The Three Daughters Puzzle

Published on 2025-02-28

Statement

A salesman is at the door of a family house and tries to convince the mother to buy some of his products. She says: "Well, I don't really need the products, but if you can guess how old my three daughters are I will do you a favor and buy one of them." The salesman asks the woman for a little help and she tells him: "The product of their ages is 36." As the salesman is not able to figure out the ages, she gives him another clue: "The sum of their ages is same as the number of my house." The salesman calculates for a little while but then says: "I'm sorry, Miss, but I still can't figure out their ages." The mother agrees to give him a final hint: "My eldest daughter plays piano." After that hint the salesman was able to tell the mother her daughters ages and sell one of his items.

How did he figure that out?

Solution

Many people get the initial impression that this puzzle is impossible to solve, as they feel that the problem fails to provide useful information, like the fact that the sum of the ages is an unknown number or the fact that the oldest daughter plays the piano, something which seems completely irrelevant to be able to figure out the ages. Still, before throwing the towel, let's try to make as much progress as we can with the information we do have.

The first key piece of information is that there are three daughters. We'll use three unknowns $d_{1}, d_{2}$ and $d_{3}$ to represent their ages. Because we usually think of ages as a whole number of years, we'll have these unknowns be natural numbers:

d_{1}, d_{2}, d_{3} \in ℕ

It's often useful to define an ordering of the unknowns, since we don't lose generality by doing so and it ends up becoming an additional piece of information that could possibly come in handy later:

d_{1} \geq d_{2} \geq d_{3}

The second key piece of information is that the product of their ages equals $36$ :

d_{1} \cdot d_{2} \cdot d_{3} = 36

From here, nothing else the mother tells the salesman seems to be very useful, so it doesn't seem like the problem has a unique solution. Even so, let's try to calculate the set of possible solutions given the information we have so far.

Let's calculate the prime factorization of $36$ to see what integers can be multiplied together to get that result:

36 = 2 \cdot 2 \cdot 3 \cdot 3

From here, we can produce the following set of candidates. For simplification purposes, we'll use the format $(d_{1}, d_{2}, d_{3})$ :

(36, 1, 1)

(18, 2, 1)

(12, 3, 1)

(9, 4, 1)

(9, 2, 2)

(6, 6, 1)

(6, 3, 2)

(4, 3, 3)

Let's now revisit the remaining pieces of information that the mother gives the salesman. The next one was that the sum of the ages is equal to the number of her house. That in itself is not very useful since we don't know the number of the house. But the salesman does! And even equipped with that critical bit of information, he still wasn't able to calculate the ages and requested additional information. So let's try to figure out why that was the case by calculating the sum of the ages for each of the 8 possibilites above. We'll use the format $(d_{1}, d_{2}, d_{3}, d_{1} + d_{2} + d_{3})$ :

(36, 1, 1, 38)

(18, 2, 1, 21)

(12, 3, 1, 16)

(9, 4, 1, 14)

(9, 2, 2, 13)

(6, 6, 1, 13)

(6, 3, 2, 11)

(4, 3, 3, 10)

Notice that if the number of the house was $38$ , the salesman would have been able to correctly guess the ages. The same thing would have happened if the number of the house was $21$ , $16$ , $14$ , $11$ or $10$ . But since he still wasn't able to do it, it means the number of the house couldn't have been any of those, which leaves us with only two possibilities:

(9, 2, 2, 13)

(6, 6, 1, 13)

At this point, the mother gave the salesman the final cryptic hint: the eldest daughter plays the piano. The important part here is not that someone plays the piano, but that there is a single eldest daughter. This rules out the possibility where the eldest daughters are twins aged 6, leaving us with a single option:

(9, 2, 2, 13)

This means that the mother has an eldest daughter aged $9$ and younger twins aged $2$ .

Conclusion

Puzzles like this one stump a lot of people as it seems they don't give enough information to solve it. It is true that this puzzle gave some useless details, like the instrument that the eldest daughter plays, but it still gave enough information to solve it, it just purposely hid it in plain sight.

The are two important lessons to take away from this puzzle. The first one is that sometimes what's useful is not to know the value of an unknown, but the fact that knowing its value wouldn't make a difference anyway. The second lesson is to never give up. Even when we believe the problem doesn't provide enough information to completely solve it, there is often some progress to be made, and making such progress will often help us realize how to keep moving forward.