Three players enter a room and a red or blue hat is placed on each
person's head. The color of each hat is determined by a coin toss,
with the outcome of one coin toss having no effect on the others.
Each person can see the other players' hats but not his own.
No communication of any sort is allowed, except for an initial
strategy session before the game begins. Once they have had a
chance to look at the other hats, the players must simultaneously
guess the color of their own hats or pass. The group shares a
hypothetical $3 million prize if at least one player guesses
correctly and no players guess incorrectly. What strategy should they use to maximize their chances of success?
(From - http://www.princeton.edu/~sjmiller/riddles/riddles.html)
well, it stands to reason that if they can see the others hats, then they can also see the others eyes. Now, since there are three players, and only two colors of hat, then at least two must share the same color. If a person sees two hats of the same color, then he closes his eyes, thus alerting the remaining players that they share a hat color, and can guess their own hat.
Of course, this might interfere with the no communication rule.
And what if all three have the same hat color. Then they all have their eyes closed and nobody will realize that they all share the same color!! This of course is easily sidestepped by choice of a different signal.