I’ve seen a number of “hat problem” logic puzzles, but this one I found the other day was new to me nonetheless. I’m stealing from http://www.relisoft.com/science/hats.html, where you can find a beautiful description of the answer.
Three people enter the room, each with a hat on their head. There are two colors of hats: red and blue; they are assigned randomly. Each person can see the hats of the two other people, but they can’t see their own hats. Each person can either try to guess the color of their own hat or pass. All three do it simultaneously, so there is no way to base their guesses on the guesses of others. If nobody guesses incorrectly and at least one person guesses correctly, they all share a big prize. Otherwise they all lose.
One more thing: before the contest, the three people have a meeting during which they decide their strategy. What is the best strategy?