AGES OF MY CHILDREN

First factorize 72 to give 2, 2, 2, 3, 3. You are now in a position to work out all the possible ages of the children. You should allow for a child aged 1 and all possible combinations, however unlikely. At the same time work out the sums of each of the possible three ages. Summarised in this table:

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

The bartender is told that the sum of the ages is on the front door of the bar. He replies "That is not enough information". Now, the bartender will know what this number is (even if we don't) so we look at our table for two sets of three ages with the same sum. There is only one such pair:-

2 6 6 14
3 3 8 14

The final piece of information is that the man has a 'youngest' child. This excludes the twins aged 3 with elder child aged 8 option. So, the ages of the 3 children are 2, 6 and 6. (And the number on the door is 14.)