Hint: Suppose the bus number is 21. Then person A could be aged 90 and still the first part of the conversation would make sense:

A. The sum of my children's ages is the number of the bus, and their product is my own age.
B. Perhaps if you told me how many children you have and also your age, I could work out their ages?
A. No, that would not be possible.

This is because Person A might have 4 children aged 1,1,9,10 or 1,2,3,15. So the children's ages are not uniquely determined. However, Person A could also be aged 96 because he might have 3 children aged 1,8,12, or 2,3,16. That means the last line doesn't make sense:

B. Aha! Now I can work out your age.

And so the bus number cannot be 21.

Adding more children of ages 1 to the above argument, it is easy to show the bus number cannot be any larger than 21.

How far can you push this argument?

Complete Solution