A New Variation of Hat Guessing Games

Several variations of hat guessing games have been popularly discussed in recreational mathematics. In a typical hat guessing game, after initially coordinating a strategy, each of n players is assigned a hat from a given color set. Simultaneously, each player tries to guess the color of his/her own hat by looking at colors of hats worn by other players. In this paper, we consider a new variation of this game, in which we require at least k correct guesses and no wrong guess for the players to win the game, but they can choose to “pass”. A strategy is called perfect if it can achieve the simple upper bound n n+k of the winning probability. We present sufficient and necessary condition on the parameters n and k for the existence of perfect strategy in the hat guessing games. In fact for any fixed parameter k, the existence of perfect strategy can be determined for every sufficiently large n. In our construction we introduce a new notion: (d1, d2)-regular partition of the boolean hypercube, which is worth to study in its own right. For example, it is related to the kdominating set of the hypercube. It also might be interesting in coding theory. The existence of (d1, d2)-regular partition is explored in the paper and the existence of perfect k-dominating set follows as a corollary.