Three logicians play a game with colored hats. (That's always how the trouble begins.)
Each logician wears a hat, and that hat has a 50% chance of being red and a 50% chance of being blue. The probabilities for each hat are independent of each other. None of the logicians is allowed to see the color of her own hat. They may make a game plan before starting, but during the game they may only look at the other hats without communicating.
And yet, the logicians guess the colors of their own hats.
They guess simultaneously, writing their guesses on paper which are only revealed after all guesses have been made. They may write "red", "blue", or "abstain". If at least one logician guesses right and none guess wrong, then they win. Abstaining counts as neither right nor wrong.
What strategy should the logicians use to have the highest probability of victory?
This puzzle was taken from Haidong. He also asks what happens if there are 2k - 1 logicians. The general case is rather difficult, but you should at least try for 7 logicians.