Alice is telling Bob about her children. Alice says she has three daughters and that the sum of their ages is the number of the house across the street.
Bob asks what their ages are.
Instead of answering directly, Alice tells Bob that the product of their ages is 36.
Bob says that this is not enough information to determine their ages.
Alice agrees and adds that the oldest daughter has the beautiful blue eyes.
Bob now knows the ages of Alice’s daughters.
How old are Alice’s daughters?
The daughters are 2, 2 and 9 years old.
Proof
The possible ages that multiply to 36 are (along with their sums):
- 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)
Bob knows the sum of the ages because he can see the house number. Since knowing the sum is not enough to determine the ages, the sum must be one of the sums that is not unique.
The only sum that is not unique is 13, which is the sum of (1, 6, 6) and also (2, 2, 9).
Next Alice tells Bob about her oldest daughter. This means that the ages cannot be (1, 6, 6), since there is no oldest daughter in this case.
Hence the ages are 2, 2 and 9.