孩子年龄

What are the children's ages?

专题: Brainteaser / 脑筋急转弯
难度: L2
来源: QuantQuestion

个人笔记

题目详情

一位人口普查员敲门，一位女士说她有三个孩子。普查员问年龄，女士说三人的年龄乘积是 36。

普查员说信息不够，女士补充：三人年龄之和等于隔壁楼的门牌号。普查员去看了门牌号，回来后仍说信息不够。

女士说不再回答更多问题，因为她最大（最年长）的孩子正在楼上睡觉，不想吵醒他。

问：三个孩子的年龄分别是多少？

英文原题

A census taker goes to a house,a woman answers the door and says she has three children. The census taker asks their ages and she says that if you multiply their ages, the result is 36. He says he needs more info so she tells him that the total of their ages is the address of the building next door. He goes and looks, then comes back and says he still needs more information. She tells him that shewon't answer any more questions because her eldest child is sleeping upstairs and she doesn't want to wake him. What are the children's ages?

解析

列出三整数乘积为 36 的所有无序三元组及其和：

$(1,1,36)$ 和为 38
$(1,2,18)$ 和为 21
$(1,3,12)$ 和为 16
$(1,4,9)$ 和为 14
$(1,6,6)$ 和为 13
$(2,2,9)$ 和为 13
$(2,3,6)$ 和为 11
$(3,3,4)$ 和为 10

普查员看了门牌号（即总和）仍不够，说明该和对应至少两组。只有和为 13 对应两组： $(1,6,6)$ 与 $(2,2,9)$ 。

女士说“最大孩子（eldest）在睡觉”，暗示最大年龄是唯一的（不是并列最大），因此排除 $(1,6,6)$ 。

所以年龄为 $\boxed{2,2,9}$ 。

英文解析

List all unordered triplets of three integers whose product is 36 and their sums:

- $(1,1,36)$ sums to 38
- $(1,2,18)$ sums to 21
- $(1,3,12)$ sums to 16
- $(1,4,9)$ sums to 14
- $(1,6,6)$ sums to 13
- $(2,2,9)$ sums to 13
- $(2,3,6)$ sums to 11
- $(3,3,4)$ sums to 10

The enumerator looked at the house number (i.e., the sum) but still wasn't sure, indicating that the sum corresponds to at least two groups. Only the sum of 13 corresponds to two groups: $(1,6,6)$ and $(2,2,9)$ .

The lady said, "The eldest child is sleeping," implying that the maximum age is unique (not tied), thus excluding $(1,6,6)$ .

Therefore, the ages are $\boxed{2,2,9}$ .