Puzzle: A man in my neighborhood has three daughters. One day when I asked their ages he said, “The product of their ages is 36.”

When I still couldn’t find their ages he said, “Ok. I’ll give you another clue: the sum of their ages is same as the number of my house.”

I knew the number but still couldn’t calculate their ages. So the man gave me a last clue, “My eldest daughter lives upstairs.”

Finally I was able to figure out their ages. How old are they?

Hint: There are only so many ways that three numbers can have a product of 36.

Answer: The ages are 2, 2, and 9.

Solution: There are only so many ways that three numbers can have a product of 36:

1 × 1 × 36 = 36

1 × 2 × 18 = 36

1 × 3 × 12 = 36

1 × 4 × 9 = 36

1 × 6 × 6 = 36

2 × 2 × 9 = 36

2 × 3 × 6 = 36

3 × 3 × 4 = 36

It’s got to be one of these, but we don’t know which one. The second clue is about the sum of their ages:

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

The narrator knew the number of the man’s house, so the only way that this clue didn’t help him is if the sum is 13, in which case it’s still ambiguous. So we know the ages are either 1, 6, and 6, or 2, 2, and 9.

The third clue makes a reference to the oldest daughter, so the ages must be 2, 2, and 9.

(It’s true that there could still be an “oldest” daughter in the case of 1, 6, and 6, since they could be 10 months apart and both be 6. But since the man gave this as a clue, we can assume that it’s suppose to push us towards the 2, 2, and 9 solution.)