Ten prisoners are locked together in a room and told that tomorrow they'll all be placed in a line, each prisoner facing the back of the man in front of him. The guards will then put red and blue hats on the prisoners, so that each man can see the hats in front of him but not his own hat. Then, from the back of the line to the front, each prisoner is forced to guess the color of his own hat. Each one who guesses correctly is set free, while the rest go back to prison -- and, critically, each of the prisoners can hear the guesses of the people behind him. Is there a good strategy for the prisoners? \\ \noindent We'll examine this problem, a hat-guessing game that's really an error-correcting code in disguise, and finally a hat-guessing game that will make you wonder whether the axiom of choice is such a great idea. The ideas in this talk are simple, and I'll spend most of the time treating these games as riddles or puzzles rather than talking about the deeper theories hidden in their solutions.