A more colorful hat problem

Abstract

The topic is the hat problem in which each of n players is randomly fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The aim is to maximize the probability of winning. We consider a generalized hat problem with $q ≥ 2$ colors. We solve the problem with three players and three colors. Next we prove some upper bounds on the chance of success of any strategy for the generalized hat problem with $n$ players and $q$ colors. We also consider the numbers of strategies that suffice to be examined to solve the hat problem, or the generalized hat problem.