The Ages of Three Children

Two old friends meet. One asks the other the ages of his three children. “The product of their ages is $36$,” he says. “That doesn’t tell me enough,” says the first. “Their sum is the number on that house across the road.” The first looks, thinks, and says “I still can’t tell.” “Ah,” says the father, “the eldest is learning the piano.” At once the first friend knows all three ages. What are they?

Solution

List the ways three whole-number ages can multiply to $36$, with the sum of each: $$\begin{array}{lc} (1,1,36)\!: 38 & (1,2,18)\!: 21 \\ (1,3,12)\!: 16 & (1,4,9)\!: 14 \\ (1,6,6)\!: 13 & (2,2,9)\!: 13 \\ (2,3,6)\!: 11 & (3,3,4)\!: 10 \end{array}$$ The first friend can see the house number, that is, the sum, yet still cannot decide. So the sum must fail to pick out a single line, which happens for one value only: the sum $13$, shared by $(1,6,6)$ and $(2,2,9)$. Every other sum appears just once and would have given the answer away.

The deciding clue is that there is an eldest child. Of the two possibilities, $(1,6,6)$ has no single eldest, its two older children being the same age, while $(2,2,9)$ does. So the ages are $2$, $2$ and $9$. The remark about the piano carries no information of its own; it matters only because the word “eldest” tells us one child stands alone at the top.