This is a puzzle rather than a theorem, but I think it fits in this wonderful list:

Conway’s Wizards, as discussed here by Tanya Khovanova.

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

A: “I have a positive integral number of children, whose ages are positive integers, the sum of which is the number of this bus, while the product is my own age.”

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

A: “No.”

B: “Aha! AT LAST I know how old you are!”

Now what was the number of the bus?