Two mathematicians, Tom and Smith are walking down the street.

Tom: I know you have three sons. What are their ages?
Smith: The product of their ages is 36.
Tom: I can’t tell their ages from that.
Smith: Well, the sum of their ages is the same as that address across the street.
Tom: I still can’t tell.
Smith: The eldest is visiting his grandfather today.
Tom: Now I know their ages.

Do you know the age of the kids?

Answer

The eldest is 9 years old and the 2 younger ones are 2 years old.

Let’s break it down. The product of their ages is 36. So the possible choices are:

1,1,36 – sum(1,1,36) = 38
1,6,6 – sum(1,6,6) = 13
1,2,18 – sum(1,2,18) = 21
1,3,12 – sum(1,3,12) = 16
1,4,9 – sum(1,4,9) = 14
2,2,9 – sum(2,2,9) = 13
2,3,6 – sum(2,3,6) = 11
3,3,4 – sum(3,3,4) = 10

Six of the sums are unique, so if it were one of those, Tom would have recognised the number across the street that matches and he would know the answer, but he could not figure out the answer. This means there are two or more combinations with the same sum. From the choices above, only two of them are possible now.

1,6,6 – sum(1,6,6) = 13
2,2,9 – sum(2,2,9) = 13

When Tom heard that the eldest is visiting his grandfather, we can eliminate combination 1 since there are two eldest ones. This leaves us with only 1 option left, that is 2, 2 and 9.

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