Three daughters and a house number
A man tells a friend that the ages of his three daughters, in whole years, multiply to 72. The friend looks at the house number in front of them, which the man says is the sum of the ages, and replies that this is still not enough to work out the ages. The man adds: “My eldest daughter loves chess.” (So there is exactly one eldest.)
How old is the eldest daughter?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
List all the ways to write 72 as a product of three whole numbers, and the sum of each.
The friend knows the sum and still cannot decide. So the sum must appear more than once in your list.
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The friend's failure tells us the house number is a sum shared by two possible triples. The last remark picks one of the two.
Step 1: the triples. The triples with product 72 (in increasing order) and their sums are:
- 1, 1, 72: 74; 1, 2, 36: 39; 1, 3, 24: 28; 1, 4, 18: 23;
- 1, 6, 12: 19; 1, 8, 9: 18; 2, 2, 18: 22; 2, 3, 12: 17;
- 2, 4, 9: 15; 2, 6, 6: 14; 3, 3, 8: 14; 3, 4, 6: 13.
Step 2: the ambiguous sum. Only one sum appears twice: 14, for 2, 6, 6 and 3, 3, 8. Since the friend could not decide, the house number is 14.
Step 3: one eldest. The triple 2, 6, 6 has two eldest daughters (twins), so the ages are 3, 3 and 8, and the eldest is 8.
A variant. With product 36, the ambiguous sum is 13 (1, 6, 6 and 2, 2, 9), and the answer is 9.