This week we take a look at a notoriously tricky math puzzle, one originally penned by English mathematician back in the 1960s. Apparently he was riding the bus home one evening when he encountered a pair of wizards.

### Problem

Last night I sat behind two wizards on a bus and overheard the following:

Blue Wizard: I have a positive integer number of children, whose ages are positive integers. The sum of their ages is the number of this bus, while the product is my own age.

Red Wizard: How interesting! Perhaps if you told me your age and the number of your children, I could work out their individual ages?

Blue Wizard: No, you could not.

Red Wizard: Aha! At last, I know how old you are!

Apparently the Red Wizard had been trying to determine the Blue Wizard's age for some time. Now, what was the number of the bus?

### Hint

Of course, the key here is that if the Blue Wizard *were *to tell the Red Wizard his age and how many children he has, then the Red Wizard would *not be able* to deduce the ages of the children. In other words, there must be multiple possible age combinations for the Blue Wizard's children that would result in the same age as well as the same bus number.

Here are the relevant variables: the **number of children (N), **the distribution of the **children's ages (C)**, the **age of the Blue Wizard (A)**, and the **b****us number (B)**, which is what we are trying to find.

We can assume the Red Wizard knows the number of the bus he is on and uses that information to determine the Blue Wizard's age. We don't know how many children the Blue Wizard has, but a good starting point might be to ask yourself if it's possible for him to only have one child. What about two?

There is only one possible number for the bus.

### Solution

Once you figure which bus the wizards take, you can check the solution here.

**See all of our riddles here.*