This is one of my favorite puzzles, both because the answer isn't easy, and because the setup is very similar to another problem a lot of people know.
You and two other prisoners are set to be executed. You're given one chance to get out, by some sadistic game invented by your captors. (Isn't that how it always is?)
You're each privately given a hat, which is either red or blue (randomly chosen, independent of the hats chosen for the other people). You can't see your own hat. Once you're all wearing the hats, you're ushered into a common room where you can see the other two people and their hats. Without being allowed to communicate, you're then sent back to your solitary confinement and your hats are removed.
Each of you is then given a piece of paper to write down your hat color. You may either write "red", write "blue", or leave the paper blank. If any of you guess wrong, or if you all leave your papers blank, then you're all executed. But if even one of you guesses right, without any wrong guesses, you're all set free.
If you are allowed to discuss strategy among the three of you before the hats are handed out, but you're allowed no communication during or afterwards, what is the optimal strategy? In other words, what should you do such that you all have the best chance of survival? (The answer is "legit", assuming no wording loopholes, no color-blindness, no sympathetic guards, and nothing else like that.)