Problem of the Week-2: The Hat Puzzle
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 $3 million prize if at least one player guesses correctly and no players guess incorrectly (a player can pass).
Does there exist a strategy for the group that maximizes its chances of winning the prize? If so, what is it? Prove that it is indeed the best strategy.
Note: If you can solve the same problem with 7 players, then you will be given preference to win the price over those who solve only the 3-player version.
Submit your answers (by e-mail or hard copy) before 4 pm on 9/22/06 to Noah Aydin.