12 球找次品
Defective Ball
题目详情
有 12 个外观相同的球，其中 1 个是次品，可能更重也可能更轻（未知）。你有一架天平，只能比较哪边更重。
如何用 3 次称重找出次品并判断它更重还是更轻？
You have 12 identical balls. One is defective, but it could be heavier or lighter; you do not know which. You have a balance scale that only tells you which side is heavier. How do you find the defective ball in 3 weighings?
解析
这是经典 12 球问题。
做法要点：
- 第一次称 4 vs 4，把情况分成“平衡”和“不平衡”两大类；
- 后两次称重通过安排球的交换位置，并用已知正常球作参照，在 3 次内把候选缩小到 1 个，同时确定是重还是轻。
完整方案有固定表格版流程（常见标准解）。
Original Explanation
A classic puzzle. Label the 12 balls 1 through 12, then:
- Weigh (1,2,3,4) vs (5,6,7,8).
- Depending on whether they balance or tilt, you deduce which subset (of size 4) contains the defective ball and whether it is heavier or lighter.
- Weigh combinations to isolate the single defective ball in a total of 3 weighings.
By carefully arranging which balls go on each side in the second and third weighings (including using known “normal” balls), you can pinpoint exactly which ball is defective and whether it is heavier or lighter.