Puzzle:

A countable infinity of prisoners are placed in a room so that they can all see each other, but are not allowed to communicate in any way and cannot see their own heads. The warden places on the head of each prisoner a red hat or a black hat. The prisoners will each guess the color of their own hat. They will all be released if at most finitely many of them guess incorrectly, and they will all be killed otherwise. The prisoners know all of this, and may collude beforehand. The prisoners are all distinguishable - think of them as being numbered 1,2,3,.... Again, once the warden has placed the hats, the prisoners receive no information other than the color of their fellow prisoners' hats. Prove that there is a strategy that guarantees a win for the prisoners.

(On my honor, this is possible.)