "options": [ + "2487465426963728", + "2487465426963698", + "2487465426963718", + "2487465426963708" + ], + "answer": "2487465426963708", + "correct_value": 2487465426963708, + "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_000821", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102000000 * 99000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 99000000?", + "options": [ + "10098000000000020", + "10097999999999990", + "10098000000000000", + "10098000000000010" + ], + "answer": "10098000000000000", + "correct_value": 10098000000000000, + "explanation": "Round to 100000000: 102000000≈100000000, 99000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98000000 * 50016424", + "ns_shortcut": "Round 98000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 50016424?", + "options": [ + "4901609552000000", + "4901609551999990", + "4901609552000020", + "4901609552000010" + ], + "answer": "4901609552000000", + "correct_value": 4901609552000000, + "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": "67714953 * 40559547", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 67714953 * 40559547?", + "options": [ + "2746487818806311", + "2746487818806281", + "2746487818806301", + "2746487818806291" + ], + "answer": "2746487818806291", + "correct_value": 2746487818806291, + "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_000822", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98000000 * 98000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 98000000?", + "options": [ + "9603999999999990", + "9604000000000020", + "9604000000000000", + "9604000000000010" + ], + "answer": "9604000000000000", + "correct_value": 9604000000000000, + "explanation": "Round to 100000000: 98000000≈100000000, 98000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98000000 * 49612627", + "ns_shortcut": "Round 98000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 49612627?", + "options": [ + "4862037445999990", + "4862037446000000", + "4862037446000010", + "4862037446000020" + ], + "answer": "4862037446000000", + "correct_value": 4862037446000000, + "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": "56444347 * 56842269", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 56444347 * 56842269?", + "options": [ + "3208424755703353", + "3208424755703343", + "3208424755703363", + "3208424755703333" + ], + "answer": "3208424755703343", + "correct_value": 3208424755703343, + "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_000823", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102000000 * 98000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 98000000?", + "options": [ + "9996000000000000", + "9996000000000020", + "9996000000000010", + "9995999999999990" + ], + "answer": "9996000000000000", + "correct_value": 9996000000000000, + "explanation": "Round to 100000000: 102000000≈100000000, 98000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 51541311", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 51541311?", + "options": [ + "5257213722000010", + "5257213721999990", + "5257213722000020", + "5257213722000000" + ], + "answer": "5257213722000000", + "correct_value": 5257213722000000, + "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": "41864449 * 53386235", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 41864449 * 53386235?", + "options": [ + "2234985312459535", + "2234985312459525", + "2234985312459515", + "2234985312459505" + ], + "answer": "2234985312459515", + "correct_value": 2234985312459515, + "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_000824", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99000000 * 98000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 98000000?", + "options": [ + "9701999999999990", + "9702000000000020", + "9702000000000010", + "9702000000000000" + ], + "answer": "9702000000000000", + "correct_value": 9702000000000000, + "explanation": "Round to 100000000: 99000000≈100000000, 98000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 51241608", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 51241608?", + "options": [ + "5226644016000000", + "5226644015999990", + "5226644016000020", + "5226644016000010" + ], + "answer": "5226644016000000", + "correct_value": 5226644016000000, + "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": "57511272 * 63176575", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 57511272 * 63176575?", + "options": [ + "3633365188853420", + "3633365188853390", + "3633365188853410", + "3633365188853400" + ], + "answer": "3633365188853400", + "correct_value": 3633365188853400, + "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_000825", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99000000 * 101000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 101000000?", + "options": [ + "9999000000000020", + "9999000000000010", + "9999000000000000", + "9998999999999990" + ], + "answer": "9999000000000000", + "correct_value": 9999000000000000, + "explanation": "Round to 100000000: 99000000≈100000000, 101000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 48733201", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 48733201?", + "options": [ + "4970786502000010", + "4970786502000020", + "4970786501999990", + "4970786502000000" + ], + "answer": "4970786502000000", + "correct_value": 4970786502000000, + "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": "63541633 * 68899862", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 63541633 * 68899862?", + "options": [ + "4378009744954656", + "4378009744954646", + "4378009744954666", + "4378009744954636" + ], + "answer": "4378009744954646", + "correct_value": 4378009744954646, + "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_000826", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99000000 * 99000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 99000000?", + "options": [ + "9801000000000020", + "9801000000000010", + "9800999999999990", + "9801000000000000" + ], + "answer": "9801000000000000", + "correct_value": 9801000000000000, + "explanation": "Round to 100000000: 99000000≈100000000, 99000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 51675212", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 51675212?", + "options": [ + "5270871624000020", + "5270871623999990", + "5270871624000010", + "5270871624000000" + ], + "answer": "5270871624000000", + "correct_value": 5270871624000000, + "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": "60363073 * 67631444", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 60363073 * 67631444?", + "options": [ + "4082441791267412", + "4082441791267402", + "4082441791267422", + "4082441791267432" + ], + "answer": "4082441791267412", + "correct_value": 4082441791267412, + "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_000827", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101000000 * 101000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 101000000?", + "options": [ + "10201000000000000", + "10201000000000020", + "10200999999999990", + "10201000000000010" + ], + "answer": "10201000000000000", + "correct_value": 10201000000000000, + "explanation": "Round to 100000000: 101000000≈100000000, 101000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 51070116", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 51070116?", + "options": [ + "5209151832000000", + "5209151832000010", + "5209151832000020", + "5209151831999990" + ], + "answer": "5209151832000000", + "correct_value": 5209151832000000, + "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": "53901652 * 61600446", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 53901652 * 61600446?", + "options": [ + "3320365803336792", + "3320365803336782", + "3320365803336802", + "3320365803336812" + ], + "answer": "3320365803336792", + "correct_value": 3320365803336792, + "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_000828", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102000000 * 99000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 99000000?", + "options": [ + "10097999999999990", + "10098000000000000", + "10098000000000010", + "10098000000000020" + ], + "answer": "10098000000000000", + "correct_value": 10098000000000000, + "explanation": "Round to 100000000: 102000000≈100000000, 99000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 49451907", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 49451907?", + "options": [ + "5044094514000010", + "5044094514000000", + "5044094514000020", + "5044094513999990" + ], + "answer": "5044094514000000", + "correct_value": 5044094514000000, + "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": "60290458 * 44587671", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 60290458 * 44587671?", + "options": [ + "2688211105743328", + "2688211105743338", + "2688211105743308", + "2688211105743318" + ], + "answer": "2688211105743318", + "correct_value": 2688211105743318, + "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_000829", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101000000 * 101000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 101000000?", + "options": [ + "10201000000000010", + "10201000000000020", + "10200999999999990", + "10201000000000000" + ], + "answer": "10201000000000000", + "correct_value": 10201000000000000, + "explanation": "Round to 100000000: 101000000≈100000000, 101000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98000000 * 50897091", + "ns_shortcut": "Round 98000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 50897091?", + "options": [ + "4987914917999990", + "4987914918000000", + "4987914918000020", + "4987914918000010" + ], + "answer": "4987914918000000", + "correct_value": 4987914918000000, + "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": "45483645 * 58190347", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 45483645 * 58190347?", + "options": [ + "2646709085374835", + "2646709085374815", + "2646709085374805", + "2646709085374825" + ], + "answer": "2646709085374815", + "correct_value": 2646709085374815, + "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_000830", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99000000 * 101000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 101000000?", + "options": [ + "9998999999999990", + "9999000000000000", + "9999000000000010", + "9999000000000020" + ], + "answer": "9999000000000000", + "correct_value": 9999000000000000, + "explanation": "Round to 100000000: 99000000≈100000000, 101000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98000000 * 49559117", + "ns_shortcut": "Round 98000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 49559117?", + "options": [ + "4856793466000020", + "4856793466000000", + "4856793466000010", + "4856793465999990" + ], + "answer": "4856793466000000", + "correct_value": 4856793466000000, + "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": "61481253 * 61115267", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 61481253 * 61115267?", + "options": [ + "3757443192589551", + "3757443192589561", + "3757443192589541", + "3757443192589571" + ], + "answer": "3757443192589551", + "correct_value": 3757443192589551, + "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_000831", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99000000 * 98000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 98000000?", + "options": [ + "9702000000000000", + "9702000000000010", + "9701999999999990", + "9702000000000020" + ], + "answer": "9702000000000000", + "correct_value": 9702000000000000, + "explanation": "Round to 100000000: 99000000≈100000000, 98000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99000000 * 48967792", + "ns_shortcut": "Round 99000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 48967792?", + "options": [ + "4847811407999990", + "4847811408000000", + "4847811408000010", + "4847811408000020" + ], + "answer": "4847811408000000", + "correct_value": 4847811408000000, + "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": "56972129 * 47671963", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 56972129 * 47671963?", + "options": [ + "2715973225719217", + "2715973225719247", + "2715973225719237", + "2715973225719227" + ], + "answer": "2715973225719227", + "correct_value": 2715973225719227, + "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_000832", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99000000 * 99000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 99000000?", + "options": [ + "9801000000000010", + "9800999999999990", + "9801000000000020", + "9801000000000000" + ], + "answer": "9801000000000000", + "correct_value": 9801000000000000, + "explanation": "Round to 100000000: 99000000≈100000000, 99000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 50170921", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 50170921?", + "options": [ + "5117433942000000", + "5117433942000010", + "5117433942000020", + "5117433941999990" + ], + "answer": "5117433942000000", + "correct_value": 5117433942000000, + "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": "57650569 * 61084638", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 57650569 * 61084638?", + "options": [ + "3521564137859032", + "3521564137859042", + "3521564137859012", + "3521564137859022" + ], + "answer": "3521564137859022", + "correct_value": 3521564137859022, + "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_000833", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101000000 * 98000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 98000000?", + "options": [ + "9898000000000000", + "9897999999999990", + "9898000000000010", + "9898000000000020" + ], + "answer": "9898000000000000", + "correct_value": 9898000000000000, + "explanation": "Round to 100000000: 101000000≈100000000, 98000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 50256695", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 50256695?", + "options": [ + "5126182889999990", + "5126182890000020", + "5126182890000010", + "5126182890000000" + ], + "answer": "5126182890000000", + "correct_value": 5126182890000000, + "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": "42403572 * 40824553", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 42403572 * 40824553?", + "options": [ + "1731106872503336", + "1731106872503306", + "1731106872503326", + "1731106872503316" + ], + "answer": "1731106872503316", + "correct_value": 1731106872503316, + "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_000834", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98000000 * 98000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 98000000?", + "options": [ + "9603999999999990", + "9604000000000000", + "9604000000000020", + "9604000000000010" + ], + "answer": "9604000000000000", + "correct_value": 9604000000000000, + "explanation": "Round to 100000000: 98000000≈100000000, 98000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101000000 * 51675324", + "ns_shortcut": "Round 101000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 51675324?", + "options": [ + "5219207724000000", + "5219207724000020", + "5219207724000010", + "5219207723999990" + ], + "answer": "5219207724000000", + "correct_value": 5219207724000000, + "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": "63018771 * 47661043", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 63018771 * 47661043?", + "options": [ + "3003540354438153", + "3003540354438143", + "3003540354438163", + "3003540354438173" + ], + "answer": "3003540354438153", + "correct_value": 3003540354438153, + "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_000835", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102000000 * 98000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 98000000?", + "options": [ + "9996000000000000", + "9996000000000020", + "9996000000000010", + "9995999999999990" + ], + "answer": "9996000000000000", + "correct_value": 9996000000000000, + "explanation": "Round to 100000000: 102000000≈100000000, 98000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 48615292", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 48615292?", + "options": [ + "4958759784000020", + "4958759784000000", + "4958759784000010", + "4958759783999990" + ], + "answer": "4958759784000000", + "correct_value": 4958759784000000, + "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": "58312157 * 49120827", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 58312157 * 49120827?", + "options": [ + "2864341375993829", + "2864341375993839", + "2864341375993859", + "2864341375993849" + ], + "answer": "2864341375993839", + "correct_value": 2864341375993839, + "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_000836", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101000000 * 99000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 99000000?", + "options": [ + "9998999999999990", + "9999000000000020", + "9999000000000000", + "9999000000000010" + ], + "answer": "9999000000000000", + "correct_value": 9999000000000000, + "explanation": "Round to 100000000: 101000000≈100000000, 99000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 50343393", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 50343393?", + "options": [ + "5135026086000000", + "5135026086000010", + "5135026085999990", + "5135026086000020" + ], + "answer": "5135026086000000", + "correct_value": 5135026086000000, + "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": "50398175 * 67937066", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 50398175 * 67937066?", + "options": [ + "3423904141254570", + "3423904141254540", + "3423904141254550", + "3423904141254560" + ], + "answer": "3423904141254550", + "correct_value": 3423904141254550, + "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_000837", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98000000 * 101000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 101000000?", + "options": [ + "9898000000000010", + "9897999999999990", + "9898000000000000", + "9898000000000020" + ], + "answer": "9898000000000000", + "correct_value": 9898000000000000, + "explanation": "Round to 100000000: 98000000≈100000000, 101000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 48990975", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 48990975?", + "options": [ + "4997079450000010", + "4997079450000000", + "4997079449999990", + "4997079450000020" + ], + "answer": "4997079450000000", + "correct_value": 4997079450000000, + "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": "62416031 * 64843655", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 62416031 * 64843655?", + "options": [ + "4047283580633305", + "4047283580633295", + "4047283580633315", + "4047283580633325" + ], + "answer": "4047283580633305", + "correct_value": 4047283580633305, + "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_000838", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102000000 * 98000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 98000000?", + "options": [ + "9996000000000010", + "9996000000000020", + "9995999999999990", + "9996000000000000" + ], + "answer": "9996000000000000", + "correct_value": 9996000000000000, + "explanation": "Round to 100000000: 102000000≈100000000, 98000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99000000 * 48434672", + "ns_shortcut": "Round 99000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 48434672?", + "options": [ + "4795032528000020", + "4795032528000010", + "4795032528000000", + "4795032527999990" + ], + "answer": "4795032528000000", + "correct_value": 4795032528000000, + "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": "48288262 * 54908436", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 48288262 * 54908436?", + "options": [ + "2651432943578232", + "2651432943578242", + "2651432943578222", + "2651432943578252" + ], + "answer": "2651432943578232", + "correct_value": 2651432943578232, + "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_000839", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99000000 * 102000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 102000000?", + "options": [ + "10097999999999990", + "10098000000000010", + "10098000000000020", + "10098000000000000" + ], + "answer": "10098000000000000", + "correct_value": 10098000000000000, + "explanation": "Round to 100000000: 99000000≈100000000, 102000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101000000 * 51938841", + "ns_shortcut": "Round 101000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 51938841?", + "options": [ + "5245822941000020", + "5245822941000010", + "5245822941000000", + "5245822940999990" + ], + "answer": "5245822941000000", + "correct_value": 5245822941000000, + "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": "60393326 * 69430136", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 60393326 * 69430136?", + "options": [ + "4193116837672336", + "4193116837672326", + "4193116837672356", + "4193116837672346" + ], + "answer": "4193116837672336", + "correct_value": 4193116837672336, + "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_000840", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102000000 * 101000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 101000000?", + "options": [ + "10302000000000020", + "10302000000000010", + "10301999999999990", + "10302000000000000" + ], + "answer": "10302000000000000", + "correct_value": 10302000000000000, + "explanation": "Round to 100000000: 102000000≈100000000, 101000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98000000 * 48441118", + "ns_shortcut": "Round 98000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 48441118?", + "options": [ + "4747229563999990", + "4747229564000000", + "4747229564000020", + "4747229564000010" + ], + "answer": "4747229564000000", + "correct_value": 4747229564000000, + "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": "66104539 * 57112764", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 66104539 * 57112764?", + "options": [ + "3775412935235816", + "3775412935235806", + "3775412935235786", + "3775412935235796" + ], + "answer": "3775412935235796", + "correct_value": 3775412935235796, + "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_000841", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98000000 * 101000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 101000000?", + "options": [ + "9897999999999990", + "9898000000000000", + "9898000000000010", + "9898000000000020" + ], + "answer": "9898000000000000", + "correct_value": 9898000000000000, + "explanation": "Round to 100000000: 98000000≈100000000, 101000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99000000 * 51857746", + "ns_shortcut": "Round 99000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 51857746?", + "options": [ + "5133916854000020", + "5133916853999990", + "5133916854000010", + "5133916854000000" + ], + "answer": "5133916854000000", + "correct_value": 5133916854000000, + "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": "64672326 * 53067453", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 64672326 * 53067453?", + "options": [ + "3431995620405698", + "3431995620405668", + "3431995620405678", + "3431995620405688" + ], + "answer": "3431995620405678", + "correct_value": 3431995620405678, + "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_000842", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101000000 * 101000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 101000000?", + "options": [ + "10201000000000000", + "10201000000000020", + "10201000000000010", + "10200999999999990" + ], + "answer": "10201000000000000", + "correct_value": 10201000000000000, + "explanation": "Round to 100000000: 101000000≈100000000, 101000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99000000 * 50013971", + "ns_shortcut": "Round 99000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 50013971?", + "options": [ + "4951383129000000", + "4951383128999990", + "4951383129000020", + "4951383129000010" + ], + "answer": "4951383129000000", + "correct_value": 4951383129000000, + "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": "47474226 * 42891154", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 47474226 * 42891154?", + "options": [ + "2036224338396814", + "2036224338396794", + "2036224338396824", + "2036224338396804" + ], + "answer": "2036224338396804", + "correct_value": 2036224338396804, + "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_000843", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99000000 * 102000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 102000000?", + "options": [ + "10098000000000000", + "10098000000000010", + "10097999999999990", + "10098000000000020" + ], + "answer": "10098000000000000", + "correct_value": 10098000000000000, + "explanation": "Round to 100000000: 99000000≈100000000, 102000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 49897950", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 49897950?", + "options": [ + "5089590900000010", + "5089590900000020", + "5089590899999990", + "5089590900000000" + ], + "answer": "5089590900000000", + "correct_value": 5089590900000000, + "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": "47111064 * 42711029", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 47111064 * 42711029?", + "options": [ + "2012162020724856", + "2012162020724846", + "2012162020724866", + "2012162020724876" + ], + "answer": "2012162020724856", + "correct_value": 2012162020724856, + "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_000844", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101000000 * 99000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 99000000?", + "options": [ + "9999000000000010", + "9999000000000020", + "9998999999999990", + "9999000000000000" + ], + "answer": "9999000000000000", + "correct_value": 9999000000000000, + "explanation": "Round to 100000000: 101000000≈100000000, 99000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99000000 * 49492188", + "ns_shortcut": "Round 99000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 49492188?", + "options": [ + "4899726612000000", + "4899726612000010", + "4899726612000020", + "4899726611999990" + ], + "answer": "4899726612000000", + "correct_value": 4899726612000000, + "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": "54790043 * 63750139", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 54790043 * 63750139?", + "options": [ + "3492872857065977", + "3492872857065967", + "3492872857065997", + "3492872857065987" + ], + "answer": "3492872857065977", + "correct_value": 3492872857065977, + "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_000845", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99000000 * 101000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 99000000 * 101000000?", + "options": [ + "9998999999999990", + "9999000000000000", + "9999000000000010", + "9999000000000020" + ], + "answer": "9999000000000000", + "correct_value": 9999000000000000, + "explanation": "Round to 100000000: 99000000≈100000000, 101000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 50090296", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 50090296?", + "options": [ + "5109210192000020", + "5109210191999990", + "5109210192000000", + "5109210192000010" + ], + "answer": "5109210192000000", + "correct_value": 5109210192000000, + "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": "60115861 * 58933756", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 60115861 * 58933756?", + "options": [ + "3542853483903936", + "3542853483903926", + "3542853483903906", + "3542853483903916" + ], + "answer": "3542853483903916", + "correct_value": 3542853483903916, + "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_000846", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102000000 * 102000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 102000000?", + "options": [ + "10404000000000010", + "10404000000000020", + "10403999999999990", + "10404000000000000" + ], + "answer": "10404000000000000", + "correct_value": 10404000000000000, + "explanation": "Round to 100000000: 102000000≈100000000, 102000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 51188670", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 51188670?", + "options": [ + "5221244340000020", + "5221244339999990", + "5221244340000010", + "5221244340000000" + ], + "answer": "5221244340000000", + "correct_value": 5221244340000000, + "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": "67647433 * 54668575", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 67647433 * 54668575?", + "options": [ + "3698188764517965", + "3698188764517995", + "3698188764517985", + "3698188764517975" + ], + "answer": "3698188764517975", + "correct_value": 3698188764517975, + "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_000847", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102000000 * 98000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 98000000?", + "options": [ + "9996000000000000", + "9995999999999990", + "9996000000000010", + "9996000000000020" + ], + "answer": "9996000000000000", + "correct_value": 9996000000000000, + "explanation": "Round to 100000000: 102000000≈100000000, 98000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101000000 * 50071178", + "ns_shortcut": "Round 101000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 50071178?", + "options": [ + "5057188978000020", + "5057188978000010", + "5057188978000000", + "5057188977999990" + ], + "answer": "5057188978000000", + "correct_value": 5057188978000000, + "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": "48780173 * 68058629", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 48780173 * 68058629?", + "options": [ + "3319911696762817", + "3319911696762837", + "3319911696762807", + "3319911696762827" + ], + "answer": "3319911696762817", + "correct_value": 3319911696762817, + "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_000848", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101000000 * 101000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 101000000?", + "options": [ + "10201000000000010", + "10201000000000000", + "10200999999999990", + "10201000000000020" + ], + "answer": "10201000000000000", + "correct_value": 10201000000000000, + "explanation": "Round to 100000000: 101000000≈100000000, 101000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102000000 * 51861848", + "ns_shortcut": "Round 102000000≈100000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 51861848?", + "options": [ + "5289908496000020", + "5289908496000000", + "5289908495999990", + "5289908496000010" + ], + "answer": "5289908496000000", + "correct_value": 5289908496000000, + "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": "47802635 * 50827933", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 47802635 * 50827933?", + "options": [ + "2429709129003465", + "2429709129003455", + "2429709129003475", + "2429709129003445" + ], + "answer": "2429709129003455", + "correct_value": 2429709129003455, + "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_000849", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102000000 * 99000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 102000000 * 99000000?", + "options": [ + "10097999999999990", + "10098000000000010", + "10098000000000020", + "10098000000000000" + ], + "answer": "10098000000000000", + "correct_value": 10098000000000000, + "explanation": "Round to 100000000: 102000000≈100000000, 99000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101000000 * 49024036", + "ns_shortcut": "Round 101000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 49024036?", + "options": [ + "4951427636000020", + "4951427635999990", + "4951427636000000", + "4951427636000010" + ], + "answer": "4951427636000000", + "correct_value": 4951427636000000, + "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": "42671657 * 63217625", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 42671657 * 63217625?", + "options": [ + "2697600810354615", + "2697600810354625", + "2697600810354635", + "2697600810354645" + ], + "answer": "2697600810354625", + "correct_value": 2697600810354625, + "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_000850", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98000000 * 102000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 98000000 * 102000000?", + "options": [ + "9996000000000020", + "9996000000000000", + "9996000000000010", + "9995999999999990" + ], + "answer": "9996000000000000", + "correct_value": 9996000000000000, + "explanation": "Round to 100000000: 98000000≈100000000, 102000000≈100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101000000 * 50668273", + "ns_shortcut": "Round 101000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 50668273?", + "options": [ + "5117495573000020", + "5117495573000010", + "5117495573000000", + "5117495572999990" + ], + "answer": "5117495573000000", + "correct_value": 5117495573000000, + "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": "47994169 * 47158756", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 47994169 * 47158756?", + "options": [ + "2263345305293784", + "2263345305293774", + "2263345305293764", + "2263345305293754" + ], + "answer": "2263345305293764", + "correct_value": 2263345305293764, + "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_000851", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101000000 * 101000000", + "ns_shortcut": "Round both to 100000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 101000000?", + "options": [ + "10201000000000000", + "10200999999999990", + "10201000000000010", + "10201000000000020" + ], + "answer": "10201000000000000", + "correct_value": 10201000000000000, + "explanation": "Round to 100000000: 101000000≈100000000, 101000000≈100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101000000 * 49515475", + "ns_shortcut": "Round 101000000≈100000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 101000000 * 49515475?", + "options": [ + "5001062975000010", + "5001062975000020", + "5001062975000000", + "5001062974999990" + ], + "answer": "5001062975000000", + "correct_value": 5001062975000000, + "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": "60272552 * 52460863", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 60272552 * 52460863?", + "options": [ + "3161950093132386", + "3161950093132366", + "3161950093132376", + "3161950093132396" + ], + "answer": "3161950093132376", + "correct_value": 3161950093132376, + "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": "structural_shortcuts_000852", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000001 * 83673108", + "ns_shortcut": "Use (100000000+1)*m = 100000000*m + 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 100000001 * 83673108?", + "options": [ + "8367310883673098", + "8367310883673128", + "8367310883673108", + "8367310883673118" + ], + "answer": "8367310883673108", + "correct_value": 8367310883673108, + "explanation": "Distributive: (100000000+1)*83673108 = 100000000*83673108 + 1*83673108." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000043 * 69736340", + "ns_shortcut": "Use (100000000+43)*m, correction is 43*69736340.", + "shortcut_strength": 0.9999914, + "pure_math": { + "question": "What is 100000043 * 69736340?", + "options": [ + "6973636998662610", + "6973636998662620", + "6973636998662630", + "6973636998662640" + ], + "answer": "6973636998662620", + "correct_value": 6973636998662620, + "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": "63447014 * 61348985", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 63447014 * 61348985?", + "options": [ + "3892409910180800", + "3892409910180780", + "3892409910180810", + "3892409910180790" + ], + "answer": "3892409910180790", + "correct_value": 3892409910180790, + "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_000853", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000002 * 19587874", + "ns_shortcut": "Use (100000000+2)*m = 100000000*m + 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 100000002 * 19587874?", + "options": [ + "1958787439175758", + "1958787439175748", + "1958787439175768", + "1958787439175738" + ], + "answer": "1958787439175748", + "correct_value": 1958787439175748, + "explanation": "Distributive: (100000000+2)*19587874 = 100000000*19587874 + 2*19587874." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999985 * 19541571", + "ns_shortcut": "Use (100000000-15)*m, correction is 15*19541571.", + "shortcut_strength": 0.999997, + "pure_math": { + "question": "What is 99999985 * 19541571?", + "options": [ + "1954156806876455", + "1954156806876445", + "1954156806876425", + "1954156806876435" + ], + "answer": "1954156806876435", + "correct_value": 1954156806876435, + "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": "68428181 * 51330709", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 68428181 * 51330709?", + "options": [ + "3512467046310339", + "3512467046310349", + "3512467046310319", + "3512467046310329" + ], + "answer": "3512467046310329", + "correct_value": 3512467046310329, + "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_000854", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999998 * 99238969", + "ns_shortcut": "Use (100000000-2)*m = 100000000*m - 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 99999998 * 99238969?", + "options": [ + "9923896701522072", + "9923896701522052", + "9923896701522082", + "9923896701522062" + ], + "answer": "9923896701522062", + "correct_value": 9923896701522062, + "explanation": "Distributive: (100000000-2)*99238969 = 100000000*99238969 - 2*99238969." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000022 * 47732244", + "ns_shortcut": "Use (100000000+22)*m, correction is 22*47732244.", + "shortcut_strength": 0.9999956, + "pure_math": { + "question": "What is 100000022 * 47732244?", + "options": [ + "4773225450109368", + "4773225450109388", + "4773225450109378", + "4773225450109358" + ], + "answer": "4773225450109368", + "correct_value": 4773225450109368, + "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": "42772689 * 63083985", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 42772689 * 63083985?", + "options": [ + "2698271671285685", + "2698271671285665", + "2698271671285655", + "2698271671285675" + ], + "answer": "2698271671285665", + "correct_value": 2698271671285665, + "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_000855", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000002 * 82347561", + "ns_shortcut": "Use (100000000+2)*m = 100000000*m + 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 100000002 * 82347561?", + "options": [ + "8234756264695132", + "8234756264695112", + "8234756264695122", + "8234756264695142" + ], + "answer": "8234756264695122", + "correct_value": 8234756264695122, + "explanation": "Distributive: (100000000+2)*82347561 = 100000000*82347561 + 2*82347561." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000030 * 94069610", + "ns_shortcut": "Use (100000000+30)*m, correction is 30*94069610.", + "shortcut_strength": 0.999994, + "pure_math": { + "question": "What is 100000030 * 94069610?", + "options": [ + "9406963822088300", + "9406963822088320", + "9406963822088290", + "9406963822088310" + ], + "answer": "9406963822088300", + "correct_value": 9406963822088300, + "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": "62554568 * 43864045", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 62554568 * 43864045?", + "options": [ + "2743896385707560", + "2743896385707550", + "2743896385707570", + "2743896385707580" + ], + "answer": "2743896385707560", + "correct_value": 2743896385707560, + "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_000856", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 21413934", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 21413934?", + "options": [ + "2141393378586066", + "2141393378586056", + "2141393378586086", + "2141393378586076" + ], + "answer": "2141393378586066", + "correct_value": 2141393378586066, + "explanation": "Distributive: (100000000-1)*21413934 = 100000000*21413934 - 1*21413934." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999970 * 12342064", + "ns_shortcut": "Use (100000000-30)*m, correction is 30*12342064.", + "shortcut_strength": 0.999994, + "pure_math": { + "question": "What is 99999970 * 12342064?", + "options": [ + "1234206029738080", + "1234206029738090", + "1234206029738070", + "1234206029738100" + ], + "answer": "1234206029738080", + "correct_value": 1234206029738080, + "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": "67660830 * 65609703", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 67660830 * 65609703?", + "options": [ + "4439206961033500", + "4439206961033490", + "4439206961033480", + "4439206961033510" + ], + "answer": "4439206961033490", + "correct_value": 4439206961033490, + "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_000857", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000001 * 60928280", + "ns_shortcut": "Use (100000000+1)*m = 100000000*m + 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 100000001 * 60928280?", + "options": [ + "6092828060928300", + "6092828060928290", + "6092828060928280", + "6092828060928270" + ], + "answer": "6092828060928280", + "correct_value": 6092828060928280, + "explanation": "Distributive: (100000000+1)*60928280 = 100000000*60928280 + 1*60928280." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999971 * 18761776", + "ns_shortcut": "Use (100000000-29)*m, correction is 29*18761776.", + "shortcut_strength": 0.9999942, + "pure_math": { + "question": "What is 99999971 * 18761776?", + "options": [ + "1876177055908486", + "1876177055908496", + "1876177055908506", + "1876177055908516" + ], + "answer": "1876177055908496", + "correct_value": 1876177055908496, + "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": "40093803 * 62214278", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 40093803 * 62214278?", + "options": [ + "2494407005919224", + "2494407005919244", + "2494407005919234", + "2494407005919254" + ], + "answer": "2494407005919234", + "correct_value": 2494407005919234, + "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_000858", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000001 * 95452121", + "ns_shortcut": "Use (100000000+1)*m = 100000000*m + 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 100000001 * 95452121?", + "options": [ + "9545212195452111", + "9545212195452131", + "9545212195452121", + "9545212195452141" + ], + "answer": "9545212195452121", + "correct_value": 9545212195452121, + "explanation": "Distributive: (100000000+1)*95452121 = 100000000*95452121 + 1*95452121." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000017 * 38315110", + "ns_shortcut": "Use (100000000+17)*m, correction is 17*38315110.", + "shortcut_strength": 0.9999966, + "pure_math": { + "question": "What is 100000017 * 38315110?", + "options": [ + "3831511651356860", + "3831511651356890", + "3831511651356870", + "3831511651356880" + ], + "answer": "3831511651356870", + "correct_value": 3831511651356870, + "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": "53595696 * 52674218", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 53595696 * 52674218?", + "options": [ + "2823111374965748", + "2823111374965738", + "2823111374965718", + "2823111374965728" + ], + "answer": "2823111374965728", + "correct_value": 2823111374965728, + "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_000859", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 46159501", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 46159501?", + "options": [ + "4615950053840499", + "4615950053840519", + "4615950053840489", + "4615950053840509" + ], + "answer": "4615950053840499", + "correct_value": 4615950053840499, + "explanation": "Distributive: (100000000-1)*46159501 = 100000000*46159501 - 1*46159501." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000048 * 46841476", + "ns_shortcut": "Use (100000000+48)*m, correction is 48*46841476.", + "shortcut_strength": 0.9999904, + "pure_math": { + "question": "What is 100000048 * 46841476?", + "options": [ + "4684149848390838", + "4684149848390858", + "4684149848390868", + "4684149848390848" + ], + "answer": "4684149848390848", + "correct_value": 4684149848390848, + "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": "66344922 * 59355040", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 66344922 * 59355040?", + "options": [ + "3937905499106890", + "3937905499106880", + "3937905499106900", + "3937905499106870" + ], + "answer": "3937905499106880", + "correct_value": 3937905499106880, + "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_000860", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999998 * 93888349", + "ns_shortcut": "Use (100000000-2)*m = 100000000*m - 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 99999998 * 93888349?", + "options": [ + "9388834712223322", + "9388834712223312", + "9388834712223302", + "9388834712223292" + ], + "answer": "9388834712223302", + "correct_value": 9388834712223302, + "explanation": "Distributive: (100000000-2)*93888349 = 100000000*93888349 - 2*93888349." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999984 * 60841427", + "ns_shortcut": "Use (100000000-16)*m, correction is 16*60841427.", + "shortcut_strength": 0.9999968, + "pure_math": { + "question": "What is 99999984 * 60841427?", + "options": [ + "6084141726537168", + "6084141726537188", + "6084141726537178", + "6084141726537158" + ], + "answer": "6084141726537168", + "correct_value": 6084141726537168, + "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": "60739694 * 51914408", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 60739694 * 51914408?", + "options": [ + "3153265256111162", + "3153265256111142", + "3153265256111172", + "3153265256111152" + ], + "answer": "3153265256111152", + "correct_value": 3153265256111152, + "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_000861", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 18658992", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 18658992?", + "options": [ + "1865899181341028", + "1865899181340998", + "1865899181341018", + "1865899181341008" + ], + "answer": "1865899181341008", + "correct_value": 1865899181341008, + "explanation": "Distributive: (100000000-1)*18658992 = 100000000*18658992 - 1*18658992." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000027 * 77888552", + "ns_shortcut": "Use (100000000+27)*m, correction is 27*77888552.", + "shortcut_strength": 0.9999946, + "pure_math": { + "question": "What is 100000027 * 77888552?", + "options": [ + "7788857302990904", + "7788857302990894", + "7788857302990914", + "7788857302990924" + ], + "answer": "7788857302990904", + "correct_value": 7788857302990904, + "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": "69870321 * 58649332", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 69870321 * 58649332?", + "options": [ + "4097847653275562", + "4097847653275592", + "4097847653275582", + "4097847653275572" + ], + "answer": "4097847653275572", + "correct_value": 4097847653275572, + "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_000862", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 83481208", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 83481208?", + "options": [ + "8348120716518812", + "8348120716518792", + "8348120716518802", + "8348120716518782" + ], + "answer": "8348120716518792", + "correct_value": 8348120716518792, + "explanation": "Distributive: (100000000-1)*83481208 = 100000000*83481208 - 1*83481208." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000023 * 47011903", + "ns_shortcut": "Use (100000000+23)*m, correction is 23*47011903.", + "shortcut_strength": 0.9999954, + "pure_math": { + "question": "What is 100000023 * 47011903?", + "options": [ + "4701191381273789", + "4701191381273759", + "4701191381273769", + "4701191381273779" + ], + "answer": "4701191381273769", + "correct_value": 4701191381273769, + "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": "57727520 * 61734632", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 57727520 * 61734632?", + "options": [ + "3563787203472630", + "3563787203472660", + "3563787203472650", + "3563787203472640" + ], + "answer": "3563787203472640", + "correct_value": 3563787203472640, + "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_000863", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 30785082", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 30785082?", + "options": [ + "3078508169214938", + "3078508169214928", + "3078508169214908", + "3078508169214918" + ], + "answer": "3078508169214918", + "correct_value": 3078508169214918, + "explanation": "Distributive: (100000000-1)*30785082 = 100000000*30785082 - 1*30785082." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000019 * 20627484", + "ns_shortcut": "Use (100000000+19)*m, correction is 19*20627484.", + "shortcut_strength": 0.9999962, + "pure_math": { + "question": "What is 100000019 * 20627484?", + "options": [ + "2062748791922216", + "2062748791922196", + "2062748791922206", + "2062748791922186" + ], + "answer": "2062748791922196", + "correct_value": 2062748791922196, + "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": "53678168 * 49985352", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 53678168 * 49985352?", + "options": [ + "2683122122195126", + "2683122122195156", + "2683122122195136", + "2683122122195146" + ], + "answer": "2683122122195136", + "correct_value": 2683122122195136, + "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_000864", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000002 * 27516806", + "ns_shortcut": "Use (100000000+2)*m = 100000000*m + 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 100000002 * 27516806?", + "options": [ + "2751680655033612", + "2751680655033622", + "2751680655033602", + "2751680655033632" + ], + "answer": "2751680655033612", + "correct_value": 2751680655033612, + "explanation": "Distributive: (100000000+2)*27516806 = 100000000*27516806 + 2*27516806." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999958 * 55530514", + "ns_shortcut": "Use (100000000-42)*m, correction is 42*55530514.", + "shortcut_strength": 0.9999916, + "pure_math": { + "question": "What is 99999958 * 55530514?", + "options": [ + "5553049067718432", + "5553049067718402", + "5553049067718422", + "5553049067718412" + ], + "answer": "5553049067718412", + "correct_value": 5553049067718412, + "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": "55500981 * 69307922", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 55500981 * 69307922?", + "options": [ + "3846657662071472", + "3846657662071502", + "3846657662071492", + "3846657662071482" + ], + "answer": "3846657662071482", + "correct_value": 3846657662071482, + "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_000865", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000002 * 31028161", + "ns_shortcut": "Use (100000000+2)*m = 100000000*m + 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 100000002 * 31028161?", + "options": [ + "3102816162056322", + "3102816162056312", + "3102816162056332", + "3102816162056342" + ], + "answer": "3102816162056322", + "correct_value": 3102816162056322, + "explanation": "Distributive: (100000000+2)*31028161 = 100000000*31028161 + 2*31028161." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999978 * 15604404", + "ns_shortcut": "Use (100000000-22)*m, correction is 22*15604404.", + "shortcut_strength": 0.9999956, + "pure_math": { + "question": "What is 99999978 * 15604404?", + "options": [ + "1560440056703102", + "1560440056703132", + "1560440056703122", + "1560440056703112" + ], + "answer": "1560440056703112", + "correct_value": 1560440056703112, + "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": "51406181 * 58235799", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 51406181 * 58235799?", + "options": [ + "2993680024073639", + "2993680024073619", + "2993680024073609", + "2993680024073629" + ], + "answer": "2993680024073619", + "correct_value": 2993680024073619, + "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_000866", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999998 * 99746613", + "ns_shortcut": "Use (100000000-2)*m = 100000000*m - 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 99999998 * 99746613?", + "options": [ + "9974661100506764", + "9974661100506794", + "9974661100506784", + "9974661100506774" + ], + "answer": "9974661100506774", + "correct_value": 9974661100506774, + "explanation": "Distributive: (100000000-2)*99746613 = 100000000*99746613 - 2*99746613." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000050 * 34874941", + "ns_shortcut": "Use (100000000+50)*m, correction is 50*34874941.", + "shortcut_strength": 0.99999, + "pure_math": { + "question": "What is 100000050 * 34874941?", + "options": [ + "3487495843747070", + "3487495843747060", + "3487495843747040", + "3487495843747050" + ], + "answer": "3487495843747050", + "correct_value": 3487495843747050, + "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": "65905258 * 47578817", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 65905258 * 47578817?", + "options": [ + "3135694209719806", + "3135694209719776", + "3135694209719796", + "3135694209719786" + ], + "answer": "3135694209719786", + "correct_value": 3135694209719786, + "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_000867", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 46968017", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 46968017?", + "options": [ + "4696801653031993", + "4696801653031983", + "4696801653031973", + "4696801653032003" + ], + "answer": "4696801653031983", + "correct_value": 4696801653031983, + "explanation": "Distributive: (100000000-1)*46968017 = 100000000*46968017 - 1*46968017." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999980 * 37820749", + "ns_shortcut": "Use (100000000-20)*m, correction is 20*37820749.", + "shortcut_strength": 0.999996, + "pure_math": { + "question": "What is 99999980 * 37820749?", + "options": [ + "3782074143585040", + "3782074143585030", + "3782074143585010", + "3782074143585020" + ], + "answer": "3782074143585020", + "correct_value": 3782074143585020, + "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": "41259665 * 49720590", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 41259665 * 49720590?", + "options": [ + "2051454887002370", + "2051454887002340", + "2051454887002360", + "2051454887002350" + ], + "answer": "2051454887002350", + "correct_value": 2051454887002350, + "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_000868", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 39008943", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 39008943?", + "options": [ + "3900894260991077", + "3900894260991057", + "3900894260991047", + "3900894260991067" + ], + "answer": "3900894260991057", + "correct_value": 3900894260991057, + "explanation": "Distributive: (100000000-1)*39008943 = 100000000*39008943 - 1*39008943." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000040 * 58432281", + "ns_shortcut": "Use (100000000+40)*m, correction is 40*58432281.", + "shortcut_strength": 0.999992, + "pure_math": { + "question": "What is 100000040 * 58432281?", + "options": [ + "5843230437291240", + "5843230437291230", + "5843230437291260", + "5843230437291250" + ], + "answer": "5843230437291240", + "correct_value": 5843230437291240, + "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": "53244604 * 64442759", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 53244604 * 64442759?", + "options": [ + "3431229183622426", + "3431229183622456", + "3431229183622436", + "3431229183622446" + ], + "answer": "3431229183622436", + "correct_value": 3431229183622436, + "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_000869", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999998 * 22693626", + "ns_shortcut": "Use (100000000-2)*m = 100000000*m - 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 99999998 * 22693626?", + "options": [ + "2269362554612748", + "2269362554612758", + "2269362554612768", + "2269362554612738" + ], + "answer": "2269362554612748", + "correct_value": 2269362554612748, + "explanation": "Distributive: (100000000-2)*22693626 = 100000000*22693626 - 2*22693626." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000021 * 28331769", + "ns_shortcut": "Use (100000000+21)*m, correction is 21*28331769.", + "shortcut_strength": 0.9999958, + "pure_math": { + "question": "What is 100000021 * 28331769?", + "options": [ + "2833177494967139", + "2833177494967169", + "2833177494967149", + "2833177494967159" + ], + "answer": "2833177494967149", + "correct_value": 2833177494967149, + "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": "48089849 * 59908437", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 48089849 * 59908437?", + "options": [ + "2880987689156023", + "2880987689156003", + "2880987689156033", + "2880987689156013" + ], + "answer": "2880987689156013", + "correct_value": 2880987689156013, + "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_000870", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000002 * 39096022", + "ns_shortcut": "Use (100000000+2)*m = 100000000*m + 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 100000002 * 39096022?", + "options": [ + "3909602278192044", + "3909602278192064", + "3909602278192034", + "3909602278192054" + ], + "answer": "3909602278192044", + "correct_value": 3909602278192044, + "explanation": "Distributive: (100000000+2)*39096022 = 100000000*39096022 + 2*39096022." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000041 * 20449039", + "ns_shortcut": "Use (100000000+41)*m, correction is 41*20449039.", + "shortcut_strength": 0.9999918, + "pure_math": { + "question": "What is 100000041 * 20449039?", + "options": [ + "2044904738410609", + "2044904738410599", + "2044904738410619", + "2044904738410589" + ], + "answer": "2044904738410599", + "correct_value": 2044904738410599, + "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": "42814051 * 65167246", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 42814051 * 65167246?", + "options": [ + "2790073793773546", + "2790073793773566", + "2790073793773556", + "2790073793773536" + ], + "answer": "2790073793773546", + "correct_value": 2790073793773546, + "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_000871", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999998 * 71970086", + "ns_shortcut": "Use (100000000-2)*m = 100000000*m - 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 99999998 * 71970086?", + "options": [ + "7197008456059838", + "7197008456059828", + "7197008456059848", + "7197008456059818" + ], + "answer": "7197008456059828", + "correct_value": 7197008456059828, + "explanation": "Distributive: (100000000-2)*71970086 = 100000000*71970086 - 2*71970086." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000026 * 19361357", + "ns_shortcut": "Use (100000000+26)*m, correction is 26*19361357.", + "shortcut_strength": 0.9999948, + "pure_math": { + "question": "What is 100000026 * 19361357?", + "options": [ + "1936136203395272", + "1936136203395302", + "1936136203395292", + "1936136203395282" + ], + "answer": "1936136203395282", + "correct_value": 1936136203395282, + "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": "59665925 * 43581793", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 59665925 * 43581793?", + "options": [ + "2600347992503515", + "2600347992503525", + "2600347992503535", + "2600347992503545" + ], + "answer": "2600347992503525", + "correct_value": 2600347992503525, + "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_000872", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 90154379", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 90154379?", + "options": [ + "9015437809845641", + "9015437809845631", + "9015437809845611", + "9015437809845621" + ], + "answer": "9015437809845621", + "correct_value": 9015437809845621, + "explanation": "Distributive: (100000000-1)*90154379 = 100000000*90154379 - 1*90154379." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000048 * 95031494", + "ns_shortcut": "Use (100000000+48)*m, correction is 48*95031494.", + "shortcut_strength": 0.9999904, + "pure_math": { + "question": "What is 100000048 * 95031494?", + "options": [ + "9503153961511712", + "9503153961511722", + "9503153961511702", + "9503153961511732" + ], + "answer": "9503153961511712", + "correct_value": 9503153961511712, + "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": "67616669 * 60000860", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 67616669 * 60000860?", + "options": [ + "4057058290335330", + "4057058290335350", + "4057058290335360", + "4057058290335340" + ], + "answer": "4057058290335340", + "correct_value": 4057058290335340, + "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_000873", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999998 * 93357007", + "ns_shortcut": "Use (100000000-2)*m = 100000000*m - 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 99999998 * 93357007?", + "options": [ + "9335700513285986", + "9335700513286006", + "9335700513285976", + "9335700513285996" + ], + "answer": "9335700513285986", + "correct_value": 9335700513285986, + "explanation": "Distributive: (100000000-2)*93357007 = 100000000*93357007 - 2*93357007." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000035 * 50303099", + "ns_shortcut": "Use (100000000+35)*m, correction is 35*50303099.", + "shortcut_strength": 0.999993, + "pure_math": { + "question": "What is 100000035 * 50303099?", + "options": [ + "5030311660608475", + "5030311660608455", + "5030311660608485", + "5030311660608465" + ], + "answer": "5030311660608465", + "correct_value": 5030311660608465, + "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": "50443072 * 60256521", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 50443072 * 60256521?", + "options": [ + "3039524027272532", + "3039524027272522", + "3039524027272502", + "3039524027272512" + ], + "answer": "3039524027272512", + "correct_value": 3039524027272512, + "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_000874", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999998 * 82374135", + "ns_shortcut": "Use (100000000-2)*m = 100000000*m - 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 99999998 * 82374135?", + "options": [ + "8237413335251740", + "8237413335251720", + "8237413335251730", + "8237413335251750" + ], + "answer": "8237413335251730", + "correct_value": 8237413335251730, + "explanation": "Distributive: (100000000-2)*82374135 = 100000000*82374135 - 2*82374135." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999965 * 47092843", + "ns_shortcut": "Use (100000000-35)*m, correction is 35*47092843.", + "shortcut_strength": 0.999993, + "pure_math": { + "question": "What is 99999965 * 47092843?", + "options": [ + "4709282651750485", + "4709282651750505", + "4709282651750495", + "4709282651750515" + ], + "answer": "4709282651750495", + "correct_value": 4709282651750495, + "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": "59681588 * 44673918", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 59681588 * 44673918?", + "options": [ + "2666210368421794", + "2666210368421804", + "2666210368421784", + "2666210368421774" + ], + "answer": "2666210368421784", + "correct_value": 2666210368421784, + "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_000875", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 38935409", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 38935409?", + "options": [ + "3893540861064581", + "3893540861064611", + "3893540861064591", + "3893540861064601" + ], + "answer": "3893540861064591", + "correct_value": 3893540861064591, + "explanation": "Distributive: (100000000-1)*38935409 = 100000000*38935409 - 1*38935409." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000044 * 54008327", + "ns_shortcut": "Use (100000000+44)*m, correction is 44*54008327.", + "shortcut_strength": 0.9999912, + "pure_math": { + "question": "What is 100000044 * 54008327?", + "options": [ + "5400835076366388", + "5400835076366408", + "5400835076366378", + "5400835076366398" + ], + "answer": "5400835076366388", + "correct_value": 5400835076366388, + "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": "42575121 * 43094547", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 42575121 * 43094547?", + "options": [ + "1834755552965177", + "1834755552965197", + "1834755552965187", + "1834755552965207" + ], + "answer": "1834755552965187", + "correct_value": 1834755552965187, + "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_000876", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 79750301", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 79750301?", + "options": [ + "7975030020249709", + "7975030020249699", + "7975030020249719", + "7975030020249689" + ], + "answer": "7975030020249699", + "correct_value": 7975030020249699, + "explanation": "Distributive: (100000000-1)*79750301 = 100000000*79750301 - 1*79750301." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999973 * 88478879", + "ns_shortcut": "Use (100000000-27)*m, correction is 27*88478879.", + "shortcut_strength": 0.9999946, + "pure_math": { + "question": "What is 99999973 * 88478879?", + "options": [ + "8847885511070257", + "8847885511070287", + "8847885511070277", + "8847885511070267" + ], + "answer": "8847885511070267", + "correct_value": 8847885511070267, + "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": "56521380 * 53270661", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 56521380 * 53270661?", + "options": [ + "3010931273232200", + "3010931273232180", + "3010931273232190", + "3010931273232170" + ], + "answer": "3010931273232180", + "correct_value": 3010931273232180, + "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_000877", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999998 * 25638062", + "ns_shortcut": "Use (100000000-2)*m = 100000000*m - 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 99999998 * 25638062?", + "options": [ + "2563806148723896", + "2563806148723876", + "2563806148723866", + "2563806148723886" + ], + "answer": "2563806148723876", + "correct_value": 2563806148723876, + "explanation": "Distributive: (100000000-2)*25638062 = 100000000*25638062 - 2*25638062." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999970 * 28377366", + "ns_shortcut": "Use (100000000-30)*m, correction is 30*28377366.", + "shortcut_strength": 0.999994, + "pure_math": { + "question": "What is 99999970 * 28377366?", + "options": [ + "2837735748679020", + "2837735748679010", + "2837735748679040", + "2837735748679030" + ], + "answer": "2837735748679020", + "correct_value": 2837735748679020, + "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": "52123956 * 53106015", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 52123956 * 53106015?", + "options": [ + "2768095589195350", + "2768095589195340", + "2768095589195360", + "2768095589195330" + ], + "answer": "2768095589195340", + "correct_value": 2768095589195340, + "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_000878", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999998 * 61846177", + "ns_shortcut": "Use (100000000-2)*m = 100000000*m - 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 99999998 * 61846177?", + "options": [ + "6184617576307646", + "6184617576307636", + "6184617576307666", + "6184617576307656" + ], + "answer": "6184617576307646", + "correct_value": 6184617576307646, + "explanation": "Distributive: (100000000-2)*61846177 = 100000000*61846177 - 2*61846177." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999961 * 36839493", + "ns_shortcut": "Use (100000000-39)*m, correction is 39*36839493.", + "shortcut_strength": 0.9999922, + "pure_math": { + "question": "What is 99999961 * 36839493?", + "options": [ + "3683947863259783", + "3683947863259793", + "3683947863259763", + "3683947863259773" + ], + "answer": "3683947863259773", + "correct_value": 3683947863259773, + "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": "65756201 * 47335134", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 65756201 * 47335134?", + "options": [ + "3112578585665934", + "3112578585665924", + "3112578585665944", + "3112578585665954" + ], + "answer": "3112578585665934", + "correct_value": 3112578585665934, + "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_000879", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999998 * 67685899", + "ns_shortcut": "Use (100000000-2)*m = 100000000*m - 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 99999998 * 67685899?", + "options": [ + "6768589764628222", + "6768589764628202", + "6768589764628192", + "6768589764628212" + ], + "answer": "6768589764628202", + "correct_value": 6768589764628202, + "explanation": "Distributive: (100000000-2)*67685899 = 100000000*67685899 - 2*67685899." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000039 * 40951243", + "ns_shortcut": "Use (100000000+39)*m, correction is 39*40951243.", + "shortcut_strength": 0.9999922, + "pure_math": { + "question": "What is 100000039 * 40951243?", + "options": [ + "4095125897098467", + "4095125897098497", + "4095125897098487", + "4095125897098477" + ], + "answer": "4095125897098477", + "correct_value": 4095125897098477, + "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": "60843296 * 47075239", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 60843296 * 47075239?", + "options": [ + "2864212700747734", + "2864212700747744", + "2864212700747754", + "2864212700747764" + ], + "answer": "2864212700747744", + "correct_value": 2864212700747744, + "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_000880", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999998 * 42494127", + "ns_shortcut": "Use (100000000-2)*m = 100000000*m - 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 99999998 * 42494127?", + "options": [ + "4249412615011756", + "4249412615011746", + "4249412615011736", + "4249412615011766" + ], + "answer": "4249412615011746", + "correct_value": 4249412615011746, + "explanation": "Distributive: (100000000-2)*42494127 = 100000000*42494127 - 2*42494127." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999952 * 92391322", + "ns_shortcut": "Use (100000000-48)*m, correction is 48*92391322.", + "shortcut_strength": 0.9999904, + "pure_math": { + "question": "What is 99999952 * 92391322?", + "options": [ + "9239127765216554", + "9239127765216544", + "9239127765216564", + "9239127765216534" + ], + "answer": "9239127765216544", + "correct_value": 9239127765216544, + "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": "47966682 * 56001667", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 47966682 * 56001667?", + "options": [ + "2686214152458914", + "2686214152458894", + "2686214152458884", + "2686214152458904" + ], + "answer": "2686214152458894", + "correct_value": 2686214152458894, + "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_000881", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 41373052", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 41373052?", + "options": [ + "4137305158626958", + "4137305158626948", + "4137305158626938", + "4137305158626968" + ], + "answer": "4137305158626948", + "correct_value": 4137305158626948, + "explanation": "Distributive: (100000000-1)*41373052 = 100000000*41373052 - 1*41373052." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000021 * 75240070", + "ns_shortcut": "Use (100000000+21)*m, correction is 21*75240070.", + "shortcut_strength": 0.9999958, + "pure_math": { + "question": "What is 100000021 * 75240070?", + "options": [ + "7524008580041490", + "7524008580041470", + "7524008580041460", + "7524008580041480" + ], + "answer": "7524008580041470", + "correct_value": 7524008580041470, + "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": "60323316 * 67590361", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 60323316 * 67590361?", + "options": [ + "4077274705157066", + "4077274705157086", + "4077274705157096", + "4077274705157076" + ], + "answer": "4077274705157076", + "correct_value": 4077274705157076, + "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_000882", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999998 * 48835846", + "ns_shortcut": "Use (100000000-2)*m = 100000000*m - 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 99999998 * 48835846?", + "options": [ + "4883584502328318", + "4883584502328308", + "4883584502328298", + "4883584502328328" + ], + "answer": "4883584502328308", + "correct_value": 4883584502328308, + "explanation": "Distributive: (100000000-2)*48835846 = 100000000*48835846 - 2*48835846." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999952 * 52278106", + "ns_shortcut": "Use (100000000-48)*m, correction is 48*52278106.", + "shortcut_strength": 0.9999904, + "pure_math": { + "question": "What is 99999952 * 52278106?", + "options": [ + "5227808090650902", + "5227808090650932", + "5227808090650912", + "5227808090650922" + ], + "answer": "5227808090650912", + "correct_value": 5227808090650912, + "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": "53517571 * 65374488", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 53517571 * 65374488?", + "options": [ + "3498683803128648", + "3498683803128658", + "3498683803128668", + "3498683803128638" + ], + "answer": "3498683803128648", + "correct_value": 3498683803128648, + "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_000883", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000001 * 84181946", + "ns_shortcut": "Use (100000000+1)*m = 100000000*m + 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 100000001 * 84181946?", + "options": [ + "8418194684181966", + "8418194684181956", + "8418194684181946", + "8418194684181936" + ], + "answer": "8418194684181946", + "correct_value": 8418194684181946, + "explanation": "Distributive: (100000000+1)*84181946 = 100000000*84181946 + 1*84181946." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000040 * 56874167", + "ns_shortcut": "Use (100000000+40)*m, correction is 40*56874167.", + "shortcut_strength": 0.999992, + "pure_math": { + "question": "What is 100000040 * 56874167?", + "options": [ + "5687418974966690", + "5687418974966700", + "5687418974966680", + "5687418974966670" + ], + "answer": "5687418974966680", + "correct_value": 5687418974966680, + "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": "61366689 * 59084458", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 61366689 * 59084458?", + "options": [ + "3625817558819582", + "3625817558819552", + "3625817558819562", + "3625817558819572" + ], + "answer": "3625817558819562", + "correct_value": 3625817558819562, + "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_000884", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999998 * 49514706", + "ns_shortcut": "Use (100000000-2)*m = 100000000*m - 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 99999998 * 49514706?", + "options": [ + "4951470500970578", + "4951470500970598", + "4951470500970608", + "4951470500970588" + ], + "answer": "4951470500970588", + "correct_value": 4951470500970588, + "explanation": "Distributive: (100000000-2)*49514706 = 100000000*49514706 - 2*49514706." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000038 * 83538571", + "ns_shortcut": "Use (100000000+38)*m, correction is 38*83538571.", + "shortcut_strength": 0.9999924, + "pure_math": { + "question": "What is 100000038 * 83538571?", + "options": [ + "8353860274465718", + "8353860274465708", + "8353860274465688", + "8353860274465698" + ], + "answer": "8353860274465698", + "correct_value": 8353860274465698, + "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": "43635785 * 46206045", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 43635785 * 46206045?", + "options": [ + "2016237045320345", + "2016237045320325", + "2016237045320315", + "2016237045320335" + ], + "answer": "2016237045320325", + "correct_value": 2016237045320325, + "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_000885", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000002 * 64252439", + "ns_shortcut": "Use (100000000+2)*m = 100000000*m + 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 100000002 * 64252439?", + "options": [ + "6425244028504898", + "6425244028504868", + "6425244028504888", + "6425244028504878" + ], + "answer": "6425244028504878", + "correct_value": 6425244028504878, + "explanation": "Distributive: (100000000+2)*64252439 = 100000000*64252439 + 2*64252439." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000034 * 52169819", + "ns_shortcut": "Use (100000000+34)*m, correction is 34*52169819.", + "shortcut_strength": 0.9999932, + "pure_math": { + "question": "What is 100000034 * 52169819?", + "options": [ + "5216983673773836", + "5216983673773856", + "5216983673773846", + "5216983673773866" + ], + "answer": "5216983673773846", + "correct_value": 5216983673773846, + "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": "55340464 * 65628592", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 55340464 * 65628592?", + "options": [ + "3631916732946708", + "3631916732946698", + "3631916732946688", + "3631916732946678" + ], + "answer": "3631916732946688", + "correct_value": 3631916732946688, + "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_000886", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 38978527", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 38978527?", + "options": [ + "3897852661021483", + "3897852661021473", + "3897852661021463", + "3897852661021493" + ], + "answer": "3897852661021473", + "correct_value": 3897852661021473, + "explanation": "Distributive: (100000000-1)*38978527 = 100000000*38978527 - 1*38978527." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000022 * 24446105", + "ns_shortcut": "Use (100000000+22)*m, correction is 22*24446105.", + "shortcut_strength": 0.9999956, + "pure_math": { + "question": "What is 100000022 * 24446105?", + "options": [ + "2444611037814310", + "2444611037814320", + "2444611037814330", + "2444611037814300" + ], + "answer": "2444611037814310", + "correct_value": 2444611037814310, + "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": "61816741 * 48185998", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 61816741 * 48185998?", + "options": [ + "2978701358192508", + "2978701358192518", + "2978701358192538", + "2978701358192528" + ], + "answer": "2978701358192518", + "correct_value": 2978701358192518, + "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_000887", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000001 * 10412348", + "ns_shortcut": "Use (100000000+1)*m = 100000000*m + 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 100000001 * 10412348?", + "options": [ + "1041234810412338", + "1041234810412348", + "1041234810412358", + "1041234810412368" + ], + "answer": "1041234810412348", + "correct_value": 1041234810412348, + "explanation": "Distributive: (100000000+1)*10412348 = 100000000*10412348 + 1*10412348." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000019 * 19181068", + "ns_shortcut": "Use (100000000+19)*m, correction is 19*19181068.", + "shortcut_strength": 0.9999962, + "pure_math": { + "question": "What is 100000019 * 19181068?", + "options": [ + "1918107164440302", + "1918107164440282", + "1918107164440312", + "1918107164440292" + ], + "answer": "1918107164440292", + "correct_value": 1918107164440292, + "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": "58429844 * 66997400", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 58429844 * 66997400?", + "options": [ + "3914647630405600", + "3914647630405620", + "3914647630405590", + "3914647630405610" + ], + "answer": "3914647630405600", + "correct_value": 3914647630405600, + "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_000888", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 68270032", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 68270032?", + "options": [ + "6827003131729958", + "6827003131729978", + "6827003131729968", + "6827003131729988" + ], + "answer": "6827003131729968", + "correct_value": 6827003131729968, + "explanation": "Distributive: (100000000-1)*68270032 = 100000000*68270032 - 1*68270032." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000034 * 47285126", + "ns_shortcut": "Use (100000000+34)*m, correction is 34*47285126.", + "shortcut_strength": 0.9999932, + "pure_math": { + "question": "What is 100000034 * 47285126?", + "options": [ + "4728514207694284", + "4728514207694294", + "4728514207694304", + "4728514207694274" + ], + "answer": "4728514207694284", + "correct_value": 4728514207694284, + "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": "67823827 * 47268966", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 67823827 * 47268966?", + "options": [ + "3205962172452872", + "3205962172452892", + "3205962172452902", + "3205962172452882" + ], + "answer": "3205962172452882", + "correct_value": 3205962172452882, + "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_000889", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000002 * 23187264", + "ns_shortcut": "Use (100000000+2)*m = 100000000*m + 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 100000002 * 23187264?", + "options": [ + "2318726446374548", + "2318726446374518", + "2318726446374528", + "2318726446374538" + ], + "answer": "2318726446374528", + "correct_value": 2318726446374528, + "explanation": "Distributive: (100000000+2)*23187264 = 100000000*23187264 + 2*23187264." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999974 * 63756040", + "ns_shortcut": "Use (100000000-26)*m, correction is 26*63756040.", + "shortcut_strength": 0.9999948, + "pure_math": { + "question": "What is 99999974 * 63756040?", + "options": [ + "6375602342342950", + "6375602342342960", + "6375602342342970", + "6375602342342980" + ], + "answer": "6375602342342960", + "correct_value": 6375602342342960, + "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": "66079128 * 48960611", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 66079128 * 48960611?", + "options": [ + "3235274481227218", + "3235274481227208", + "3235274481227228", + "3235274481227198" + ], + "answer": "3235274481227208", + "correct_value": 3235274481227208, + "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_000890", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 88867841", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 88867841?", + "options": [ + "8886784011132159", + "8886784011132169", + "8886784011132149", + "8886784011132179" + ], + "answer": "8886784011132159", + "correct_value": 8886784011132159, + "explanation": "Distributive: (100000000-1)*88867841 = 100000000*88867841 - 1*88867841." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000016 * 36550670", + "ns_shortcut": "Use (100000000+16)*m, correction is 16*36550670.", + "shortcut_strength": 0.9999968, + "pure_math": { + "question": "What is 100000016 * 36550670?", + "options": [ + "3655067584810730", + "3655067584810720", + "3655067584810740", + "3655067584810710" + ], + "answer": "3655067584810720", + "correct_value": 3655067584810720, + "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": "65244100 * 47507289", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 65244100 * 47507289?", + "options": [ + "3099570314244920", + "3099570314244910", + "3099570314244890", + "3099570314244900" + ], + "answer": "3099570314244900", + "correct_value": 3099570314244900, + "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_000891", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000002 * 62249121", + "ns_shortcut": "Use (100000000+2)*m = 100000000*m + 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 100000002 * 62249121?", + "options": [ + "6224912224498252", + "6224912224498262", + "6224912224498232", + "6224912224498242" + ], + "answer": "6224912224498242", + "correct_value": 6224912224498242, + "explanation": "Distributive: (100000000+2)*62249121 = 100000000*62249121 + 2*62249121." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999974 * 67991971", + "ns_shortcut": "Use (100000000-26)*m, correction is 26*67991971.", + "shortcut_strength": 0.9999948, + "pure_math": { + "question": "What is 99999974 * 67991971?", + "options": [ + "6799195332208764", + "6799195332208774", + "6799195332208754", + "6799195332208744" + ], + "answer": "6799195332208754", + "correct_value": 6799195332208754, + "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": "62424930 * 46487174", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 62424930 * 46487174?", + "options": [ + "2901958582847830", + "2901958582847810", + "2901958582847820", + "2901958582847840" + ], + "answer": "2901958582847820", + "correct_value": 2901958582847820, + "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_000892", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999998 * 60357342", + "ns_shortcut": "Use (100000000-2)*m = 100000000*m - 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 99999998 * 60357342?", + "options": [ + "6035734079285306", + "6035734079285326", + "6035734079285336", + "6035734079285316" + ], + "answer": "6035734079285316", + "correct_value": 6035734079285316, + "explanation": "Distributive: (100000000-2)*60357342 = 100000000*60357342 - 2*60357342." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999970 * 40776961", + "ns_shortcut": "Use (100000000-30)*m, correction is 30*40776961.", + "shortcut_strength": 0.999994, + "pure_math": { + "question": "What is 99999970 * 40776961?", + "options": [ + "4077694876691160", + "4077694876691170", + "4077694876691190", + "4077694876691180" + ], + "answer": "4077694876691170", + "correct_value": 4077694876691170, + "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": "41822794 * 55204750", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 41822794 * 55204750?", + "options": [ + "2308816887071510", + "2308816887071520", + "2308816887071500", + "2308816887071490" + ], + "answer": "2308816887071500", + "correct_value": 2308816887071500, + "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_000893", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000002 * 96177608", + "ns_shortcut": "Use (100000000+2)*m = 100000000*m + 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 100000002 * 96177608?", + "options": [ + "9617760992355226", + "9617760992355216", + "9617760992355236", + "9617760992355206" + ], + "answer": "9617760992355216", + "correct_value": 9617760992355216, + "explanation": "Distributive: (100000000+2)*96177608 = 100000000*96177608 + 2*96177608." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000048 * 13055975", + "ns_shortcut": "Use (100000000+48)*m, correction is 48*13055975.", + "shortcut_strength": 0.9999904, + "pure_math": { + "question": "What is 100000048 * 13055975?", + "options": [ + "1305598126686820", + "1305598126686790", + "1305598126686810", + "1305598126686800" + ], + "answer": "1305598126686800", + "correct_value": 1305598126686800, + "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": "49063772 * 40564314", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 49063772 * 40564314?", + "options": [ + "1990238253432408", + "1990238253432428", + "1990238253432418", + "1990238253432398" + ], + "answer": "1990238253432408", + "correct_value": 1990238253432408, + "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_000894", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000001 * 62443962", + "ns_shortcut": "Use (100000000+1)*m = 100000000*m + 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 100000001 * 62443962?", + "options": [ + "6244396262443982", + "6244396262443952", + "6244396262443962", + "6244396262443972" + ], + "answer": "6244396262443962", + "correct_value": 6244396262443962, + "explanation": "Distributive: (100000000+1)*62443962 = 100000000*62443962 + 1*62443962." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000050 * 11439649", + "ns_shortcut": "Use (100000000+50)*m, correction is 50*11439649.", + "shortcut_strength": 0.99999, + "pure_math": { + "question": "What is 100000050 * 11439649?", + "options": [ + "1143965471982470", + "1143965471982450", + "1143965471982460", + "1143965471982440" + ], + "answer": "1143965471982450", + "correct_value": 1143965471982450, + "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": "60008909 * 44603435", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 60008909 * 44603435?", + "options": [ + "2676603472002425", + "2676603472002405", + "2676603472002415", + "2676603472002435" + ], + "answer": "2676603472002415", + "correct_value": 2676603472002415, + "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_000895", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000002 * 93938382", + "ns_shortcut": "Use (100000000+2)*m = 100000000*m + 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 100000002 * 93938382?", + "options": [ + "9393838387876784", + "9393838387876764", + "9393838387876774", + "9393838387876754" + ], + "answer": "9393838387876764", + "correct_value": 9393838387876764, + "explanation": "Distributive: (100000000+2)*93938382 = 100000000*93938382 + 2*93938382." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000034 * 57543316", + "ns_shortcut": "Use (100000000+34)*m, correction is 34*57543316.", + "shortcut_strength": 0.9999932, + "pure_math": { + "question": "What is 100000034 * 57543316?", + "options": [ + "5754333556472744", + "5754333556472734", + "5754333556472754", + "5754333556472764" + ], + "answer": "5754333556472744", + "correct_value": 5754333556472744, + "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": "53330466 * 57581898", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 53330466 * 57581898?", + "options": [ + "3070869453504478", + "3070869453504458", + "3070869453504468", + "3070869453504488" + ], + "answer": "3070869453504468", + "correct_value": 3070869453504468, + "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_000896", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 37201429", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 37201429?", + "options": [ + "3720142862798581", + "3720142862798591", + "3720142862798571", + "3720142862798561" + ], + "answer": "3720142862798571", + "correct_value": 3720142862798571, + "explanation": "Distributive: (100000000-1)*37201429 = 100000000*37201429 - 1*37201429." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000022 * 85255181", + "ns_shortcut": "Use (100000000+22)*m, correction is 22*85255181.", + "shortcut_strength": 0.9999956, + "pure_math": { + "question": "What is 100000022 * 85255181?", + "options": [ + "8525519975613992", + "8525519975613982", + "8525519975613972", + "8525519975614002" + ], + "answer": "8525519975613982", + "correct_value": 8525519975613982, + "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": "46798750 * 51567591", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 46798750 * 51567591?", + "options": [ + "2413298799311270", + "2413298799311260", + "2413298799311250", + "2413298799311240" + ], + "answer": "2413298799311250", + "correct_value": 2413298799311250, + "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_000897", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999998 * 72311256", + "ns_shortcut": "Use (100000000-2)*m = 100000000*m - 2*m.", + "shortcut_strength": 0.99999998, + "pure_math": { + "question": "What is 99999998 * 72311256?", + "options": [ + "7231125455377498", + "7231125455377508", + "7231125455377478", + "7231125455377488" + ], + "answer": "7231125455377488", + "correct_value": 7231125455377488, + "explanation": "Distributive: (100000000-2)*72311256 = 100000000*72311256 - 2*72311256." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99999956 * 30195196", + "ns_shortcut": "Use (100000000-44)*m, correction is 44*30195196.", + "shortcut_strength": 0.9999912, + "pure_math": { + "question": "What is 99999956 * 30195196?", + "options": [ + "3019518271411386", + "3019518271411366", + "3019518271411396", + "3019518271411376" + ], + "answer": "3019518271411376", + "correct_value": 3019518271411376, + "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": "40015583 * 52744963", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 40015583 * 52744963?", + "options": [ + "2110620444758449", + "2110620444758439", + "2110620444758419", + "2110620444758429" + ], + "answer": "2110620444758429", + "correct_value": 2110620444758429, + "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_000898", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 78893211", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 78893211?", + "options": [ + "7889321021106779", + "7889321021106799", + "7889321021106789", + "7889321021106809" + ], + "answer": "7889321021106789", + "correct_value": 7889321021106789, + "explanation": "Distributive: (100000000-1)*78893211 = 100000000*78893211 - 1*78893211." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000028 * 94209689", + "ns_shortcut": "Use (100000000+28)*m, correction is 28*94209689.", + "shortcut_strength": 0.9999944, + "pure_math": { + "question": "What is 100000028 * 94209689?", + "options": [ + "9420971537871312", + "9420971537871292", + "9420971537871282", + "9420971537871302" + ], + "answer": "9420971537871292", + "correct_value": 9420971537871292, + "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": "46388333 * 47323378", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 46388333 * 47323378?", + "options": [ + "2195252617348894", + "2195252617348884", + "2195252617348874", + "2195252617348864" + ], + "answer": "2195252617348874", + "correct_value": 2195252617348874, + "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_000899", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000001 * 45513832", + "ns_shortcut": "Use (100000000+1)*m = 100000000*m + 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 100000001 * 45513832?", + "options": [ + "4551383245513842", + "4551383245513822", + "4551383245513852", + "4551383245513832" + ], + "answer": "4551383245513832", + "correct_value": 4551383245513832, + "explanation": "Distributive: (100000000+1)*45513832 = 100000000*45513832 + 1*45513832." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000047 * 19369522", + "ns_shortcut": "Use (100000000+47)*m, correction is 47*19369522.", + "shortcut_strength": 0.9999906, + "pure_math": { + "question": "What is 100000047 * 19369522?", + "options": [ + "1936953110367554", + "1936953110367534", + "1936953110367544", + "1936953110367524" + ], + "answer": "1936953110367534", + "correct_value": 1936953110367534, + "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": "49865555 * 57121772", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 49865555 * 57121772?", + "options": [ + "2848408863363480", + "2848408863363470", + "2848408863363450", + "2848408863363460" + ], + "answer": "2848408863363460", + "correct_value": 2848408863363460, + "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_000900", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "100000001 * 48219923", + "ns_shortcut": "Use (100000000+1)*m = 100000000*m + 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 100000001 * 48219923?", + "options": [ + "4821992348219943", + "4821992348219933", + "4821992348219913", + "4821992348219923" + ], + "answer": "4821992348219923", + "correct_value": 4821992348219923, + "explanation": "Distributive: (100000000+1)*48219923 = 100000000*48219923 + 1*48219923." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000045 * 10563752", + "ns_shortcut": "Use (100000000+45)*m, correction is 45*10563752.", + "shortcut_strength": 0.999991, + "pure_math": { + "question": "What is 100000045 * 10563752?", + "options": [ + "1056375675368840", + "1056375675368830", + "1056375675368860", + "1056375675368850" + ], + "answer": "1056375675368840", + "correct_value": 1056375675368840, + "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": "57924166 * 43633360", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 57924166 * 43633360?", + "options": [ + "2527425987777760", + "2527425987777780", + "2527425987777750", + "2527425987777770" + ], + "answer": "2527425987777760", + "correct_value": 2527425987777760, + "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_000901", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99999999 * 35051514", + "ns_shortcut": "Use (100000000-1)*m = 100000000*m - 1*m.", + "shortcut_strength": 0.99999999, + "pure_math": { + "question": "What is 99999999 * 35051514?", + "options": [ + "3505151364948496", + "3505151364948506", + "3505151364948476", + "3505151364948486" + ], + "answer": "3505151364948486", + "correct_value": 3505151364948486, + "explanation": "Distributive: (100000000-1)*35051514 = 100000000*35051514 - 1*35051514." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "100000035 * 32212036", + "ns_shortcut": "Use (100000000+35)*m, correction is 35*32212036.", + "shortcut_strength": 0.999993, + "pure_math": { + "question": "What is 100000035 * 32212036?", + "options": [ + "3221204727421250", + "3221204727421270", + "3221204727421260", + "3221204727421280" + ], + "answer": "3221204727421260", + "correct_value": 3221204727421260, + "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": "43433891 * 61081095", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 43433891 * 61081095?", + "options": [ + "2652989622390645", + "2652989622390665", + "2652989622390635", + "2652989622390655" + ], + "answer": "2652989622390645", + "correct_value": 2652989622390645, + "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_000902", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 21730905/45226705 or 6140251/11581687?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.0496806110938216, + "pure_math": { + "question": "Which is larger: 21730905/45226705 or 6140251/11581687?", + "options": [ + "21730905/45226705", + "6140251/11581687", + "46832308/99422811", + "55569007/97130302" + ], + "answer": "6140251/11581687", + "correct_value": 0.5301689641586756, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 36612190/73761490 or 39779970/78891140?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.007879608781458103, + "pure_math": { + "question": "Which is larger: 36612190/73761490 or 39779970/78891140?", + "options": [ + "26208466/50706675", + "31889334/68473662", + "39779970/78891140", + "36612190/73761490" + ], + "answer": "39779970/78891140", + "correct_value": 0.5042387522857447, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 5644534/10262788 or 35994084/62058765?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 5644534/10262788 or 35994084/62058765?", + "options": [ + "33300018/51561908", + "35994084/62058765", + "48141832/92953127", + "5644534/10262788" + ], + "answer": "35994084/62058765", + "correct_value": 0.5800000048341278, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000903", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 14076765/28732895 or 21139658/41111775?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.02428150501110682, + "pure_math": { + "question": "Which is larger: 14076765/28732895 or 21139658/41111775?", + "options": [ + "14388205/34015448", + "12166034/20824106", + "14076765/28732895", + "21139658/41111775" + ], + "answer": "21139658/41111775", + "correct_value": 0.5141995936687238, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 35319697/71467901 or 14860659/28911627?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.019799218646269323, + "pure_math": { + "question": "Which is larger: 35319697/71467901 or 14860659/28911627?", + "options": [ + "35319697/71467901", + "24130564/46069767", + "46137077/90408505", + "14860659/28911627" + ], + "answer": "14860659/28911627", + "correct_value": 0.5140028612018273, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 38908731/70743146 or 27473714/47368471?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 38908731/70743146 or 27473714/47368471?", + "options": [ + "14359562/22274894", + "27473714/47368471", + "38908731/70743146", + "41763046/71773393" + ], + "answer": "27473714/47368471", + "correct_value": 0.5800000173110929, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000904", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 27263074/56510920 or 6622529/12840760?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.033303719673775434, + "pure_math": { + "question": "Which is larger: 27263074/56510920 or 6622529/12840760?", + "options": [ + "27263074/56510920", + "6551349/11891228", + "15359349/30836721", + "6622529/12840760" + ], + "answer": "6622529/12840760", + "correct_value": 0.5157427597743436, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 37611160/75658751 or 39596482/78586575?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.006742302365318309, + "pure_math": { + "question": "Which is larger: 37611160/75658751 or 39596482/78586575?", + "options": [ + "37611160/75658751", + "39596482/78586575", + "16976980/41213685", + "10813537/18360232" + ], + "answer": "39596482/78586575", + "correct_value": 0.5038580953553453, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 37004251/67280455 or 49150389/84742049?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 37004251/67280455 or 49150389/84742049?", + "options": [ + "22559818/38092843", + "49150389/84742049", + "37004251/67280455", + "37809608/58907136" + ], + "answer": "49150389/84742049", + "correct_value": 0.5800000068443, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000905", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 18914372/39008071 or 45471416/88562878?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.028552969931401495, + "pure_math": { + "question": "Which is larger: 18914372/39008071 or 45471416/88562878?", + "options": [ + "42193499/76937138", + "45471416/88562878", + "16305225/37804257", + "18914372/39008071" + ], + "answer": "45471416/88562878", + "correct_value": 0.5134365213379809, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 22580860/45410764 or 40920617/80705709?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.009777098394263217, + "pure_math": { + "question": "Which is larger: 22580860/45410764 or 40920617/80705709?", + "options": [ + "22580860/45410764", + "15074431/31159614", + "40920617/80705709", + "9702459/23289113" + ], + "answer": "40920617/80705709", + "correct_value": 0.5070349732012143, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 8504259/15462289 or 7100702/12242589?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 8504259/15462289 or 7100702/12242589?", + "options": [ + "51385362/96768874", + "8504259/15462289", + "7117082/13335064", + "7100702/12242589" + ], + "answer": "7100702/12242589", + "correct_value": 0.5800000310391863, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000906", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 26923516/55609756 or 30503684/59359604?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.02972855987560208, + "pure_math": { + "question": "Which is larger: 26923516/55609756 or 30503684/59359604?", + "options": [ + "23302024/38575979", + "30503684/59359604", + "22364764/45977355", + "26923516/55609756" + ], + "answer": "30503684/59359604", + "correct_value": 0.5138795063390248, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 44488499/90804234 or 42447251/83285095?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01972343804947363, + "pure_math": { + "question": "Which is larger: 44488499/90804234 or 42447251/83285095?", + "options": [ + "34641915/60662864", + "24762547/44224047", + "42447251/83285095", + "44488499/90804234" + ], + "answer": "42447251/83285095", + "correct_value": 0.5096620349655602, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 31530244/57327715 or 22128576/38152716?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 31530244/57327715 or 22128576/38152716?", + "options": [ + "10963562/16331897", + "22128576/38152716", + "31530244/57327715", + "19251024/29198571" + ], + "answer": "22128576/38152716", + "correct_value": 0.5800000188715267, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000907", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 26575920/55200638 or 38555814/74303297?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.037455493376471805, + "pure_math": { + "question": "Which is larger: 26575920/55200638 or 38555814/74303297?", + "options": [ + "38555814/74303297", + "26575920/55200638", + "17055304/39002516", + "29001510/66729990" + ], + "answer": "38555814/74303297", + "correct_value": 0.5188977549677237, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 15482040/31918706 or 32503458/64512822?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.018783445798224274, + "pure_math": { + "question": "Which is larger: 15482040/31918706 or 32503458/64512822?", + "options": [ + "19739591/34126986", + "15482040/31918706", + "32503458/64512822", + "47306217/86087320" + ], + "answer": "32503458/64512822", + "correct_value": 0.5038294247924855, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 9198209/16724016 or 36213294/62436713?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 9198209/16724016 or 36213294/62436713?", + "options": [ + "9198209/16724016", + "45322332/89096599", + "36213294/62436713", + "7014200/11630948" + ], + "answer": "36213294/62436713", + "correct_value": 0.5800000073674603, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000908", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 33228026/67953233 or 20545861/39772545?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.027600275687265807, + "pure_math": { + "question": "Which is larger: 33228026/67953233 or 20545861/39772545?", + "options": [ + "33228026/67953233", + "27646828/64137663", + "20545861/39772545", + "8243975/17217823" + ], + "answer": "20545861/39772545", + "correct_value": 0.516584015430745, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 19731020/40244460 or 40218868/79074363?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.018341669299375307, + "pure_math": { + "question": "Which is larger: 19731020/40244460 or 40218868/79074363?", + "options": [ + "40218868/79074363", + "22429670/45132145", + "11465712/23915581", + "19731020/40244460" + ], + "answer": "40218868/79074363", + "correct_value": 0.5086208282196342, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 16153862/29370657 or 17552001/30262069?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 16153862/29370657 or 17552001/30262069?", + "options": [ + "26849747/42098978", + "16153862/29370657", + "17552001/30262069", + "30415440/47773651" + ], + "answer": "17552001/30262069", + "correct_value": 0.5800000323837738, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000909", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 39739094/82101905 or 6547687/12642778?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03387780619387193, + "pure_math": { + "question": "Which is larger: 39739094/82101905 or 6547687/12642778?", + "options": [ + "13302919/23148146", + "39739094/82101905", + "6547687/12642778", + "44967437/91888694" + ], + "answer": "6547687/12642778", + "correct_value": 0.5178993888843101, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 10771795/21838289 or 46234341/90247140?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01905540253039545, + "pure_math": { + "question": "Which is larger: 10771795/21838289 or 46234341/90247140?", + "options": [ + "46234341/90247140", + "33739039/77885601", + "10771795/21838289", + "31500361/53812390" + ], + "answer": "46234341/90247140", + "correct_value": 0.5123081019520397, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 39396717/71630393 or 15730591/27121707?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 39396717/71630393 or 15730591/27121707?", + "options": [ + "60456773/93295320", + "15730591/27121707", + "58608159/97621507", + "39396717/71630393" + ], + "answer": "15730591/27121707", + "correct_value": 0.5800000346585855, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000910", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 48511188/99188409 or 43737381/84409057?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.0290785915842649, + "pure_math": { + "question": "Which is larger: 48511188/99188409 or 43737381/84409057?", + "options": [ + "48511188/99188409", + "43737381/84409057", + "31376648/51776905", + "29941863/48509224" + ], + "answer": "43737381/84409057", + "correct_value": 0.5181598107416364, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 24105578/49089607 or 41448129/82006644?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.014371459563769007, + "pure_math": { + "question": "Which is larger: 24105578/49089607 or 41448129/82006644?", + "options": [ + "41448129/82006644", + "42827237/90333051", + "40701399/95874951", + "24105578/49089607" + ], + "answer": "41448129/82006644", + "correct_value": 0.505424036130536, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 37595035/68354608 or 48402178/83452030?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 37595035/68354608 or 48402178/83452030?", + "options": [ + "25787152/47882891", + "37595035/68354608", + "32295722/58336727", + "48402178/83452030" + ], + "answer": "48402178/83452030", + "correct_value": 0.5800000071897592, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000911", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 8565978/17880110 or 11897514/23173814?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03432476495225417, + "pure_math": { + "question": "Which is larger: 8565978/17880110 or 11897514/23173814?", + "options": [ + "11897514/23173814", + "9914781/18404696", + "8565978/17880110", + "43100648/88733821" + ], + "answer": "11897514/23173814", + "correct_value": 0.5134033612248722, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 12267983/24791226 or 37136409/72786323?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.015359554447266721, + "pure_math": { + "question": "Which is larger: 12267983/24791226 or 37136409/72786323?", + "options": [ + "43445103/71926505", + "37136409/72786323", + "12267983/24791226", + "27374310/49734416" + ], + "answer": "37136409/72786323", + "correct_value": 0.5102113620988932, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 52981537/96330067 or 20702915/35694680?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 52981537/96330067 or 20702915/35694680?", + "options": [ + "21321947/38270880", + "20702915/35694680", + "52981537/96330067", + "10495791/17445489" + ], + "answer": "20702915/35694680", + "correct_value": 0.5800000168092276, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000912", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 17395014/36195460 or 26042788/50602468?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03406898167573352, + "pure_math": { + "question": "Which is larger: 17395014/36195460 or 26042788/50602468?", + "options": [ + "26042788/50602468", + "40011599/78940560", + "17395014/36195460", + "46357619/82469530" + ], + "answer": "26042788/50602468", + "correct_value": 0.5146545026222832, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 34509862/70559245 or 33776202/66704892?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.017262118360675205, + "pure_math": { + "question": "Which is larger: 34509862/70559245 or 33776202/66704892?", + "options": [ + "44017806/79161193", + "34509862/70559245", + "34685922/85339631", + "33776202/66704892" + ], + "answer": "33776202/66704892", + "correct_value": 0.506352697490313, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 23533068/42787395 or 30297862/52237693?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 23533068/42787395 or 30297862/52237693?", + "options": [ + "23533068/42787395", + "22234007/40915628", + "30297862/52237693", + "19149482/29184889" + ], + "answer": "30297862/52237693", + "correct_value": 0.5800000011485958, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000913", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 37630560/77309264 or 7799005/14771042?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.041239312764511205, + "pure_math": { + "question": "Which is larger: 37630560/77309264 or 7799005/14771042?", + "options": [ + "7799005/14771042", + "33245387/56904862", + "37630560/77309264", + "34090044/58284435" + ], + "answer": "7799005/14771042", + "correct_value": 0.5279928795815488, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 45700529/93505749 or 16459153/32295883?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.020890599303986124, + "pure_math": { + "question": "Which is larger: 45700529/93505749 or 16459153/32295883?", + "options": [ + "26075008/53524484", + "45700529/93505749", + "16027941/31878921", + "16459153/32295883" + ], + "answer": "16459153/32295883", + "correct_value": 0.5096362592098813, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 7306081/13283782 or 54720512/94345709?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 7306081/13283782 or 54720512/94345709?", + "options": [ + "7306081/13283782", + "33136429/64854183", + "54720512/94345709", + "35946240/55273395" + ], + "answer": "54720512/94345709", + "correct_value": 0.5800000082674667, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000914", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 44640349/93194769 or 36297810/69545316?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04292974748917028, + "pure_math": { + "question": "Which is larger: 44640349/93194769 or 36297810/69545316?", + "options": [ + "44640349/93194769", + "23762783/55294740", + "36297810/69545316", + "22979825/52817593" + ], + "answer": "36297810/69545316", + "correct_value": 0.5219303338847436, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 17434698/35332602 or 7196170/13984883?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.021122703341295845, + "pure_math": { + "question": "Which is larger: 17434698/35332602 or 7196170/13984883?", + "options": [ + "10103502/17862668", + "17434698/35332602", + "7196170/13984883", + "10299960/18541368" + ], + "answer": "7196170/13984883", + "correct_value": 0.5145677657796637, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 53790587/97801066 or 46652227/80434874?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 53790587/97801066 or 46652227/80434874?", + "options": [ + "53790587/97801066", + "46652227/80434874", + "30213493/58672778", + "14037842/22621606" + ], + "answer": "46652227/80434874", + "correct_value": 0.5800000009945935, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000915", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 21095319/43908555 or 29706622/57804537?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03347749888151558, + "pure_math": { + "question": "Which is larger: 21095319/43908555 or 29706622/57804537?", + "options": [ + "53665131/90066267", + "7537802/13886212", + "29706622/57804537", + "21095319/43908555" + ], + "answer": "29706622/57804537", + "correct_value": 0.5139150582591813, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 39555566/79667670 or 6352934/12651752?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.005631548457859126, + "pure_math": { + "question": "Which is larger: 39555566/79667670 or 6352934/12651752?", + "options": [ + "39555566/79667670", + "45958040/80179000", + "6352934/12651752", + "39723789/71460594" + ], + "answer": "6352934/12651752", + "correct_value": 0.5021386761295985, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 15942079/28985598 or 34537450/59547327?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 15942079/28985598 or 34537450/59547327?", + "options": [ + "41174224/79946331", + "34537450/59547327", + "43695812/86039762", + "15942079/28985598" + ], + "answer": "34537450/59547327", + "correct_value": 0.5800000057097441, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000916", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 20829623/43188260 or 31900816/62233437?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.030301028004501795, + "pure_math": { + "question": "Which is larger: 20829623/43188260 or 31900816/62233437?", + "options": [ + "29855588/53295395", + "24114946/40345324", + "20829623/43188260", + "31900816/62233437" + ], + "answer": "31900816/62233437", + "correct_value": 0.5125992960986552, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 17850422/36348053 or 13664797/26619901?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.022233059987488646, + "pure_math": { + "question": "Which is larger: 17850422/36348053 or 13664797/26619901?", + "options": [ + "29188515/49659160", + "17850422/36348053", + "33474098/78965016", + "13664797/26619901" + ], + "answer": "13664797/26619901", + "correct_value": 0.5133301209497361, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 47421460/86220836 or 39047603/67323452?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 47421460/86220836 or 39047603/67323452?", + "options": [ + "58510880/94246537", + "47421460/86220836", + "39047603/67323452", + "20416211/38853462" + ], + "answer": "39047603/67323452", + "correct_value": 0.5800000124770786, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000917", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 25719386/52695452 or 12116052/23587167?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.025595342769488882, + "pure_math": { + "question": "Which is larger: 25719386/52695452 or 12116052/23587167?", + "options": [ + "12116052/23587167", + "23041749/55063109", + "25719386/52695452", + "47699728/94220377" + ], + "answer": "12116052/23587167", + "correct_value": 0.5136713535796817, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 21284179/42931652 or 10764048/21312412?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.009291130164011951, + "pure_math": { + "question": "Which is larger: 21284179/42931652 or 10764048/21312412?", + "options": [ + "21284179/42931652", + "31294751/68830390", + "34263932/64236374", + "10764048/21312412" + ], + "answer": "10764048/21312412", + "correct_value": 0.5050600560837506, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 36212824/65841498 or 11642755/20073714?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 36212824/65841498 or 11642755/20073714?", + "options": [ + "36212824/65841498", + "34366125/59578972", + "45281842/67192842", + "11642755/20073714" + ], + "answer": "11642755/20073714", + "correct_value": 0.5800000438384247, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000918", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 8873575/18856641 or 26640854/50990015?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.05189108301123879, + "pure_math": { + "question": "Which is larger: 8873575/18856641 or 26640854/50990015?", + "options": [ + "8873575/18856641", + "11548677/23150107", + "26640854/50990015", + "25273490/51336884" + ], + "answer": "26640854/50990015", + "correct_value": 0.5224719780921814, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 19599518/40154243 or 39889682/79151178?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.015862489389142642, + "pure_math": { + "question": "Which is larger: 19599518/40154243 or 39889682/79151178?", + "options": [ + "24745126/43805584", + "39889682/79151178", + "19599518/40154243", + "23412194/49397450" + ], + "answer": "39889682/79151178", + "correct_value": 0.5039682669031155, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 23143869/42079760 or 54255678/93544272?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 23143869/42079760 or 54255678/93544272?", + "options": [ + "45801108/88895529", + "24940585/36970889", + "23143869/42079760", + "54255678/93544272" + ], + "answer": "54255678/93544272", + "correct_value": 0.5800000025656301, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000919", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 41295759/85852239 or 46958709/89921845?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04120722915797187, + "pure_math": { + "question": "Which is larger: 41295759/85852239 or 46958709/89921845?", + "options": [ + "46958709/89921845", + "26800523/45047629", + "41295759/85852239", + "50128758/93455535" + ], + "answer": "46958709/89921845", + "correct_value": 0.5222169207048633, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 47963460/97103888 or 39870757/77901793?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.017868304958188086, + "pure_math": { + "question": "Which is larger: 47963460/97103888 or 39870757/77901793?", + "options": [ + "47963460/97103888", + "39870757/77901793", + "51164851/97315338", + "45110889/87814577" + ], + "answer": "39870757/77901793", + "correct_value": 0.5118079502996805, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 11176012/20320021 or 55216549/95200945?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 11176012/20320021 or 55216549/95200945?", + "options": [ + "51983843/85639868", + "46201897/76265274", + "11176012/20320021", + "55216549/95200945" + ], + "answer": "55216549/95200945", + "correct_value": 0.5800000094536877, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000920", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 7674936/16220744 or 40885989/79757422?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03947366114225681, + "pure_math": { + "question": "Which is larger: 7674936/16220744 or 40885989/79757422?", + "options": [ + "46237736/99906195", + "36284885/60346889", + "40885989/79757422", + "7674936/16220744" + ], + "answer": "40885989/79757422", + "correct_value": 0.5126292697875816, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 41977069/86519078 or 41086711/81396925?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01959279127367536, + "pure_math": { + "question": "Which is larger: 41977069/86519078 or 41086711/81396925?", + "options": [ + "41977069/86519078", + "41086711/81396925", + "8187599/16754856", + "33291752/66496596" + ], + "answer": "41086711/81396925", + "correct_value": 0.5047698177787429, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 7165826/13028774 or 21852667/37677011?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 7165826/13028774 or 21852667/37677011?", + "options": [ + "7165826/13028774", + "44854774/66232049", + "21852667/37677011", + "17035000/33811165" + ], + "answer": "21852667/37677011", + "correct_value": 0.5800000164556578, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000921", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 15345577/31899946 or 12896957/24602190?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.0431664915053408, + "pure_math": { + "question": "Which is larger: 15345577/31899946 or 12896957/24602190?", + "options": [ + "15345577/31899946", + "41596073/80290636", + "12896957/24602190", + "16106182/36920124" + ], + "answer": "12896957/24602190", + "correct_value": 0.5242198763606004, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 16436401/33181953 or 47379662/93278389?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01259667432073841, + "pure_math": { + "question": "Which is larger: 16436401/33181953 or 47379662/93278389?", + "options": [ + "15468543/30783070", + "47379662/93278389", + "7183392/16813572", + "16436401/33181953" + ], + "answer": "47379662/93278389", + "correct_value": 0.5079382535219386, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 41641012/75710930 or 56725371/97802363?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 41641012/75710930 or 56725371/97802363?", + "options": [ + "43090053/75100948", + "34696157/61453662", + "41641012/75710930", + "56725371/97802363" + ], + "answer": "56725371/97802363", + "correct_value": 0.5800000047033629, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000922", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 12145287/24879041 or 47817953/92168746?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03063532308068473, + "pure_math": { + "question": "Which is larger: 12145287/24879041 or 47817953/92168746?", + "options": [ + "13626197/25914482", + "60040425/99092790", + "12145287/24879041", + "47817953/92168746" + ], + "answer": "47817953/92168746", + "correct_value": 0.5188087619202284, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 12982565/26452379 or 37037902/73475219?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.013296917972526356, + "pure_math": { + "question": "Which is larger: 12982565/26452379 or 37037902/73475219?", + "options": [ + "25851458/52246583", + "37037902/73475219", + "12982565/26452379", + "22931637/46394254" + ], + "answer": "37037902/73475219", + "correct_value": 0.5040869902000564, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 47470612/86310202 or 43043419/74212790?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 47470612/86310202 or 43043419/74212790?", + "options": [ + "43043419/74212790", + "14670474/26290105", + "41633674/65352119", + "47470612/86310202" + ], + "answer": "43043419/74212790", + "correct_value": 0.580000010779813, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000923", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 30532691/63989110 or 16795664/32211594?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04426224194758993, + "pure_math": { + "question": "Which is larger: 30532691/63989110 or 16795664/32211594?", + "options": [ + "4854041/11423367", + "43345524/97193798", + "30532691/63989110", + "16795664/32211594" + ], + "answer": "16795664/32211594", + "correct_value": 0.5214167296408865, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 19230615/38895928 or 25782798/50761663?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.0135066651807767, + "pure_math": { + "question": "Which is larger: 19230615/38895928 or 25782798/50761663?", + "options": [ + "19230615/38895928", + "25782798/50761663", + "38335286/80899595", + "6255379/13318328" + ], + "answer": "25782798/50761663", + "correct_value": 0.5079187023482663, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 32534004/59152733 or 55587967/95841321?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 32534004/59152733 or 55587967/95841321?", + "options": [ + "55587967/95841321", + "37831093/65312493", + "44500406/90288937", + "32534004/59152733" + ], + "answer": "55587967/95841321", + "correct_value": 0.5800000085558086, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000924", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 15161689/30980287 or 9478419/18426192?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.025001279596531145, + "pure_math": { + "question": "Which is larger: 15161689/30980287 or 9478419/18426192?", + "options": [ + "22793327/48346644", + "9478419/18426192", + "15161689/30980287", + "47236177/85373591" + ], + "answer": "9478419/18426192", + "correct_value": 0.5143992312681861, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 23519733/48451421 or 42087479/83149200?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.020739008883363774, + "pure_math": { + "question": "Which is larger: 23519733/48451421 or 42087479/83149200?", + "options": [ + "23519733/48451421", + "4985947/10112004", + "42087479/83149200", + "46912006/77967689" + ], + "answer": "42087479/83149200", + "correct_value": 0.5061681772043507, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 24362653/44295731 or 31431541/54192311?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 24362653/44295731 or 31431541/54192311?", + "options": [ + "31431541/54192311", + "40181267/59104095", + "24362653/44295731", + "48593019/78416763" + ], + "answer": "31431541/54192311", + "correct_value": 0.5800000114407374, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000925", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 12927708/27136778 or 13831773/26818257?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.0393689058287337, + "pure_math": { + "question": "Which is larger: 12927708/27136778 or 13831773/26818257?", + "options": [ + "12927708/27136778", + "10366518/23347889", + "13831773/26818257", + "13879746/32849418" + ], + "answer": "13831773/26818257", + "correct_value": 0.515759581243479, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 23497802/48090298 or 38201003/74499200?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.024152298444415632, + "pure_math": { + "question": "Which is larger: 23497802/48090298 or 38201003/74499200?", + "options": [ + "27633129/55926182", + "23497802/48090298", + "14104057/31418219", + "38201003/74499200" + ], + "answer": "38201003/74499200", + "correct_value": 0.5127706472015807, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 43357907/78832558 or 33488094/57738092?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 43357907/78832558 or 33488094/57738092?", + "options": [ + "43357907/78832558", + "41126873/81180475", + "17125544/35241011", + "33488094/57738092" + ], + "answer": "33488094/57738092", + "correct_value": 0.5800000110845367, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000926", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 40448197/83669655 or 22045233/42781925?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03186583340334054, + "pure_math": { + "question": "Which is larger: 40448197/83669655 or 22045233/42781925?", + "options": [ + "22045233/42781925", + "40448197/83669655", + "23586348/49513473", + "23250137/41921249" + ], + "answer": "22045233/42781925", + "correct_value": 0.5152931524235994, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 10190194/20743057 or 49870803/97194402?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.021845593029502564, + "pure_math": { + "question": "Which is larger: 10190194/20743057 or 49870803/97194402?", + "options": [ + "49870803/97194402", + "21465362/41177719", + "42824338/96107106", + "10190194/20743057" + ], + "answer": "49870803/97194402", + "correct_value": 0.513103655908084, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 20051233/36456787 or 36774609/63404497?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 20051233/36456787 or 36774609/63404497?", + "options": [ + "36774609/63404497", + "11723037/17717616", + "20051233/36456787", + "29999942/59802468" + ], + "answer": "36774609/63404497", + "correct_value": 0.5800000116710965, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000927", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 35799159/73169720 or 49946575/96314756?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.029314623760875735, + "pure_math": { + "question": "Which is larger: 35799159/73169720 or 49946575/96314756?", + "options": [ + "17619413/30043912", + "49946575/96314756", + "35799159/73169720", + "14560182/25121771" + ], + "answer": "49946575/96314756", + "correct_value": 0.5185765616226032, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 25393763/51154409 or 10902745/21243293?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.016818376123017065, + "pure_math": { + "question": "Which is larger: 25393763/51154409 or 10902745/21243293?", + "options": [ + "25393763/51154409", + "18014159/36378538", + "49812822/99597639", + "10902745/21243293" + ], + "answer": "10902745/21243293", + "correct_value": 0.5132323411440967, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 33937988/61705432 or 25063455/43212853?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 33937988/61705432 or 25063455/43212853?", + "options": [ + "25063455/43212853", + "17703607/28060018", + "33937988/61705432", + "24879534/44632506" + ], + "answer": "25063455/43212853", + "correct_value": 0.5800000060167284, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000928", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 27199094/56522411 or 11176460/21352422?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04221918884769016, + "pure_math": { + "question": "Which is larger: 27199094/56522411 or 11176460/21352422?", + "options": [ + "24395974/46299729", + "6486460/15024849", + "11176460/21352422", + "27199094/56522411" + ], + "answer": "11176460/21352422", + "correct_value": 0.5234282087530866, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 32053798/65908831 or 7900734/15462222?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.024634752076808486, + "pure_math": { + "question": "Which is larger: 32053798/65908831 or 7900734/15462222?", + "options": [ + "21322867/36939895", + "12706535/29689345", + "32053798/65908831", + "7900734/15462222" + ], + "answer": "7900734/15462222", + "correct_value": 0.5109701568118735, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 15928247/28960448 or 19652214/33883126?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 15928247/28960448 or 19652214/33883126?", + "options": [ + "16639639/27314337", + "15928247/28960448", + "14039308/24268416", + "19652214/33883126" + ], + "answer": "19652214/33883126", + "correct_value": 0.5800000271521583, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000929", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 24631860/50742907 or 16436379/32224936?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.024626871827899577, + "pure_math": { + "question": "Which is larger: 24631860/50742907 or 16436379/32224936?", + "options": [ + "24631860/50742907", + "16436379/32224936", + "44354872/86576135", + "21121770/39798274" + ], + "answer": "16436379/32224936", + "correct_value": 0.5100515637951927, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 36655520/75031367 or 35533960/69516608?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.022621903754799, + "pure_math": { + "question": "Which is larger: 36655520/75031367 or 35533960/69516608?", + "options": [ + "48000422/93727852", + "36655520/75031367", + "36942897/84844364", + "35533960/69516608" + ], + "answer": "35533960/69516608", + "correct_value": 0.5111578516604263, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 14326573/26048314 or 30000744/51725420?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 14326573/26048314 or 30000744/51725420?", + "options": [ + "12199862/23516526", + "14326573/26048314", + "22992698/34232877", + "30000744/51725420" + ], + "answer": "30000744/51725420", + "correct_value": 0.5800000077331416, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000930", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 15354223/31392583 or 37041549/71186940?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.031238394591180607, + "pure_math": { + "question": "Which is larger: 15354223/31392583 or 37041549/71186940?", + "options": [ + "37041549/71186940", + "15354223/31392583", + "10079359/22409180", + "48710611/92769240" + ], + "answer": "37041549/71186940", + "correct_value": 0.5203419194588221, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 26418351/53661529 or 20679266/40449692?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.018919661087791806, + "pure_math": { + "question": "Which is larger: 26418351/53661529 or 20679266/40449692?", + "options": [ + "20679266/40449692", + "20764837/43288501", + "27363363/46081570", + "26418351/53661529" + ], + "answer": "20679266/40449692", + "correct_value": 0.5112342017338476, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 27001737/49094067 or 18033822/31092796?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 27001737/49094067 or 18033822/31092796?", + "options": [ + "18033822/31092796", + "27001737/49094067", + "21694920/41148901", + "33274537/54275847" + ], + "answer": "18033822/31092796", + "correct_value": 0.5800000102917731, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000931", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 47248187/98100309 or 18837646/36683537?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.031886334378843695, + "pure_math": { + "question": "Which is larger: 47248187/98100309 or 18837646/36683537?", + "options": [ + "39877625/90672407", + "22750874/47198568", + "47248187/98100309", + "18837646/36683537" + ], + "answer": "18837646/36683537", + "correct_value": 0.513517712318744, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 17797672/36266636 or 26092538/51159617?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01927710550299011, + "pure_math": { + "question": "Which is larger: 17797672/36266636 or 26092538/51159617?", + "options": [ + "34735432/78521798", + "17797672/36266636", + "7326166/12583921", + "26092538/51159617" + ], + "answer": "26092538/51159617", + "correct_value": 0.5100221528241698, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 39335302/71518730 or 22772245/39262491?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 39335302/71518730 or 22772245/39262491?", + "options": [ + "22772245/39262491", + "15784023/24569846", + "41547770/70283414", + "39335302/71518730" + ], + "answer": "22772245/39262491", + "correct_value": 0.5800000056033123, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000932", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 5885199/12178645 or 6431112/12294686?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03984139463332309, + "pure_math": { + "question": "Which is larger: 5885199/12178645 or 6431112/12294686?", + "options": [ + "5587076/12412751", + "6431112/12294686", + "5885199/12178645", + "31120515/73124930" + ], + "answer": "6431112/12294686", + "correct_value": 0.5230806219857913, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 46737577/94813507 or 8178048/16083121?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.015544199717336504, + "pure_math": { + "question": "Which is larger: 46737577/94813507 or 8178048/16083121?", + "options": [ + "46737577/94813507", + "28260992/46581753", + "8178048/16083121", + "29812810/64821935" + ], + "answer": "8178048/16083121", + "correct_value": 0.5084863814678755, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 14080318/25600577 or 53322838/91935927?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 14080318/25600577 or 53322838/91935927?", + "options": [ + "8142580/16362253", + "53322838/91935927", + "14080318/25600577", + "30537799/48194993" + ], + "answer": "53322838/91935927", + "correct_value": 0.5800000036982278, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000933", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 45761610/94200877 or 35167869/68551033?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.027229830232893992, + "pure_math": { + "question": "Which is larger: 45761610/94200877 or 35167869/68551033?", + "options": [ + "35167869/68551033", + "45761610/94200877", + "33995115/76081176", + "29159152/55407428" + ], + "answer": "35167869/68551033", + "correct_value": 0.5130173457779987, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 21347831/43442312 or 30248601/59456169?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.017348205991868593, + "pure_math": { + "question": "Which is larger: 21347831/43442312 or 30248601/59456169?", + "options": [ + "30248601/59456169", + "26585420/64132917", + "21347831/43442312", + "48307192/82900650" + ], + "answer": "30248601/59456169", + "correct_value": 0.5087546256133657, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 18571638/33766614 or 26424789/45559980?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 18571638/33766614 or 26424789/45559980?", + "options": [ + "18571638/33766614", + "25972692/41170365", + "26424789/45559980", + "16831043/34968767" + ], + "answer": "26424789/45559980", + "correct_value": 0.5800000131694527, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000934", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 10910437/22303275 or 6152731/11842695?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03035265579455637, + "pure_math": { + "question": "Which is larger: 10910437/22303275 or 6152731/11842695?", + "options": [ + "9807166/16436520", + "6152731/11842695", + "10910437/22303275", + "19227857/41331034" + ], + "answer": "6152731/11842695", + "correct_value": 0.519538078114821, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 22407591/45583372 or 43056771/84757098?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.016428157862547865, + "pure_math": { + "question": "Which is larger: 22407591/45583372 or 43056771/84757098?", + "options": [ + "30795612/53581274", + "37981048/69689744", + "43056771/84757098", + "22407591/45583372" + ], + "answer": "43056771/84757098", + "correct_value": 0.5080019492880703, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 11776293/21411441 or 32786225/56527974?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 11776293/21411441 or 32786225/56527974?", + "options": [ + "11776293/21411441", + "32786225/56527974", + "6869028/14258066", + "21284661/31317654" + ], + "answer": "32786225/56527974", + "correct_value": 0.5800000014152284, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000935", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 34756637/71062720 or 31816356/61204317?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.030740390016088748, + "pure_math": { + "question": "Which is larger: 34756637/71062720 or 31816356/61204317?", + "options": [ + "34756637/71062720", + "31746228/62798552", + "31816356/61204317", + "40141910/79559670" + ], + "answer": "31816356/61204317", + "correct_value": 0.5198384290441473, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 28207558/56665162 or 37664157/73858003?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.012159980416369065, + "pure_math": { + "question": "Which is larger: 28207558/56665162 or 37664157/73858003?", + "options": [ + "28207558/56665162", + "37664157/73858003", + "20897835/46339710", + "4722780/11420454" + ], + "answer": "37664157/73858003", + "correct_value": 0.5099536335960776, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 22571910/41039835 or 23778146/40996803?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 22571910/41039835 or 23778146/40996803?", + "options": [ + "11229997/18418141", + "23778146/40996803", + "20457919/34266904", + "22571910/41039835" + ], + "answer": "23778146/40996803", + "correct_value": 0.5800000063419579, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000936", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 9455956/19472798 or 47146318/92226773?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.02560169180479205, + "pure_math": { + "question": "Which is larger: 9455956/19472798 or 47146318/92226773?", + "options": [ + "47146318/92226773", + "46309966/77205846", + "9455956/19472798", + "6257255/13555481" + ], + "answer": "47146318/92226773", + "correct_value": 0.5111999093798935, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 47590021/96399834 or 11099798/21763764?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01633952295329505, + "pure_math": { + "question": "Which is larger: 47590021/96399834 or 11099798/21763764?", + "options": [ + "39728412/88996595", + "48831941/96065660", + "11099798/21763764", + "47590021/96399834" + ], + "answer": "11099798/21763764", + "correct_value": 0.5100127900670123, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 32084035/58334609 or 34141262/58864244?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 32084035/58334609 or 34141262/58864244?", + "options": [ + "42167436/63316418", + "24071443/36663356", + "32084035/58334609", + "34141262/58864244" + ], + "answer": "34141262/58864244", + "correct_value": 0.5800000081543559, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000937", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 12551516/25829310 or 10223976/19968758?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.026057776154911227, + "pure_math": { + "question": "Which is larger: 12551516/25829310 or 10223976/19968758?", + "options": [ + "10223976/19968758", + "13608380/28865730", + "15321602/27316628", + "12551516/25829310" + ], + "answer": "10223976/19968758", + "correct_value": 0.5119985930021286, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 6287519/12711909 or 24643416/48580297?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.012655400464240796, + "pure_math": { + "question": "Which is larger: 6287519/12711909 or 24643416/48580297?", + "options": [ + "6287519/12711909", + "43698427/98606945", + "33694003/60992255", + "24643416/48580297" + ], + "answer": "24643416/48580297", + "correct_value": 0.507271826683151, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 32085779/58337779 or 46896998/80856892?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 32085779/58337779 or 46896998/80856892?", + "options": [ + "42049461/72746623", + "32394377/53192484", + "46896998/80856892", + "32085779/58337779" + ], + "answer": "46896998/80856892", + "correct_value": 0.5800000079152189, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000938", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 16040800/33608773 or 40358369/77949394?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.040470757818017955, + "pure_math": { + "question": "Which is larger: 16040800/33608773 or 40358369/77949394?", + "options": [ + "36853865/66730056", + "29926855/64166472", + "16040800/33608773", + "40358369/77949394" + ], + "answer": "40358369/77949394", + "correct_value": 0.5177509013091237, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 12179892/24822630 or 26786534/52536029?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.019192853881414063, + "pure_math": { + "question": "Which is larger: 12179892/24822630 or 26786534/52536029?", + "options": [ + "26786534/52536029", + "47165737/96949899", + "39084108/88423139", + "12179892/24822630" + ], + "answer": "26786534/52536029", + "correct_value": 0.5098697885978402, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 53989900/98163453 or 50443962/86972348?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 53989900/98163453 or 50443962/86972348?", + "options": [ + "9326911/15412319", + "53989900/98163453", + "49650882/93239187", + "50443962/86972348" + ], + "answer": "50443962/86972348", + "correct_value": 0.5800000018396652, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000939", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 43847524/89561144 or 15988215/30696189?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03127141931598332, + "pure_math": { + "question": "Which is larger: 43847524/89561144 or 15988215/30696189?", + "options": [ + "15988215/30696189", + "11293970/22467542", + "43847524/89561144", + "20731910/48335753" + ], + "answer": "15988215/30696189", + "correct_value": 0.520853419295796, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 15526456/31387022 or 37690250/73573276?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.017604219201784232, + "pure_math": { + "question": "Which is larger: 15526456/31387022 or 37690250/73573276?", + "options": [ + "32017673/67230323", + "15526456/31387022", + "37690250/73573276", + "29429944/49734672" + ], + "answer": "37690250/73573276", + "correct_value": 0.5122817964501132, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 52563618/95570214 or 53795666/92751148?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 52563618/95570214 or 53795666/92751148?", + "options": [ + "15198209/23743686", + "61228103/99769075", + "52563618/95570214", + "53795666/92751148" + ], + "answer": "53795666/92751148", + "correct_value": 0.5800000017250461, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000940", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 34183639/70957875 or 37583075/73275829?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03115313816519638, + "pure_math": { + "question": "Which is larger: 34183639/70957875 or 37583075/73275829?", + "options": [ + "27505058/62892993", + "34183639/70957875", + "9437593/19038251", + "37583075/73275829" + ], + "answer": "37583075/73275829", + "correct_value": 0.5128986667622689, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 29291996/59316399 or 41411172/81870575?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01198637506641581, + "pure_math": { + "question": "Which is larger: 29291996/59316399 or 41411172/81870575?", + "options": [ + "10160244/17653908", + "36199911/87363850", + "41411172/81870575", + "29291996/59316399" + ], + "answer": "41411172/81870575", + "correct_value": 0.505812643919992, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 8101516/14730029 or 43562664/75108040?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 8101516/14730029 or 43562664/75108040?", + "options": [ + "43562664/75108040", + "53433344/88133841", + "8101516/14730029", + "52371525/99457804" + ], + "answer": "43562664/75108040", + "correct_value": 0.5800000106513231, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000941", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 42146857/88192385 or 16371270/31892466?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03543040931242458, + "pure_math": { + "question": "Which is larger: 42146857/88192385 or 16371270/31892466?", + "options": [ + "16371270/31892466", + "37002556/77767856", + "42146857/88192385", + "27655998/46585585" + ], + "answer": "16371270/31892466", + "correct_value": 0.5133271914438977, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 13742767/28146735 or 9704640/18986451?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.02288062682731512, + "pure_math": { + "question": "Which is larger: 13742767/28146735 or 9704640/18986451?", + "options": [ + "26424083/55344564", + "9704640/18986451", + "11722935/26677344", + "13742767/28146735" + ], + "answer": "9704640/18986451", + "correct_value": 0.5111350193882995, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 25333795/46061444 or 43953242/75781451?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 25333795/46061444 or 43953242/75781451?", + "options": [ + "17900335/30388230", + "43953242/75781451", + "25333795/46061444", + "30756157/46552106" + ], + "answer": "43953242/75781451", + "correct_value": 0.5800000055422534, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000942", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 7612963/15804798 or 16512008/31755955?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03827889281512925, + "pure_math": { + "question": "Which is larger: 7612963/15804798 or 16512008/31755955?", + "options": [ + "7612963/15804798", + "16512008/31755955", + "31107201/70453107", + "12274698/19913234" + ], + "answer": "16512008/31755955", + "correct_value": 0.5199657198153858, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 6187139/12693857 or 42258081/82718980?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.02345111714484599, + "pure_math": { + "question": "Which is larger: 6187139/12693857 or 42258081/82718980?", + "options": [ + "6187139/12693857", + "30701011/68826987", + "57419053/94264966", + "42258081/82718980" + ], + "answer": "42258081/82718980", + "correct_value": 0.5108631779550473, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 12069752/21945002 or 35850001/61810346?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 12069752/21945002 or 35850001/61810346?", + "options": [ + "35850001/61810346", + "34649104/58275874", + "12069752/21945002", + "15505596/30465940" + ], + "answer": "35850001/61810346", + "correct_value": 0.5800000051771268, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000943", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 37570400/77403260 or 21490323/41961484?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.02675877671476673, + "pure_math": { + "question": "Which is larger: 37570400/77403260 or 21490323/41961484?", + "options": [ + "37570400/77403260", + "35509826/73599732", + "21490323/41961484", + "12721548/30334390" + ], + "answer": "21490323/41961484", + "correct_value": 0.512144017594802, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 21139686/43373182 or 49834164/98688420?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.017573944734585123, + "pure_math": { + "question": "Which is larger: 21139686/43373182 or 49834164/98688420?", + "options": [ + "7293568/14290819", + "33482661/61904520", + "49834164/98688420", + "21139686/43373182" + ], + "answer": "49834164/98688420", + "correct_value": 0.5049646554276581, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 11879479/21599051 or 10151071/17501845?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 11879479/21599051 or 10151071/17501845?", + "options": [ + "10151071/17501845", + "13393395/20645962", + "56318583/90910830", + "11879479/21599051" + ], + "answer": "10151071/17501845", + "correct_value": 0.5800000514231499, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000944", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 48439295/99763821 or 10259245/19825660?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03193336989640949, + "pure_math": { + "question": "Which is larger: 48439295/99763821 or 10259245/19825660?", + "options": [ + "48439295/99763821", + "25393562/43983546", + "10259245/19825660", + "22113418/51864138" + ], + "answer": "10259245/19825660", + "correct_value": 0.5174730626874464, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 11320929/22869325 or 24582384/48348831?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01341121207729673, + "pure_math": { + "question": "Which is larger: 11320929/22869325 or 24582384/48348831?", + "options": [ + "24582384/48348831", + "11320929/22869325", + "13853827/29080666", + "37833472/82344150" + ], + "answer": "24582384/48348831", + "correct_value": 0.5084380220071919, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 8155453/14828096 or 29804917/51387787?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 8155453/14828096 or 29804917/51387787?", + "options": [ + "29804917/51387787", + "8916874/16689309", + "23738906/37388061", + "8155453/14828096" + ], + "answer": "29804917/51387787", + "correct_value": 0.5800000105083334, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000945", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 35447645/73141612 or 7805096/14925116?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03830641959861458, + "pure_math": { + "question": "Which is larger: 35447645/73141612 or 7805096/14925116?", + "options": [ + "13244772/28278426", + "35447645/73141612", + "7805096/14925116", + "19071893/43517099" + ], + "answer": "7805096/14925116", + "correct_value": 0.5229504413902043, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 19683069/39589748 or 16079252/31872203?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.007315479904823563, + "pure_math": { + "question": "Which is larger: 19683069/39589748 or 16079252/31872203?", + "options": [ + "11626539/26643390", + "19683069/39589748", + "24299054/56094107", + "16079252/31872203" + ], + "answer": "16079252/31872203", + "correct_value": 0.5044913901935175, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 6467498/11759087 or 29334981/50577552?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 6467498/11759087 or 29334981/50577552?", + "options": [ + "29334981/50577552", + "6467498/11759087", + "25022714/43226860", + "11706190/22763709" + ], + "answer": "29334981/50577552", + "correct_value": 0.5800000166081585, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000946", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 38734328/79953928 or 23573689/45585262?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.032675903232541825, + "pure_math": { + "question": "Which is larger: 38734328/79953928 or 23573689/45585262?", + "options": [ + "23573689/45585262", + "38734328/79953928", + "14068353/28795821", + "31254403/65220197" + ], + "answer": "23573689/45585262", + "correct_value": 0.5171340026519975, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 28579894/58838092 or 36188099/72051818?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01651307207218755, + "pure_math": { + "question": "Which is larger: 28579894/58838092 or 36188099/72051818?", + "options": [ + "36188099/72051818", + "28579894/58838092", + "39176498/87695532", + "30851747/57608327" + ], + "answer": "36188099/72051818", + "correct_value": 0.5022510188431332, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 9388555/17070099 or 13882966/23936148?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 9388555/17070099 or 13882966/23936148?", + "options": [ + "9388555/17070099", + "13882966/23936148", + "18479079/32445114", + "43453691/82953318" + ], + "answer": "13882966/23936148", + "correct_value": 0.5800000066844506, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000947", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 44100431/90754656 or 36206433/69894696?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.032083796343153836, + "pure_math": { + "question": "Which is larger: 44100431/90754656 or 36206433/69894696?", + "options": [ + "36206433/69894696", + "44100431/90754656", + "46757752/99228424", + "26004188/53158541" + ], + "answer": "36206433/69894696", + "correct_value": 0.5180140278455464, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 27633706/56632403 or 44690663/87060333?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.0253811213504761, + "pure_math": { + "question": "Which is larger: 27633706/56632403 or 44690663/87060333?", + "options": [ + "23399665/38511738", + "44690663/87060333", + "37079447/61118611", + "27633706/56632403" + ], + "answer": "44690663/87060333", + "correct_value": 0.5133297962460125, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 44866820/81576035 or 24964875/43042887?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 44866820/81576035 or 24964875/43042887?", + "options": [ + "24964875/43042887", + "44866820/81576035", + "18617446/35138822", + "58514439/98507523" + ], + "answer": "24964875/43042887", + "correct_value": 0.5800000125456268, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000948", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 17268385/36259370 or 13757652/26686981?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03927309392834161, + "pure_math": { + "question": "Which is larger: 17268385/36259370 or 13757652/26686981?", + "options": [ + "13757652/26686981", + "22701174/38757051", + "17268385/36259370", + "28419012/57671803" + ], + "answer": "13757652/26686981", + "correct_value": 0.5155192338916118, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 14490749/29371186 or 16480285/32466435?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.014243784792712544, + "pure_math": { + "question": "Which is larger: 14490749/29371186 or 16480285/32466435?", + "options": [ + "16480285/32466435", + "42335262/84275393", + "14490749/29371186", + "12401411/28924049" + ], + "answer": "16480285/32466435", + "correct_value": 0.5076099362310644, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 17447900/31723454 or 25139636/43344199?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 17447900/31723454 or 25139636/43344199?", + "options": [ + "14203522/26632035", + "25139636/43344199", + "31590058/52127309", + "17447900/31723454" + ], + "answer": "25139636/43344199", + "correct_value": 0.5800000133812601, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000949", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 29135386/60978827 or 12302393/23888193?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03720377446922485, + "pure_math": { + "question": "Which is larger: 29135386/60978827 or 12302393/23888193?", + "options": [ + "10517740/18091248", + "33218055/67825105", + "29135386/60978827", + "12302393/23888193" + ], + "answer": "12302393/23888193", + "correct_value": 0.5149988950608361, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 31648030/64860537 or 43952587/87103964?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.016659474117615924, + "pure_math": { + "question": "Which is larger: 31648030/64860537 or 43952587/87103964?", + "options": [ + "20483177/48447350", + "43952587/87103964", + "31648030/64860537", + "19279311/32152576" + ], + "answer": "43952587/87103964", + "correct_value": 0.5045991592299979, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 26939761/48981382 or 21211168/36570979?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 26939761/48981382 or 21211168/36570979?", + "options": [ + "52328160/77032240", + "26939761/48981382", + "21211168/36570979", + "11566429/20682482" + ], + "answer": "21211168/36570979", + "correct_value": 0.5800000049219355, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000950", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 37366811/78246449 or 13975840/27132919?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03753516138438545, + "pure_math": { + "question": "Which is larger: 37366811/78246449 or 13975840/27132919?", + "options": [ + "13674154/30128989", + "37366811/78246449", + "13975840/27132919", + "33053806/70564631" + ], + "answer": "13975840/27132919", + "correct_value": 0.5150879638125186, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 43240451/88871038 or 29218150/57516419?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.02144389728572793, + "pure_math": { + "question": "Which is larger: 43240451/88871038 or 29218150/57516419?", + "options": [ + "39289886/67122715", + "29218150/57516419", + "23290424/41736099", + "43240451/88871038" + ], + "answer": "29218150/57516419", + "correct_value": 0.5079966817822925, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 35799276/65089591 or 27867179/48046859?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 35799276/65089591 or 27867179/48046859?", + "options": [ + "27867179/48046859", + "29988682/57474269", + "35799276/65089591", + "55862415/97290100" + ], + "answer": "27867179/48046859", + "correct_value": 0.5800000162341518, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000951", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 34969122/72759912 or 16627104/32319779?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.033846258374204796, + "pure_math": { + "question": "Which is larger: 34969122/72759912 or 16627104/32319779?", + "options": [ + "34969122/72759912", + "23539816/48348722", + "35799972/59676114", + "16627104/32319779" + ], + "answer": "16627104/32319779", + "correct_value": 0.514455993031388, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 24946362/50108555 or 20175654/39465041?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.013382141520282442, + "pure_math": { + "question": "Which is larger: 24946362/50108555 or 20175654/39465041?", + "options": [ + "20175654/39465041", + "29406677/58825292", + "24946362/50108555", + "11072867/18120209" + ], + "answer": "20175654/39465041", + "correct_value": 0.5112285072755911, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 39575302/71955094 or 25726826/44356595?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 39575302/71955094 or 25726826/44356595?", + "options": [ + "25726826/44356595", + "7290646/10907464", + "39575302/71955094", + "13193654/24672106" + ], + "answer": "25726826/44356595", + "correct_value": 0.5800000202901057, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "cancellation_identity_000952", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "66201522 + 55165254 - 55165253", + "ns_shortcut": "Near-cancellation: 55165254 - 55165253 = 1, so answer ≈ 66201522 + 1.", + "shortcut_strength": 0.9999999818726476, + "pure_math": { + "question": "Evaluate: 66201522 + 55165254 - 55165253", + "options": [ + "66201543", + "66201533", + "66201513", + "66201523" + ], + "answer": "66201523", + "correct_value": 66201523, + "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": "35147748 + 88404920 - 90853520", + "ns_shortcut": "Near-cancellation: 88404920 - 90853520 = -2448600, offset is manageable.", + "shortcut_strength": 0.9723024465154202, + "pure_math": { + "question": "Evaluate: 35147748 + 88404920 - 90853520", + "options": [ + "32699158", + "32699138", + "32699168", + "32699148" + ], + "answer": "32699148", + "correct_value": 32699148, + "explanation": "B-C = -2448600; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "66201522 + 55165254 - 72022057", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 66201522 + 55165254 - 72022057", + "options": [ + "49344709", + "49344739", + "49344719", + "49344729" + ], + "answer": "49344719", + "correct_value": 49344719, + "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_000953", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "33475742 + 30422014 - 30422013", + "ns_shortcut": "Near-cancellation: 30422014 - 30422013 = 1, so answer ≈ 33475742 + 1.", + "shortcut_strength": 0.9999999671290665, + "pure_math": { + "question": "Evaluate: 33475742 + 30422014 - 30422013", + "options": [ + "33475743", + "33475763", + "33475733", + "33475753" + ], + "answer": "33475743", + "correct_value": 33475743, + "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": "60147749 + 53926650 - 52569636", + "ns_shortcut": "Near-cancellation: 53926650 - 52569636 = 1357014, offset is manageable.", + "shortcut_strength": 0.9748359299159136, + "pure_math": { + "question": "Evaluate: 60147749 + 53926650 - 52569636", + "options": [ + "61504763", + "61504753", + "61504783", + "61504773" + ], + "answer": "61504763", + "correct_value": 61504763, + "explanation": "B-C = 1357014; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "33475742 + 30422014 - 41802411", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 33475742 + 30422014 - 41802411", + "options": [ + "22095365", + "22095335", + "22095355", + "22095345" + ], + "answer": "22095345", + "correct_value": 22095345, + "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_000954", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "23912513 + 20611940 - 20611941", + "ns_shortcut": "Near-cancellation: 20611940 - 20611941 = -1, so answer ≈ 23912513 + -1.", + "shortcut_strength": 0.9999999514844309, + "pure_math": { + "question": "Evaluate: 23912513 + 20611940 - 20611941", + "options": [ + "23912522", + "23912512", + "23912502", + "23912532" + ], + "answer": "23912512", + "correct_value": 23912512, + "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": "88622220 + 78125318 - 75446033", + "ns_shortcut": "Near-cancellation: 78125318 - 75446033 = 2679285, offset is manageable.", + "shortcut_strength": 0.9657052915931811, + "pure_math": { + "question": "Evaluate: 88622220 + 78125318 - 75446033", + "options": [ + "91301515", + "91301505", + "91301495", + "91301525" + ], + "answer": "91301505", + "correct_value": 91301505, + "explanation": "B-C = 2679285; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "23912513 + 20611940 - 13925305", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 23912513 + 20611940 - 13925305", + "options": [ + "30599168", + "30599158", + "30599138", + "30599148" + ], + "answer": "30599148", + "correct_value": 30599148, + "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_000955", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "89028730 + 79292641 - 79292639", + "ns_shortcut": "Near-cancellation: 79292641 - 79292639 = 2, so answer ≈ 89028730 + 2.", + "shortcut_strength": 0.9999999747769783, + "pure_math": { + "question": "Evaluate: 89028730 + 79292641 - 79292639", + "options": [ + "89028732", + "89028742", + "89028752", + "89028722" + ], + "answer": "89028732", + "correct_value": 89028732, + "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": "61228877 + 81910263 - 85445111", + "ns_shortcut": "Near-cancellation: 81910263 - 85445111 = -3534848, offset is manageable.", + "shortcut_strength": 0.9568448705872181, + "pure_math": { + "question": "Evaluate: 61228877 + 81910263 - 85445111", + "options": [ + "57694039", + "57694019", + "57694049", + "57694029" + ], + "answer": "57694029", + "correct_value": 57694029, + "explanation": "B-C = -3534848; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "89028730 + 79292641 - 40603718", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 89028730 + 79292641 - 40603718", + "options": [ + "127717643", + "127717653", + "127717663", + "127717673" + ], + "answer": "127717653", + "correct_value": 127717653, + "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_000956", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "64834475 + 45315671 - 45315672", + "ns_shortcut": "Near-cancellation: 45315671 - 45315672 = -1, so answer ≈ 64834475 + -1.", + "shortcut_strength": 0.9999999779325788, + "pure_math": { + "question": "Evaluate: 64834475 + 45315671 - 45315672", + "options": [ + "64834484", + "64834494", + "64834474", + "64834464" + ], + "answer": "64834474", + "correct_value": 64834474, + "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": "39829819 + 12430348 - 13030184", + "ns_shortcut": "Near-cancellation: 12430348 - 13030184 = -599836, offset is manageable.", + "shortcut_strength": 0.951744231134961, + "pure_math": { + "question": "Evaluate: 39829819 + 12430348 - 13030184", + "options": [ + "39230003", + "39229993", + "39229973", + "39229983" + ], + "answer": "39229983", + "correct_value": 39229983, + "explanation": "B-C = -599836; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "64834475 + 45315671 - 62749125", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 64834475 + 45315671 - 62749125", + "options": [ + "47401011", + "47401031", + "47401021", + "47401041" + ], + "answer": "47401021", + "correct_value": 47401021, + "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_000957", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "56401649 + 89019645 - 89019643", + "ns_shortcut": "Near-cancellation: 89019645 - 89019643 = 2, so answer ≈ 56401649 + 2.", + "shortcut_strength": 0.999999977533049, + "pure_math": { + "question": "Evaluate: 56401649 + 89019645 - 89019643", + "options": [ + "56401641", + "56401671", + "56401661", + "56401651" + ], + "answer": "56401651", + "correct_value": 56401651, + "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": "86177812 + 10045310 - 10475188", + "ns_shortcut": "Near-cancellation: 10045310 - 10475188 = -429878, offset is manageable.", + "shortcut_strength": 0.9572060991646848, + "pure_math": { + "question": "Evaluate: 86177812 + 10045310 - 10475188", + "options": [ + "85747954", + "85747934", + "85747924", + "85747944" + ], + "answer": "85747934", + "correct_value": 85747934, + "explanation": "B-C = -429878; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "56401649 + 89019645 - 51919833", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 56401649 + 89019645 - 51919833", + "options": [ + "93501471", + "93501451", + "93501481", + "93501461" + ], + "answer": "93501461", + "correct_value": 93501461, + "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_000958", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "77573618 + 53526663 - 53526662", + "ns_shortcut": "Near-cancellation: 53526663 - 53526662 = 1, so answer ≈ 77573618 + 1.", + "shortcut_strength": 0.9999999813177219, + "pure_math": { + "question": "Evaluate: 77573618 + 53526663 - 53526662", + "options": [ + "77573609", + "77573619", + "77573629", + "77573639" + ], + "answer": "77573619", + "correct_value": 77573619, + "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": "69941736 + 31194468 - 32689255", + "ns_shortcut": "Near-cancellation: 31194468 - 32689255 = -1494787, offset is manageable.", + "shortcut_strength": 0.952081663966829, + "pure_math": { + "question": "Evaluate: 69941736 + 31194468 - 32689255", + "options": [ + "68446959", + "68446949", + "68446969", + "68446939" + ], + "answer": "68446949", + "correct_value": 68446949, + "explanation": "B-C = -1494787; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "77573618 + 53526663 - 81913473", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 77573618 + 53526663 - 81913473", + "options": [ + "49186828", + "49186808", + "49186818", + "49186798" + ], + "answer": "49186808", + "correct_value": 49186808, + "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_000959", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "36415300 + 20295013 - 20295015", + "ns_shortcut": "Near-cancellation: 20295013 - 20295015 = -2, so answer ≈ 36415300 + -2.", + "shortcut_strength": 0.9999999014536232, + "pure_math": { + "question": "Evaluate: 36415300 + 20295013 - 20295015", + "options": [ + "36415288", + "36415318", + "36415298", + "36415308" + ], + "answer": "36415298", + "correct_value": 36415298, + "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": "55777979 + 97146121 - 93333683", + "ns_shortcut": "Near-cancellation: 97146121 - 93333683 = 3812438, offset is manageable.", + "shortcut_strength": 0.9607556332588926, + "pure_math": { + "question": "Evaluate: 55777979 + 97146121 - 93333683", + "options": [ + "59590417", + "59590427", + "59590407", + "59590437" + ], + "answer": "59590417", + "correct_value": 59590417, + "explanation": "B-C = 3812438; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "36415300 + 20295013 - 30164682", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 36415300 + 20295013 - 30164682", + "options": [ + "26545641", + "26545651", + "26545631", + "26545621" + ], + "answer": "26545631", + "correct_value": 26545631, + "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_000960", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99187418 + 49388858 - 49388859", + "ns_shortcut": "Near-cancellation: 49388858 - 49388859 = -1, so answer ≈ 99187418 + -1.", + "shortcut_strength": 0.9999999797525183, + "pure_math": { + "question": "Evaluate: 99187418 + 49388858 - 49388859", + "options": [ + "99187427", + "99187437", + "99187417", + "99187407" + ], + "answer": "99187417", + "correct_value": 99187417, + "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": "69192575 + 95929960 - 99465192", + "ns_shortcut": "Near-cancellation: 95929960 - 99465192 = -3535232, offset is manageable.", + "shortcut_strength": 0.9631477799010861, + "pure_math": { + "question": "Evaluate: 69192575 + 95929960 - 99465192", + "options": [ + "65657333", + "65657353", + "65657343", + "65657363" + ], + "answer": "65657343", + "correct_value": 65657343, + "explanation": "B-C = -3535232; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "99187418 + 49388858 - 20820794", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 99187418 + 49388858 - 20820794", + "options": [ + "127755502", + "127755482", + "127755492", + "127755472" + ], + "answer": "127755482", + "correct_value": 127755482, + "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_000961", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "97919966 + 82867076 - 82867078", + "ns_shortcut": "Near-cancellation: 82867076 - 82867078 = -2, so answer ≈ 97919966 + -2.", + "shortcut_strength": 0.9999999758649624, + "pure_math": { + "question": "Evaluate: 97919966 + 82867076 - 82867078", + "options": [ + "97919954", + "97919974", + "97919984", + "97919964" + ], + "answer": "97919964", + "correct_value": 97919964, + "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": "61821026 + 60731476 - 62797611", + "ns_shortcut": "Near-cancellation: 60731476 - 62797611 = -2066135, offset is manageable.", + "shortcut_strength": 0.9659791736331256, + "pure_math": { + "question": "Evaluate: 61821026 + 60731476 - 62797611", + "options": [ + "59754891", + "59754901", + "59754911", + "59754881" + ], + "answer": "59754891", + "correct_value": 59754891, + "explanation": "B-C = -2066135; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "97919966 + 82867076 - 108841209", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 97919966 + 82867076 - 108841209", + "options": [ + "71945853", + "71945833", + "71945823", + "71945843" + ], + "answer": "71945833", + "correct_value": 71945833, + "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_000962", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "45804636 + 93769175 - 93769176", + "ns_shortcut": "Near-cancellation: 93769175 - 93769176 = -1, so answer ≈ 45804636 + -1.", + "shortcut_strength": 0.9999999893355146, + "pure_math": { + "question": "Evaluate: 45804636 + 93769175 - 93769176", + "options": [ + "45804625", + "45804645", + "45804635", + "45804655" + ], + "answer": "45804635", + "correct_value": 45804635, + "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": "67658484 + 90668739 - 88451333", + "ns_shortcut": "Near-cancellation: 90668739 - 88451333 = 2217406, offset is manageable.", + "shortcut_strength": 0.9755438751607651, + "pure_math": { + "question": "Evaluate: 67658484 + 90668739 - 88451333", + "options": [ + "69875910", + "69875880", + "69875900", + "69875890" + ], + "answer": "69875890", + "correct_value": 69875890, + "explanation": "B-C = 2217406; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "45804636 + 93769175 - 145896559", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 45804636 + 93769175 - 145896559", + "options": [ + "-6322748", + "-6322732", + "-6322742", + "-6322762" + ], + "answer": "-6322748", + "correct_value": -6322748, + "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_000963", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "79013223 + 18987340 - 18987338", + "ns_shortcut": "Near-cancellation: 18987340 - 18987338 = 2, so answer ≈ 79013223 + 2.", + "shortcut_strength": 0.9999998946666568, + "pure_math": { + "question": "Evaluate: 79013223 + 18987340 - 18987338", + "options": [ + "79013215", + "79013235", + "79013225", + "79013245" + ], + "answer": "79013225", + "correct_value": 79013225, + "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": "12241356 + 88970299 - 90998018", + "ns_shortcut": "Near-cancellation: 88970299 - 90998018 = -2027719, offset is manageable.", + "shortcut_strength": 0.9772090346689742, + "pure_math": { + "question": "Evaluate: 12241356 + 88970299 - 90998018", + "options": [ + "10213657", + "10213647", + "10213637", + "10213627" + ], + "answer": "10213637", + "correct_value": 10213637, + "explanation": "B-C = -2027719; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "79013223 + 18987340 - 28269650", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 79013223 + 18987340 - 28269650", + "options": [ + "69730913", + "69730903", + "69730933", + "69730923" + ], + "answer": "69730913", + "correct_value": 69730913, + "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_000964", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "18852650 + 73768382 - 73768383", + "ns_shortcut": "Near-cancellation: 73768382 - 73768383 = -1, so answer ≈ 18852650 + -1.", + "shortcut_strength": 0.9999999864440567, + "pure_math": { + "question": "Evaluate: 18852650 + 73768382 - 73768383", + "options": [ + "18852659", + "18852649", + "18852669", + "18852639" + ], + "answer": "18852649", + "correct_value": 18852649, + "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": "46165764 + 25939315 - 24708977", + "ns_shortcut": "Near-cancellation: 25939315 - 24708977 = 1230338, offset is manageable.", + "shortcut_strength": 0.9525686009827168, + "pure_math": { + "question": "Evaluate: 46165764 + 25939315 - 24708977", + "options": [ + "47396122", + "47396102", + "47396092", + "47396112" + ], + "answer": "47396102", + "correct_value": 47396102, + "explanation": "B-C = 1230338; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "18852650 + 73768382 - 100834201", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 18852650 + 73768382 - 100834201", + "options": [ + "-8213169", + "-8213161", + "-8213181", + "-8213151" + ], + "answer": "-8213169", + "correct_value": -8213169, + "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_000965", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "93496702 + 86222056 - 86222054", + "ns_shortcut": "Near-cancellation: 86222056 - 86222054 = 2, so answer ≈ 93496702 + 2.", + "shortcut_strength": 0.999999976804079, + "pure_math": { + "question": "Evaluate: 93496702 + 86222056 - 86222054", + "options": [ + "93496694", + "93496704", + "93496724", + "93496714" + ], + "answer": "93496704", + "correct_value": 93496704, + "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": "34145842 + 60860348 - 62350916", + "ns_shortcut": "Near-cancellation: 60860348 - 62350916 = -1490568, offset is manageable.", + "shortcut_strength": 0.9755083884830892, + "pure_math": { + "question": "Evaluate: 34145842 + 60860348 - 62350916", + "options": [ + "32655294", + "32655264", + "32655284", + "32655274" + ], + "answer": "32655274", + "correct_value": 32655274, + "explanation": "B-C = -1490568; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "93496702 + 86222056 - 140190859", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 93496702 + 86222056 - 140190859", + "options": [ + "39527889", + "39527919", + "39527909", + "39527899" + ], + "answer": "39527899", + "correct_value": 39527899, + "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_000966", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "28459003 + 44063540 - 44063538", + "ns_shortcut": "Near-cancellation: 44063540 - 44063538 = 2, so answer ≈ 28459003 + 2.", + "shortcut_strength": 0.9999999546110004, + "pure_math": { + "question": "Evaluate: 28459003 + 44063540 - 44063538", + "options": [ + "28458995", + "28459025", + "28459005", + "28459015" + ], + "answer": "28459005", + "correct_value": 28459005, + "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": "49896014 + 57614677 - 59091564", + "ns_shortcut": "Near-cancellation: 57614677 - 59091564 = -1476887, offset is manageable.", + "shortcut_strength": 0.9743661324353168, + "pure_math": { + "question": "Evaluate: 49896014 + 57614677 - 59091564", + "options": [ + "48419147", + "48419117", + "48419127", + "48419137" + ], + "answer": "48419127", + "correct_value": 48419127, + "explanation": "B-C = -1476887; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "28459003 + 44063540 - 27426371", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 28459003 + 44063540 - 27426371", + "options": [ + "45096182", + "45096192", + "45096172", + "45096162" + ], + "answer": "45096172", + "correct_value": 45096172, + "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_000967", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "41979677 + 86237442 - 86237444", + "ns_shortcut": "Near-cancellation: 86237442 - 86237444 = -2, so answer ≈ 41979677 + -2.", + "shortcut_strength": 0.9999999768082175, + "pure_math": { + "question": "Evaluate: 41979677 + 86237442 - 86237444", + "options": [ + "41979695", + "41979665", + "41979675", + "41979685" + ], + "answer": "41979675", + "correct_value": 41979675, + "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": "60216565 + 18220836 - 17600431", + "ns_shortcut": "Near-cancellation: 18220836 - 17600431 = 620405, offset is manageable.", + "shortcut_strength": 0.9659507939152737, + "pure_math": { + "question": "Evaluate: 60216565 + 18220836 - 17600431", + "options": [ + "60836990", + "60836980", + "60836970", + "60836960" + ], + "answer": "60836970", + "correct_value": 60836970, + "explanation": "B-C = 620405; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "41979677 + 86237442 - 54581343", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 41979677 + 86237442 - 54581343", + "options": [ + "73635766", + "73635796", + "73635776", + "73635786" + ], + "answer": "73635776", + "correct_value": 73635776, + "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_000968", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26257335 + 16010556 - 16010558", + "ns_shortcut": "Near-cancellation: 16010556 - 16010558 = -2, so answer ≈ 26257335 + -2.", + "shortcut_strength": 0.9999998750824144, + "pure_math": { + "question": "Evaluate: 26257335 + 16010556 - 16010558", + "options": [ + "26257343", + "26257353", + "26257323", + "26257333" + ], + "answer": "26257333", + "correct_value": 26257333, + "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": "54644928 + 52601433 - 54618271", + "ns_shortcut": "Near-cancellation: 52601433 - 54618271 = -2016838, offset is manageable.", + "shortcut_strength": 0.9616581168045365, + "pure_math": { + "question": "Evaluate: 54644928 + 52601433 - 54618271", + "options": [ + "52628100", + "52628090", + "52628080", + "52628110" + ], + "answer": "52628090", + "correct_value": 52628090, + "explanation": "B-C = -2016838; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "26257335 + 16010556 - 24683463", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 26257335 + 16010556 - 24683463", + "options": [ + "17584438", + "17584448", + "17584428", + "17584418" + ], + "answer": "17584428", + "correct_value": 17584428, + "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_000969", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "13999753 + 78172300 - 78172298", + "ns_shortcut": "Near-cancellation: 78172300 - 78172298 = 2, so answer ≈ 13999753 + 2.", + "shortcut_strength": 0.9999999744154899, + "pure_math": { + "question": "Evaluate: 13999753 + 78172300 - 78172298", + "options": [ + "13999765", + "13999775", + "13999755", + "13999745" + ], + "answer": "13999755", + "correct_value": 13999755, + "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": "51482349 + 56000056 - 58487839", + "ns_shortcut": "Near-cancellation: 56000056 - 58487839 = -2487783, offset is manageable.", + "shortcut_strength": 0.9555753479960806, + "pure_math": { + "question": "Evaluate: 51482349 + 56000056 - 58487839", + "options": [ + "48994566", + "48994556", + "48994586", + "48994576" + ], + "answer": "48994566", + "correct_value": 48994566, + "explanation": "B-C = -2487783; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "13999753 + 78172300 - 104742554", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 13999753 + 78172300 - 104742554", + "options": [ + "-12570499", + "-12570519", + "-12570489", + "-12570501" + ], + "answer": "-12570501", + "correct_value": -12570501, + "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_000970", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "84226768 + 48837648 - 48837647", + "ns_shortcut": "Near-cancellation: 48837648 - 48837647 = 1, so answer ≈ 84226768 + 1.", + "shortcut_strength": 0.9999999795239934, + "pure_math": { + "question": "Evaluate: 84226768 + 48837648 - 48837647", + "options": [ + "84226759", + "84226769", + "84226789", + "84226779" + ], + "answer": "84226769", + "correct_value": 84226769, + "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": "47919889 + 62896186 - 65496737", + "ns_shortcut": "Near-cancellation: 62896186 - 65496737 = -2600551, offset is manageable.", + "shortcut_strength": 0.9586532798666043, + "pure_math": { + "question": "Evaluate: 47919889 + 62896186 - 65496737", + "options": [ + "45319358", + "45319328", + "45319338", + "45319348" + ], + "answer": "45319338", + "correct_value": 45319338, + "explanation": "B-C = -2600551; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "84226768 + 48837648 - 66760287", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 84226768 + 48837648 - 66760287", + "options": [ + "66304129", + "66304119", + "66304139", + "66304149" + ], + "answer": "66304129", + "correct_value": 66304129, + "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_000971", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "23107454 + 14680071 - 14680073", + "ns_shortcut": "Near-cancellation: 14680071 - 14680073 = -2, so answer ≈ 23107454 + -2.", + "shortcut_strength": 0.9999998637608769, + "pure_math": { + "question": "Evaluate: 23107454 + 14680071 - 14680073", + "options": [ + "23107442", + "23107452", + "23107472", + "23107462" + ], + "answer": "23107452", + "correct_value": 23107452, + "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": "70165002 + 38987376 - 40546188", + "ns_shortcut": "Near-cancellation: 38987376 - 40546188 = -1558812, offset is manageable.", + "shortcut_strength": 0.9600175195170868, + "pure_math": { + "question": "Evaluate: 70165002 + 38987376 - 40546188", + "options": [ + "68606210", + "68606190", + "68606200", + "68606180" + ], + "answer": "68606190", + "correct_value": 68606190, + "explanation": "B-C = -1558812; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "23107454 + 14680071 - 21424074", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 23107454 + 14680071 - 21424074", + "options": [ + "16363471", + "16363451", + "16363461", + "16363441" + ], + "answer": "16363451", + "correct_value": 16363451, + "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_000972", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "54480754 + 12623311 - 12623313", + "ns_shortcut": "Near-cancellation: 12623311 - 12623313 = -2, so answer ≈ 54480754 + -2.", + "shortcut_strength": 0.9999998415629624, + "pure_math": { + "question": "Evaluate: 54480754 + 12623311 - 12623313", + "options": [ + "54480752", + "54480772", + "54480762", + "54480742" + ], + "answer": "54480752", + "correct_value": 54480752, + "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": "22350412 + 52877255 - 50582156", + "ns_shortcut": "Near-cancellation: 52877255 - 50582156 = 2295099, offset is manageable.", + "shortcut_strength": 0.9565957234353409, + "pure_math": { + "question": "Evaluate: 22350412 + 52877255 - 50582156", + "options": [ + "24645501", + "24645531", + "24645511", + "24645521" + ], + "answer": "24645511", + "correct_value": 24645511, + "explanation": "B-C = 2295099; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "54480754 + 12623311 - 19706921", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 54480754 + 12623311 - 19706921", + "options": [ + "47397154", + "47397134", + "47397144", + "47397164" + ], + "answer": "47397144", + "correct_value": 47397144, + "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_000973", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "40171534 + 47875246 - 47875248", + "ns_shortcut": "Near-cancellation: 47875246 - 47875248 = -2, so answer ≈ 40171534 + -2.", + "shortcut_strength": 0.9999999582247577, + "pure_math": { + "question": "Evaluate: 40171534 + 47875246 - 47875248", + "options": [ + "40171532", + "40171542", + "40171552", + "40171522" + ], + "answer": "40171532", + "correct_value": 40171532, + "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": "28266661 + 99621828 - 95951447", + "ns_shortcut": "Near-cancellation: 99621828 - 95951447 = 3670381, offset is manageable.", + "shortcut_strength": 0.9631568595589312, + "pure_math": { + "question": "Evaluate: 28266661 + 99621828 - 95951447", + "options": [ + "31937042", + "31937062", + "31937052", + "31937032" + ], + "answer": "31937042", + "correct_value": 31937042, + "explanation": "B-C = 3670381; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "40171534 + 47875246 - 32808741", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 40171534 + 47875246 - 32808741", + "options": [ + "55238049", + "55238039", + "55238029", + "55238059" + ], + "answer": "55238039", + "correct_value": 55238039, + "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_000974", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "67699732 + 66329899 - 66329900", + "ns_shortcut": "Near-cancellation: 66329899 - 66329900 = -1, so answer ≈ 67699732 + -1.", + "shortcut_strength": 0.9999999849238426, + "pure_math": { + "question": "Evaluate: 67699732 + 66329899 - 66329900", + "options": [ + "67699741", + "67699731", + "67699721", + "67699751" + ], + "answer": "67699731", + "correct_value": 67699731, + "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": "26246515 + 73827458 - 77394147", + "ns_shortcut": "Near-cancellation: 73827458 - 77394147 = -3566689, offset is manageable.", + "shortcut_strength": 0.9516888553849436, + "pure_math": { + "question": "Evaluate: 26246515 + 73827458 - 77394147", + "options": [ + "22679836", + "22679826", + "22679846", + "22679816" + ], + "answer": "22679826", + "correct_value": 22679826, + "explanation": "B-C = -3566689; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "67699732 + 66329899 - 106594168", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 67699732 + 66329899 - 106594168", + "options": [ + "27435473", + "27435453", + "27435483", + "27435463" + ], + "answer": "27435463", + "correct_value": 27435463, + "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_000975", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "82468931 + 68176549 - 68176551", + "ns_shortcut": "Near-cancellation: 68176549 - 68176551 = -2, so answer ≈ 82468931 + -2.", + "shortcut_strength": 0.9999999706643996, + "pure_math": { + "question": "Evaluate: 82468931 + 68176549 - 68176551", + "options": [ + "82468929", + "82468939", + "82468949", + "82468919" + ], + "answer": "82468929", + "correct_value": 82468929, + "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": "51252524 + 14927546 - 14438852", + "ns_shortcut": "Near-cancellation: 14927546 - 14438852 = 488694, offset is manageable.", + "shortcut_strength": 0.9672622680244964, + "pure_math": { + "question": "Evaluate: 51252524 + 14927546 - 14438852", + "options": [ + "51741208", + "51741218", + "51741228", + "51741238" + ], + "answer": "51741218", + "correct_value": 51741218, + "explanation": "B-C = 488694; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "82468931 + 68176549 - 94689781", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 82468931 + 68176549 - 94689781", + "options": [ + "55955719", + "55955709", + "55955699", + "55955689" + ], + "answer": "55955699", + "correct_value": 55955699, + "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_000976", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "60607407 + 13137160 - 13137161", + "ns_shortcut": "Near-cancellation: 13137160 - 13137161 = -1, so answer ≈ 60607407 + -1.", + "shortcut_strength": 0.9999999238800471, + "pure_math": { + "question": "Evaluate: 60607407 + 13137160 - 13137161", + "options": [ + "60607396", + "60607426", + "60607406", + "60607416" + ], + "answer": "60607406", + "correct_value": 60607406, + "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": "21252551 + 56030775 - 58028365", + "ns_shortcut": "Near-cancellation: 56030775 - 58028365 = -1997590, offset is manageable.", + "shortcut_strength": 0.9643483424957089, + "pure_math": { + "question": "Evaluate: 21252551 + 56030775 - 58028365", + "options": [ + "19254981", + "19254951", + "19254971", + "19254961" + ], + "answer": "19254961", + "correct_value": 19254961, + "explanation": "B-C = -1997590; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "60607407 + 13137160 - 20267157", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 60607407 + 13137160 - 20267157", + "options": [ + "53477420", + "53477400", + "53477410", + "53477430" + ], + "answer": "53477410", + "correct_value": 53477410, + "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_000977", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "86724177 + 59465112 - 59465113", + "ns_shortcut": "Near-cancellation: 59465112 - 59465113 = -1, so answer ≈ 86724177 + -1.", + "shortcut_strength": 0.9999999831834169, + "pure_math": { + "question": "Evaluate: 86724177 + 59465112 - 59465113", + "options": [ + "86724166", + "86724196", + "86724186", + "86724176" + ], + "answer": "86724176", + "correct_value": 86724176, + "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": "34237787 + 10871700 - 11366599", + "ns_shortcut": "Near-cancellation: 10871700 - 11366599 = -494899, offset is manageable.", + "shortcut_strength": 0.9544782324751419, + "pure_math": { + "question": "Evaluate: 34237787 + 10871700 - 11366599", + "options": [ + "33742888", + "33742878", + "33742908", + "33742898" + ], + "answer": "33742888", + "correct_value": 33742888, + "explanation": "B-C = -494899; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "86724177 + 59465112 - 93634174", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 86724177 + 59465112 - 93634174", + "options": [ + "52555125", + "52555115", + "52555105", + "52555135" + ], + "answer": "52555115", + "correct_value": 52555115, + "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_000978", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "28517985 + 75535535 - 75535534", + "ns_shortcut": "Near-cancellation: 75535535 - 75535534 = 1, so answer ≈ 28517985 + 1.", + "shortcut_strength": 0.999999986761198, + "pure_math": { + "question": "Evaluate: 28517985 + 75535535 - 75535534", + "options": [ + "28517996", + "28517986", + "28518006", + "28517976" + ], + "answer": "28517986", + "correct_value": 28517986, + "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": "29124651 + 69213327 - 72012340", + "ns_shortcut": "Near-cancellation: 69213327 - 72012340 = -2799013, offset is manageable.", + "shortcut_strength": 0.9595596235389754, + "pure_math": { + "question": "Evaluate: 29124651 + 69213327 - 72012340", + "options": [ + "26325628", + "26325638", + "26325658", + "26325648" + ], + "answer": "26325638", + "correct_value": 26325638, + "explanation": "B-C = -2799013; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "28517985 + 75535535 - 118615800", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 28517985 + 75535535 - 118615800", + "options": [ + "-14562280", + "-14562260", + "-14562290", + "-14562270" + ], + "answer": "-14562280", + "correct_value": -14562280, + "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_000979", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "33760048 + 31839751 - 31839752", + "ns_shortcut": "Near-cancellation: 31839751 - 31839752 = -1, so answer ≈ 33760048 + -1.", + "shortcut_strength": 0.9999999685927192, + "pure_math": { + "question": "Evaluate: 33760048 + 31839751 - 31839752", + "options": [ + "33760067", + "33760047", + "33760057", + "33760037" + ], + "answer": "33760047", + "correct_value": 33760047, + "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": "56739001 + 83047461 - 80962669", + "ns_shortcut": "Near-cancellation: 83047461 - 80962669 = 2084792, offset is manageable.", + "shortcut_strength": 0.9748963788308953, + "pure_math": { + "question": "Evaluate: 56739001 + 83047461 - 80962669", + "options": [ + "58823813", + "58823803", + "58823793", + "58823783" + ], + "answer": "58823793", + "correct_value": 58823793, + "explanation": "B-C = 2084792; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "33760048 + 31839751 - 17818359", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 33760048 + 31839751 - 17818359", + "options": [ + "47781430", + "47781450", + "47781460", + "47781440" + ], + "answer": "47781440", + "correct_value": 47781440, + "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_000980", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49412831 + 60042871 - 60042869", + "ns_shortcut": "Near-cancellation: 60042871 - 60042869 = 2, so answer ≈ 49412831 + 2.", + "shortcut_strength": 0.9999999666904669, + "pure_math": { + "question": "Evaluate: 49412831 + 60042871 - 60042869", + "options": [ + "49412823", + "49412853", + "49412843", + "49412833" + ], + "answer": "49412833", + "correct_value": 49412833, + "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": "92408766 + 89226839 - 91817959", + "ns_shortcut": "Near-cancellation: 89226839 - 91817959 = -2591120, offset is manageable.", + "shortcut_strength": 0.9709603071335968, + "pure_math": { + "question": "Evaluate: 92408766 + 89226839 - 91817959", + "options": [ + "89817646", + "89817636", + "89817656", + "89817666" + ], + "answer": "89817646", + "correct_value": 89817646, + "explanation": "B-C = -2591120; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "49412831 + 60042871 - 25355305", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 49412831 + 60042871 - 25355305", + "options": [ + "84100407", + "84100387", + "84100417", + "84100397" + ], + "answer": "84100397", + "correct_value": 84100397, + "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_000981", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "18505287 + 94841482 - 94841480", + "ns_shortcut": "Near-cancellation: 94841482 - 94841480 = 2, so answer ≈ 18505287 + 2.", + "shortcut_strength": 0.9999999789121811, + "pure_math": { + "question": "Evaluate: 18505287 + 94841482 - 94841480", + "options": [ + "18505289", + "18505279", + "18505309", + "18505299" + ], + "answer": "18505289", + "correct_value": 18505289, + "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": "81086060 + 46963465 - 44799020", + "ns_shortcut": "Near-cancellation: 46963465 - 44799020 = 2164445, offset is manageable.", + "shortcut_strength": 0.9539121527766318, + "pure_math": { + "question": "Evaluate: 81086060 + 46963465 - 44799020", + "options": [ + "83250505", + "83250515", + "83250525", + "83250495" + ], + "answer": "83250505", + "correct_value": 83250505, + "explanation": "B-C = 2164445; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "18505287 + 94841482 - 59299923", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 18505287 + 94841482 - 59299923", + "options": [ + "54046836", + "54046846", + "54046856", + "54046866" + ], + "answer": "54046846", + "correct_value": 54046846, + "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_000982", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "77524762 + 31871454 - 31871453", + "ns_shortcut": "Near-cancellation: 31871454 - 31871453 = 1, so answer ≈ 77524762 + 1.", + "shortcut_strength": 0.9999999686239605, + "pure_math": { + "question": "Evaluate: 77524762 + 31871454 - 31871453", + "options": [ + "77524773", + "77524753", + "77524763", + "77524783" + ], + "answer": "77524763", + "correct_value": 77524763, + "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": "17875461 + 62571371 - 64087536", + "ns_shortcut": "Near-cancellation: 62571371 - 64087536 = -1516165, offset is manageable.", + "shortcut_strength": 0.9757690302167105, + "pure_math": { + "question": "Evaluate: 17875461 + 62571371 - 64087536", + "options": [ + "16359316", + "16359296", + "16359286", + "16359306" + ], + "answer": "16359296", + "correct_value": 16359296, + "explanation": "B-C = -1516165; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "77524762 + 31871454 - 14716248", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 77524762 + 31871454 - 14716248", + "options": [ + "94679988", + "94679978", + "94679958", + "94679968" + ], + "answer": "94679968", + "correct_value": 94679968, + "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_000983", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "19729173 + 62275940 - 62275939", + "ns_shortcut": "Near-cancellation: 62275940 - 62275939 = 1, so answer ≈ 19729173 + 1.", + "shortcut_strength": 0.9999999839424343, + "pure_math": { + "question": "Evaluate: 19729173 + 62275940 - 62275939", + "options": [ + "19729194", + "19729164", + "19729184", + "19729174" + ], + "answer": "19729174", + "correct_value": 19729174, + "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": "32617641 + 12300328 - 12681685", + "ns_shortcut": "Near-cancellation: 12300328 - 12681685 = -381357, offset is manageable.", + "shortcut_strength": 0.9689961926218553, + "pure_math": { + "question": "Evaluate: 32617641 + 12300328 - 12681685", + "options": [ + "32236304", + "32236284", + "32236274", + "32236294" + ], + "answer": "32236284", + "correct_value": 32236284, + "explanation": "B-C = -381357; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "19729173 + 62275940 - 100231077", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 19729173 + 62275940 - 100231077", + "options": [ + "-18225956", + "-18225976", + "-18225946", + "-18225964" + ], + "answer": "-18225964", + "correct_value": -18225964, + "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_000984", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "72253115 + 24657560 - 24657562", + "ns_shortcut": "Near-cancellation: 24657560 - 24657562 = -2, so answer ≈ 72253115 + -2.", + "shortcut_strength": 0.9999999188889737, + "pure_math": { + "question": "Evaluate: 72253115 + 24657560 - 24657562", + "options": [ + "72253103", + "72253123", + "72253113", + "72253133" + ], + "answer": "72253113", + "correct_value": 72253113, + "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": "13223200 + 11614866 - 11282962", + "ns_shortcut": "Near-cancellation: 11614866 - 11282962 = 331904, offset is manageable.", + "shortcut_strength": 0.9714242075629629, + "pure_math": { + "question": "Evaluate: 13223200 + 11614866 - 11282962", + "options": [ + "13555124", + "13555104", + "13555114", + "13555094" + ], + "answer": "13555104", + "correct_value": 13555104, + "explanation": "B-C = 331904; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "72253115 + 24657560 - 16813682", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 72253115 + 24657560 - 16813682", + "options": [ + "80096983", + "80097013", + "80096993", + "80097003" + ], + "answer": "80096993", + "correct_value": 80096993, + "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_000985", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "72058983 + 19513380 - 19513378", + "ns_shortcut": "Near-cancellation: 19513380 - 19513378 = 2, so answer ≈ 72058983 + 2.", + "shortcut_strength": 0.999999897506224, + "pure_math": { + "question": "Evaluate: 72058983 + 19513380 - 19513378", + "options": [ + "72058995", + "72058985", + "72058975", + "72059005" + ], + "answer": "72058985", + "correct_value": 72058985, + "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": "97074737 + 39455531 - 40786796", + "ns_shortcut": "Near-cancellation: 39455531 - 40786796 = -1331265, offset is manageable.", + "shortcut_strength": 0.9662591031913877, + "pure_math": { + "question": "Evaluate: 97074737 + 39455531 - 40786796", + "options": [ + "95743482", + "95743492", + "95743462", + "95743472" + ], + "answer": "95743472", + "correct_value": 95743472, + "explanation": "B-C = -1331265; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "72058983 + 19513380 - 29596762", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 72058983 + 19513380 - 29596762", + "options": [ + "61975601", + "61975611", + "61975591", + "61975621" + ], + "answer": "61975601", + "correct_value": 61975601, + "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_000986", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "40467872 + 96685830 - 96685831", + "ns_shortcut": "Near-cancellation: 96685830 - 96685831 = -1, so answer ≈ 40467872 + -1.", + "shortcut_strength": 0.9999999896572228, + "pure_math": { + "question": "Evaluate: 40467872 + 96685830 - 96685831", + "options": [ + "40467891", + "40467871", + "40467881", + "40467861" + ], + "answer": "40467871", + "correct_value": 40467871, + "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": "46213263 + 14571882 - 15224211", + "ns_shortcut": "Near-cancellation: 14571882 - 15224211 = -652329, offset is manageable.", + "shortcut_strength": 0.9552337165508202, + "pure_math": { + "question": "Evaluate: 46213263 + 14571882 - 15224211", + "options": [ + "45560954", + "45560934", + "45560924", + "45560944" + ], + "answer": "45560934", + "correct_value": 45560934, + "explanation": "B-C = -652329; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "40467872 + 96685830 - 135054223", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 40467872 + 96685830 - 135054223", + "options": [ + "2099499", + "2099469", + "2099489", + "2099479" + ], + "answer": "2099479", + "correct_value": 2099479, + "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_000987", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "93806250 + 79077896 - 79077897", + "ns_shortcut": "Near-cancellation: 79077896 - 79077897 = -1, so answer ≈ 93806250 + -1.", + "shortcut_strength": 0.9999999873542412, + "pure_math": { + "question": "Evaluate: 93806250 + 79077896 - 79077897", + "options": [ + "93806269", + "93806259", + "93806249", + "93806239" + ], + "answer": "93806249", + "correct_value": 93806249, + "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": "50110387 + 25828859 - 26953779", + "ns_shortcut": "Near-cancellation: 25828859 - 26953779 = -1124920, offset is manageable.", + "shortcut_strength": 0.9564471663266272, + "pure_math": { + "question": "Evaluate: 50110387 + 25828859 - 26953779", + "options": [ + "48985477", + "48985487", + "48985457", + "48985467" + ], + "answer": "48985467", + "correct_value": 48985467, + "explanation": "B-C = -1124920; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "93806250 + 79077896 - 44204721", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 93806250 + 79077896 - 44204721", + "options": [ + "128679425", + "128679445", + "128679415", + "128679435" + ], + "answer": "128679425", + "correct_value": 128679425, + "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_000988", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "39766678 + 34585456 - 34585458", + "ns_shortcut": "Near-cancellation: 34585456 - 34585458 = -2, so answer ≈ 39766678 + -2.", + "shortcut_strength": 0.9999999421722241, + "pure_math": { + "question": "Evaluate: 39766678 + 34585456 - 34585458", + "options": [ + "39766696", + "39766676", + "39766686", + "39766666" + ], + "answer": "39766676", + "correct_value": 39766676, + "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": "30318497 + 90075221 - 92826385", + "ns_shortcut": "Near-cancellation: 90075221 - 92826385 = -2751164, offset is manageable.", + "shortcut_strength": 0.9694570385789006, + "pure_math": { + "question": "Evaluate: 30318497 + 90075221 - 92826385", + "options": [ + "27567353", + "27567343", + "27567323", + "27567333" + ], + "answer": "27567333", + "correct_value": 27567333, + "explanation": "B-C = -2751164; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "39766678 + 34585456 - 18549270", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 39766678 + 34585456 - 18549270", + "options": [ + "55802874", + "55802864", + "55802884", + "55802854" + ], + "answer": "55802864", + "correct_value": 55802864, + "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_000989", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "64442947 + 46642584 - 46642582", + "ns_shortcut": "Near-cancellation: 46642584 - 46642582 = 2, so answer ≈ 64442947 + 2.", + "shortcut_strength": 0.999999957120729, + "pure_math": { + "question": "Evaluate: 64442947 + 46642584 - 46642582", + "options": [ + "64442939", + "64442959", + "64442969", + "64442949" + ], + "answer": "64442949", + "correct_value": 64442949, + "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": "45949703 + 19533904 - 19004877", + "ns_shortcut": "Near-cancellation: 19533904 - 19004877 = 529027, offset is manageable.", + "shortcut_strength": 0.9729174977004085, + "pure_math": { + "question": "Evaluate: 45949703 + 19533904 - 19004877", + "options": [ + "46478750", + "46478730", + "46478740", + "46478720" + ], + "answer": "46478730", + "correct_value": 46478730, + "explanation": "B-C = 529027; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "64442947 + 46642584 - 75583787", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 64442947 + 46642584 - 75583787", + "options": [ + "35501744", + "35501734", + "35501764", + "35501754" + ], + "answer": "35501744", + "correct_value": 35501744, + "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_000990", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "33739994 + 52670436 - 52670437", + "ns_shortcut": "Near-cancellation: 52670436 - 52670437 = -1, so answer ≈ 33739994 + -1.", + "shortcut_strength": 0.999999981014017, + "pure_math": { + "question": "Evaluate: 33739994 + 52670436 - 52670437", + "options": [ + "33739993", + "33740003", + "33740013", + "33739983" + ], + "answer": "33739993", + "correct_value": 33739993, + "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": "95550747 + 19691506 - 18986040", + "ns_shortcut": "Near-cancellation: 19691506 - 18986040 = 705466, offset is manageable.", + "shortcut_strength": 0.9641740961813687, + "pure_math": { + "question": "Evaluate: 95550747 + 19691506 - 18986040", + "options": [ + "96256203", + "96256233", + "96256213", + "96256223" + ], + "answer": "96256213", + "correct_value": 96256213, + "explanation": "B-C = 705466; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "33739994 + 52670436 - 70252399", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 33739994 + 52670436 - 70252399", + "options": [ + "16158051", + "16158031", + "16158041", + "16158021" + ], + "answer": "16158031", + "correct_value": 16158031, + "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_000991", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "62811125 + 58332625 - 58332623", + "ns_shortcut": "Near-cancellation: 58332625 - 58332623 = 2, so answer ≈ 62811125 + 2.", + "shortcut_strength": 0.9999999657138694, + "pure_math": { + "question": "Evaluate: 62811125 + 58332625 - 58332623", + "options": [ + "62811117", + "62811127", + "62811137", + "62811147" + ], + "answer": "62811127", + "correct_value": 62811127, + "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": "33359652 + 28777766 - 27628855", + "ns_shortcut": "Near-cancellation: 28777766 - 27628855 = 1148911, offset is manageable.", + "shortcut_strength": 0.9600764353980779, + "pure_math": { + "question": "Evaluate: 33359652 + 28777766 - 27628855", + "options": [ + "34508553", + "34508563", + "34508573", + "34508583" + ], + "answer": "34508563", + "correct_value": 34508563, + "explanation": "B-C = 1148911; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "62811125 + 58332625 - 40506248", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 62811125 + 58332625 - 40506248", + "options": [ + "80637492", + "80637502", + "80637512", + "80637522" + ], + "answer": "80637502", + "correct_value": 80637502, + "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_000992", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "15704947 + 76220952 - 76220951", + "ns_shortcut": "Near-cancellation: 76220952 - 76220951 = 1, so answer ≈ 15704947 + 1.", + "shortcut_strength": 0.9999999868802478, + "pure_math": { + "question": "Evaluate: 15704947 + 76220952 - 76220951", + "options": [ + "15704958", + "15704938", + "15704968", + "15704948" + ], + "answer": "15704948", + "correct_value": 15704948, + "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": "87461081 + 53469706 - 51199113", + "ns_shortcut": "Near-cancellation: 53469706 - 51199113 = 2270593, offset is manageable.", + "shortcut_strength": 0.9575349638167078, + "pure_math": { + "question": "Evaluate: 87461081 + 53469706 - 51199113", + "options": [ + "89731674", + "89731684", + "89731664", + "89731694" + ], + "answer": "89731674", + "correct_value": 89731674, + "explanation": "B-C = 2270593; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "15704947 + 76220952 - 116116459", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 15704947 + 76220952 - 116116459", + "options": [ + "-24190550", + "-24190570", + "-24190540", + "-24190560" + ], + "answer": "-24190560", + "correct_value": -24190560, + "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_000993", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "29102432 + 54095086 - 54095085", + "ns_shortcut": "Near-cancellation: 54095086 - 54095085 = 1, so answer ≈ 29102432 + 1.", + "shortcut_strength": 0.9999999815140326, + "pure_math": { + "question": "Evaluate: 29102432 + 54095086 - 54095085", + "options": [ + "29102433", + "29102443", + "29102453", + "29102423" + ], + "answer": "29102433", + "correct_value": 29102433, + "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": "30316031 + 81803369 - 83468188", + "ns_shortcut": "Near-cancellation: 81803369 - 83468188 = -1664819, offset is manageable.", + "shortcut_strength": 0.9796485276786094, + "pure_math": { + "question": "Evaluate: 30316031 + 81803369 - 83468188", + "options": [ + "28651212", + "28651202", + "28651222", + "28651232" + ], + "answer": "28651212", + "correct_value": 28651212, + "explanation": "B-C = -1664819; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "29102432 + 54095086 - 83950834", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 29102432 + 54095086 - 83950834", + "options": [ + "-753316", + "-753304", + "-753334", + "-753314" + ], + "answer": "-753316", + "correct_value": -753316, + "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_000994", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "21161254 + 33773518 - 33773520", + "ns_shortcut": "Near-cancellation: 33773518 - 33773520 = -2, so answer ≈ 21161254 + -2.", + "shortcut_strength": 0.9999999407820056, + "pure_math": { + "question": "Evaluate: 21161254 + 33773518 - 33773520", + "options": [ + "21161262", + "21161242", + "21161272", + "21161252" + ], + "answer": "21161252", + "correct_value": 21161252, + "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": "11882834 + 70643957 - 67134470", + "ns_shortcut": "Near-cancellation: 70643957 - 67134470 = 3509487, offset is manageable.", + "shortcut_strength": 0.9503214832657236, + "pure_math": { + "question": "Evaluate: 11882834 + 70643957 - 67134470", + "options": [ + "15392321", + "15392311", + "15392331", + "15392341" + ], + "answer": "15392321", + "correct_value": 15392321, + "explanation": "B-C = 3509487; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "21161254 + 33773518 - 22642866", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 21161254 + 33773518 - 22642866", + "options": [ + "32291926", + "32291906", + "32291896", + "32291916" + ], + "answer": "32291906", + "correct_value": 32291906, + "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_000995", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "20595135 + 52883581 - 52883583", + "ns_shortcut": "Near-cancellation: 52883581 - 52883583 = -2, so answer ≈ 20595135 + -2.", + "shortcut_strength": 0.9999999621810785, + "pure_math": { + "question": "Evaluate: 20595135 + 52883581 - 52883583", + "options": [ + "20595143", + "20595123", + "20595133", + "20595153" + ], + "answer": "20595133", + "correct_value": 20595133, + "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": "11867704 + 42038126 - 40987099", + "ns_shortcut": "Near-cancellation: 42038126 - 40987099 = 1051027, offset is manageable.", + "shortcut_strength": 0.9749982432613671, + "pure_math": { + "question": "Evaluate: 11867704 + 42038126 - 40987099", + "options": [ + "12918751", + "12918721", + "12918741", + "12918731" + ], + "answer": "12918731", + "correct_value": 12918731, + "explanation": "B-C = 1051027; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "20595135 + 52883581 - 69822082", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 20595135 + 52883581 - 69822082", + "options": [ + "3656624", + "3656644", + "3656634", + "3656654" + ], + "answer": "3656634", + "correct_value": 3656634, + "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_000996", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "23407419 + 69450706 - 69450707", + "ns_shortcut": "Near-cancellation: 69450706 - 69450707 = -1, so answer ≈ 23407419 + -1.", + "shortcut_strength": 0.9999999856012983, + "pure_math": { + "question": "Evaluate: 23407419 + 69450706 - 69450707", + "options": [ + "23407438", + "23407418", + "23407408", + "23407428" + ], + "answer": "23407418", + "correct_value": 23407418, + "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": "33514770 + 28413767 - 29064089", + "ns_shortcut": "Near-cancellation: 28413767 - 29064089 = -650322, offset is manageable.", + "shortcut_strength": 0.9771124328569316, + "pure_math": { + "question": "Evaluate: 33514770 + 28413767 - 29064089", + "options": [ + "32864448", + "32864438", + "32864458", + "32864468" + ], + "answer": "32864448", + "correct_value": 32864448, + "explanation": "B-C = -650322; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "23407419 + 69450706 - 40877856", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 23407419 + 69450706 - 40877856", + "options": [ + "51980289", + "51980279", + "51980259", + "51980269" + ], + "answer": "51980269", + "correct_value": 51980269, + "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_000997", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "48633985 + 93488894 - 93488895", + "ns_shortcut": "Near-cancellation: 93488894 - 93488895 = -1, so answer ≈ 48633985 + -1.", + "shortcut_strength": 0.9999999893035423, + "pure_math": { + "question": "Evaluate: 48633985 + 93488894 - 93488895", + "options": [ + "48633994", + "48633984", + "48633974", + "48634004" + ], + "answer": "48633984", + "correct_value": 48633984, + "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": "28281941 + 10195849 - 10577147", + "ns_shortcut": "Near-cancellation: 10195849 - 10577147 = -381298, offset is manageable.", + "shortcut_strength": 0.9626026238717345, + "pure_math": { + "question": "Evaluate: 28281941 + 10195849 - 10577147", + "options": [ + "27900633", + "27900663", + "27900653", + "27900643" + ], + "answer": "27900643", + "correct_value": 27900643, + "explanation": "B-C = -381298; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "48633985 + 93488894 - 53758244", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 48633985 + 93488894 - 53758244", + "options": [ + "88364655", + "88364645", + "88364625", + "88364635" + ], + "answer": "88364635", + "correct_value": 88364635, + "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_000998", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "16121285 + 77978906 - 77978905", + "ns_shortcut": "Near-cancellation: 77978906 - 77978905 = 1, so answer ≈ 16121285 + 1.", + "shortcut_strength": 0.9999999871760191, + "pure_math": { + "question": "Evaluate: 16121285 + 77978906 - 77978905", + "options": [ + "16121306", + "16121296", + "16121276", + "16121286" + ], + "answer": "16121286", + "correct_value": 16121286, + "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": "49117936 + 84471197 - 81727089", + "ns_shortcut": "Near-cancellation: 84471197 - 81727089 = 2744108, offset is manageable.", + "shortcut_strength": 0.9675142759016425, + "pure_math": { + "question": "Evaluate: 49117936 + 84471197 - 81727089", + "options": [ + "51862054", + "51862034", + "51862044", + "51862064" + ], + "answer": "51862044", + "correct_value": 51862044, + "explanation": "B-C = 2744108; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "16121285 + 77978906 - 119674619", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 16121285 + 77978906 - 119674619", + "options": [ + "-25574442", + "-25574422", + "-25574428", + "-25574412" + ], + "answer": "-25574428", + "correct_value": -25574428, + "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_000999", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "86112485 + 55696794 - 55696795", + "ns_shortcut": "Near-cancellation: 55696794 - 55696795 = -1, so answer ≈ 86112485 + -1.", + "shortcut_strength": 0.9999999820456452, + "pure_math": { + "question": "Evaluate: 86112485 + 55696794 - 55696795", + "options": [ + "86112474", + "86112504", + "86112494", + "86112484" + ], + "answer": "86112484", + "correct_value": 86112484, + "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": "96949864 + 49403592 - 47594499", + "ns_shortcut": "Near-cancellation: 49403592 - 47594499 = 1809093, offset is manageable.", + "shortcut_strength": 0.9633813468461969, + "pure_math": { + "question": "Evaluate: 96949864 + 49403592 - 47594499", + "options": [ + "98758977", + "98758957", + "98758947", + "98758967" + ], + "answer": "98758957", + "correct_value": 98758957, + "explanation": "B-C = 1809093; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "86112485 + 55696794 - 38549931", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 86112485 + 55696794 - 38549931", + "options": [ + "103259348", + "103259358", + "103259368", + "103259338" + ], + "answer": "103259348", + "correct_value": 103259348, + "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_001000", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "83483941 + 17890640 - 17890642", + "ns_shortcut": "Near-cancellation: 17890640 - 17890642 = -2, so answer ≈ 83483941 + -2.", + "shortcut_strength": 0.9999998882097008, + "pure_math": { + "question": "Evaluate: 83483941 + 17890640 - 17890642", + "options": [ + "83483939", + "83483949", + "83483959", + "83483929" + ], + "answer": "83483939", + "correct_value": 83483939, + "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": "68710632 + 57024482 - 58384174", + "ns_shortcut": "Near-cancellation: 57024482 - 58384174 = -1359692, offset is manageable.", + "shortcut_strength": 0.9761559955950148, + "pure_math": { + "question": "Evaluate: 68710632 + 57024482 - 58384174", + "options": [ + "67350950", + "67350930", + "67350960", + "67350940" + ], + "answer": "67350940", + "correct_value": 67350940, + "explanation": "B-C = -1359692; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "83483941 + 17890640 - 27275824", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 83483941 + 17890640 - 27275824", + "options": [ + "74098747", + "74098757", + "74098777", + "74098767" + ], + "answer": "74098757", + "correct_value": 74098757, + "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_001001", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98880638 + 36017496 - 36017495", + "ns_shortcut": "Near-cancellation: 36017496 - 36017495 = 1, so answer ≈ 98880638 + 1.", + "shortcut_strength": 0.9999999722357157, + "pure_math": { + "question": "Evaluate: 98880638 + 36017496 - 36017495", + "options": [ + "98880639", + "98880649", + "98880629", + "98880659" + ], + "answer": "98880639", + "correct_value": 98880639, + "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": "53309502 + 53462068 - 55996213", + "ns_shortcut": "Near-cancellation: 53462068 - 55996213 = -2534145, offset is manageable.", + "shortcut_strength": 0.9525991961253725, + "pure_math": { + "question": "Evaluate: 53309502 + 53462068 - 55996213", + "options": [ + "50775377", + "50775347", + "50775357", + "50775367" + ], + "answer": "50775357", + "correct_value": 50775357, + "explanation": "B-C = -2534145; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "98880638 + 36017496 - 54514699", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 98880638 + 36017496 - 54514699", + "options": [ + "80383435", + "80383455", + "80383425", + "80383445" + ], + "answer": "80383435", + "correct_value": 80383435, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001002", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2490 * 40", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.998, + "pure_math": { + "question": "Estimate 2490 × 40.", + "options": [ + "99620", + "99590", + "99610", + "99600" + ], + "answer": "99600", + "correct_value": 99600, + "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": "2520 * 49", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8835, + "pure_math": { + "question": "Estimate 2520 × 49.", + "options": [ + "123500", + "123490", + "123480", + "123470" + ], + "answer": "123480", + "correct_value": 123480, + "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": "2804 * 60", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2804 × 60.", + "options": [ + "168250", + "168230", + "168240", + "168260" + ], + "answer": "168240", + "correct_value": 168240, + "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_001003", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4972 * 19", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9722, + "pure_math": { + "question": "Estimate 4972 × 19.", + "options": [ + "94478", + "94458", + "94468", + "94488" + ], + "answer": "94468", + "correct_value": 94468, + "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": "4906 * 14", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8406, + "pure_math": { + "question": "Estimate 4906 × 14.", + "options": [ + "68674", + "68694", + "68704", + "68684" + ], + "answer": "68684", + "correct_value": 68684, + "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": "6193 * 26", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6193 × 26.", + "options": [ + "161028", + "161008", + "161018", + "161038" + ], + "answer": "161018", + "correct_value": 161018, + "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_001004", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12254 * 8444", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.96241, + "pure_math": { + "question": "Estimate 12254 × 8444.", + "options": [ + "103472766", + "103472796", + "103472786", + "103472776" + ], + "answer": "103472776", + "correct_value": 103472776, + "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": "12276 * 10647", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8256025, + "pure_math": { + "question": "Estimate 12276 × 10647.", + "options": [ + "130702582", + "130702562", + "130702592", + "130702572" + ], + "answer": "130702572", + "correct_value": 130702572, + "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": "86110044 * 47245729", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 86110044 × 47245729.", + "options": [ + "4068331803002066", + "4068331803002096", + "4068331803002076", + "4068331803002086" + ], + "answer": "4068331803002076", + "correct_value": 4068331803002076, + "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_001005", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1248 * 74", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.9617, + "pure_math": { + "question": "Estimate 1248 × 74.", + "options": [ + "92342", + "92362", + "92352", + "92372" + ], + "answer": "92352", + "correct_value": 92352, + "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": "1243 * 58", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8597, + "pure_math": { + "question": "Estimate 1243 × 58.", + "options": [ + "72084", + "72114", + "72094", + "72104" + ], + "answer": "72094", + "correct_value": 72094, + "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": "1492 * 111", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1492 × 111.", + "options": [ + "165622", + "165612", + "165602", + "165632" + ], + "answer": "165612", + "correct_value": 165612, + "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_001006", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2529 * 40", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.9942, + "pure_math": { + "question": "Estimate 2529 × 40.", + "options": [ + "101180", + "101150", + "101160", + "101170" + ], + "answer": "101160", + "correct_value": 101160, + "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": "2495 * 49", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8865, + "pure_math": { + "question": "Estimate 2495 × 49.", + "options": [ + "122245", + "122255", + "122275", + "122265" + ], + "answer": "122255", + "correct_value": 122255, + "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": "2842 * 56", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2842 × 56.", + "options": [ + "159162", + "159152", + "159172", + "159142" + ], + "answer": "159152", + "correct_value": 159152, + "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_001007", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2496 * 44", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.9492, + "pure_math": { + "question": "Estimate 2496 × 44.", + "options": [ + "109814", + "109834", + "109844", + "109824" + ], + "answer": "109824", + "correct_value": 109824, + "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": "2457 * 52", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8414, + "pure_math": { + "question": "Estimate 2457 × 52.", + "options": [ + "127784", + "127754", + "127774", + "127764" + ], + "answer": "127764", + "correct_value": 127764, + "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": "2807 * 59", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2807 × 59.", + "options": [ + "165613", + "165603", + "165633", + "165623" + ], + "answer": "165613", + "correct_value": 165613, + "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_001008", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1227 * 87", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.94705, + "pure_math": { + "question": "Estimate 1227 × 87.", + "options": [ + "106759", + "106769", + "106749", + "106739" + ], + "answer": "106749", + "correct_value": 106749, + "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": "1267 * 64", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8932, + "pure_math": { + "question": "Estimate 1267 × 64.", + "options": [ + "81078", + "81098", + "81088", + "81108" + ], + "answer": "81088", + "correct_value": 81088, + "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": "1511 * 118", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1511 × 118.", + "options": [ + "178318", + "178288", + "178298", + "178308" + ], + "answer": "178298", + "correct_value": 178298, + "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_001009", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5022 * 21", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9728, + "pure_math": { + "question": "Estimate 5022 × 21.", + "options": [ + "105472", + "105452", + "105462", + "105482" + ], + "answer": "105462", + "correct_value": 105462, + "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": "4979 * 24", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8979, + "pure_math": { + "question": "Estimate 4979 × 24.", + "options": [ + "119506", + "119516", + "119486", + "119496" + ], + "answer": "119496", + "correct_value": 119496, + "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": "6064 * 27", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6064 × 27.", + "options": [ + "163738", + "163728", + "163718", + "163748" + ], + "answer": "163728", + "correct_value": 163728, + "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_001010", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4916 * 20", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9916, + "pure_math": { + "question": "Estimate 4916 × 20.", + "options": [ + "98310", + "98330", + "98320", + "98340" + ], + "answer": "98320", + "correct_value": 98320, + "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": "5062 * 26", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8438, + "pure_math": { + "question": "Estimate 5062 × 26.", + "options": [ + "131602", + "131612", + "131632", + "131622" + ], + "answer": "131612", + "correct_value": 131612, + "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": "6037 * 26", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6037 × 26.", + "options": [ + "156952", + "156982", + "156972", + "156962" + ], + "answer": "156962", + "correct_value": 156962, + "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_001011", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4914 * 18", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9414, + "pure_math": { + "question": "Estimate 4914 × 18.", + "options": [ + "88462", + "88472", + "88442", + "88452" + ], + "answer": "88452", + "correct_value": 88452, + "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": "4994 * 26", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8494, + "pure_math": { + "question": "Estimate 4994 × 26.", + "options": [ + "129834", + "129864", + "129844", + "129854" + ], + "answer": "129844", + "correct_value": 129844, + "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": "5633 * 29", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5633 × 29.", + "options": [ + "163377", + "163357", + "163367", + "163347" + ], + "answer": "163357", + "correct_value": 163357, + "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_001012", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2477 * 40", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.9954, + "pure_math": { + "question": "Estimate 2477 × 40.", + "options": [ + "99070", + "99080", + "99100", + "99090" + ], + "answer": "99080", + "correct_value": 99080, + "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": "2521 * 29", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8583, + "pure_math": { + "question": "Estimate 2521 × 29.", + "options": [ + "73099", + "73109", + "73119", + "73129" + ], + "answer": "73109", + "correct_value": 73109, + "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": "2751 * 53", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2751 × 53.", + "options": [ + "145813", + "145803", + "145793", + "145823" + ], + "answer": "145803", + "correct_value": 145803, + "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_001013", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49498 * 2105", + "ns_shortcut": "Round to 50000×2000 for easy mental computation.", + "shortcut_strength": 0.96873, + "pure_math": { + "question": "Estimate 49498 × 2105.", + "options": [ + "104193300", + "104193310", + "104193290", + "104193280" + ], + "answer": "104193290", + "correct_value": 104193290, + "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": "49220 * 1410", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8447, + "pure_math": { + "question": "Estimate 49220 × 1410.", + "options": [ + "69400220", + "69400200", + "69400210", + "69400190" + ], + "answer": "69400200", + "correct_value": 69400200, + "explanation": "One number rounds to 50000, other is farther from 2000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "11008473 * 14547539", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 11008473 × 14547539.", + "options": [ + "160146190297957", + "160146190297937", + "160146190297967", + "160146190297947" + ], + "answer": "160146190297947", + "correct_value": 160146190297947, + "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_001014", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2463 * 40", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.9926, + "pure_math": { + "question": "Estimate 2463 × 40.", + "options": [ + "98540", + "98510", + "98530", + "98520" + ], + "answer": "98520", + "correct_value": 98520, + "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": "2504 * 51", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8617, + "pure_math": { + "question": "Estimate 2504 × 51.", + "options": [ + "127694", + "127724", + "127704", + "127714" + ], + "answer": "127704", + "correct_value": 127704, + "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": "2853 * 54", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2853 × 54.", + "options": [ + "154082", + "154062", + "154072", + "154052" + ], + "answer": "154062", + "correct_value": 154062, + "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_001015", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2525 * 43", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.9575, + "pure_math": { + "question": "Estimate 2525 × 43.", + "options": [ + "108595", + "108575", + "108585", + "108565" + ], + "answer": "108575", + "correct_value": 108575, + "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": "2514 * 49", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8847, + "pure_math": { + "question": "Estimate 2514 × 49.", + "options": [ + "123176", + "123196", + "123206", + "123186" + ], + "answer": "123186", + "correct_value": 123186, + "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": "2935 * 58", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2935 × 58.", + "options": [ + "170220", + "170250", + "170240", + "170230" + ], + "answer": "170230", + "correct_value": 170230, + "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_001016", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1270 * 73", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.94825, + "pure_math": { + "question": "Estimate 1270 × 73.", + "options": [ + "92730", + "92710", + "92720", + "92700" + ], + "answer": "92710", + "correct_value": 92710, + "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": "1269 * 61", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.87365, + "pure_math": { + "question": "Estimate 1269 × 61.", + "options": [ + "77429", + "77409", + "77419", + "77399" + ], + "answer": "77409", + "correct_value": 77409, + "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": "1452 * 116", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1452 × 116.", + "options": [ + "168452", + "168442", + "168432", + "168422" + ], + "answer": "168432", + "correct_value": 168432, + "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_001017", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4979 * 20", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9979, + "pure_math": { + "question": "Estimate 4979 × 20.", + "options": [ + "99590", + "99600", + "99580", + "99570" + ], + "answer": "99580", + "correct_value": 99580, + "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": "5018 * 24", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8982, + "pure_math": { + "question": "Estimate 5018 × 24.", + "options": [ + "120442", + "120432", + "120452", + "120422" + ], + "answer": "120432", + "correct_value": 120432, + "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": "5851 * 30", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5851 × 30.", + "options": [ + "175520", + "175530", + "175540", + "175550" + ], + "answer": "175530", + "correct_value": 175530, + "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_001018", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2466 * 44", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.9432, + "pure_math": { + "question": "Estimate 2466 × 44.", + "options": [ + "108494", + "108514", + "108524", + "108504" + ], + "answer": "108504", + "correct_value": 108504, + "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": "2460 * 27", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8295, + "pure_math": { + "question": "Estimate 2460 × 27.", + "options": [ + "66430", + "66420", + "66410", + "66440" + ], + "answer": "66420", + "correct_value": 66420, + "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": "3086 * 59", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 3086 × 59.", + "options": [ + "182084", + "182064", + "182074", + "182094" + ], + "answer": "182074", + "correct_value": 182074, + "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_001019", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1246 * 87", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.95465, + "pure_math": { + "question": "Estimate 1246 × 87.", + "options": [ + "108422", + "108402", + "108392", + "108412" + ], + "answer": "108402", + "correct_value": 108402, + "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": "1267 * 61", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.87445, + "pure_math": { + "question": "Estimate 1267 × 61.", + "options": [ + "77307", + "77287", + "77277", + "77297" + ], + "answer": "77287", + "correct_value": 77287, + "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": "1383 * 118", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1383 × 118.", + "options": [ + "163184", + "163214", + "163194", + "163204" + ], + "answer": "163194", + "correct_value": 163194, + "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_001020", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1262 * 75", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.96395, + "pure_math": { + "question": "Estimate 1262 × 75.", + "options": [ + "94650", + "94640", + "94660", + "94670" + ], + "answer": "94650", + "correct_value": 94650, + "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 * 103", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.84745, + "pure_math": { + "question": "Estimate 1272 × 103.", + "options": [ + "131016", + "131006", + "131026", + "131036" + ], + "answer": "131016", + "correct_value": 131016, + "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": "1525 * 117", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1525 × 117.", + "options": [ + "178425", + "178435", + "178415", + "178445" + ], + "answer": "178425", + "correct_value": 178425, + "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_001021", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5030 * 21", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.972, + "pure_math": { + "question": "Estimate 5030 × 21.", + "options": [ + "105630", + "105650", + "105640", + "105620" + ], + "answer": "105630", + "correct_value": 105630, + "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": "5093 * 24", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8907, + "pure_math": { + "question": "Estimate 5093 × 24.", + "options": [ + "122252", + "122242", + "122232", + "122222" + ], + "answer": "122232", + "correct_value": 122232, + "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": "5629 * 26", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5629 × 26.", + "options": [ + "146354", + "146374", + "146344", + "146364" + ], + "answer": "146354", + "correct_value": 146354, + "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_001022", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1233 * 75", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.96195, + "pure_math": { + "question": "Estimate 1233 × 75.", + "options": [ + "92485", + "92475", + "92465", + "92495" + ], + "answer": "92475", + "correct_value": 92475, + "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": "1255 * 105", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.84175, + "pure_math": { + "question": "Estimate 1255 × 105.", + "options": [ + "131775", + "131795", + "131765", + "131785" + ], + "answer": "131775", + "correct_value": 131775, + "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": "1522 * 112", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1522 × 112.", + "options": [ + "170464", + "170484", + "170474", + "170454" + ], + "answer": "170464", + "correct_value": 170464, + "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_001023", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4970 * 19", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.972, + "pure_math": { + "question": "Estimate 4970 × 19.", + "options": [ + "94440", + "94420", + "94430", + "94450" + ], + "answer": "94430", + "correct_value": 94430, + "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": "5099 * 15", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8651, + "pure_math": { + "question": "Estimate 5099 × 15.", + "options": [ + "76505", + "76485", + "76475", + "76495" + ], + "answer": "76485", + "correct_value": 76485, + "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": "5792 * 26", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5792 × 26.", + "options": [ + "150602", + "150592", + "150582", + "150612" + ], + "answer": "150592", + "correct_value": 150592, + "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_001024", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4947 * 19", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9697, + "pure_math": { + "question": "Estimate 4947 × 19.", + "options": [ + "93983", + "93993", + "94003", + "94013" + ], + "answer": "93993", + "correct_value": 93993, + "explanation": "Round to compatible pair 5000×20 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4943 * 24", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8943, + "pure_math": { + "question": "Estimate 4943 × 24.", + "options": [ + "118642", + "118652", + "118632", + "118622" + ], + "answer": "118632", + "correct_value": 118632, + "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": "6197 * 26", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6197 × 26.", + "options": [ + "161132", + "161112", + "161142", + "161122" + ], + "answer": "161122", + "correct_value": 161122, + "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_001025", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12709 * 8657", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.9505775, + "pure_math": { + "question": "Estimate 12709 × 8657.", + "options": [ + "110021813", + "110021823", + "110021833", + "110021803" + ], + "answer": "110021813", + "correct_value": 110021813, + "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": "12563 * 10295", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8540425, + "pure_math": { + "question": "Estimate 12563 × 10295.", + "options": [ + "129336095", + "129336075", + "129336085", + "129336105" + ], + "answer": "129336085", + "correct_value": 129336085, + "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": "90136357 * 74054651", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 90136357 × 74054651.", + "options": [ + "6675016460046407", + "6675016460046417", + "6675016460046427", + "6675016460046397" + ], + "answer": "6675016460046407", + "correct_value": 6675016460046407, + "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_001026", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1271 * 80", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.9916, + "pure_math": { + "question": "Estimate 1271 × 80.", + "options": [ + "101700", + "101670", + "101690", + "101680" + ], + "answer": "101680", + "correct_value": 101680, + "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": "1230 * 56", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.842, + "pure_math": { + "question": "Estimate 1230 × 56.", + "options": [ + "68900", + "68880", + "68890", + "68870" + ], + "answer": "68880", + "correct_value": 68880, + "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": "1431 * 120", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1431 × 120.", + "options": [ + "171720", + "171740", + "171730", + "171710" + ], + "answer": "171720", + "correct_value": 171720, + "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_001027", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5051 * 21", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9699, + "pure_math": { + "question": "Estimate 5051 × 21.", + "options": [ + "106091", + "106061", + "106071", + "106081" + ], + "answer": "106071", + "correct_value": 106071, + "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": "4986 * 24", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8986, + "pure_math": { + "question": "Estimate 4986 × 24.", + "options": [ + "119654", + "119674", + "119684", + "119664" + ], + "answer": "119664", + "correct_value": 119664, + "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": "5656 * 29", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5656 × 29.", + "options": [ + "164044", + "164034", + "164014", + "164024" + ], + "answer": "164024", + "correct_value": 164024, + "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_001028", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1249 * 78", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.9871, + "pure_math": { + "question": "Estimate 1249 × 78.", + "options": [ + "97432", + "97422", + "97442", + "97412" + ], + "answer": "97422", + "correct_value": 97422, + "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": "1248 * 98", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8867, + "pure_math": { + "question": "Estimate 1248 × 98.", + "options": [ + "122294", + "122304", + "122314", + "122324" + ], + "answer": "122304", + "correct_value": 122304, + "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": "1405 * 112", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1405 × 112.", + "options": [ + "157350", + "157370", + "157380", + "157360" + ], + "answer": "157360", + "correct_value": 157360, + "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_001029", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4902 * 19", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9652000000000001, + "pure_math": { + "question": "Estimate 4902 × 19.", + "options": [ + "93158", + "93128", + "93148", + "93138" + ], + "answer": "93138", + "correct_value": 93138, + "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 * 16", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8975, + "pure_math": { + "question": "Estimate 4975 × 16.", + "options": [ + "79600", + "79610", + "79620", + "79590" + ], + "answer": "79600", + "correct_value": 79600, + "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": "5859 * 27", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5859 × 27.", + "options": [ + "158203", + "158213", + "158193", + "158183" + ], + "answer": "158193", + "correct_value": 158193, + "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_001030", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5100 * 19", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.965, + "pure_math": { + "question": "Estimate 5100 × 19.", + "options": [ + "96900", + "96890", + "96920", + "96910" + ], + "answer": "96900", + "correct_value": 96900, + "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": "5084 * 16", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8916, + "pure_math": { + "question": "Estimate 5084 × 16.", + "options": [ + "81334", + "81364", + "81344", + "81354" + ], + "answer": "81344", + "correct_value": 81344, + "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": "6155 * 29", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6155 × 29.", + "options": [ + "178485", + "178495", + "178515", + "178505" + ], + "answer": "178495", + "correct_value": 178495, + "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_001031", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4964 * 21", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9714, + "pure_math": { + "question": "Estimate 4964 × 21.", + "options": [ + "104264", + "104244", + "104234", + "104254" + ], + "answer": "104244", + "correct_value": 104244, + "explanation": "Round to compatible pair 5000×20 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4971 * 25", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8721, + "pure_math": { + "question": "Estimate 4971 × 25.", + "options": [ + "124265", + "124285", + "124295", + "124275" + ], + "answer": "124275", + "correct_value": 124275, + "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": "5535 * 29", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5535 × 29.", + "options": [ + "160525", + "160535", + "160515", + "160505" + ], + "answer": "160515", + "correct_value": 160515, + "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_001032", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12439 * 8107", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.9908725, + "pure_math": { + "question": "Estimate 12439 × 8107.", + "options": [ + "100842983", + "100842963", + "100842973", + "100842993" + ], + "answer": "100842973", + "correct_value": 100842973, + "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": "12296 * 9653", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8885275, + "pure_math": { + "question": "Estimate 12296 × 9653.", + "options": [ + "118693288", + "118693308", + "118693298", + "118693278" + ], + "answer": "118693288", + "correct_value": 118693288, + "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": "51922734 * 84085258", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 51922734 × 84085258.", + "options": [ + "4365936484455382", + "4365936484455362", + "4365936484455392", + "4365936484455372" + ], + "answer": "4365936484455372", + "correct_value": 4365936484455372, + "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_001033", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4937 * 22", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9437, + "pure_math": { + "question": "Estimate 4937 × 22.", + "options": [ + "108614", + "108624", + "108604", + "108634" + ], + "answer": "108614", + "correct_value": 108614, + "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": "5061 * 15", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8689, + "pure_math": { + "question": "Estimate 5061 × 15.", + "options": [ + "75935", + "75915", + "75925", + "75905" + ], + "answer": "75915", + "correct_value": 75915, + "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": "5648 * 27", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5648 × 27.", + "options": [ + "152486", + "152496", + "152516", + "152506" + ], + "answer": "152496", + "correct_value": 152496, + "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_001034", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4909 * 18", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9409, + "pure_math": { + "question": "Estimate 4909 × 18.", + "options": [ + "88352", + "88372", + "88382", + "88362" + ], + "answer": "88362", + "correct_value": 88362, + "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": "5019 * 26", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8481, + "pure_math": { + "question": "Estimate 5019 × 26.", + "options": [ + "130484", + "130494", + "130514", + "130504" + ], + "answer": "130494", + "correct_value": 130494, + "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": "5682 * 28", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5682 × 28.", + "options": [ + "159116", + "159106", + "159096", + "159086" + ], + "answer": "159096", + "correct_value": 159096, + "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_001035", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5027 * 22", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9473, + "pure_math": { + "question": "Estimate 5027 × 22.", + "options": [ + "110614", + "110584", + "110604", + "110594" + ], + "answer": "110594", + "correct_value": 110594, + "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": "5031 * 14", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8469, + "pure_math": { + "question": "Estimate 5031 × 14.", + "options": [ + "70444", + "70424", + "70454", + "70434" + ], + "answer": "70434", + "correct_value": 70434, + "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": "5778 * 29", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5778 × 29.", + "options": [ + "167582", + "167562", + "167572", + "167552" + ], + "answer": "167562", + "correct_value": 167562, + "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_001036", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4938 * 20", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9938, + "pure_math": { + "question": "Estimate 4938 × 20.", + "options": [ + "98750", + "98770", + "98760", + "98780" + ], + "answer": "98760", + "correct_value": 98760, + "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": "4999 * 16", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8999, + "pure_math": { + "question": "Estimate 4999 × 16.", + "options": [ + "80004", + "79994", + "79984", + "79974" + ], + "answer": "79984", + "correct_value": 79984, + "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": "6176 * 26", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6176 × 26.", + "options": [ + "160576", + "160586", + "160566", + "160596" + ], + "answer": "160576", + "correct_value": 160576, + "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_001037", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5084 * 21", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9666, + "pure_math": { + "question": "Estimate 5084 × 21.", + "options": [ + "106754", + "106784", + "106774", + "106764" + ], + "answer": "106764", + "correct_value": 106764, + "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": "5005 * 25", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8745, + "pure_math": { + "question": "Estimate 5005 × 25.", + "options": [ + "125135", + "125125", + "125115", + "125145" + ], + "answer": "125125", + "correct_value": 125125, + "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": "5678 * 26", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5678 × 26.", + "options": [ + "147618", + "147638", + "147628", + "147648" + ], + "answer": "147628", + "correct_value": 147628, + "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_001038", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1248 * 85", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.96795, + "pure_math": { + "question": "Estimate 1248 × 85.", + "options": [ + "106100", + "106080", + "106090", + "106070" + ], + "answer": "106080", + "correct_value": 106080, + "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", + "129350", + "129380", + "129370" + ], + "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": "1490 * 114", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1490 × 114.", + "options": [ + "169880", + "169860", + "169870", + "169850" + ], + "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_001039", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2500 * 40", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Estimate 2500 × 40.", + "options": [ + "99990", + "100010", + "100000", + "100020" + ], + "answer": "100000", + "correct_value": 100000, + "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": "2458 * 32", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8916, + "pure_math": { + "question": "Estimate 2458 × 32.", + "options": [ + "78666", + "78676", + "78656", + "78646" + ], + "answer": "78656", + "correct_value": 78656, + "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": "3123 * 54", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 3123 × 54.", + "options": [ + "168662", + "168642", + "168632", + "168652" + ], + "answer": "168642", + "correct_value": 168642, + "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_001040", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4904 * 22", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9404, + "pure_math": { + "question": "Estimate 4904 × 22.", + "options": [ + "107888", + "107908", + "107898", + "107878" + ], + "answer": "107888", + "correct_value": 107888, + "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": "5004 * 24", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8996, + "pure_math": { + "question": "Estimate 5004 × 24.", + "options": [ + "120116", + "120106", + "120086", + "120096" + ], + "answer": "120096", + "correct_value": 120096, + "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": "6002 * 29", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6002 × 29.", + "options": [ + "174048", + "174078", + "174058", + "174068" + ], + "answer": "174058", + "correct_value": 174058, + "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_001041", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5083 * 22", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9417, + "pure_math": { + "question": "Estimate 5083 × 22.", + "options": [ + "111816", + "111846", + "111836", + "111826" + ], + "answer": "111826", + "correct_value": 111826, + "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": "5083 * 25", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8667, + "pure_math": { + "question": "Estimate 5083 × 25.", + "options": [ + "127065", + "127075", + "127095", + "127085" + ], + "answer": "127075", + "correct_value": 127075, + "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": "6146 * 30", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6146 × 30.", + "options": [ + "184390", + "184370", + "184380", + "184400" + ], + "answer": "184380", + "correct_value": 184380, + "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_001042", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2521 * 41", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.9833, + "pure_math": { + "question": "Estimate 2521 × 41.", + "options": [ + "103351", + "103371", + "103381", + "103361" + ], + "answer": "103361", + "correct_value": 103361, + "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": "2486 * 31", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8847, + "pure_math": { + "question": "Estimate 2486 × 31.", + "options": [ + "77076", + "77086", + "77056", + "77066" + ], + "answer": "77066", + "correct_value": 77066, + "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": "3050 * 54", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 3050 × 54.", + "options": [ + "164690", + "164710", + "164700", + "164720" + ], + "answer": "164700", + "correct_value": 164700, + "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_001043", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49121 * 2031", + "ns_shortcut": "Round to 50000×2000 for easy mental computation.", + "shortcut_strength": 0.98346, + "pure_math": { + "question": "Estimate 49121 × 2031.", + "options": [ + "99764741", + "99764751", + "99764771", + "99764761" + ], + "answer": "99764751", + "correct_value": 99764751, + "explanation": "Round to compatible pair 50000×2000 = 100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "50981 * 2445", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.87894, + "pure_math": { + "question": "Estimate 50981 × 2445.", + "options": [ + "124648535", + "124648555", + "124648545", + "124648565" + ], + "answer": "124648545", + "correct_value": 124648545, + "explanation": "One number rounds to 50000, other is farther from 2000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "86549252 * 37647263", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 86549252 × 37647263.", + "options": [ + "3258342452497266", + "3258342452497286", + "3258342452497276", + "3258342452497296" + ], + "answer": "3258342452497276", + "correct_value": 3258342452497276, + "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_001044", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49808 * 2179", + "ns_shortcut": "Round to 50000×2000 for easy mental computation.", + "shortcut_strength": 0.95333, + "pure_math": { + "question": "Estimate 49808 × 2179.", + "options": [ + "108531642", + "108531622", + "108531632", + "108531652" + ], + "answer": "108531632", + "correct_value": 108531632, + "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": "50295 * 1428", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.85405, + "pure_math": { + "question": "Estimate 50295 × 1428.", + "options": [ + "71821270", + "71821280", + "71821260", + "71821250" + ], + "answer": "71821260", + "correct_value": 71821260, + "explanation": "One number rounds to 50000, other is farther from 2000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "14194873 * 57393956", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 14194873 × 57393956.", + "options": [ + "814699916387598", + "814699916387588", + "814699916387578", + "814699916387608" + ], + "answer": "814699916387588", + "correct_value": 814699916387588, + "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_001045", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49626 * 2150", + "ns_shortcut": "Round to 50000×2000 for easy mental computation.", + "shortcut_strength": 0.9587600000000001, + "pure_math": { + "question": "Estimate 49626 × 2150.", + "options": [ + "106695900", + "106695890", + "106695910", + "106695920" + ], + "answer": "106695900", + "correct_value": 106695900, + "explanation": "Round to compatible pair 50000×2000 = 100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "49991 * 1531", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.88266, + "pure_math": { + "question": "Estimate 49991 × 1531.", + "options": [ + "76536231", + "76536241", + "76536221", + "76536211" + ], + "answer": "76536221", + "correct_value": 76536221, + "explanation": "One number rounds to 50000, other is farther from 2000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "39764868 * 23929226", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 39764868 × 23929226.", + "options": [ + "951542513232188", + "951542513232178", + "951542513232168", + "951542513232158" + ], + "answer": "951542513232168", + "correct_value": 951542513232168, + "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_001046", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12430 * 7211", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.9478875, + "pure_math": { + "question": "Estimate 12430 × 7211.", + "options": [ + "89632730", + "89632750", + "89632720", + "89632740" + ], + "answer": "89632730", + "correct_value": 89632730, + "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": "12284 * 5525", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8366724999999999, + "pure_math": { + "question": "Estimate 12284 × 5525.", + "options": [ + "67869090", + "67869120", + "67869100", + "67869110" + ], + "answer": "67869100", + "correct_value": 67869100, + "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": "22348064 * 24742952", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 22348064 × 24742952.", + "options": [ + "552957074844918", + "552957074844928", + "552957074844938", + "552957074844948" + ], + "answer": "552957074844928", + "correct_value": 552957074844928, + "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_001047", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49475 * 1807", + "ns_shortcut": "Round to 50000×2000 for easy mental computation.", + "shortcut_strength": 0.9465, + "pure_math": { + "question": "Estimate 49475 × 1807.", + "options": [ + "89401335", + "89401345", + "89401315", + "89401325" + ], + "answer": "89401325", + "correct_value": 89401325, + "explanation": "Round to compatible pair 50000×2000 = 100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "50763 * 1402", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.84287, + "pure_math": { + "question": "Estimate 50763 × 1402.", + "options": [ + "71169726", + "71169746", + "71169716", + "71169736" + ], + "answer": "71169726", + "correct_value": 71169726, + "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": "54074175 * 59942071", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 54074175 × 59942071.", + "options": [ + "3241318037116435", + "3241318037116415", + "3241318037116445", + "3241318037116425" + ], + "answer": "3241318037116425", + "correct_value": 3241318037116425, + "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_001048", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12321 * 8723", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.9476525, + "pure_math": { + "question": "Estimate 12321 × 8723.", + "options": [ + "107476093", + "107476103", + "107476083", + "107476073" + ], + "answer": "107476083", + "correct_value": 107476083, + "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": "12680 * 6203", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8804875, + "pure_math": { + "question": "Estimate 12680 × 6203.", + "options": [ + "78654050", + "78654060", + "78654040", + "78654030" + ], + "answer": "78654040", + "correct_value": 78654040, + "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": "87122241 * 58400342", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 87122241 × 58400342.", + "options": [ + "5087968670206442", + "5087968670206432", + "5087968670206412", + "5087968670206422" + ], + "answer": "5087968670206422", + "correct_value": 5087968670206422, + "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_001049", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24832 * 4386", + "ns_shortcut": "Round to 25000×4000 for easy mental computation.", + "shortcut_strength": 0.94839, + "pure_math": { + "question": "Estimate 24832 × 4386.", + "options": [ + "108913152", + "108913162", + "108913142", + "108913172" + ], + "answer": "108913152", + "correct_value": 108913152, + "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": "25120 * 3068", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8811, + "pure_math": { + "question": "Estimate 25120 × 3068.", + "options": [ + "77068160", + "77068180", + "77068150", + "77068170" + ], + "answer": "77068160", + "correct_value": 77068160, + "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": "97434275 * 20291255", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 97434275 × 20291255.", + "options": [ + "1977063719765145", + "1977063719765135", + "1977063719765125", + "1977063719765115" + ], + "answer": "1977063719765125", + "correct_value": 1977063719765125, + "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_001050", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1256 * 86", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.9601, + "pure_math": { + "question": "Estimate 1256 × 86.", + "options": [ + "108016", + "108006", + "108026", + "108036" + ], + "answer": "108016", + "correct_value": 108016, + "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": "1226 * 64", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8904, + "pure_math": { + "question": "Estimate 1226 × 64.", + "options": [ + "78454", + "78484", + "78464", + "78474" + ], + "answer": "78464", + "correct_value": 78464, + "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": "1452 * 110", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1452 × 110.", + "options": [ + "159720", + "159740", + "159730", + "159710" + ], + "answer": "159720", + "correct_value": 159720, + "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_001051", + "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", + "88364", + "88334", + "88344" + ], + "answer": "88344", + "correct_value": 88344, + "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": "2506 * 32", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8988, + "pure_math": { + "question": "Estimate 2506 × 32.", + "options": [ + "80192", + "80182", + "80212", + "80202" + ], + "answer": "80192", + "correct_value": 80192, + "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": "2920 * 59", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2920 × 59.", + "options": [ + "172280", + "172290", + "172270", + "172300" + ], + "answer": "172280", + "correct_value": 172280, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "landmark_comparison_001052", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "23% of 694675950 vs 328713977", + "ns_shortcut": "Use 25% as landmark: 25% of 694675950 = 173668987.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 23% of 694675950 greater than 328713977?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 159775468.5, + "explanation": "23% ≈ 25%. 25% of 694675950 = 173668987. Compare to 328713977." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20% of 694675950 vs 328713977", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 20% of 694675950 greater than 328713977?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 138935190.0, + "explanation": "20% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 694675950 vs 328713977", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 694675950 greater than 328713977?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 576581038.5, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001053", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26% of 603441050 vs 216904107", + "ns_shortcut": "Use 25% as landmark: 25% of 603441050 = 150860262.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 26% of 603441050 greater than 216904107?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 156894673.0, + "explanation": "26% ≈ 25%. 25% of 603441050 = 150860262. Compare to 216904107." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20% of 603441050 vs 216904107", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 20% of 603441050 greater than 216904107?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 120688210.0, + "explanation": "20% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "63% of 603441050 vs 216904107", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 63% of 603441050 greater than 216904107?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 380167861.5, + "explanation": "63% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001054", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26% of 500431390 vs 150861534", + "ns_shortcut": "Use 25% as landmark: 25% of 500431390 = 125107847.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 26% of 500431390 greater than 150861534?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 130112161.4, + "explanation": "26% ≈ 25%. 25% of 500431390 = 125107847. Compare to 150861534." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "29% of 500431390 vs 150861534", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 29% of 500431390 greater than 150861534?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 145125103.1, + "explanation": "29% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 500431390 vs 150861534", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 500431390 greater than 150861534?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 185159614.3, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001055", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 197279640 vs 54409278", + "ns_shortcut": "Use 10% as landmark: 10% of 197279640 = 19727964.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 197279640 greater than 54409278?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 19727964.0, + "explanation": "10% ≈ 10%. 10% of 197279640 = 19727964. Compare to 54409278." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5% of 197279640 vs 54409278", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 5% of 197279640 greater than 54409278?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 9863982.0, + "explanation": "5% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 197279640 vs 54409278", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 197279640 greater than 54409278?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 84830245.2, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001056", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 496088350 vs 127655569", + "ns_shortcut": "Use 10% as landmark: 10% of 496088350 = 49608835.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 496088350 greater than 127655569?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 54569718.5, + "explanation": "11% ≈ 10%. 10% of 496088350 = 49608835. Compare to 127655569." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5% of 496088350 vs 127655569", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 5% of 496088350 greater than 127655569?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 24804417.5, + "explanation": "5% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "87% of 496088350 vs 127655569", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 87% of 496088350 greater than 127655569?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 431596864.5, + "explanation": "87% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001057", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51% of 714072000 vs 221400247", + "ns_shortcut": "Use 50% as landmark: 50% of 714072000 = 357036000.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 51% of 714072000 greater than 221400247?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 364176720.0, + "explanation": "51% ≈ 50%. 50% of 714072000 = 357036000. Compare to 221400247." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "47% of 714072000 vs 221400247", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 47% of 714072000 greater than 221400247?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 335613840.0, + "explanation": "47% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 714072000 vs 221400247", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 714072000 greater than 221400247?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 264206640.0, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001058", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 321753880 vs 84055552", + "ns_shortcut": "Use 10% as landmark: 10% of 321753880 = 32175388.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 321753880 greater than 84055552?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 32175388.0, + "explanation": "10% ≈ 10%. 10% of 321753880 = 32175388. Compare to 84055552." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7% of 321753880 vs 84055552", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 7% of 321753880 greater than 84055552?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 22522771.6, + "explanation": "7% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "63% of 321753880 vs 84055552", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 63% of 321753880 greater than 84055552?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 202704944.4, + "explanation": "63% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001059", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 141656140 vs 34759939", + "ns_shortcut": "Use 25% as landmark: 25% of 141656140 = 35414035.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 141656140 greater than 34759939?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 35414035.0, + "explanation": "25% ≈ 25%. 25% of 141656140 = 35414035. Compare to 34759939." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "22% of 141656140 vs 34759939", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 22% of 141656140 greater than 34759939?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 31164350.8, + "explanation": "22% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 141656140 vs 34759939", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 141656140 greater than 34759939?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 60912140.2, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001060", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9% of 304032390 vs 106858439", + "ns_shortcut": "Use 10% as landmark: 10% of 304032390 = 30403239.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 9% of 304032390 greater than 106858439?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 27362915.1, + "explanation": "9% ≈ 10%. 10% of 304032390 = 30403239. Compare to 106858439." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "14% of 304032390 vs 106858439", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 14% of 304032390 greater than 106858439?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 42564534.6, + "explanation": "14% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 304032390 vs 106858439", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 304032390 greater than 106858439?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 112491984.3, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001061", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 154426490 vs 45770088", + "ns_shortcut": "Use 10% as landmark: 10% of 154426490 = 15442649.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 154426490 greater than 45770088?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 15442649.0, + "explanation": "10% ≈ 10%. 10% of 154426490 = 15442649. Compare to 45770088." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "15% of 154426490 vs 45770088", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 15% of 154426490 greater than 45770088?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 23163973.5, + "explanation": "15% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 154426490 vs 45770088", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 154426490 greater than 45770088?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 66403390.7, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001062", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "48% of 527533070 vs 121195419", + "ns_shortcut": "Use 50% as landmark: 50% of 527533070 = 263766535.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 48% of 527533070 greater than 121195419?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 253215873.6, + "explanation": "48% ≈ 50%. 50% of 527533070 = 263766535. Compare to 121195419." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "53% of 527533070 vs 121195419", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 53% of 527533070 greater than 121195419?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 279592527.1, + "explanation": "53% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 527533070 vs 121195419", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 527533070 greater than 121195419?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 226839220.1, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001063", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 317415310 vs 138522118", + "ns_shortcut": "Use 10% as landmark: 10% of 317415310 = 31741531.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 317415310 greater than 138522118?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 34915684.1, + "explanation": "11% ≈ 10%. 10% of 317415310 = 31741531. Compare to 138522118." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "15% of 317415310 vs 138522118", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 15% of 317415310 greater than 138522118?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 47612296.5, + "explanation": "15% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 317415310 vs 138522118", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 317415310 greater than 138522118?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 136488583.3, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001064", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 625400140 vs 203271602", + "ns_shortcut": "Use 50% as landmark: 50% of 625400140 = 312700070.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 625400140 greater than 203271602?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 312700070.0, + "explanation": "50% ≈ 50%. 50% of 625400140 = 312700070. Compare to 203271602." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "43% of 625400140 vs 203271602", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 43% of 625400140 greater than 203271602?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 268922060.2, + "explanation": "43% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "87% of 625400140 vs 203271602", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 87% of 625400140 greater than 203271602?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 544098121.8, + "explanation": "87% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001065", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9% of 167257020 vs 33713982", + "ns_shortcut": "Use 10% as landmark: 10% of 167257020 = 16725702.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 9% of 167257020 greater than 33713982?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 15053131.8, + "explanation": "9% ≈ 10%. 10% of 167257020 = 16725702. Compare to 33713982." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4% of 167257020 vs 33713982", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 4% of 167257020 greater than 33713982?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6690280.8, + "explanation": "4% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "87% of 167257020 vs 33713982", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 87% of 167257020 greater than 33713982?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 145513607.4, + "explanation": "87% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001066", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 638579050 vs 281694925", + "ns_shortcut": "Use 50% as landmark: 50% of 638579050 = 319289525.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 638579050 greater than 281694925?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 319289525.0, + "explanation": "50% ≈ 50%. 50% of 638579050 = 319289525. Compare to 281694925." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "46% of 638579050 vs 281694925", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 46% of 638579050 greater than 281694925?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 293746363.0, + "explanation": "46% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 638579050 vs 281694925", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 638579050 greater than 281694925?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 236274248.5, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001067", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 542686270 vs 148019114", + "ns_shortcut": "Use 10% as landmark: 10% of 542686270 = 54268627.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 542686270 greater than 148019114?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 54268627.0, + "explanation": "10% ≈ 10%. 10% of 542686270 = 54268627. Compare to 148019114." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "16% of 542686270 vs 148019114", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 16% of 542686270 greater than 148019114?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 86829803.2, + "explanation": "16% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 542686270 vs 148019114", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 542686270 greater than 148019114?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 233355096.1, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001068", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 999698360 vs 429600313", + "ns_shortcut": "Use 25% as landmark: 25% of 999698360 = 249924590.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 999698360 greater than 429600313?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 249924590.0, + "explanation": "25% ≈ 25%. 25% of 999698360 = 249924590. Compare to 429600313." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "31% of 999698360 vs 429600313", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 31% of 999698360 greater than 429600313?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 309906491.6, + "explanation": "31% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 999698360 vs 429600313", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 999698360 greater than 429600313?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 429870294.8, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001069", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 742081660 vs 312074051", + "ns_shortcut": "Use 10% as landmark: 10% of 742081660 = 74208166.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 742081660 greater than 312074051?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 81628982.6, + "explanation": "11% ≈ 10%. 10% of 742081660 = 74208166. Compare to 312074051." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7% of 742081660 vs 312074051", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 7% of 742081660 greater than 312074051?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 51945716.2, + "explanation": "7% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 742081660 vs 312074051", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 742081660 greater than 312074051?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 319095113.8, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001070", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27% of 779721940 vs 317002179", + "ns_shortcut": "Use 25% as landmark: 25% of 779721940 = 194930485.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 27% of 779721940 greater than 317002179?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 210524923.8, + "explanation": "27% ≈ 25%. 25% of 779721940 = 194930485. Compare to 317002179." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "18% of 779721940 vs 317002179", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 18% of 779721940 greater than 317002179?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 140349949.2, + "explanation": "18% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "87% of 779721940 vs 317002179", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 87% of 779721940 greater than 317002179?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 678358087.8, + "explanation": "87% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001071", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 846423180 vs 375867858", + "ns_shortcut": "Use 25% as landmark: 25% of 846423180 = 211605795.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 846423180 greater than 375867858?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 211605795.0, + "explanation": "25% ≈ 25%. 25% of 846423180 = 211605795. Compare to 375867858." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "22% of 846423180 vs 375867858", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 22% of 846423180 greater than 375867858?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 186213099.6, + "explanation": "22% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 846423180 vs 375867858", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 846423180 greater than 375867858?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 363961967.4, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001072", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "48% of 463533050 vs 128089084", + "ns_shortcut": "Use 50% as landmark: 50% of 463533050 = 231766525.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 48% of 463533050 greater than 128089084?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 222495864.0, + "explanation": "48% ≈ 50%. 50% of 463533050 = 231766525. Compare to 128089084." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "56% of 463533050 vs 128089084", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 56% of 463533050 greater than 128089084?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 259578508.0, + "explanation": "56% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 463533050 vs 128089084", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 463533050 greater than 128089084?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 171507228.5, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001073", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 880513230 vs 393400992", + "ns_shortcut": "Use 10% as landmark: 10% of 880513230 = 88051323.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 880513230 greater than 393400992?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 88051323.0, + "explanation": "10% ≈ 10%. 10% of 880513230 = 88051323. Compare to 393400992." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "14% of 880513230 vs 393400992", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 14% of 880513230 greater than 393400992?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 123271852.2, + "explanation": "14% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 880513230 vs 393400992", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 880513230 greater than 393400992?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 378620688.9, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001074", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 992529920 vs 437519403", + "ns_shortcut": "Use 10% as landmark: 10% of 992529920 = 99252992.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 992529920 greater than 437519403?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 99252992.0, + "explanation": "10% ≈ 10%. 10% of 992529920 = 99252992. Compare to 437519403." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "13% of 992529920 vs 437519403", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 13% of 992529920 greater than 437519403?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 129028889.6, + "explanation": "13% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 992529920 vs 437519403", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 992529920 greater than 437519403?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 823799833.6, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001075", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9% of 447548080 vs 213679790", + "ns_shortcut": "Use 10% as landmark: 10% of 447548080 = 44754808.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 9% of 447548080 greater than 213679790?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 40279327.2, + "explanation": "9% ≈ 10%. 10% of 447548080 = 44754808. Compare to 213679790." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "15% of 447548080 vs 213679790", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 15% of 447548080 greater than 213679790?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 67132212.0, + "explanation": "15% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 447548080 vs 213679790", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 447548080 greater than 213679790?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 299857213.6, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001076", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51% of 197452000 vs 89649200", + "ns_shortcut": "Use 50% as landmark: 50% of 197452000 = 98726000.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 51% of 197452000 greater than 89649200?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 100700520.0, + "explanation": "51% ≈ 50%. 50% of 197452000 = 98726000. Compare to 89649200." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "46% of 197452000 vs 89649200", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 46% of 197452000 greater than 89649200?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 90827920.0, + "explanation": "46% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 197452000 vs 89649200", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 197452000 greater than 89649200?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 163885160.0, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001077", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 881963990 vs 219141084", + "ns_shortcut": "Use 25% as landmark: 25% of 881963990 = 220490997.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 881963990 greater than 219141084?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 220490997.5, + "explanation": "25% ≈ 25%. 25% of 881963990 = 220490997. Compare to 219141084." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "30% of 881963990 vs 219141084", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 30% of 881963990 greater than 219141084?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 264589197.0, + "explanation": "30% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 881963990 vs 219141084", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 881963990 greater than 219141084?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 326326676.3, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001078", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 487313390 vs 176287187", + "ns_shortcut": "Use 25% as landmark: 25% of 487313390 = 121828347.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 487313390 greater than 176287187?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 121828347.5, + "explanation": "25% ≈ 25%. 25% of 487313390 = 121828347. Compare to 176287187." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20% of 487313390 vs 176287187", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 20% of 487313390 greater than 176287187?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 97462678.0, + "explanation": "20% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 487313390 vs 176287187", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 487313390 greater than 176287187?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 180305954.3, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001079", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12% of 617110550 vs 229240151", + "ns_shortcut": "Use 10% as landmark: 10% of 617110550 = 61711055.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 12% of 617110550 greater than 229240151?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 74053266.0, + "explanation": "12% ≈ 10%. 10% of 617110550 = 61711055. Compare to 229240151." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "17% of 617110550 vs 229240151", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 17% of 617110550 greater than 229240151?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 104908793.5, + "explanation": "17% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 617110550 vs 229240151", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 617110550 greater than 229240151?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 265357536.5, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001080", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49% of 432061940 vs 151570771", + "ns_shortcut": "Use 50% as landmark: 50% of 432061940 = 216030970.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 49% of 432061940 greater than 151570771?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 211710350.6, + "explanation": "49% ≈ 50%. 50% of 432061940 = 216030970. Compare to 151570771." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "46% of 432061940 vs 151570771", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 46% of 432061940 greater than 151570771?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 198748492.4, + "explanation": "46% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 432061940 vs 151570771", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 432061940 greater than 151570771?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 159862917.8, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001081", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 274470600 vs 66240394", + "ns_shortcut": "Use 50% as landmark: 50% of 274470600 = 137235300.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 274470600 greater than 66240394?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 137235300.0, + "explanation": "50% ≈ 50%. 50% of 274470600 = 137235300. Compare to 66240394." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "57% of 274470600 vs 66240394", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 57% of 274470600 greater than 66240394?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 156448242.0, + "explanation": "57% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 274470600 vs 66240394", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 274470600 greater than 66240394?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 101554122.0, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001082", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49% of 807228480 vs 226078055", + "ns_shortcut": "Use 50% as landmark: 50% of 807228480 = 403614240.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 49% of 807228480 greater than 226078055?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 395541955.2, + "explanation": "49% ≈ 50%. 50% of 807228480 = 403614240. Compare to 226078055." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "53% of 807228480 vs 226078055", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 53% of 807228480 greater than 226078055?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 427831094.4, + "explanation": "53% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "87% of 807228480 vs 226078055", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 87% of 807228480 greater than 226078055?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 702288777.6, + "explanation": "87% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001083", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "23% of 166415960 vs 65812956", + "ns_shortcut": "Use 25% as landmark: 25% of 166415960 = 41603990.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 23% of 166415960 greater than 65812956?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 38275670.8, + "explanation": "23% ≈ 25%. 25% of 166415960 = 41603990. Compare to 65812956." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "18% of 166415960 vs 65812956", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 18% of 166415960 greater than 65812956?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 29954872.8, + "explanation": "18% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 166415960 vs 65812956", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 166415960 greater than 65812956?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 71558862.8, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001084", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 230904470 vs 92968006", + "ns_shortcut": "Use 25% as landmark: 25% of 230904470 = 57726117.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 230904470 greater than 92968006?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 57726117.5, + "explanation": "25% ≈ 25%. 25% of 230904470 = 57726117. Compare to 92968006." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "31% of 230904470 vs 92968006", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 31% of 230904470 greater than 92968006?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 71580385.7, + "explanation": "31% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 230904470 vs 92968006", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 230904470 greater than 92968006?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 99288922.1, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001085", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 662992380 vs 327455537", + "ns_shortcut": "Use 25% as landmark: 25% of 662992380 = 165748095.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 662992380 greater than 327455537?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 165748095.0, + "explanation": "25% ≈ 25%. 25% of 662992380 = 165748095. Compare to 327455537." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "29% of 662992380 vs 327455537", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 29% of 662992380 greater than 327455537?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 192267790.2, + "explanation": "29% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "63% of 662992380 vs 327455537", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 63% of 662992380 greater than 327455537?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 417685199.4, + "explanation": "63% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001086", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "48% of 620510340 vs 192426183", + "ns_shortcut": "Use 50% as landmark: 50% of 620510340 = 310255170.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 48% of 620510340 greater than 192426183?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 297844963.2, + "explanation": "48% ≈ 50%. 50% of 620510340 = 310255170. Compare to 192426183." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "54% of 620510340 vs 192426183", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 54% of 620510340 greater than 192426183?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 335075583.6, + "explanation": "54% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 620510340 vs 192426183", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 620510340 greater than 192426183?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 266819446.2, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001087", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 128808510 vs 49968980", + "ns_shortcut": "Use 25% as landmark: 25% of 128808510 = 32202127.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 128808510 greater than 49968980?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 32202127.5, + "explanation": "25% ≈ 25%. 25% of 128808510 = 32202127. Compare to 49968980." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "19% of 128808510 vs 49968980", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 19% of 128808510 greater than 49968980?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 24473616.9, + "explanation": "19% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 128808510 vs 49968980", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 128808510 greater than 49968980?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 55387659.3, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001088", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12% of 605845370 vs 207203366", + "ns_shortcut": "Use 10% as landmark: 10% of 605845370 = 60584537.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 12% of 605845370 greater than 207203366?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 72701444.4, + "explanation": "12% ≈ 10%. 10% of 605845370 = 60584537. Compare to 207203366." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "14% of 605845370 vs 207203366", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 14% of 605845370 greater than 207203366?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 84818351.8, + "explanation": "14% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 605845370 vs 207203366", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 605845370 greater than 207203366?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 260513509.1, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001089", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 102023630 vs 43636340", + "ns_shortcut": "Use 10% as landmark: 10% of 102023630 = 10202363.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 102023630 greater than 43636340?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 10202363.0, + "explanation": "10% ≈ 10%. 10% of 102023630 = 10202363. Compare to 43636340." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "15% of 102023630 vs 43636340", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 15% of 102023630 greater than 43636340?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 15303544.5, + "explanation": "15% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 102023630 vs 43636340", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 102023630 greater than 43636340?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 68355832.1, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001090", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 352224060 vs 154329671", + "ns_shortcut": "Use 50% as landmark: 50% of 352224060 = 176112030.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 352224060 greater than 154329671?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 176112030.0, + "explanation": "50% ≈ 50%. 50% of 352224060 = 176112030. Compare to 154329671." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "53% of 352224060 vs 154329671", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 53% of 352224060 greater than 154329671?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 186678751.8, + "explanation": "53% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "87% of 352224060 vs 154329671", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 87% of 352224060 greater than 154329671?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 306434932.2, + "explanation": "87% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001091", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 664347260 vs 322244138", + "ns_shortcut": "Use 10% as landmark: 10% of 664347260 = 66434726.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 664347260 greater than 322244138?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 66434726.0, + "explanation": "10% ≈ 10%. 10% of 664347260 = 66434726. Compare to 322244138." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3% of 664347260 vs 322244138", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 3% of 664347260 greater than 322244138?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 19930417.8, + "explanation": "3% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "87% of 664347260 vs 322244138", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 87% of 664347260 greater than 322244138?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 577982116.2, + "explanation": "87% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001092", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "23% of 195793120 vs 80234445", + "ns_shortcut": "Use 25% as landmark: 25% of 195793120 = 48948280.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 23% of 195793120 greater than 80234445?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 45032417.6, + "explanation": "23% ≈ 25%. 25% of 195793120 = 48948280. Compare to 80234445." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "31% of 195793120 vs 80234445", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 31% of 195793120 greater than 80234445?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 60695867.2, + "explanation": "31% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 195793120 vs 80234445", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 195793120 greater than 80234445?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 131181390.4, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001093", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 668859480 vs 211471118", + "ns_shortcut": "Use 50% as landmark: 50% of 668859480 = 334429740.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 668859480 greater than 211471118?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 334429740.0, + "explanation": "50% ≈ 50%. 50% of 668859480 = 334429740. Compare to 211471118." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "45% of 668859480 vs 211471118", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 45% of 668859480 greater than 211471118?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 300986766.0, + "explanation": "45% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "63% of 668859480 vs 211471118", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 63% of 668859480 greater than 211471118?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 421381472.4, + "explanation": "63% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001094", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 687866970 vs 192717633", + "ns_shortcut": "Use 50% as landmark: 50% of 687866970 = 343933485.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 687866970 greater than 192717633?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 343933485.0, + "explanation": "50% ≈ 50%. 50% of 687866970 = 343933485. Compare to 192717633." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "44% of 687866970 vs 192717633", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 44% of 687866970 greater than 192717633?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 302661466.8, + "explanation": "44% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 687866970 vs 192717633", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 687866970 greater than 192717633?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 254510778.9, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001095", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51% of 663551310 vs 162215068", + "ns_shortcut": "Use 50% as landmark: 50% of 663551310 = 331775655.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 51% of 663551310 greater than 162215068?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 338411168.1, + "explanation": "51% ≈ 50%. 50% of 663551310 = 331775655. Compare to 162215068." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "56% of 663551310 vs 162215068", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 56% of 663551310 greater than 162215068?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 371588733.6, + "explanation": "56% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "63% of 663551310 vs 162215068", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 63% of 663551310 greater than 162215068?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 418037325.3, + "explanation": "63% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001096", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49% of 722361770 vs 224224946", + "ns_shortcut": "Use 50% as landmark: 50% of 722361770 = 361180885.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 49% of 722361770 greater than 224224946?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 353957267.3, + "explanation": "49% ≈ 50%. 50% of 722361770 = 361180885. Compare to 224224946." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "57% of 722361770 vs 224224946", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 57% of 722361770 greater than 224224946?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 411746208.9, + "explanation": "57% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 722361770 vs 224224946", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 722361770 greater than 224224946?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 599560269.1, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001097", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24% of 505476060 vs 135734925", + "ns_shortcut": "Use 25% as landmark: 25% of 505476060 = 126369015.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 24% of 505476060 greater than 135734925?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 121314254.4, + "explanation": "24% ≈ 25%. 25% of 505476060 = 126369015. Compare to 135734925." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "22% of 505476060 vs 135734925", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 22% of 505476060 greater than 135734925?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 111204733.2, + "explanation": "22% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "63% of 505476060 vs 135734925", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 63% of 505476060 greater than 135734925?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 318449917.8, + "explanation": "63% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001098", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51% of 483644640 vs 128142752", + "ns_shortcut": "Use 50% as landmark: 50% of 483644640 = 241822320.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 51% of 483644640 greater than 128142752?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 246658766.4, + "explanation": "51% ≈ 50%. 50% of 483644640 = 241822320. Compare to 128142752." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "47% of 483644640 vs 128142752", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 47% of 483644640 greater than 128142752?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 227312980.8, + "explanation": "47% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 483644640 vs 128142752", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 483644640 greater than 128142752?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 207967195.2, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001099", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 556128860 vs 265268366", + "ns_shortcut": "Use 50% as landmark: 50% of 556128860 = 278064430.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 556128860 greater than 265268366?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 278064430.0, + "explanation": "50% ≈ 50%. 50% of 556128860 = 278064430. Compare to 265268366." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "45% of 556128860 vs 265268366", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 45% of 556128860 greater than 265268366?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 250257987.0, + "explanation": "45% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 556128860 vs 265268366", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 556128860 greater than 265268366?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 461586953.8, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001100", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 997077820 vs 282184589", + "ns_shortcut": "Use 10% as landmark: 10% of 997077820 = 99707782.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 997077820 greater than 282184589?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 99707782.0, + "explanation": "10% ≈ 10%. 10% of 997077820 = 99707782. Compare to 282184589." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "14% of 997077820 vs 282184589", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 14% of 997077820 greater than 282184589?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 139590894.8, + "explanation": "14% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 997077820 vs 282184589", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 997077820 greater than 282184589?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 368918793.4, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001101", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52% of 466760210 vs 227024251", + "ns_shortcut": "Use 50% as landmark: 50% of 466760210 = 233380105.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 52% of 466760210 greater than 227024251?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 242715309.2, + "explanation": "52% ≈ 50%. 50% of 466760210 = 233380105. Compare to 227024251." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "45% of 466760210 vs 227024251", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 45% of 466760210 greater than 227024251?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 210042094.5, + "explanation": "45% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 466760210 vs 227024251", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 466760210 greater than 227024251?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 200706890.3, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "equation_reasoning_001102", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "96745874 + 99767928 + ___ = 99767928 + 83984145 + 96745874", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 96745874 + 99767928 + ___ = 99767928 + 83984145 + 96745874", + "options": [ + "83984155", + "83984135", + "83984145", + "83984165" + ], + "answer": "83984145", + "correct_value": 83984145, + "explanation": "A and B appear on both sides and cancel; blank = C = 83984145." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "37482786 + 24357156 + ___ = 24357156 + 39935290 + 97872518", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 37482786 + 24357156 + ___ = 24357156 + 39935290 + 97872518", + "options": [ + "100325042", + "100325032", + "100325012", + "100325022" + ], + "answer": "100325022", + "correct_value": 100325022, + "explanation": "B cancels; blank = C + D − A = 39935290 + 97872518 − 37482786 = 100325022." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "87027101 + 81730341 + ___ = 40125106 + 31076736", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 87027101 + 81730341 + ___ = 40125106 + 31076736", + "options": [ + "-97555590", + "-97555610", + "-97555580", + "-97555600" + ], + "answer": "-97555600", + "correct_value": -97555600, + "explanation": "No matching terms; blank = 40125106 + 31076736 − 87027101 − 81730341 = -97555600." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001103", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "30379621 × 22664947 × 33205676 = ___ × 33205676 × 30379621", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 30379621 × 22664947 × 33205676 = ___ × 33205676 × 30379621", + "options": [ + "22664937", + "22664967", + "22664957", + "22664947" + ], + "answer": "22664947", + "correct_value": 22664947, + "explanation": "A and C appear on both sides; blank = B = 22664947." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "78633818 + 98952299 + ___ = 98952299 + 25119992 + 59994691", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 78633818 + 98952299 + ___ = 98952299 + 25119992 + 59994691", + "options": [ + "6480865", + "6480885", + "6480855", + "6480875" + ], + "answer": "6480865", + "correct_value": 6480865, + "explanation": "B cancels; blank = C + D − A = 25119992 + 59994691 − 78633818 = 6480865." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "81911907 + 77042229 + ___ = 63883504 + 49990742", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 81911907 + 77042229 + ___ = 63883504 + 49990742", + "options": [ + "-45079880", + "-45079870", + "-45079900", + "-45079890" + ], + "answer": "-45079890", + "correct_value": -45079890, + "explanation": "No matching terms; blank = 63883504 + 49990742 − 81911907 − 77042229 = -45079890." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001104", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "45184355 × (84661566 + 82106291) = 45184355 × ___ + 45184355 × 82106291", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 45184355 × (84661566 + 82106291) = 45184355 × ___ + 45184355 × 82106291", + "options": [ + "84661556", + "84661586", + "84661576", + "84661566" + ], + "answer": "84661566", + "correct_value": 84661566, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 84661566." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "51016226 + 27926819 + ___ = 27926819 + 10128848 + 15715022", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 51016226 + 27926819 + ___ = 27926819 + 10128848 + 15715022", + "options": [ + "-25172344", + "-25172354", + "-25172356", + "-25172374" + ], + "answer": "-25172356", + "correct_value": -25172356, + "explanation": "B cancels; blank = C + D − A = 10128848 + 15715022 − 51016226 = -25172356." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "69357007 + 95668618 + ___ = 80386112 + 74483772", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 69357007 + 95668618 + ___ = 80386112 + 74483772", + "options": [ + "-10155741", + "-10155729", + "-10155759", + "-10155739" + ], + "answer": "-10155741", + "correct_value": -10155741, + "explanation": "No matching terms; blank = 80386112 + 74483772 − 69357007 − 95668618 = -10155741." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001105", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "65426423 × 25518268 × 32553850 = ___ × 32553850 × 65426423", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 65426423 × 25518268 × 32553850 = ___ × 32553850 × 65426423", + "options": [ + "25518288", + "25518258", + "25518268", + "25518278" + ], + "answer": "25518268", + "correct_value": 25518268, + "explanation": "A and C appear on both sides; blank = B = 25518268." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99754223 + 50911590 + ___ = 50911590 + 81597116 + 66027351", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 99754223 + 50911590 + ___ = 50911590 + 81597116 + 66027351", + "options": [ + "47870244", + "47870234", + "47870264", + "47870254" + ], + "answer": "47870244", + "correct_value": 47870244, + "explanation": "B cancels; blank = C + D − A = 81597116 + 66027351 − 99754223 = 47870244." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "43542505 + 63920359 + ___ = 35792124 + 35733790", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 43542505 + 63920359 + ___ = 35792124 + 35733790", + "options": [ + "-35936930", + "-35936950", + "-35936960", + "-35936940" + ], + "answer": "-35936950", + "correct_value": -35936950, + "explanation": "No matching terms; blank = 35792124 + 35733790 − 43542505 − 63920359 = -35936950." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001106", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "34564761 × 12296163 × 51022074 = ___ × 51022074 × 34564761", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 34564761 × 12296163 × 51022074 = ___ × 51022074 × 34564761", + "options": [ + "12296183", + "12296173", + "12296153", + "12296163" + ], + "answer": "12296163", + "correct_value": 12296163, + "explanation": "A and C appear on both sides; blank = B = 12296163." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "31890260 + 92105579 + ___ = 92105579 + 43948447 + 81390079", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 31890260 + 92105579 + ___ = 92105579 + 43948447 + 81390079", + "options": [ + "93448286", + "93448256", + "93448266", + "93448276" + ], + "answer": "93448266", + "correct_value": 93448266, + "explanation": "B cancels; blank = C + D − A = 43948447 + 81390079 − 31890260 = 93448266." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "75175799 + 78595982 + ___ = 86282908 + 25021199", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 75175799 + 78595982 + ___ = 86282908 + 25021199", + "options": [ + "-42467686", + "-42467656", + "-42467666", + "-42467674" + ], + "answer": "-42467674", + "correct_value": -42467674, + "explanation": "No matching terms; blank = 86282908 + 25021199 − 75175799 − 78595982 = -42467674." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001107", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "64526122 × 88937138 × 91793674 = ___ × 91793674 × 64526122", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 64526122 × 88937138 × 91793674 = ___ × 91793674 × 64526122", + "options": [ + "88937148", + "88937138", + "88937128", + "88937158" + ], + "answer": "88937138", + "correct_value": 88937138, + "explanation": "A and C appear on both sides; blank = B = 88937138." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "97468040 + 31593437 + ___ = 31593437 + 78945937 + 36731136", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 97468040 + 31593437 + ___ = 31593437 + 78945937 + 36731136", + "options": [ + "18209043", + "18209023", + "18209053", + "18209033" + ], + "answer": "18209033", + "correct_value": 18209033, + "explanation": "B cancels; blank = C + D − A = 78945937 + 36731136 − 97468040 = 18209033." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "17652163 + 39439338 + ___ = 67980848 + 66673212", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 17652163 + 39439338 + ___ = 67980848 + 66673212", + "options": [ + "77562569", + "77562579", + "77562559", + "77562549" + ], + "answer": "77562559", + "correct_value": 77562559, + "explanation": "No matching terms; blank = 67980848 + 66673212 − 17652163 − 39439338 = 77562559." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001108", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "18373570 + 10722446 + ___ = 10722446 + 56685693 + 18373570", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 18373570 + 10722446 + ___ = 10722446 + 56685693 + 18373570", + "options": [ + "56685683", + "56685693", + "56685713", + "56685703" + ], + "answer": "56685693", + "correct_value": 56685693, + "explanation": "A and B appear on both sides and cancel; blank = C = 56685693." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "82375751 + 55185815 + ___ = 55185815 + 35956606 + 83010662", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 82375751 + 55185815 + ___ = 55185815 + 35956606 + 83010662", + "options": [ + "36591527", + "36591537", + "36591517", + "36591507" + ], + "answer": "36591517", + "correct_value": 36591517, + "explanation": "B cancels; blank = C + D − A = 35956606 + 83010662 − 82375751 = 36591517." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "10516583 + 53186631 + ___ = 94799875 + 16229481", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 10516583 + 53186631 + ___ = 94799875 + 16229481", + "options": [ + "47326132", + "47326152", + "47326142", + "47326162" + ], + "answer": "47326142", + "correct_value": 47326142, + "explanation": "No matching terms; blank = 94799875 + 16229481 − 10516583 − 53186631 = 47326142." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001109", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "81062939 × 41533557 × 83797377 = ___ × 83797377 × 81062939", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 81062939 × 41533557 × 83797377 = ___ × 83797377 × 81062939", + "options": [ + "41533577", + "41533547", + "41533567", + "41533557" + ], + "answer": "41533557", + "correct_value": 41533557, + "explanation": "A and C appear on both sides; blank = B = 41533557." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "66093594 + 92670664 + ___ = 92670664 + 32939969 + 98581654", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 66093594 + 92670664 + ___ = 92670664 + 32939969 + 98581654", + "options": [ + "65428049", + "65428039", + "65428019", + "65428029" + ], + "answer": "65428029", + "correct_value": 65428029, + "explanation": "B cancels; blank = C + D − A = 32939969 + 98581654 − 66093594 = 65428029." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "63433094 + 81669833 + ___ = 68008304 + 32013599", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 63433094 + 81669833 + ___ = 68008304 + 32013599", + "options": [ + "-45081006", + "-45081024", + "-45081016", + "-45081036" + ], + "answer": "-45081024", + "correct_value": -45081024, + "explanation": "No matching terms; blank = 68008304 + 32013599 − 63433094 − 81669833 = -45081024." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001110", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "33798199 × 84599555 × 37757450 = ___ × 37757450 × 33798199", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 33798199 × 84599555 × 37757450 = ___ × 37757450 × 33798199", + "options": [ + "84599565", + "84599555", + "84599575", + "84599545" + ], + "answer": "84599555", + "correct_value": 84599555, + "explanation": "A and C appear on both sides; blank = B = 84599555." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "38788381 + 74914697 + ___ = 74914697 + 64694937 + 21180576", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 38788381 + 74914697 + ___ = 74914697 + 64694937 + 21180576", + "options": [ + "47087142", + "47087132", + "47087122", + "47087152" + ], + "answer": "47087132", + "correct_value": 47087132, + "explanation": "B cancels; blank = C + D − A = 64694937 + 21180576 − 38788381 = 47087132." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "94705607 + 82790437 + ___ = 64021354 + 33760984", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 94705607 + 82790437 + ___ = 64021354 + 33760984", + "options": [ + "-79713694", + "-79713706", + "-79713704", + "-79713724" + ], + "answer": "-79713706", + "correct_value": -79713706, + "explanation": "No matching terms; blank = 64021354 + 33760984 − 94705607 − 82790437 = -79713706." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001111", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "68257666 × 84108315 × 50896516 = ___ × 50896516 × 68257666", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 68257666 × 84108315 × 50896516 = ___ × 50896516 × 68257666", + "options": [ + "84108325", + "84108335", + "84108305", + "84108315" + ], + "answer": "84108315", + "correct_value": 84108315, + "explanation": "A and C appear on both sides; blank = B = 84108315." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "35484177 + 30810799 + ___ = 30810799 + 99096950 + 77892895", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 35484177 + 30810799 + ___ = 30810799 + 99096950 + 77892895", + "options": [ + "141505658", + "141505688", + "141505668", + "141505678" + ], + "answer": "141505668", + "correct_value": 141505668, + "explanation": "B cancels; blank = C + D − A = 99096950 + 77892895 − 35484177 = 141505668." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "91708236 + 30448683 + ___ = 23307600 + 82553306", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 91708236 + 30448683 + ___ = 23307600 + 82553306", + "options": [ + "-16295997", + "-16296013", + "-16296027", + "-16296007" + ], + "answer": "-16296013", + "correct_value": -16296013, + "explanation": "No matching terms; blank = 23307600 + 82553306 − 91708236 − 30448683 = -16296013." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001112", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "14817067 × 36511747 × 56798543 = ___ × 56798543 × 14817067", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 14817067 × 36511747 × 56798543 = ___ × 56798543 × 14817067", + "options": [ + "36511767", + "36511747", + "36511757", + "36511737" + ], + "answer": "36511747", + "correct_value": 36511747, + "explanation": "A and C appear on both sides; blank = B = 36511747." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "49682802 + 13202151 + ___ = 13202151 + 22300549 + 67500011", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 49682802 + 13202151 + ___ = 13202151 + 22300549 + 67500011", + "options": [ + "40117768", + "40117748", + "40117778", + "40117758" + ], + "answer": "40117758", + "correct_value": 40117758, + "explanation": "B cancels; blank = C + D − A = 22300549 + 67500011 − 49682802 = 40117758." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "33371573 + 78269158 + ___ = 70696781 + 34213032", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 33371573 + 78269158 + ___ = 70696781 + 34213032", + "options": [ + "-6730912", + "-6730918", + "-6730902", + "-6730932" + ], + "answer": "-6730918", + "correct_value": -6730918, + "explanation": "No matching terms; blank = 70696781 + 34213032 − 33371573 − 78269158 = -6730918." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001113", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "43200403 × (20074430 + 93372014) = 43200403 × ___ + 43200403 × 93372014", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 43200403 × (20074430 + 93372014) = 43200403 × ___ + 43200403 × 93372014", + "options": [ + "20074430", + "20074450", + "20074420", + "20074440" + ], + "answer": "20074430", + "correct_value": 20074430, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 20074430." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "64995923 + 25673796 + ___ = 25673796 + 71826632 + 77534062", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 64995923 + 25673796 + ___ = 25673796 + 71826632 + 77534062", + "options": [ + "84364761", + "84364791", + "84364781", + "84364771" + ], + "answer": "84364771", + "correct_value": 84364771, + "explanation": "B cancels; blank = C + D − A = 71826632 + 77534062 − 64995923 = 84364771." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "87786608 + 81570396 + ___ = 96335755 + 43145043", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 87786608 + 81570396 + ___ = 96335755 + 43145043", + "options": [ + "-29876206", + "-29876224", + "-29876204", + "-29876194" + ], + "answer": "-29876206", + "correct_value": -29876206, + "explanation": "No matching terms; blank = 96335755 + 43145043 − 87786608 − 81570396 = -29876206." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001114", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12232886 × 30223295 × 62450843 = ___ × 62450843 × 12232886", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 12232886 × 30223295 × 62450843 = ___ × 62450843 × 12232886", + "options": [ + "30223295", + "30223315", + "30223305", + "30223285" + ], + "answer": "30223295", + "correct_value": 30223295, + "explanation": "A and C appear on both sides; blank = B = 30223295." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "81654359 + 61602986 + ___ = 61602986 + 32926391 + 92092334", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 81654359 + 61602986 + ___ = 61602986 + 32926391 + 92092334", + "options": [ + "43364376", + "43364366", + "43364356", + "43364386" + ], + "answer": "43364366", + "correct_value": 43364366, + "explanation": "B cancels; blank = C + D − A = 32926391 + 92092334 − 81654359 = 43364366." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "28571145 + 53442625 + ___ = 70158717 + 37399209", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 28571145 + 53442625 + ___ = 70158717 + 37399209", + "options": [ + "25544166", + "25544156", + "25544146", + "25544176" + ], + "answer": "25544156", + "correct_value": 25544156, + "explanation": "No matching terms; blank = 70158717 + 37399209 − 28571145 − 53442625 = 25544156." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001115", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "53775151 × (39999769 + 90845239) = 53775151 × ___ + 53775151 × 90845239", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 53775151 × (39999769 + 90845239) = 53775151 × ___ + 53775151 × 90845239", + "options": [ + "39999769", + "39999779", + "39999759", + "39999789" + ], + "answer": "39999769", + "correct_value": 39999769, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 39999769." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "94325586 + 95424475 + ___ = 95424475 + 11212316 + 69637989", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 94325586 + 95424475 + ___ = 95424475 + 11212316 + 69637989", + "options": [ + "-13475279", + "-13475299", + "-13475281", + "-13475269" + ], + "answer": "-13475281", + "correct_value": -13475281, + "explanation": "B cancels; blank = C + D − A = 11212316 + 69637989 − 94325586 = -13475281." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "69733842 + 45986064 + ___ = 94208414 + 73777515", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 69733842 + 45986064 + ___ = 94208414 + 73777515", + "options": [ + "52266023", + "52266013", + "52266043", + "52266033" + ], + "answer": "52266023", + "correct_value": 52266023, + "explanation": "No matching terms; blank = 94208414 + 73777515 − 69733842 − 45986064 = 52266023." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001116", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "16694558 × (54294118 + 60517626) = 16694558 × ___ + 16694558 × 60517626", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 16694558 × (54294118 + 60517626) = 16694558 × ___ + 16694558 × 60517626", + "options": [ + "54294108", + "54294118", + "54294138", + "54294128" + ], + "answer": "54294118", + "correct_value": 54294118, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 54294118." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "92485460 + 49862571 + ___ = 49862571 + 54491468 + 21218518", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 92485460 + 49862571 + ___ = 49862571 + 54491468 + 21218518", + "options": [ + "-16775456", + "-16775466", + "-16775486", + "-16775474" + ], + "answer": "-16775474", + "correct_value": -16775474, + "explanation": "B cancels; blank = C + D − A = 54491468 + 21218518 − 92485460 = -16775474." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "82123814 + 47627247 + ___ = 16902576 + 22272998", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 82123814 + 47627247 + ___ = 16902576 + 22272998", + "options": [ + "-90575487", + "-90575483", + "-90575473", + "-90575503" + ], + "answer": "-90575487", + "correct_value": -90575487, + "explanation": "No matching terms; blank = 16902576 + 22272998 − 82123814 − 47627247 = -90575487." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001117", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "78324734 + 35976666 + ___ = 35976666 + 91639621 + 78324734", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 78324734 + 35976666 + ___ = 35976666 + 91639621 + 78324734", + "options": [ + "91639611", + "91639631", + "91639621", + "91639641" + ], + "answer": "91639621", + "correct_value": 91639621, + "explanation": "A and B appear on both sides and cancel; blank = C = 91639621." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "72014406 + 42552076 + ___ = 42552076 + 48785602 + 71516755", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 72014406 + 42552076 + ___ = 42552076 + 48785602 + 71516755", + "options": [ + "48287941", + "48287961", + "48287951", + "48287971" + ], + "answer": "48287951", + "correct_value": 48287951, + "explanation": "B cancels; blank = C + D − A = 48785602 + 71516755 − 72014406 = 48287951." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "88050960 + 81999061 + ___ = 81567304 + 24336847", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 88050960 + 81999061 + ___ = 81567304 + 24336847", + "options": [ + "-64145850", + "-64145870", + "-64145860", + "-64145880" + ], + "answer": "-64145870", + "correct_value": -64145870, + "explanation": "No matching terms; blank = 81567304 + 24336847 − 88050960 − 81999061 = -64145870." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001118", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "48925256 × (52417781 + 11325577) = 48925256 × ___ + 48925256 × 11325577", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 48925256 × (52417781 + 11325577) = 48925256 × ___ + 48925256 × 11325577", + "options": [ + "52417781", + "52417771", + "52417801", + "52417791" + ], + "answer": "52417781", + "correct_value": 52417781, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 52417781." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "11330468 + 53844504 + ___ = 53844504 + 32937991 + 25873604", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 11330468 + 53844504 + ___ = 53844504 + 32937991 + 25873604", + "options": [ + "47481147", + "47481137", + "47481117", + "47481127" + ], + "answer": "47481127", + "correct_value": 47481127, + "explanation": "B cancels; blank = C + D − A = 32937991 + 25873604 − 11330468 = 47481127." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "28728874 + 71563112 + ___ = 15245835 + 63245195", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 28728874 + 71563112 + ___ = 15245835 + 63245195", + "options": [ + "-21800944", + "-21800956", + "-21800954", + "-21800974" + ], + "answer": "-21800956", + "correct_value": -21800956, + "explanation": "No matching terms; blank = 15245835 + 63245195 − 28728874 − 71563112 = -21800956." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001119", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "83063326 + 34043964 + ___ = 34043964 + 46313891 + 83063326", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 83063326 + 34043964 + ___ = 34043964 + 46313891 + 83063326", + "options": [ + "46313881", + "46313891", + "46313901", + "46313911" + ], + "answer": "46313891", + "correct_value": 46313891, + "explanation": "A and B appear on both sides and cancel; blank = C = 46313891." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "39721110 + 10073117 + ___ = 10073117 + 84662900 + 82427103", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 39721110 + 10073117 + ___ = 10073117 + 84662900 + 82427103", + "options": [ + "127368883", + "127368893", + "127368913", + "127368903" + ], + "answer": "127368893", + "correct_value": 127368893, + "explanation": "B cancels; blank = C + D − A = 84662900 + 82427103 − 39721110 = 127368893." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "54275746 + 74755706 + ___ = 50952683 + 99490648", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 54275746 + 74755706 + ___ = 50952683 + 99490648", + "options": [ + "21411879", + "21411869", + "21411899", + "21411889" + ], + "answer": "21411879", + "correct_value": 21411879, + "explanation": "No matching terms; blank = 50952683 + 99490648 − 54275746 − 74755706 = 21411879." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001120", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24225241 × (84266992 + 56658509) = 24225241 × ___ + 24225241 × 56658509", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 24225241 × (84266992 + 56658509) = 24225241 × ___ + 24225241 × 56658509", + "options": [ + "84267002", + "84267012", + "84266992", + "84266982" + ], + "answer": "84266992", + "correct_value": 84266992, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 84266992." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "30049084 + 70005818 + ___ = 70005818 + 59735453 + 11707969", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 30049084 + 70005818 + ___ = 70005818 + 59735453 + 11707969", + "options": [ + "41394348", + "41394358", + "41394328", + "41394338" + ], + "answer": "41394338", + "correct_value": 41394338, + "explanation": "B cancels; blank = C + D − A = 59735453 + 11707969 − 30049084 = 41394338." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "16163850 + 56293495 + ___ = 62395558 + 64129615", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 16163850 + 56293495 + ___ = 62395558 + 64129615", + "options": [ + "54067818", + "54067838", + "54067848", + "54067828" + ], + "answer": "54067828", + "correct_value": 54067828, + "explanation": "No matching terms; blank = 62395558 + 64129615 − 16163850 − 56293495 = 54067828." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001121", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12320274 × 57974300 × 61996292 = ___ × 61996292 × 12320274", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 12320274 × 57974300 × 61996292 = ___ × 61996292 × 12320274", + "options": [ + "57974290", + "57974310", + "57974300", + "57974320" + ], + "answer": "57974300", + "correct_value": 57974300, + "explanation": "A and C appear on both sides; blank = B = 57974300." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "85877745 + 35713686 + ___ = 35713686 + 68824263 + 82660418", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 85877745 + 35713686 + ___ = 35713686 + 68824263 + 82660418", + "options": [ + "65606936", + "65606926", + "65606946", + "65606956" + ], + "answer": "65606936", + "correct_value": 65606936, + "explanation": "B cancels; blank = C + D − A = 68824263 + 82660418 − 85877745 = 65606936." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "86434610 + 16950641 + ___ = 77465025 + 74404694", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 86434610 + 16950641 + ___ = 77465025 + 74404694", + "options": [ + "48484478", + "48484458", + "48484488", + "48484468" + ], + "answer": "48484468", + "correct_value": 48484468, + "explanation": "No matching terms; blank = 77465025 + 74404694 − 86434610 − 16950641 = 48484468." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001122", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "36947546 + 55637310 = ___ + 72091039", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 36947546 + 55637310 = ___ + 72091039", + "options": [ + "20493827", + "20493807", + "20493817", + "20493837" + ], + "answer": "20493817", + "correct_value": 20493817, + "explanation": "Blank = A + B − C = 36947546 + 55637310 − 72091039 = 20493817." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "23912826 + 23075236 + ___ = 23075236 + 51730161 + 26622278", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 23912826 + 23075236 + ___ = 23075236 + 51730161 + 26622278", + "options": [ + "54439613", + "54439623", + "54439633", + "54439603" + ], + "answer": "54439613", + "correct_value": 54439613, + "explanation": "B cancels; blank = C + D − A = 51730161 + 26622278 − 23912826 = 54439613." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "37515219 + 25108782 + ___ = 33466979 + 18695053", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 37515219 + 25108782 + ___ = 33466979 + 18695053", + "options": [ + "-10461961", + "-10461969", + "-10461981", + "-10461951" + ], + "answer": "-10461969", + "correct_value": -10461969, + "explanation": "No matching terms; blank = 33466979 + 18695053 − 37515219 − 25108782 = -10461969." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001123", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "90023273 + 28076682 = ___ + 35962277", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 90023273 + 28076682 = ___ + 35962277", + "options": [ + "82137678", + "82137668", + "82137698", + "82137688" + ], + "answer": "82137678", + "correct_value": 82137678, + "explanation": "Blank = A + B − C = 90023273 + 28076682 − 35962277 = 82137678." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "15040071 + 84719224 + ___ = 84719224 + 98993615 + 76881774", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 15040071 + 84719224 + ___ = 84719224 + 98993615 + 76881774", + "options": [ + "160835328", + "160835318", + "160835338", + "160835308" + ], + "answer": "160835318", + "correct_value": 160835318, + "explanation": "B cancels; blank = C + D − A = 98993615 + 76881774 − 15040071 = 160835318." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "87345724 + 48620641 + ___ = 81162068 + 64721057", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 87345724 + 48620641 + ___ = 81162068 + 64721057", + "options": [ + "9916750", + "9916770", + "9916760", + "9916780" + ], + "answer": "9916760", + "correct_value": 9916760, + "explanation": "No matching terms; blank = 81162068 + 64721057 − 87345724 − 48620641 = 9916760." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001124", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "67105644 × (33925710 + 34727844) = 67105644 × ___ + 67105644 × 34727844", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 67105644 × (33925710 + 34727844) = 67105644 × ___ + 67105644 × 34727844", + "options": [ + "33925700", + "33925730", + "33925710", + "33925720" + ], + "answer": "33925710", + "correct_value": 33925710, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 33925710." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "54531458 + 41663551 + ___ = 41663551 + 38917860 + 95255405", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 54531458 + 41663551 + ___ = 41663551 + 38917860 + 95255405", + "options": [ + "79641797", + "79641807", + "79641817", + "79641827" + ], + "answer": "79641807", + "correct_value": 79641807, + "explanation": "B cancels; blank = C + D − A = 38917860 + 95255405 − 54531458 = 79641807." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "44711361 + 76961753 + ___ = 51245534 + 74478238", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 44711361 + 76961753 + ___ = 51245534 + 74478238", + "options": [ + "4050658", + "4050678", + "4050668", + "4050648" + ], + "answer": "4050658", + "correct_value": 4050658, + "explanation": "No matching terms; blank = 51245534 + 74478238 − 44711361 − 76961753 = 4050658." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001125", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "59978040 × (90166601 + 39908720) = 59978040 × ___ + 59978040 × 39908720", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 59978040 × (90166601 + 39908720) = 59978040 × ___ + 59978040 × 39908720", + "options": [ + "90166621", + "90166601", + "90166611", + "90166591" + ], + "answer": "90166601", + "correct_value": 90166601, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 90166601." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "22937099 + 41106335 + ___ = 41106335 + 81722941 + 40229076", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 22937099 + 41106335 + ___ = 41106335 + 81722941 + 40229076", + "options": [ + "99014908", + "99014928", + "99014918", + "99014938" + ], + "answer": "99014918", + "correct_value": 99014918, + "explanation": "B cancels; blank = C + D − A = 81722941 + 40229076 − 22937099 = 99014918." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "20983796 + 28411972 + ___ = 48583551 + 54696776", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 20983796 + 28411972 + ___ = 48583551 + 54696776", + "options": [ + "53884569", + "53884549", + "53884559", + "53884579" + ], + "answer": "53884559", + "correct_value": 53884559, + "explanation": "No matching terms; blank = 48583551 + 54696776 − 20983796 − 28411972 = 53884559." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001126", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "89984645 × 17564055 × 10446180 = ___ × 10446180 × 89984645", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 89984645 × 17564055 × 10446180 = ___ × 10446180 × 89984645", + "options": [ + "17564075", + "17564055", + "17564045", + "17564065" + ], + "answer": "17564055", + "correct_value": 17564055, + "explanation": "A and C appear on both sides; blank = B = 17564055." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98370567 + 34808116 + ___ = 34808116 + 22268114 + 19125390", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 98370567 + 34808116 + ___ = 34808116 + 22268114 + 19125390", + "options": [ + "-56977057", + "-56977047", + "-56977063", + "-56977077" + ], + "answer": "-56977063", + "correct_value": -56977063, + "explanation": "B cancels; blank = C + D − A = 22268114 + 19125390 − 98370567 = -56977063." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "95001070 + 72185010 + ___ = 76391266 + 31079288", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 95001070 + 72185010 + ___ = 76391266 + 31079288", + "options": [ + "-59715514", + "-59715526", + "-59715524", + "-59715544" + ], + "answer": "-59715526", + "correct_value": -59715526, + "explanation": "No matching terms; blank = 76391266 + 31079288 − 95001070 − 72185010 = -59715526." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001127", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "84582785 × (53081879 + 53114614) = 84582785 × ___ + 84582785 × 53114614", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 84582785 × (53081879 + 53114614) = 84582785 × ___ + 84582785 × 53114614", + "options": [ + "53081889", + "53081869", + "53081879", + "53081899" + ], + "answer": "53081879", + "correct_value": 53081879, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 53081879." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "88444347 + 99737085 + ___ = 99737085 + 50973161 + 32858779", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 88444347 + 99737085 + ___ = 99737085 + 50973161 + 32858779", + "options": [ + "-4612423", + "-4612403", + "-4612407", + "-4612393" + ], + "answer": "-4612407", + "correct_value": -4612407, + "explanation": "B cancels; blank = C + D − A = 50973161 + 32858779 − 88444347 = -4612407." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "23610100 + 64503164 + ___ = 25523106 + 77425226", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 23610100 + 64503164 + ___ = 25523106 + 77425226", + "options": [ + "14835068", + "14835058", + "14835078", + "14835088" + ], + "answer": "14835068", + "correct_value": 14835068, + "explanation": "No matching terms; blank = 25523106 + 77425226 − 23610100 − 64503164 = 14835068." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001128", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "70213653 + 18933547 + ___ = 18933547 + 17346961 + 70213653", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 70213653 + 18933547 + ___ = 18933547 + 17346961 + 70213653", + "options": [ + "17346971", + "17346951", + "17346961", + "17346981" + ], + "answer": "17346961", + "correct_value": 17346961, + "explanation": "A and B appear on both sides and cancel; blank = C = 17346961." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "58757651 + 51408024 + ___ = 51408024 + 90434851 + 22448176", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 58757651 + 51408024 + ___ = 51408024 + 90434851 + 22448176", + "options": [ + "54125376", + "54125366", + "54125386", + "54125396" + ], + "answer": "54125376", + "correct_value": 54125376, + "explanation": "B cancels; blank = C + D − A = 90434851 + 22448176 − 58757651 = 54125376." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "51239244 + 46137452 + ___ = 73299218 + 40913171", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 51239244 + 46137452 + ___ = 73299218 + 40913171", + "options": [ + "16835703", + "16835693", + "16835683", + "16835713" + ], + "answer": "16835693", + "correct_value": 16835693, + "explanation": "No matching terms; blank = 73299218 + 40913171 − 51239244 − 46137452 = 16835693." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001129", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "28944305 × (49938244 + 68835737) = 28944305 × ___ + 28944305 × 68835737", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 28944305 × (49938244 + 68835737) = 28944305 × ___ + 28944305 × 68835737", + "options": [ + "49938234", + "49938254", + "49938244", + "49938264" + ], + "answer": "49938244", + "correct_value": 49938244, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 49938244." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25982981 + 57836153 + ___ = 57836153 + 87924257 + 50302933", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 25982981 + 57836153 + ___ = 57836153 + 87924257 + 50302933", + "options": [ + "112244229", + "112244219", + "112244209", + "112244199" + ], + "answer": "112244209", + "correct_value": 112244209, + "explanation": "B cancels; blank = C + D − A = 87924257 + 50302933 − 25982981 = 112244209." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "48840591 + 90937930 + ___ = 37269624 + 14918680", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 48840591 + 90937930 + ___ = 37269624 + 14918680", + "options": [ + "-87590233", + "-87590213", + "-87590217", + "-87590203" + ], + "answer": "-87590217", + "correct_value": -87590217, + "explanation": "No matching terms; blank = 37269624 + 14918680 − 48840591 − 90937930 = -87590217." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001130", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "39829291 + 36204095 + ___ = 36204095 + 26751951 + 39829291", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 39829291 + 36204095 + ___ = 36204095 + 26751951 + 39829291", + "options": [ + "26751971", + "26751951", + "26751941", + "26751961" + ], + "answer": "26751951", + "correct_value": 26751951, + "explanation": "A and B appear on both sides and cancel; blank = C = 26751951." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "19213734 + 36983030 + ___ = 36983030 + 42164731 + 96477790", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 19213734 + 36983030 + ___ = 36983030 + 42164731 + 96477790", + "options": [ + "119428807", + "119428787", + "119428777", + "119428797" + ], + "answer": "119428787", + "correct_value": 119428787, + "explanation": "B cancels; blank = C + D − A = 42164731 + 96477790 − 19213734 = 119428787." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "78850990 + 73677489 + ___ = 77917874 + 81799490", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 78850990 + 73677489 + ___ = 77917874 + 81799490", + "options": [ + "7188875", + "7188885", + "7188905", + "7188895" + ], + "answer": "7188885", + "correct_value": 7188885, + "explanation": "No matching terms; blank = 77917874 + 81799490 − 78850990 − 73677489 = 7188885." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001131", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "33365734 × (74413640 + 82921170) = 33365734 × ___ + 33365734 × 82921170", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 33365734 × (74413640 + 82921170) = 33365734 × ___ + 33365734 × 82921170", + "options": [ + "74413650", + "74413660", + "74413640", + "74413630" + ], + "answer": "74413640", + "correct_value": 74413640, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 74413640." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "48284400 + 84470967 + ___ = 84470967 + 68357188 + 34689953", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 48284400 + 84470967 + ___ = 84470967 + 68357188 + 34689953", + "options": [ + "54762751", + "54762741", + "54762731", + "54762761" + ], + "answer": "54762741", + "correct_value": 54762741, + "explanation": "B cancels; blank = C + D − A = 68357188 + 34689953 − 48284400 = 54762741." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "52072014 + 34365789 + ___ = 26405048 + 69749751", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 52072014 + 34365789 + ___ = 26405048 + 69749751", + "options": [ + "9716996", + "9716986", + "9717006", + "9717016" + ], + "answer": "9716996", + "correct_value": 9716996, + "explanation": "No matching terms; blank = 26405048 + 69749751 − 52072014 − 34365789 = 9716996." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001132", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "53947932 × 40390409 × 74777625 = ___ × 74777625 × 53947932", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 53947932 × 40390409 × 74777625 = ___ × 74777625 × 53947932", + "options": [ + "40390419", + "40390429", + "40390399", + "40390409" + ], + "answer": "40390409", + "correct_value": 40390409, + "explanation": "A and C appear on both sides; blank = B = 40390409." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "57813783 + 67023177 + ___ = 67023177 + 74205221 + 35231795", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 57813783 + 67023177 + ___ = 67023177 + 74205221 + 35231795", + "options": [ + "51623233", + "51623223", + "51623243", + "51623253" + ], + "answer": "51623233", + "correct_value": 51623233, + "explanation": "B cancels; blank = C + D − A = 74205221 + 35231795 − 57813783 = 51623233." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "76647695 + 21760471 + ___ = 74315902 + 69417259", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 76647695 + 21760471 + ___ = 74315902 + 69417259", + "options": [ + "45325015", + "45325005", + "45324995", + "45324985" + ], + "answer": "45324995", + "correct_value": 45324995, + "explanation": "No matching terms; blank = 74315902 + 69417259 − 76647695 − 21760471 = 45324995." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001133", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51660060 + 40129718 + ___ = 40129718 + 38172320 + 51660060", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 51660060 + 40129718 + ___ = 40129718 + 38172320 + 51660060", + "options": [ + "38172340", + "38172330", + "38172320", + "38172310" + ], + "answer": "38172320", + "correct_value": 38172320, + "explanation": "A and B appear on both sides and cancel; blank = C = 38172320." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "60475340 + 94437041 + ___ = 94437041 + 67715183 + 45164121", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 60475340 + 94437041 + ___ = 94437041 + 67715183 + 45164121", + "options": [ + "52403974", + "52403954", + "52403964", + "52403984" + ], + "answer": "52403964", + "correct_value": 52403964, + "explanation": "B cancels; blank = C + D − A = 67715183 + 45164121 − 60475340 = 52403964." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "14956822 + 25054603 + ___ = 16472544 + 81439530", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 14956822 + 25054603 + ___ = 16472544 + 81439530", + "options": [ + "57900639", + "57900649", + "57900659", + "57900669" + ], + "answer": "57900649", + "correct_value": 57900649, + "explanation": "No matching terms; blank = 16472544 + 81439530 − 14956822 − 25054603 = 57900649." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001134", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "81308766 + 41400578 + ___ = 41400578 + 55685117 + 81308766", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 81308766 + 41400578 + ___ = 41400578 + 55685117 + 81308766", + "options": [ + "55685107", + "55685137", + "55685117", + "55685127" + ], + "answer": "55685117", + "correct_value": 55685117, + "explanation": "A and B appear on both sides and cancel; blank = C = 55685117." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "82518184 + 89785859 + ___ = 89785859 + 64975413 + 99063008", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 82518184 + 89785859 + ___ = 89785859 + 64975413 + 99063008", + "options": [ + "81520227", + "81520237", + "81520247", + "81520257" + ], + "answer": "81520237", + "correct_value": 81520237, + "explanation": "B cancels; blank = C + D − A = 64975413 + 99063008 − 82518184 = 81520237." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "97315490 + 49568600 + ___ = 62326451 + 47169633", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 97315490 + 49568600 + ___ = 62326451 + 47169633", + "options": [ + "-37388006", + "-37388004", + "-37388024", + "-37387994" + ], + "answer": "-37388006", + "correct_value": -37388006, + "explanation": "No matching terms; blank = 62326451 + 47169633 − 97315490 − 49568600 = -37388006." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001135", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "68792790 × (32910504 + 85301514) = 68792790 × ___ + 68792790 × 85301514", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 68792790 × (32910504 + 85301514) = 68792790 × ___ + 68792790 × 85301514", + "options": [ + "32910504", + "32910494", + "32910514", + "32910524" + ], + "answer": "32910504", + "correct_value": 32910504, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 32910504." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "95242946 + 86640291 + ___ = 86640291 + 56778834 + 79779701", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 95242946 + 86640291 + ___ = 86640291 + 56778834 + 79779701", + "options": [ + "41315579", + "41315589", + "41315609", + "41315599" + ], + "answer": "41315589", + "correct_value": 41315589, + "explanation": "B cancels; blank = C + D − A = 56778834 + 79779701 − 95242946 = 41315589." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "89501960 + 16891024 + ___ = 34045121 + 29254771", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 89501960 + 16891024 + ___ = 34045121 + 29254771", + "options": [ + "-43093108", + "-43093078", + "-43093088", + "-43093092" + ], + "answer": "-43093092", + "correct_value": -43093092, + "explanation": "No matching terms; blank = 34045121 + 29254771 − 89501960 − 16891024 = -43093092." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001136", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "28476277 + 50163276 + ___ = 50163276 + 80976519 + 28476277", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 28476277 + 50163276 + ___ = 50163276 + 80976519 + 28476277", + "options": [ + "80976519", + "80976509", + "80976539", + "80976529" + ], + "answer": "80976519", + "correct_value": 80976519, + "explanation": "A and B appear on both sides and cancel; blank = C = 80976519." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20999047 + 72024978 + ___ = 72024978 + 11931219 + 54045721", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 20999047 + 72024978 + ___ = 72024978 + 11931219 + 54045721", + "options": [ + "44977893", + "44977883", + "44977903", + "44977913" + ], + "answer": "44977893", + "correct_value": 44977893, + "explanation": "B cancels; blank = C + D − A = 11931219 + 54045721 − 20999047 = 44977893." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "23271654 + 45685256 + ___ = 37663707 + 67103476", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 23271654 + 45685256 + ___ = 37663707 + 67103476", + "options": [ + "35810293", + "35810283", + "35810273", + "35810263" + ], + "answer": "35810273", + "correct_value": 35810273, + "explanation": "No matching terms; blank = 37663707 + 67103476 − 23271654 − 45685256 = 35810273." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001137", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "21140095 + 34457853 + ___ = 34457853 + 39210851 + 21140095", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 21140095 + 34457853 + ___ = 34457853 + 39210851 + 21140095", + "options": [ + "39210871", + "39210861", + "39210841", + "39210851" + ], + "answer": "39210851", + "correct_value": 39210851, + "explanation": "A and B appear on both sides and cancel; blank = C = 39210851." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "68306519 + 81991176 + ___ = 81991176 + 25311147 + 20994583", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 68306519 + 81991176 + ___ = 81991176 + 25311147 + 20994583", + "options": [ + "-22000789", + "-22000801", + "-22000771", + "-22000781" + ], + "answer": "-22000789", + "correct_value": -22000789, + "explanation": "B cancels; blank = C + D − A = 25311147 + 20994583 − 68306519 = -22000789." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "82312564 + 20708745 + ___ = 46923597 + 55172233", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 82312564 + 20708745 + ___ = 46923597 + 55172233", + "options": [ + "-925471", + "-925461", + "-925491", + "-925479" + ], + "answer": "-925479", + "correct_value": -925479, + "explanation": "No matching terms; blank = 46923597 + 55172233 − 82312564 − 20708745 = -925479." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001138", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "69766468 + 72815282 + ___ = 72815282 + 87456003 + 69766468", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 69766468 + 72815282 + ___ = 72815282 + 87456003 + 69766468", + "options": [ + "87455993", + "87456023", + "87456013", + "87456003" + ], + "answer": "87456003", + "correct_value": 87456003, + "explanation": "A and B appear on both sides and cancel; blank = C = 87456003." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "29838314 + 85198705 + ___ = 85198705 + 67072792 + 66768523", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 29838314 + 85198705 + ___ = 85198705 + 67072792 + 66768523", + "options": [ + "104003021", + "104002991", + "104003001", + "104003011" + ], + "answer": "104003001", + "correct_value": 104003001, + "explanation": "B cancels; blank = C + D − A = 67072792 + 66768523 − 29838314 = 104003001." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "22003127 + 94289287 + ___ = 98180101 + 52021016", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 22003127 + 94289287 + ___ = 98180101 + 52021016", + "options": [ + "33908693", + "33908713", + "33908723", + "33908703" + ], + "answer": "33908703", + "correct_value": 33908703, + "explanation": "No matching terms; blank = 98180101 + 52021016 − 22003127 − 94289287 = 33908703." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001139", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99783064 + 52737194 + ___ = 52737194 + 36193780 + 99783064", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 99783064 + 52737194 + ___ = 52737194 + 36193780 + 99783064", + "options": [ + "36193780", + "36193770", + "36193800", + "36193790" + ], + "answer": "36193780", + "correct_value": 36193780, + "explanation": "A and B appear on both sides and cancel; blank = C = 36193780." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "12681024 + 26312537 + ___ = 26312537 + 26355844 + 87265558", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 12681024 + 26312537 + ___ = 26312537 + 26355844 + 87265558", + "options": [ + "100940368", + "100940378", + "100940398", + "100940388" + ], + "answer": "100940378", + "correct_value": 100940378, + "explanation": "B cancels; blank = C + D − A = 26355844 + 87265558 − 12681024 = 100940378." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "11400593 + 43369182 + ___ = 81489497 + 26864487", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 11400593 + 43369182 + ___ = 81489497 + 26864487", + "options": [ + "53584229", + "53584209", + "53584219", + "53584199" + ], + "answer": "53584209", + "correct_value": 53584209, + "explanation": "No matching terms; blank = 81489497 + 26864487 − 11400593 − 43369182 = 53584209." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001140", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "56917382 × (26275588 + 42661791) = 56917382 × ___ + 56917382 × 42661791", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 56917382 × (26275588 + 42661791) = 56917382 × ___ + 56917382 × 42661791", + "options": [ + "26275598", + "26275608", + "26275588", + "26275578" + ], + "answer": "26275588", + "correct_value": 26275588, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 26275588." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "81285557 + 54035507 + ___ = 54035507 + 81808026 + 23564624", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 81285557 + 54035507 + ___ = 54035507 + 81808026 + 23564624", + "options": [ + "24087113", + "24087103", + "24087083", + "24087093" + ], + "answer": "24087093", + "correct_value": 24087093, + "explanation": "B cancels; blank = C + D − A = 81808026 + 23564624 − 81285557 = 24087093." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "53988555 + 18186816 + ___ = 28203817 + 50176313", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 53988555 + 18186816 + ___ = 28203817 + 50176313", + "options": [ + "6204759", + "6204749", + "6204779", + "6204769" + ], + "answer": "6204759", + "correct_value": 6204759, + "explanation": "No matching terms; blank = 28203817 + 50176313 − 53988555 − 18186816 = 6204759." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001141", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "63748955 × 19352129 × 56070372 = ___ × 56070372 × 63748955", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 63748955 × 19352129 × 56070372 = ___ × 56070372 × 63748955", + "options": [ + "19352149", + "19352139", + "19352129", + "19352119" + ], + "answer": "19352129", + "correct_value": 19352129, + "explanation": "A and C appear on both sides; blank = B = 19352129." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "63963175 + 28771812 + ___ = 28771812 + 69038459 + 20585201", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 63963175 + 28771812 + ___ = 28771812 + 69038459 + 20585201", + "options": [ + "25660485", + "25660475", + "25660505", + "25660495" + ], + "answer": "25660485", + "correct_value": 25660485, + "explanation": "B cancels; blank = C + D − A = 69038459 + 20585201 − 63963175 = 25660485." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "78405004 + 90474782 + ___ = 63813689 + 86282637", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 78405004 + 90474782 + ___ = 63813689 + 86282637", + "options": [ + "-18783440", + "-18783470", + "-18783460", + "-18783450" + ], + "answer": "-18783460", + "correct_value": -18783460, + "explanation": "No matching terms; blank = 63813689 + 86282637 − 78405004 − 90474782 = -18783460." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001142", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "93038086 × 18100528 × 10954878 = ___ × 10954878 × 93038086", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 93038086 × 18100528 × 10954878 = ___ × 10954878 × 93038086", + "options": [ + "18100528", + "18100538", + "18100518", + "18100548" + ], + "answer": "18100528", + "correct_value": 18100528, + "explanation": "A and C appear on both sides; blank = B = 18100528." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "67740709 + 22126246 + ___ = 22126246 + 59220063 + 94942254", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 67740709 + 22126246 + ___ = 22126246 + 59220063 + 94942254", + "options": [ + "86421598", + "86421608", + "86421618", + "86421628" + ], + "answer": "86421608", + "correct_value": 86421608, + "explanation": "B cancels; blank = C + D − A = 59220063 + 94942254 − 67740709 = 86421608." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "21535774 + 97330311 + ___ = 54870630 + 61785761", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 21535774 + 97330311 + ___ = 54870630 + 61785761", + "options": [ + "-2209676", + "-2209686", + "-2209694", + "-2209706" + ], + "answer": "-2209694", + "correct_value": -2209694, + "explanation": "No matching terms; blank = 54870630 + 61785761 − 21535774 − 97330311 = -2209694." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001143", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "57924903 × 56535615 × 25527671 = ___ × 25527671 × 57924903", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 57924903 × 56535615 × 25527671 = ___ × 25527671 × 57924903", + "options": [ + "56535605", + "56535635", + "56535625", + "56535615" + ], + "answer": "56535615", + "correct_value": 56535615, + "explanation": "A and C appear on both sides; blank = B = 56535615." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "43030893 + 62732451 + ___ = 62732451 + 92335103 + 15715975", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 43030893 + 62732451 + ___ = 62732451 + 92335103 + 15715975", + "options": [ + "65020175", + "65020185", + "65020205", + "65020195" + ], + "answer": "65020185", + "correct_value": 65020185, + "explanation": "B cancels; blank = C + D − A = 92335103 + 15715975 − 43030893 = 65020185." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "73288596 + 75251163 + ___ = 34388806 + 71328876", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 73288596 + 75251163 + ___ = 34388806 + 71328876", + "options": [ + "-42822063", + "-42822073", + "-42822093", + "-42822077" + ], + "answer": "-42822077", + "correct_value": -42822077, + "explanation": "No matching terms; blank = 34388806 + 71328876 − 73288596 − 75251163 = -42822077." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001144", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "37898170 + 28287832 + ___ = 28287832 + 57858443 + 37898170", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 37898170 + 28287832 + ___ = 28287832 + 57858443 + 37898170", + "options": [ + "57858453", + "57858433", + "57858463", + "57858443" + ], + "answer": "57858443", + "correct_value": 57858443, + "explanation": "A and B appear on both sides and cancel; blank = C = 57858443." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "50845680 + 94688127 + ___ = 94688127 + 40036584 + 89267437", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 50845680 + 94688127 + ___ = 94688127 + 40036584 + 89267437", + "options": [ + "78458331", + "78458351", + "78458341", + "78458361" + ], + "answer": "78458341", + "correct_value": 78458341, + "explanation": "B cancels; blank = C + D − A = 40036584 + 89267437 − 50845680 = 78458341." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "97220095 + 42700083 + ___ = 46313698 + 78086624", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 97220095 + 42700083 + ___ = 46313698 + 78086624", + "options": [ + "-15519856", + "-15519854", + "-15519844", + "-15519874" + ], + "answer": "-15519856", + "correct_value": -15519856, + "explanation": "No matching terms; blank = 46313698 + 78086624 − 97220095 − 42700083 = -15519856." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001145", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "81861944 × (95203223 + 50213388) = 81861944 × ___ + 81861944 × 50213388", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 81861944 × (95203223 + 50213388) = 81861944 × ___ + 81861944 × 50213388", + "options": [ + "95203243", + "95203223", + "95203233", + "95203213" + ], + "answer": "95203223", + "correct_value": 95203223, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 95203223." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "60104145 + 26702017 + ___ = 26702017 + 15029596 + 35354987", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 60104145 + 26702017 + ___ = 26702017 + 15029596 + 35354987", + "options": [ + "-9719548", + "-9719558", + "-9719578", + "-9719562" + ], + "answer": "-9719562", + "correct_value": -9719562, + "explanation": "B cancels; blank = C + D − A = 15029596 + 35354987 − 60104145 = -9719562." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "38721683 + 11110324 + ___ = 45441367 + 29154201", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 38721683 + 11110324 + ___ = 45441367 + 29154201", + "options": [ + "24763581", + "24763561", + "24763571", + "24763551" + ], + "answer": "24763561", + "correct_value": 24763561, + "explanation": "No matching terms; blank = 45441367 + 29154201 − 38721683 − 11110324 = 24763561." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001146", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "86629824 × 74267059 × 62918909 = ___ × 62918909 × 86629824", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 86629824 × 74267059 × 62918909 = ___ × 62918909 × 86629824", + "options": [ + "74267069", + "74267079", + "74267059", + "74267049" + ], + "answer": "74267059", + "correct_value": 74267059, + "explanation": "A and C appear on both sides; blank = B = 74267059." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "77733411 + 52773721 + ___ = 52773721 + 93236871 + 87757057", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 77733411 + 52773721 + ___ = 52773721 + 93236871 + 87757057", + "options": [ + "103260527", + "103260517", + "103260537", + "103260507" + ], + "answer": "103260517", + "correct_value": 103260517, + "explanation": "B cancels; blank = C + D − A = 93236871 + 87757057 − 77733411 = 103260517." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "66852311 + 96166590 + ___ = 17930882 + 80840735", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 66852311 + 96166590 + ___ = 17930882 + 80840735", + "options": [ + "-64247296", + "-64247276", + "-64247266", + "-64247284" + ], + "answer": "-64247284", + "correct_value": -64247284, + "explanation": "No matching terms; blank = 17930882 + 80840735 − 66852311 − 96166590 = -64247284." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001147", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "82839132 + 39999890 = ___ + 53348890", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 82839132 + 39999890 = ___ + 53348890", + "options": [ + "69490122", + "69490142", + "69490152", + "69490132" + ], + "answer": "69490132", + "correct_value": 69490132, + "explanation": "Blank = A + B − C = 82839132 + 39999890 − 53348890 = 69490132." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "22726697 + 43322769 + ___ = 43322769 + 11325902 + 66232580", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 22726697 + 43322769 + ___ = 43322769 + 11325902 + 66232580", + "options": [ + "54831775", + "54831805", + "54831785", + "54831795" + ], + "answer": "54831785", + "correct_value": 54831785, + "explanation": "B cancels; blank = C + D − A = 11325902 + 66232580 − 22726697 = 54831785." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "74329489 + 94028911 + ___ = 32698155 + 64807144", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 74329489 + 94028911 + ___ = 32698155 + 64807144", + "options": [ + "-70853089", + "-70853099", + "-70853119", + "-70853101" + ], + "answer": "-70853101", + "correct_value": -70853101, + "explanation": "No matching terms; blank = 32698155 + 64807144 − 74329489 − 94028911 = -70853101." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001148", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "70869451 + 45686084 + ___ = 45686084 + 36379956 + 70869451", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 70869451 + 45686084 + ___ = 45686084 + 36379956 + 70869451", + "options": [ + "36379946", + "36379966", + "36379976", + "36379956" + ], + "answer": "36379956", + "correct_value": 36379956, + "explanation": "A and B appear on both sides and cancel; blank = C = 36379956." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "84194995 + 85155304 + ___ = 85155304 + 26081719 + 24181221", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 84194995 + 85155304 + ___ = 85155304 + 26081719 + 24181221", + "options": [ + "-33932035", + "-33932055", + "-33932045", + "-33932065" + ], + "answer": "-33932055", + "correct_value": -33932055, + "explanation": "B cancels; blank = C + D − A = 26081719 + 24181221 − 84194995 = -33932055." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "37258959 + 50945382 + ___ = 37952799 + 89275381", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 37258959 + 50945382 + ___ = 37952799 + 89275381", + "options": [ + "39023829", + "39023849", + "39023859", + "39023839" + ], + "answer": "39023839", + "correct_value": 39023839, + "explanation": "No matching terms; blank = 37952799 + 89275381 − 37258959 − 50945382 = 39023839." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001149", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "83601876 × 73068008 × 69483614 = ___ × 69483614 × 83601876", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 83601876 × 73068008 × 69483614 = ___ × 69483614 × 83601876", + "options": [ + "73068018", + "73068008", + "73067998", + "73068028" + ], + "answer": "73068008", + "correct_value": 73068008, + "explanation": "A and C appear on both sides; blank = B = 73068008." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "41411837 + 38782865 + ___ = 38782865 + 64318314 + 48113094", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 41411837 + 38782865 + ___ = 38782865 + 64318314 + 48113094", + "options": [ + "71019591", + "71019571", + "71019581", + "71019561" + ], + "answer": "71019571", + "correct_value": 71019571, + "explanation": "B cancels; blank = C + D − A = 64318314 + 48113094 − 41411837 = 71019571." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "43697896 + 22655804 + ___ = 41212087 + 11131286", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 43697896 + 22655804 + ___ = 41212087 + 11131286", + "options": [ + "-14010323", + "-14010343", + "-14010313", + "-14010327" + ], + "answer": "-14010327", + "correct_value": -14010327, + "explanation": "No matching terms; blank = 41212087 + 11131286 − 43697896 − 22655804 = -14010327." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001150", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "35904207 × (78806863 + 95485363) = 35904207 × ___ + 35904207 × 95485363", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 35904207 × (78806863 + 95485363) = 35904207 × ___ + 35904207 × 95485363", + "options": [ + "78806873", + "78806883", + "78806863", + "78806853" + ], + "answer": "78806863", + "correct_value": 78806863, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 78806863." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "83142099 + 27141790 + ___ = 27141790 + 63136530 + 27883205", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 83142099 + 27141790 + ___ = 27141790 + 63136530 + 27883205", + "options": [ + "7877626", + "7877636", + "7877646", + "7877656" + ], + "answer": "7877636", + "correct_value": 7877636, + "explanation": "B cancels; blank = C + D − A = 63136530 + 27883205 − 83142099 = 7877636." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "15304180 + 45160491 + ___ = 19102656 + 97940633", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 15304180 + 45160491 + ___ = 19102656 + 97940633", + "options": [ + "56578608", + "56578638", + "56578618", + "56578628" + ], + "answer": "56578618", + "correct_value": 56578618, + "explanation": "No matching terms; blank = 19102656 + 97940633 − 15304180 − 45160491 = 56578618." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001151", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "97560003 + 57794863 + ___ = 57794863 + 70426196 + 97560003", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 97560003 + 57794863 + ___ = 57794863 + 70426196 + 97560003", + "options": [ + "70426206", + "70426216", + "70426196", + "70426186" + ], + "answer": "70426196", + "correct_value": 70426196, + "explanation": "A and B appear on both sides and cancel; blank = C = 70426196." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "46041613 + 47513650 + ___ = 47513650 + 18896589 + 12641440", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 46041613 + 47513650 + ___ = 47513650 + 18896589 + 12641440", + "options": [ + "-14503596", + "-14503576", + "-14503566", + "-14503584" + ], + "answer": "-14503584", + "correct_value": -14503584, + "explanation": "B cancels; blank = C + D − A = 18896589 + 12641440 − 46041613 = -14503584." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "89398528 + 46522173 + ___ = 23382149 + 85836526", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 89398528 + 46522173 + ___ = 23382149 + 85836526", + "options": [ + "-26702014", + "-26702044", + "-26702026", + "-26702024" + ], + "answer": "-26702026", + "correct_value": -26702026, + "explanation": "No matching terms; blank = 23382149 + 85836526 − 89398528 − 46522173 = -26702026." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "option_elimination_001152", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "96151995 × 99531360", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 96151995 × 99531360?", + "options": [ + "9570138829063202", + "9570138829063199", + "9570138829063201", + "9570138829063200" + ], + "answer": "9570138829063200", + "correct_value": 9570138829063200, + "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": "10546370 × 19737148", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 10546370 × 19737148?", + "options": [ + "208155265552761", + "208155265552750", + "208155265552760", + "208155265552759" + ], + "answer": "208155265552760", + "correct_value": 208155265552760, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "74429451 × 92303875", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 74429451 × 92303875?", + "options": [ + "6450929641422625", + "6915282641422625", + "8236710141422625", + "6870126741422625" + ], + "answer": "6870126741422625", + "correct_value": 6870126741422625, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001153", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "96151995 × 99531360", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 96151995 × 99531360?", + "options": [ + "9570138829063202", + "9570138829063199", + "9570138829063201", + "9570138829063200" + ], + "answer": "9570138829063200", + "correct_value": 9570138829063200, + "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": "10546370 × 19737148", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 10546370 × 19737148?", + "options": [ + "208155265552761", + "208155265552750", + "208155265552760", + "208155265552759" + ], + "answer": "208155265552760", + "correct_value": 208155265552760, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "74429451 × 92303875", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 74429451 × 92303875?", + "options": [ + "6450929641422625", + "6915282641422625", + "8236710141422625", + "6870126741422625" + ], + "answer": "6870126741422625", + "correct_value": 6870126741422625, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001154", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "82443149 × 27972511", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 82443149 × 27972511?", + "options": [ + "2306141892277142", + "2306141892277138", + "2306141892277139", + "2306141892277140" + ], + "answer": "2306141892277139", + "correct_value": 2306141892277139, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55984720 × 60959426", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 55984720 × 60959426?", + "options": [ + "3412796395970721", + "3412796395970719", + "3412796395970720", + "3412796395970740" + ], + "answer": "3412796395970720", + "correct_value": 3412796395970720, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "80811671 × 97344041", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 80811671 × 97344041?", + "options": [ + "7866534615102511", + "7087751215102511", + "7294283515102511", + "8641025715102511" + ], + "answer": "7866534615102511", + "correct_value": 7866534615102511, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001155", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "22158245 × 93274994", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 22158245 × 93274994?", + "options": [ + "2066810169425531", + "2066810169425533", + "2066810169425530", + "2066810169425529" + ], + "answer": "2066810169425530", + "correct_value": 2066810169425530, + "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": "85504420 × 91031384", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 85504420 × 91031384?", + "options": [ + "7783585690717300", + "7783585690717280", + "7783585690717279", + "7783585690717281" + ], + "answer": "7783585690717280", + "correct_value": 7783585690717280, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "93750967 × 75309325", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 93750967 × 75309325?", + "options": [ + "7239250542867275", + "6899205642867275", + "6627572342867275", + "7060322042867275" + ], + "answer": "7060322042867275", + "correct_value": 7060322042867275, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001156", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "82443149 × 27972511", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 82443149 × 27972511?", + "options": [ + "2306141892277142", + "2306141892277138", + "2306141892277139", + "2306141892277140" + ], + "answer": "2306141892277139", + "correct_value": 2306141892277139, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55984720 × 60959426", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 55984720 × 60959426?", + "options": [ + "3412796395970721", + "3412796395970719", + "3412796395970720", + "3412796395970740" + ], + "answer": "3412796395970720", + "correct_value": 3412796395970720, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "80811671 × 97344041", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 80811671 × 97344041?", + "options": [ + "7866534615102511", + "7087751215102511", + "7294283515102511", + "8641025715102511" + ], + "answer": "7866534615102511", + "correct_value": 7866534615102511, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001157", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "22158245 × 93274994", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 22158245 × 93274994?", + "options": [ + "2066810169425531", + "2066810169425533", + "2066810169425530", + "2066810169425529" + ], + "answer": "2066810169425530", + "correct_value": 2066810169425530, + "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": "85504420 × 91031384", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 85504420 × 91031384?", + "options": [ + "7783585690717300", + "7783585690717280", + "7783585690717279", + "7783585690717281" + ], + "answer": "7783585690717280", + "correct_value": 7783585690717280, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "93750967 × 75309325", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 93750967 × 75309325?", + "options": [ + "7239250542867275", + "6899205642867275", + "6627572342867275", + "7060322042867275" + ], + "answer": "7060322042867275", + "correct_value": 7060322042867275, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001158", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "16600285 × 68254988", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 16600285 × 68254988?", + "options": [ + "1133052253471581", + "1133052253471580", + "1133052253471583", + "1133052253471579" + ], + "answer": "1133052253471580", + "correct_value": 1133052253471580, + "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": "59400180 × 83679884", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 59400180 × 83679884?", + "options": [ + "4970600171979119", + "4970600171979120", + "4970600171979140", + "4970600171979121" + ], + "answer": "4970600171979120", + "correct_value": 4970600171979120, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "82231527 × 63675933", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 82231527 × 63675933?", + "options": [ + "5236169203739691", + "5055992403739691", + "6212861503739691", + "5974206603739691" + ], + "answer": "5236169203739691", + "correct_value": 5236169203739691, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001159", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "16600285 × 68254988", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 16600285 × 68254988?", + "options": [ + "1133052253471581", + "1133052253471580", + "1133052253471583", + "1133052253471579" + ], + "answer": "1133052253471580", + "correct_value": 1133052253471580, + "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": "59400180 × 83679884", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 59400180 × 83679884?", + "options": [ + "4970600171979119", + "4970600171979120", + "4970600171979140", + "4970600171979121" + ], + "answer": "4970600171979120", + "correct_value": 4970600171979120, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "82231527 × 63675933", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 82231527 × 63675933?", + "options": [ + "5236169203739691", + "5055992403739691", + "6212861503739691", + "5974206603739691" + ], + "answer": "5236169203739691", + "correct_value": 5236169203739691, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001160", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "60337259 × 84810445", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 60337259 × 84810445?", + "options": [ + "5117229785870256", + "5117229785870255", + "5117229785870254", + "5117229785870258" + ], + "answer": "5117229785870255", + "correct_value": 5117229785870255, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "44343090 × 35681176", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 44343090 × 35681176?", + "options": [ + "1582213598673840", + "1582213598673839", + "1582213598673841", + "1582213598673830" + ], + "answer": "1582213598673840", + "correct_value": 1582213598673840, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "99775529 × 32362065", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 99775529 × 32362065?", + "options": [ + "3361501054907385", + "3000767754907385", + "3228942154907385", + "3311604154907385" + ], + "answer": "3228942154907385", + "correct_value": 3228942154907385, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001161", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "71300445 × 83081852", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 71300445 × 83081852?", + "options": [ + "5923773019024140", + "11847546038048280", + "12439923339950694", + "592377301902414" + ], + "answer": "5923773019024140", + "correct_value": 5923773019024140, + "explanation": "Product has 16 digits; only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "53216210 × 80508010", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 53216210 × 80508010?", + "options": [ + "4284331166842101", + "4284331166842099", + "4284331166842110", + "4284331166842100" + ], + "answer": "4284331166842100", + "correct_value": 4284331166842100, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "80707171 × 96777475", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 80707171 × 96777475?", + "options": [ + "9287495723773225", + "8221532023773225", + "7810636223773225", + "6350016723773225" + ], + "answer": "7810636223773225", + "correct_value": 7810636223773225, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001162", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49573275 × 23483176", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 49573275 × 23483176?", + "options": [ + "1164137941721399", + "1164137941721401", + "1164137941721403", + "1164137941721400" + ], + "answer": "1164137941721400", + "correct_value": 1164137941721400, + "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": "43497880 × 23144580", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 43497880 × 23144580?", + "options": [ + "1006740163490400", + "1006740163490420", + "1006740163490401", + "1006740163490399" + ], + "answer": "1006740163490400", + "correct_value": 1006740163490400, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "68070459 × 11015767", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 68070459 × 11015767?", + "options": [ + "797567995927053", + "749848315927053", + "646319005927053", + "840989775927053" + ], + "answer": "749848315927053", + "correct_value": 749848315927053, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001163", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "41029332 × 14244858", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 41029332 × 14244858?", + "options": [ + "1285805417984676", + "584457008174856", + "1110468315532221", + "1227359717167191" + ], + "answer": "584457008174856", + "correct_value": 584457008174856, + "explanation": "Product has 15 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "79080205 × 87490194", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 79080205 × 87490194?", + "options": [ + "6918742477009771", + "6918742477009750", + "6918742477009769", + "6918742477009770" + ], + "answer": "6918742477009770", + "correct_value": 6918742477009770, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "92367765 × 70795947", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 92367765 × 70795947?", + "options": [ + "7422099595448455", + "6539263395448455", + "7579114095448455", + "6389281795448455" + ], + "answer": "6539263395448455", + "correct_value": 6539263395448455, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001164", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "35451245 × 74144545", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 35451245 × 74144545?", + "options": [ + "2628516430208525", + "2628516430208528", + "2628516430208524", + "2628516430208526" + ], + "answer": "2628516430208525", + "correct_value": 2628516430208525, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "34226910 × 94289148", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 34226910 × 94289148?", + "options": [ + "3227226182572681", + "3227226182572680", + "3227226182572700", + "3227226182572679" + ], + "answer": "3227226182572680", + "correct_value": 3227226182572680, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "54073835 × 93510633", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 54073835 × 93510633?", + "options": [ + "5345529539587555", + "5475423939587555", + "5056478539587555", + "4618025339587555" + ], + "answer": "5056478539587555", + "correct_value": 5056478539587555, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001165", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "20254465 × 25246285", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 20254465 × 25246285?", + "options": [ + "511349995912524", + "511349995912526", + "511349995912528", + "511349995912525" + ], + "answer": "511349995912525", + "correct_value": 511349995912525, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "54833800 × 55635904", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 54833800 × 55635904?", + "options": [ + "3050728032755201", + "3050728032755220", + "3050728032755200", + "3050728032755199" + ], + "answer": "3050728032755200", + "correct_value": 3050728032755200, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "54748853 × 19213537", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 54748853 × 19213537?", + "options": [ + "1126575612823061", + "1071034612823061", + "1051919112823061", + "1245876912823061" + ], + "answer": "1051919112823061", + "correct_value": 1051919112823061, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001166", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "60337259 × 84810445", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 60337259 × 84810445?", + "options": [ + "5117229785870256", + "5117229785870255", + "5117229785870254", + "5117229785870258" + ], + "answer": "5117229785870255", + "correct_value": 5117229785870255, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "44343090 × 35681176", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 44343090 × 35681176?", + "options": [ + "1582213598673840", + "1582213598673839", + "1582213598673841", + "1582213598673830" + ], + "answer": "1582213598673840", + "correct_value": 1582213598673840, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "99775529 × 32362065", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 99775529 × 32362065?", + "options": [ + "3361501054907385", + "3000767754907385", + "3228942154907385", + "3311604154907385" + ], + "answer": "3228942154907385", + "correct_value": 3228942154907385, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001167", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "82665459 × 87112379", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 82665459 × 87112379?", + "options": [ + "7201184794616962", + "7201184794616960", + "7201184794616961", + "7201184794616964" + ], + "answer": "7201184794616961", + "correct_value": 7201184794616961, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "26256405 × 53875572", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 26256405 × 53875572?", + "options": [ + "1414578838038660", + "1414578838038661", + "1414578838038670", + "1414578838038659" + ], + "answer": "1414578838038660", + "correct_value": 1414578838038660, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "33624653 × 51558675", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 33624653 × 51558675?", + "options": [ + "1432970956014775", + "1733642556014775", + "1552264156014775", + "1905537656014775" + ], + "answer": "1733642556014775", + "correct_value": 1733642556014775, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001168", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "88765820 × 55347882", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 88765820 × 55347882?", + "options": [ + "4913000130993243", + "4913000130993239", + "4913000130993240", + "4913000130993241" + ], + "answer": "4913000130993240", + "correct_value": 4913000130993240, + "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": "68208610 × 29474816", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 68208610 × 29474816?", + "options": [ + "2010436229365760", + "2010436229365740", + "2010436229365759", + "2010436229365761" + ], + "answer": "2010436229365760", + "correct_value": 2010436229365760, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "31233651 × 61258627", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 31233651 × 61258627?", + "options": [ + "1913330576457177", + "2285870776457177", + "1777697476457177", + "2112230676457177" + ], + "answer": "1913330576457177", + "correct_value": 1913330576457177, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001169", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "71300445 × 83081852", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 71300445 × 83081852?", + "options": [ + "5923773019024140", + "11847546038048280", + "12439923339950694", + "592377301902414" + ], + "answer": "5923773019024140", + "correct_value": 5923773019024140, + "explanation": "Product has 16 digits; only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "53216210 × 80508010", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 53216210 × 80508010?", + "options": [ + "4284331166842101", + "4284331166842099", + "4284331166842110", + "4284331166842100" + ], + "answer": "4284331166842100", + "correct_value": 4284331166842100, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "80707171 × 96777475", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 80707171 × 96777475?", + "options": [ + "9287495723773225", + "8221532023773225", + "7810636223773225", + "6350016723773225" + ], + "answer": "7810636223773225", + "correct_value": 7810636223773225, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001170", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "57774671 × 25174799", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 57774671 × 25174799?", + "options": [ + "1454465729716129", + "1454465729716130", + "1454465729716132", + "1454465729716128" + ], + "answer": "1454465729716129", + "correct_value": 1454465729716129, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99990085 × 50942414", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 99990085 × 50942414?", + "options": [ + "5093736305965191", + "5093736305965190", + "5093736305965200", + "5093736305965189" + ], + "answer": "5093736305965190", + "correct_value": 5093736305965190, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "74462471 × 59670245", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 74462471 × 59670245?", + "options": [ + "3570804987875395", + "4379460787875395", + "4443193887875395", + "3847806387875395" + ], + "answer": "4443193887875395", + "correct_value": 4443193887875395, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001171", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49573275 × 23483176", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 49573275 × 23483176?", + "options": [ + "1164137941721399", + "1164137941721401", + "1164137941721403", + "1164137941721400" + ], + "answer": "1164137941721400", + "correct_value": 1164137941721400, + "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": "43497880 × 23144580", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 43497880 × 23144580?", + "options": [ + "1006740163490400", + "1006740163490420", + "1006740163490401", + "1006740163490399" + ], + "answer": "1006740163490400", + "correct_value": 1006740163490400, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "68070459 × 11015767", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 68070459 × 11015767?", + "options": [ + "797567995927053", + "749848315927053", + "646319005927053", + "840989775927053" + ], + "answer": "749848315927053", + "correct_value": 749848315927053, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001172", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12941020 × 90780173", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 12941020 × 90780173?", + "options": [ + "117478803439646", + "234957606879292", + "11747880343964698", + "1174788034396460" + ], + "answer": "1174788034396460", + "correct_value": 1174788034396460, + "explanation": "Product has 16 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "69904315 × 67466244", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 69904315 × 67466244?", + "options": [ + "4716181572442861", + "4716181572442840", + "4716181572442859", + "4716181572442860" + ], + "answer": "4716181572442860", + "correct_value": 4716181572442860, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "14777953 × 69074935", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 14777953 × 69074935?", + "options": [ + "1077399442908055", + "1020786142908055", + "1100041142908055", + "1078955742908055" + ], + "answer": "1020786142908055", + "correct_value": 1020786142908055, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001173", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "41029332 × 14244858", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 41029332 × 14244858?", + "options": [ + "1285805417984676", + "584457008174856", + "1110468315532221", + "1227359717167191" + ], + "answer": "584457008174856", + "correct_value": 584457008174856, + "explanation": "Product has 15 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "79080205 × 87490194", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 79080205 × 87490194?", + "options": [ + "6918742477009771", + "6918742477009750", + "6918742477009769", + "6918742477009770" + ], + "answer": "6918742477009770", + "correct_value": 6918742477009770, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "92367765 × 70795947", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 92367765 × 70795947?", + "options": [ + "7422099595448455", + "6539263395448455", + "7579114095448455", + "6389281795448455" + ], + "answer": "6539263395448455", + "correct_value": 6539263395448455, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001174", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52936378 × 69124950", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 52936378 × 69124950?", + "options": [ + "365922448243110", + "36592244824311049", + "3659224482431100", + "731844896486220" + ], + "answer": "3659224482431100", + "correct_value": 3659224482431100, + "explanation": "Product has 16 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "85240070 × 45355772", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 85240070 × 45355772?", + "options": [ + "3866129180184041", + "3866129180184050", + "3866129180184039", + "3866129180184040" + ], + "answer": "3866129180184040", + "correct_value": 3866129180184040, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "25015959 × 14337455", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 25015959 × 14337455?", + "options": [ + "320321226444345", + "378443166444345", + "413560206444345", + "358665186444345" + ], + "answer": "358665186444345", + "correct_value": 358665186444345, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001175", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "35451245 × 74144545", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 35451245 × 74144545?", + "options": [ + "2628516430208525", + "2628516430208528", + "2628516430208524", + "2628516430208526" + ], + "answer": "2628516430208525", + "correct_value": 2628516430208525, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "34226910 × 94289148", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 34226910 × 94289148?", + "options": [ + "3227226182572681", + "3227226182572680", + "3227226182572700", + "3227226182572679" + ], + "answer": "3227226182572680", + "correct_value": 3227226182572680, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "54073835 × 93510633", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 54073835 × 93510633?", + "options": [ + "5345529539587555", + "5475423939587555", + "5056478539587555", + "4618025339587555" + ], + "answer": "5056478539587555", + "correct_value": 5056478539587555, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001176", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "30930000 × 50580940", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 30930000 × 50580940?", + "options": [ + "1564468474200003", + "1564468474199999", + "1564468474200000", + "1564468474200001" + ], + "answer": "1564468474200000", + "correct_value": 1564468474200000, + "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": "73717460 × 65235716", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 73717460 × 65235716?", + "options": [ + "4809011284801360", + "4809011284801359", + "4809011284801350", + "4809011284801361" + ], + "answer": "4809011284801360", + "correct_value": 4809011284801360, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "56442073 × 89433661", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 56442073 × 89433661?", + "options": [ + "5727798322819253", + "5047821222819253", + "4452632322819253", + "4463803422819253" + ], + "answer": "5047821222819253", + "correct_value": 5047821222819253, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001177", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "20254465 × 25246285", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 20254465 × 25246285?", + "options": [ + "511349995912524", + "511349995912526", + "511349995912528", + "511349995912525" + ], + "answer": "511349995912525", + "correct_value": 511349995912525, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "54833800 × 55635904", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 54833800 × 55635904?", + "options": [ + "3050728032755201", + "3050728032755220", + "3050728032755200", + "3050728032755199" + ], + "answer": "3050728032755200", + "correct_value": 3050728032755200, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "54748853 × 19213537", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 54748853 × 19213537?", + "options": [ + "1126575612823061", + "1071034612823061", + "1051919112823061", + "1245876912823061" + ], + "answer": "1051919112823061", + "correct_value": 1051919112823061, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001178", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "17607481 × 52652963", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 17607481 × 52652963?", + "options": [ + "927086045616204", + "927086045616202", + "927086045616206", + "927086045616203" + ], + "answer": "927086045616203", + "correct_value": 927086045616203, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "29093840 × 83919164", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 29093840 × 83919164?", + "options": [ + "2441530730349760", + "2441530730349759", + "2441530730349780", + "2441530730349761" + ], + "answer": "2441530730349760", + "correct_value": 2441530730349760, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "85367871 × 50914141", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 85367871 × 50914141?", + "options": [ + "4420724920963811", + "4179783420963811", + "4550365120963811", + "4346431820963811" + ], + "answer": "4346431820963811", + "correct_value": 4346431820963811, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001179", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "94204820 × 11312368", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 94204820 × 11312368?", + "options": [ + "937798040268104", + "1065679591213760", + "213135918242752", + "969768428004518" + ], + "answer": "1065679591213760", + "correct_value": 1065679591213760, + "explanation": "Product has 16 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "77811775 × 63060270", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 77811775 × 63060270?", + "options": [ + "4906831540679250", + "4906831540679249", + "4906831540679251", + "4906831540679260" + ], + "answer": "4906831540679250", + "correct_value": 4906831540679250, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "29510525 × 96960825", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 29510525 × 96960825?", + "options": [ + "2294450550183125", + "2861364850183125", + "2911598450183125", + "2659846350183125" + ], + "answer": "2861364850183125", + "correct_value": 2861364850183125, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001180", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "38026899 × 90054365", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 38026899 × 90054365?", + "options": [ + "3424488242364136", + "3424488242364138", + "3424488242364135", + "3424488242364134" + ], + "answer": "3424488242364135", + "correct_value": 3424488242364135, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "51770130 × 24015880", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 51770130 × 24015880?", + "options": [ + "1243305229664380", + "1243305229664399", + "1243305229664401", + "1243305229664400" + ], + "answer": "1243305229664400", + "correct_value": 1243305229664400, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "52494757 × 49426051", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 52494757 × 49426051?", + "options": [ + "2754642836714607", + "2937125136714607", + "2441626436714607", + "2594608536714607" + ], + "answer": "2594608536714607", + "correct_value": 2594608536714607, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001181", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "82665459 × 87112379", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 82665459 × 87112379?", + "options": [ + "7201184794616962", + "7201184794616960", + "7201184794616961", + "7201184794616964" + ], + "answer": "7201184794616961", + "correct_value": 7201184794616961, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "26256405 × 53875572", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 26256405 × 53875572?", + "options": [ + "1414578838038660", + "1414578838038661", + "1414578838038670", + "1414578838038659" + ], + "answer": "1414578838038660", + "correct_value": 1414578838038660, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "33624653 × 51558675", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 33624653 × 51558675?", + "options": [ + "1432970956014775", + "1733642556014775", + "1552264156014775", + "1905537656014775" + ], + "answer": "1733642556014775", + "correct_value": 1733642556014775, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001182", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "35724479 × 11095443", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 35724479 × 11095443?", + "options": [ + "396378920449200", + "396378920449197", + "396378920449198", + "396378920449196" + ], + "answer": "396378920449197", + "correct_value": 396378920449197, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99861710 × 97437376", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 99861710 × 97437376?", + "options": [ + "9730262985272970", + "9730262985272959", + "9730262985272960", + "9730262985272961" + ], + "answer": "9730262985272960", + "correct_value": 9730262985272960, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "33104859 × 12119753", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 33104859 × 12119753?", + "options": [ + "381504924179827", + "452913514179827", + "376378454179827", + "401222714179827" + ], + "answer": "401222714179827", + "correct_value": 401222714179827, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001183", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "88765820 × 55347882", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 88765820 × 55347882?", + "options": [ + "4913000130993243", + "4913000130993239", + "4913000130993240", + "4913000130993241" + ], + "answer": "4913000130993240", + "correct_value": 4913000130993240, + "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": "68208610 × 29474816", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 68208610 × 29474816?", + "options": [ + "2010436229365760", + "2010436229365740", + "2010436229365759", + "2010436229365761" + ], + "answer": "2010436229365760", + "correct_value": 2010436229365760, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "31233651 × 61258627", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 31233651 × 61258627?", + "options": [ + "1913330576457177", + "2285870776457177", + "1777697476457177", + "2112230676457177" + ], + "answer": "1913330576457177", + "correct_value": 1913330576457177, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001184", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50244599 × 78320361", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 50244599 × 78320361?", + "options": [ + "3935175131980238", + "3935175131980239", + "3935175131980240", + "3935175131980242" + ], + "answer": "3935175131980239", + "correct_value": 3935175131980239, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "85816075 × 10890424", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 85816075 × 10890424?", + "options": [ + "934573442765799", + "934573442765801", + "934573442765800", + "934573442765780" + ], + "answer": "934573442765800", + "correct_value": 934573442765800, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "58602053 × 94052737", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 58602053 × 94052737?", + "options": [ + "5511683478469061", + "5847266078469061", + "4899746278469061", + "4786528078469061" + ], + "answer": "5511683478469061", + "correct_value": 5511683478469061, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001185", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "57774671 × 25174799", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 57774671 × 25174799?", + "options": [ + "1454465729716129", + "1454465729716130", + "1454465729716132", + "1454465729716128" + ], + "answer": "1454465729716129", + "correct_value": 1454465729716129, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99990085 × 50942414", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 99990085 × 50942414?", + "options": [ + "5093736305965191", + "5093736305965190", + "5093736305965200", + "5093736305965189" + ], + "answer": "5093736305965190", + "correct_value": 5093736305965190, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "74462471 × 59670245", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 74462471 × 59670245?", + "options": [ + "3570804987875395", + "4379460787875395", + "4443193887875395", + "3847806387875395" + ], + "answer": "4443193887875395", + "correct_value": 4443193887875395, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001186", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "75354609 × 20458341", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 75354609 × 20458341?", + "options": [ + "1541630286843670", + "1541630286843671", + "1541630286843668", + "1541630286843669" + ], + "answer": "1541630286843669", + "correct_value": 1541630286843669, + "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": "26848830 × 79427138", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 26848830 × 79427138?", + "options": [ + "2132525725548541", + "2132525725548550", + "2132525725548540", + "2132525725548539" + ], + "answer": "2132525725548540", + "correct_value": 2132525725548540, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "93379043 × 87070949", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 93379043 × 87070949?", + "options": [ + "8130601890721807", + "6679641590721807", + "7381638990721807", + "7288990890721807" + ], + "answer": "8130601890721807", + "correct_value": 8130601890721807, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001187", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "83970981 × 12895211", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 83970981 × 12895211?", + "options": [ + "1082823517871994", + "1082823517871991", + "1082823517871992", + "1082823517871990" + ], + "answer": "1082823517871991", + "correct_value": 1082823517871991, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "79476770 × 36605524", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 79476770 × 36605524?", + "options": [ + "2909288811677481", + "2909288811677480", + "2909288811677500", + "2909288811677479" + ], + "answer": "2909288811677480", + "correct_value": 2909288811677480, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "26161935 × 86990855", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 26161935 × 86990855?", + "options": [ + "1911248694104425", + "2275849094104425", + "1937679294104425", + "1836779894104425" + ], + "answer": "2275849094104425", + "correct_value": 2275849094104425, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001188", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12941020 × 90780173", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 12941020 × 90780173?", + "options": [ + "117478803439646", + "234957606879292", + "11747880343964698", + "1174788034396460" + ], + "answer": "1174788034396460", + "correct_value": 1174788034396460, + "explanation": "Product has 16 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "69904315 × 67466244", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 69904315 × 67466244?", + "options": [ + "4716181572442861", + "4716181572442840", + "4716181572442859", + "4716181572442860" + ], + "answer": "4716181572442860", + "correct_value": 4716181572442860, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "14777953 × 69074935", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 14777953 × 69074935?", + "options": [ + "1077399442908055", + "1020786142908055", + "1100041142908055", + "1078955742908055" + ], + "answer": "1020786142908055", + "correct_value": 1020786142908055, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001189", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "59194305 × 34711839", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 59194305 × 34711839?", + "options": [ + "2054743184876897", + "2054743184876895", + "2054743184876896", + "2054743184876894" + ], + "answer": "2054743184876895", + "correct_value": 2054743184876895, + "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": "73484870 × 13084906", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 73484870 × 13084906?", + "options": [ + "961542616372230", + "961542616372221", + "961542616372220", + "961542616372219" + ], + "answer": "961542616372220", + "correct_value": 961542616372220, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "59623463 × 31713051", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 59623463 × 31713051?", + "options": [ + "1635139322915613", + "2101410522915613", + "1842388522915613", + "1890841922915613" + ], + "answer": "1890841922915613", + "correct_value": 1890841922915613, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001190", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "56606951 × 63924755", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 56606951 × 63924755?", + "options": [ + "723717094794397", + "36185854739720085", + "36185854739720068", + "3618585473972005" + ], + "answer": "3618585473972005", + "correct_value": 3618585473972005, + "explanation": "Product has 16 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55585150 × 34013156", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 55585150 × 34013156?", + "options": [ + "1890626378233399", + "1890626378233401", + "1890626378233420", + "1890626378233400" + ], + "answer": "1890626378233400", + "correct_value": 1890626378233400, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "16315655 × 76055343", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 16315655 × 76055343?", + "options": [ + "1240892737294665", + "1074353137294665", + "1418829337294665", + "1295747137294665" + ], + "answer": "1240892737294665", + "correct_value": 1240892737294665, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001191", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "91153242 × 81002661", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 91153242 × 81002661?", + "options": [ + "7383655160776963", + "7383655160776964", + "7383655160776962", + "7383655160776961" + ], + "answer": "7383655160776962", + "correct_value": 7383655160776962, + "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": "31316160 × 16921372", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 31316160 × 16921372?", + "options": [ + "529912392971530", + "529912392971521", + "529912392971519", + "529912392971520" + ], + "answer": "529912392971520", + "correct_value": 529912392971520, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "65312241 × 39793837", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 65312241 × 39793837?", + "options": [ + "2715864972458717", + "2823081272458717", + "2770398572458717", + "2599024672458717" + ], + "answer": "2599024672458717", + "correct_value": 2599024672458717, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001192", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "21111248 × 32022333", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 21111248 × 32022333?", + "options": [ + "1419665968353322", + "676031413501584", + "1216856544302848", + "1352062827003164" + ], + "answer": "676031413501584", + "correct_value": 676031413501584, + "explanation": "Product has 15 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "24223720 × 25076580", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 24223720 × 25076580?", + "options": [ + "607448052477601", + "607448052477600", + "607448052477599", + "607448052477610" + ], + "answer": "607448052477600", + "correct_value": 607448052477600, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "66146945 × 23199367", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 66146945 × 23199367?", + "options": [ + "1526911152983815", + "1443154752983815", + "1534567252983815", + "1290809752983815" + ], + "answer": "1534567252983815", + "correct_value": 1534567252983815, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001193", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52936378 × 69124950", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 52936378 × 69124950?", + "options": [ + "365922448243110", + "36592244824311049", + "3659224482431100", + "731844896486220" + ], + "answer": "3659224482431100", + "correct_value": 3659224482431100, + "explanation": "Product has 16 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "85240070 × 45355772", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 85240070 × 45355772?", + "options": [ + "3866129180184041", + "3866129180184050", + "3866129180184039", + "3866129180184040" + ], + "answer": "3866129180184040", + "correct_value": 3866129180184040, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "25015959 × 14337455", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 25015959 × 14337455?", + "options": [ + "320321226444345", + "378443166444345", + "413560206444345", + "358665186444345" + ], + "answer": "358665186444345", + "correct_value": 358665186444345, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001194", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "30930000 × 50580940", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 30930000 × 50580940?", + "options": [ + "1564468474200003", + "1564468474199999", + "1564468474200000", + "1564468474200001" + ], + "answer": "1564468474200000", + "correct_value": 1564468474200000, + "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": "73717460 × 65235716", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 73717460 × 65235716?", + "options": [ + "4809011284801360", + "4809011284801359", + "4809011284801350", + "4809011284801361" + ], + "answer": "4809011284801360", + "correct_value": 4809011284801360, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "56442073 × 89433661", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 56442073 × 89433661?", + "options": [ + "5727798322819253", + "5047821222819253", + "4452632322819253", + "4463803422819253" + ], + "answer": "5047821222819253", + "correct_value": 5047821222819253, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001195", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "17607481 × 52652963", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 17607481 × 52652963?", + "options": [ + "927086045616204", + "927086045616202", + "927086045616206", + "927086045616203" + ], + "answer": "927086045616203", + "correct_value": 927086045616203, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "29093840 × 83919164", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 29093840 × 83919164?", + "options": [ + "2441530730349760", + "2441530730349759", + "2441530730349780", + "2441530730349761" + ], + "answer": "2441530730349760", + "correct_value": 2441530730349760, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "85367871 × 50914141", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 85367871 × 50914141?", + "options": [ + "4420724920963811", + "4179783420963811", + "4550365120963811", + "4346431820963811" + ], + "answer": "4346431820963811", + "correct_value": 4346431820963811, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001196", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "94204820 × 11312368", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 94204820 × 11312368?", + "options": [ + "937798040268104", + "1065679591213760", + "213135918242752", + "969768428004518" + ], + "answer": "1065679591213760", + "correct_value": 1065679591213760, + "explanation": "Product has 16 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "77811775 × 63060270", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 77811775 × 63060270?", + "options": [ + "4906831540679250", + "4906831540679249", + "4906831540679251", + "4906831540679260" + ], + "answer": "4906831540679250", + "correct_value": 4906831540679250, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "29510525 × 96960825", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 29510525 × 96960825?", + "options": [ + "2294450550183125", + "2861364850183125", + "2911598450183125", + "2659846350183125" + ], + "answer": "2861364850183125", + "correct_value": 2861364850183125, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001197", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "38026899 × 90054365", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 38026899 × 90054365?", + "options": [ + "3424488242364136", + "3424488242364138", + "3424488242364135", + "3424488242364134" + ], + "answer": "3424488242364135", + "correct_value": 3424488242364135, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "51770130 × 24015880", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 51770130 × 24015880?", + "options": [ + "1243305229664380", + "1243305229664399", + "1243305229664401", + "1243305229664400" + ], + "answer": "1243305229664400", + "correct_value": 1243305229664400, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "52494757 × 49426051", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 52494757 × 49426051?", + "options": [ + "2754642836714607", + "2937125136714607", + "2441626436714607", + "2594608536714607" + ], + "answer": "2594608536714607", + "correct_value": 2594608536714607, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001198", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "13850517 × 19419355", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 13850517 × 19419355?", + "options": [ + "268968106556536", + "268968106556534", + "268968106556535", + "268968106556538" + ], + "answer": "268968106556535", + "correct_value": 268968106556535, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99063525 × 73147670", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 99063525 × 73147670?", + "options": [ + "7246266035736770", + "7246266035736749", + "7246266035736751", + "7246266035736750" + ], + "answer": "7246266035736750", + "correct_value": 7246266035736750, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "81831971 × 59299865", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 81831971 × 59299865?", + "options": [ + "4771375132983915", + "5027895132983915", + "4251368432983915", + "4852624832983915" + ], + "answer": "4852624832983915", + "correct_value": 4852624832983915, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001199", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "19489611 × 73995048", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 19489611 × 73995048?", + "options": [ + "1442134701446329", + "1442134701446327", + "1442134701446330", + "1442134701446328" + ], + "answer": "1442134701446328", + "correct_value": 1442134701446328, + "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": "76620785 × 20191316", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 76620785 × 20191316?", + "options": [ + "1547074482103061", + "1547074482103070", + "1547074482103060", + "1547074482103059" + ], + "answer": "1547074482103060", + "correct_value": 1547074482103060, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "25754531 × 93632669", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 25754531 × 93632669?", + "options": [ + "2463488076373239", + "2411465476373239", + "2306625576373239", + "2274931276373239" + ], + "answer": "2411465476373239", + "correct_value": 2411465476373239, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001200", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "22260663 × 22311249", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 22260663 × 22311249?", + "options": [ + "496663195098087", + "496663195098088", + "496663195098086", + "496663195098089" + ], + "answer": "496663195098087", + "correct_value": 496663195098087, + "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": "32833470 × 65143968", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 32833470 × 65143968?", + "options": [ + "2138902519008970", + "2138902519008959", + "2138902519008961", + "2138902519008960" + ], + "answer": "2138902519008960", + "correct_value": 2138902519008960, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "92849641 × 49492159", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 92849641 × 49492159?", + "options": [ + "3875076395464919", + "4172036995464919", + "5393013495464919", + "4595329195464919" + ], + "answer": "4595329195464919", + "correct_value": 4595329195464919, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001201", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "35724479 × 11095443", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 35724479 × 11095443?", + "options": [ + "396378920449200", + "396378920449197", + "396378920449198", + "396378920449196" + ], + "answer": "396378920449197", + "correct_value": 396378920449197, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99861710 × 97437376", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 99861710 × 97437376?", + "options": [ + "9730262985272970", + "9730262985272959", + "9730262985272960", + "9730262985272961" + ], + "answer": "9730262985272960", + "correct_value": 9730262985272960, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "33104859 × 12119753", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 33104859 × 12119753?", + "options": [ + "381504924179827", + "452913514179827", + "376378454179827", + "401222714179827" + ], + "answer": "401222714179827", + "correct_value": 401222714179827, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + } +] \ No newline at end of file