From New York Times, Science Times, D5,  April 10, 2001:"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 [an independent] coin toss. 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 [but not their own], the Players   must simultaneously guess the color of their own hats or pass. The puzzle is to find a group strategy that maximizes the probability that at least one person guesses correctly and no-one guesses incorrectly."

The naive strategy would be for the group to agree that one person should guess and the Others   pass. This would have probability 1/2 of success. Find a strategy with a greater chance for success. (The solution is given in the article.)For a different problem, allow every one of n people to place an even bet on the color of his hat. The bet can either be on red or on blue and the amount of each bet is arbitrary. The group wins if their combined wins are strictly greater than their losses. Find, with proof, a strategy with maximal winning probability.