Age of 3 children – Mathematical Puzzle

Problem:
Two old friends, Jack and Bill, meet after a long time.
Jack: Hey, how are you man?
Bill: Not bad, got married and I have three kids now.
Jack: That’s awesome. How old are they?
Bill: The product of their ages is 72 and the sum of their ages is the same as your birth date.
Jack: Cool… But I still don’t know.
Bill: My eldest kid just started taking piano lessons.
Jack: Oh now I get it.

How old are Bill’s kids?

Solution:

This is a very good logical problem. To do it, first write down all the real possibilities that the number on that building might have been. Assuming integer ages one get get the following which equal 72 when multiplied:

2, 2, 18 – sum = 22
2, 4, 9 – sum = 15
2, 6, 6 – sum = 14
2, 3, 12 – sum = 17
3, 4, 6 – sum = 13
3, 3, 8 – sum = 14
1, 8, 9 – sum = 18
1, 3, 24 – sum = 28
1, 4, 18 – sum = 23
1, 2, 36 – sum = 39
1, 6, 12 – sum = 19

The sum of their ages is the same as your birth date. That could be anything from 1 to 31 but the fact that Jack was unable to find out the ages, it means there are two or more combinations with the same sum. From the choices above, only two of them are possible now. For any other number, the answer is unique and the Jack would have known after the second clue. So he asked for a third clue. The clue that the eldest kid just started taking piano lessons is really just saying that there is an “oldest”, meaning that the younger two are not twins.

2, 6, 6 – sum(2, 6, 6) = 14
3, 3, 8 – sum(3, 3, 8 ) = 14

Hence, the answer is that the elder is 8 years old, and the younger two are both 3 years old.
The answer is 3, 3 and 8.

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