返回题库

12 球找异球

how can you determine which ball is the defective one with 3 measurements?

专题
Brainteaser / 脑筋急转弯
难度
L6

题目详情

有 12 个看起来完全相同的球，其中 1 个比其他球重或轻（你不知道是重还是轻）。

用一架只会告诉你哪边更重（或平衡）的天平，如何在 3 次称重内找出异球并判断它是重还是轻？

英文原题

You have 12 identical balls. One of the balls is heavier OR lighter

than the rest (you don't know which). Using just a balance that can only show you which side of the tray is heavier, how can you determine which ball is the defective one with 3 measurements?

解析

给球编号 1..12。

第 1 次称重：称 1,2,3,41,2,3,4（左）对 5,6,7,85,6,7,8（右）。

分三种情况：

A. 第 1 次平衡：异球在 9,10,11,129,10,11,12 中。

  • 第 2 次：称 9,10,119,10,11（左）对 1,2,31,2,3（右，已知正常）。
    • 若平衡：异球是 12。第 3 次称 1212 对 11，即可判断重/轻。
    • 若左重：异球在 9,10,119,10,11 中且为“重”。第 3 次称 99 对 1010：
      • 哪个更重就是异球；若平衡则 11 重。
    • 若右重：异球在 9,10,119,10,11 中且为“轻”。第 3 次称 99 对 1010：
      • 哪个更轻就是异球；若平衡则 11 轻。

B. 第 1 次左重：异球要么在 1,2,3,41,2,3,4 中且为重，要么在 5,6,7,85,6,7,8 中且为轻。

  • 第 2 次：称 1,2,51,2,5（左）对 3,6,93,6,9（右，9 为正常球）。
    • 若左重：异球为 11 重 或 22 重 或 66 轻。
      • 第 3 次称 11 对 22：
        • 哪个更重即该球重；若平衡则 66 轻。
    • 若右重：异球为 33 重 或 55 轻。
      • 第 3 次称 33 对 99：若更重则 33 重，否则 55 轻。
    • 若平衡：异球为 44 重 或 77 轻 或 88 轻。
      • 第 3 次称 77 对 88：哪边更轻则该球轻；若平衡则 44 重。

C. 第 1 次右重：与 B 情况对称（把左右互换即可）。

该流程保证 3 次称重内确定异球编号并判断其重/轻。

英文解析

Number the balls 1 through 12.

First Weighing: Weigh {1,2,3,4}\{1,2,3,4\} (Left) against {5,6,7,8}\{5,6,7,8\} (Right).

There are three cases:

A. First Weighing is Balanced: The odd ball is among {9,10,11,12}\{9,10,11,12\}.

  • Second Weighing: Weigh {9,10,11}\{9,10,11\} (Left) against {1,2,3}\{1,2,3\} (Right, known good).
    • If Balanced: The odd ball is 12. Third Weighing: Weigh 12 against 1 to determine if it is heavy or light.
    • If Left is Heavy: The odd ball is among {9,10,11}\{9,10,11\} and is Heavy. Third Weighing: Weigh 9 against 10:
      • The heavier one is the odd ball; if balanced, 11 is Heavy.
    • If Right is Heavy: The odd ball is among {9,10,11}\{9,10,11\} and is Light. Third Weighing: Weigh 9 against 10:
      • The lighter one is the odd ball; if balanced, 11 is Light.

B. First Weighing is Left Heavy: The odd ball is either among {1,2,3,4}\{1,2,3,4\} and is Heavy, or among {5,6,7,8}\{5,6,7,8\} and is Light.

  • Second Weighing: Weigh {1,2,5}\{1,2,5\} (Left) against {3,6,9}\{3,6,9\} (Right, where 9 is a known good ball).
    • If Left is Heavy: The odd ball is either 1 Heavy, 2 Heavy, or 6 Light.
      • Third Weighing: Weigh 1 against 2:
        • The heavier one is the odd ball; if balanced, 6 is Light.
    • If Right is Heavy: The odd ball is either 3 Heavy or 5 Light.
      • Third Weighing: Weigh 3 against 9: If 3 is heavier, it is 3 Heavy; otherwise, 5 is Light.
    • If Balanced: The odd ball is either 4 Heavy or 7 Light or 8 Light.
      • Third Weighing: Weigh 7 against 8: The lighter side indicates the odd ball is Light; if balanced, 4 is Heavy.

C. First Weighing is Right Heavy: Symmetric to Case B (swap Left and Right).

This procedure guarantees identifying the odd ball's number and determining if it is heavy or light within three weighings.