Two days' ago we posed the following puzzle:
A door-to-door salesman is trying to persuade a woman to buy his product. Eventually she says "I will buy one if you can answer this question. I have three daughters and you have not seen any of them. The product of their ages is 36 and the sum of their ages in the same as my house number. How old are they?"
The salesman immediately goes outside to find out the house number, and returns saying "It is not possible to find their ages from the information you have given me - I need something more."
She replies "You are correct, you do need more information my eldest daughter plays the piano".
The salesman then tells her the daughters' ages and makes the sale.
What are their ages, and incidentally what is the house number?
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There are eight combinations of ages with a product of 36:
36, 1, 1 (Sum = 38)
18, 2, 1 (Sum = 21)
12, 3, 1 (Sum = 16)
9, 4, 1 (Sum = 14)
9, 2, 2 (Sum = 13)
6, 6, 1 (Sum = 13)
6, 3, 2 (Sum = 11)
4, 3, 3 (Sum = 10)
If the sum of the ages is anything other than 13, the salesman would have known the answer when he saw the house number. But he needed more information, so the sum must be 13.
Knowing that the woman has an eldest daughter means that the daughters' ages must be 9, 2 and 2, and not 6, 6 and 2.