毒酒桶 I

Poisoned Kegs I

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

题目详情

国王有 8 桶酒，其中 1 桶有毒。任何人只要喝到毒桶酒，就会在 1 个月后准时死亡。

有 3 名仆人自愿试酒。问：最少需要多少个月才能确定哪一桶有毒？

A king has 8 kegs of liquor. 1 of the kegs is poisonous. If someone drinks any amount of liquor from the poisoned keg, they will die in exactly 1 month. 3 servants have volunteered to risk their lives for the king and test the kegs. What is the minimum number of months needed before the poisoned keg can be identified?

解析

把 3 名仆人的生死看作 3 位二进制指示器，因此可区分 $2^3=8$ 桶。

只需 1 个月：按二进制给每名仆人分配要尝的桶，1 个月后根据死亡模式唯一确定毒桶。

Original Explanation

Each patient can serve as a binary indicator for each keg. Hence, only 1 month is needed to determine the poisoned keg. As an example, suppose our kegs are labeled $K_1, K_2, \ldots, K_8$ . Then, we could have servant 1 try $K_1, K_3, K_5, K_7$ , servant 2 try $K_2, K_3, K_6, K_7$ , and servant 3 try $K_4, K_5, K_6, K_7$ . Each possible configuration of how the 3 servants die corresponds to one possible poisoned keg.