找出假币袋

Counterfeit Coins I

专题: Brainteaser / 脑筋急转弯
难度: L4
来源: QuantQuestion

题目详情

有 10 袋硬币，每袋 10 枚。只有 1 袋是同重量的假币：要么每枚 9g，要么每枚 11g。真币每枚 10g。

你有电子秤（读数精确）。能否只称一次就找出哪一袋是假币？如果能，怎么做？

There are 10 bags, each with 10 coins. One bag has counterfeit coins, all of which weigh 9 g or 11 g. The real coins weigh 10 g. You have a digital scale that gives exact weight. Can you identify the counterfeit bag in one weighing?

解析

可以。

把袋子编号 1~10。从第 $i$ 袋取 $i$ 枚硬币混在一起称重。

若全是真币，期望重量为 $10\times(1+2+\cdots+10)=10\times55=550$ 克。

实际读数为 $550\pm k$ ：

若少了 $k$ 克，则第 $k$ 袋是 9g 假币袋；
若多了 $k$ 克，则第 $k$ 袋是 11g 假币袋。

Original Explanation

Yes. Label the bags 1 through 10. From bag $i$ , take $i$ coins. Combine them into one weighing. If the total is under by $k$ grams, the bag is the one you took $k$ coins from (the bag with 9 g coins). If the total is over by $k$ grams, that is also the bag (the bag with 11 g coins). For example, expected weight is $55 \times 10 = 550$ g. If the scale reads 548 g, that is 2 g under, so bag #2 has 9 g coins, etc.