The Penguin Book of Curious and Interesting Numbers: Revised Edition (Penguin Press Science) [Paperback]
David Wells (Author)
and other books by him.
June 7: Here is a puzzle with a solution by Dirac.
5 sailors are shipwrecked on a South Sea Island, and before they go to sleep they collect a pile of bananas to share out in the morning. One of then wakes up, does not trust the others, so divides the pile into 5 equal parts, with one left over, which he gives to the monkey, and then takes his own part for himself. This process repeats with the other sailors. In the morning, they find the pile divides into 5 equal parts, with one left for the monkey.
Problem: How many bananas were there in the initial pile?
Easy solution by Dirac: Start with $-4$ bananas. With one for the monkey, that makes $-5$ which divides into $5$ equal parts; take away $-1$ for the sailor leaves $-4$, as before. So the process repeats. To get a positive solution, add $5^5$. (No wonder the problem was difficult to solve directly!)
As an exercise for students, you can ask them to generalise the problem, since that is what mathematicians do.