### Brain Teasers

# Bus Numbers

You are on a bus with just two other mathematicians when you overhear them talking.

The first mathematicians says, "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."

The other person replies, "How interesting! Perhaps if you told me your age and the number of your children, I could work out their individual ages?"

"No, you could not."

"Aha! At last, I know how old you are!"

Apparently the second mathematician had been trying to determine the first mathematician's age for some time. Now, what was the number of the bus?

The bus number is a positive integer and the mathematicians know the bus number.

The first mathematicians says, "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."

The other person replies, "How interesting! Perhaps if you told me your age and the number of your children, I could work out their individual ages?"

"No, you could not."

"Aha! At last, I know how old you are!"

Apparently the second mathematician had been trying to determine the first mathematician's age for some time. Now, what was the number of the bus?

The bus number is a positive integer and the mathematicians know the bus number.

### Hint

There are multiple possible age combinations for the mathematician's children that would result in the same age as well as the same bus number otherwise the second mathematician can determine the ages of the children given the first mathematician's age and number of children.Also, there must be at least three children because the second mathematician could work out the ages of the children if there are only one or two children.

### Answer

The bus number is 12. First, we know that the age of the first mathematician (A), the number of children (N), and the bus number (B) cannot be used by the second mathematician to determine the children's ages (C) based on the problem. At this point, we know that C must be at least 3; A, N, and B cannot be used to find C; and that there are multiple combinations for C. This is all the information that can be gathered from the problem, so we just plug in numbers for B and see if the results don't violate the parameters we have established. 1 and 2 are automatically ruled out because that would mean less than three children are present. If we take a random number like 5 for B, we then plot out all possible values for A, N, and C. The A and N must be between 1 and 5. The children's ages must be 1, 1, 1, 1, 1; 1, 1, 1, 2; 1, 1, 3; 1, 2, 2; 1, 4; or 5. However, the second mathematician can easily determine the ages of the children given A and N. This will be true for all of the numbers lower than the correct answer. As we work our way into higher bus numbers this uniqueness disappears, but it's replaced by another problem — the second mathematician must be able to deduce the first mathematician's age despite the ambiguity. For example, if the bus number is 21 and the first mathematician tells us that he's 96 years old and has three children, then it's true that we can't work out the children's ages: They might be 1, 8, and 12 or 2, 3, and 16. But when the mathematician informs us of this, we can't declare triumphantly that at last we know how old he is, because we don't — he might be 96, but he might also be 240, with children aged 4, 5, and 12 or 3, 8, and 10. After enough trial and error, the perfect balance between the two is reached with the number 12.Hide Hint Show Hint Hide Answer Show Answer

## What Next?

View a Similar Brain Teaser...

If you become a registered user you can vote on this brain teaser, keep track of which ones you have seen, and even make your own.

### Solve a Puzzle

Comments hidden to avoid spoilers.

## Follow Braingle!