What are the ages?

Question

Once, I met a man called Ibhan which asked me to guess the ages of his three daughters. He gave me a sheet of paper and kindly requested me, “Here, write your guess on that sheet.”. “I can’t guess, can you give me some hints?”, I suggested. The man then proudly spoke to me, “The product of my daughters’ ages is 36,” “And the sum of my daughters’ ages is the same as the number of my house,”, he uttered. I knew the number secretly, but I was still unable to guess. “And also, my eldest daughter can play the piano.”, proudly informed Ibhan. At last, I finally guessed them correctly! What are the ages? Hint: There are only so many ways that three numbers can have a product of 36 (4 × 3 × 3 is one of these).

Comments

The three daughters’ ages are 2, 2 and 9.

There are only so many ways (currently 8 unique number combinations of them) that three numbers can have a product of 36 (here are the examples of these):

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. So, the second hint refers to addition, right?

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 visitor knew the number of Ibhan’s house so that the only possible way that this clue didn’t aid him is if the sum is 13, in which case is 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 sister (who can play the piano)*, 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 supposed to push us towards the 2, 2, and 9 solution.