Hats: all or nothing

Abstract: N players are randomly fitted with a colored hat (q different colors). All players guess simultaneously the color of their own hat observing only the hat colors of the other N-1 players. The team wins if all players guess right. No communication of any sort is allowed, except for an initial strategy session before the game begins. In the first part of our investigation we have q different colors with equal probability. Up to 4 colors we construct optimal strategies for any number of players using Hamming Complete Sets. For 5 colors we find optimal strategies up to 5 players using Optimal Hamming Sets. In the second part we have two colors where the probabilities may differ. We construct optimal strategies and maximal probability of winning the game for any number of players.