John Allen Paulos at ABC News:
Here's the situation. Three people enter a room sequentially and a red or a blue hat is placed on each of their heads depending upon whether a coin lands heads or tails.
Once in the room, they can see the hat color of each of the other two people but not their own hat color. They can't communicate with each other in any way, but each has the option of guessing the color of his or her own hat. If at least one person guesses right and no one guesses wrong, they'll each win a million dollars. If no one guesses correctly or at least one person guesses wrong, they win nothing.
The three people are allowed to confer about a possible strategy before entering the room, however. They may decide, for example, that only one designated person will guess his own hat color and the other two will remain silent, a strategy that will result in a 50 percent chance of winning the money. Can they come up with a strategy that works more frequently?
Most observers think that this is impossible because the hat colors are independent of each other and none of the three people can learn anything about his or her hat color by looking at the hat colors of the others. Any guess is as likely to be wrong as right.
Is there a strategy the group can follow that results in its winning the money more than 50 percent of the time? The solution and a discussion are below, but you might want to think about the problem before reading on.
More here. [Thanks to Tunku Varadarajan.]