Infinitely many prisoners, invisible hats, and a strategy that shouldn't exist
Imagine infinitely many prisoners, one for every whole number, each wearing a red or blue hat they cannot see. Each prisoner can see every other prisoner's hat but not their own, and all must simultaneously guess their own hat's colour with no communication once the hats are placed. Is there any strategy the prisoners can agree on in advance that guarantees only finitely many of them guess wrong?
Reveal the answer
Astonishingly, yes. Using the Axiom of Choice, the prisoners agree in advance to sort every possible infinite hat-sequence into groups, where two sequences belong to the same group if they differ in only finitely many places, then each prisoner commits to a single representative sequence for their group. Guessing as if the real sequence matched that representative guarantees that all but finitely many guesses are correct, even though no one saw their own hat and no information was ever exchanged.