# 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.

### 2 Thoughts on “Age of 3 children – Mathematical Puzzle”

1. unknown on April 12, 2017 at 3:14 pm said:

Nice puzzle
Now i got it

2. Riya Ojha on August 16, 2017 at 8:42 pm said:

for 72
The prime factors of 72 are 2, 2, 2, 3, 3; in other words, 2 × 2 × 2 × 3 × 3 = 72

This gives the following triplets of possible solutions;

Age one Age two Age three Total (Sum)
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
Because the census taker said that knowing the total (from the number on the gate) did not help, we know that knowing the sum of the ages does not give a definitive answer; thus, there must be more than one solution with the same total.

Only two sets of possible ages add up to the same totals:

A. 2 + 6 + 6 = 14

B. 3 + 3 + 8 = 14

In case ‘A’, there is no ‘eldest child’ – two children are aged six (although one could be a few minutes or around 9 to 12 months older and they still both be 6). Therefore, when told that one child is the eldest, the census-taker concludes that the correct solution is ‘B’.[2]

for 36
The prime factors of 36 are 2, 2, 3, 3 This gives the following triplets of possible solutions;

Age one Age two Age three Total (Sum)
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
3 3 4 10
Using the same argument as before it becomes clear that the number on the gate is 13, and the ages 9, 2 and 2.