Solved by Daniel Giaimo

Question:

Two men are talking. One says to the other, "I have three sons whose ages I want you to figure out from the following clues. Stop me when you know their ages.

"1. The sum of their ages is thirteen.
"2. The product of their ages is the same as your age.
"3. My oldest son weighs sixty-one pounds."

"Stop," says the second man. "I know their ages."

What are their ages?

Answer:

The first clue implies that the three ages add to 13. The only triplets of integers that add to 13 are:

     (1,1,11), (1,2,10), (1,3,9), (1,4,8), (1,5,7), (1,6,6), (2,2,9), (2,3,8), (2,4,7), (2,5,6), (3,3,7), (3,4,6), (3,5,5), (4,4,5)

Because the second man didn't get the answer after the second clue, at least 2 of these triplets must have the same product. The only two that have the same product are:

     (1,6,6), (2,2,9)

Because the third clue mentions an oldest child, the answer must be*:

     (2,2,9)

Actually, the answer could still be 1,6,6 since the two six-year olds were most likely not born at the same instant." This is technically true, but it's unlikely that a father would refer to one twin as his "oldest child"

-- Daniel Giaimo