[
  {
    "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"
      ]
    }
  }
]