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盲人分硬币

Coin Piles

专题
Probability / 概率
难度
L2

题目详情

你被蒙眼，地上有 1000 枚硬币，其中 980 枚反面朝上，20 枚正面朝上。

你无法通过触摸判断某枚硬币朝向，但你可以随意翻硬币。

如何把硬币分成两堆，使两堆中正面朝上的硬币数量相同？

You are blindfolded in a room with 1000 coins on the floor, 980 of which are tails-up, 20 of which are heads-up. You cannot feel which side is up by touch, but you can flip any coins you want. How can you separate the coins into two piles that each contain the same number of heads-up coins?

解析

随便拿出 20 枚硬币作为 A 堆，并把 A 堆 20 枚全部翻面；其余 980 枚作为 B 堆。

设 A 堆原本有 mm 枚正面，则 B 堆原本有 20−m20-m 枚正面（总正面为 20）。翻转 A 堆后，A 堆正面数变为 20−m20-m，与 B 堆相同。

Original Explanation

Take any 20 coins as one pile and flip them all. Leave the other 980 coins as the second pile. Both piles will have the same number of heads.

Reason: Suppose the chosen 20 coins had mm heads initially. Then the other pile has (20−m)(20 - m) heads (because total heads is 20). Flipping those 20 coins turns the mm heads into mm tails, and (20−m)(20 - m) tails into (20−m)(20 - m) heads. Now both piles have (20−m)(20 - m) heads.