A New Variation of Hat Guessing Games

TL;DR

This paper introduces a new variation of hat guessing games where players aim for at least k correct guesses without errors, establishing conditions for perfect strategies and exploring related combinatorial structures like regular partitions of hypercubes.

Contribution

It provides necessary and sufficient conditions for the existence of perfect strategies in the new hat guessing game variation and introduces the concept of (d_1,d_2)-regular partitions of hypercubes.

Findings

Conditions for perfect strategies depending on n and k.

Introduction of (d_1,d_2)-regular partitions and their relation to hypercube domination.

Existence results for perfect k-dominating sets in hypercubes.

Abstract

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 {\em perfect} if it can achieve the simple upper bound $\frac{n}{n+k}$ 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: $(d_1,d_2)$-regular partition of the boolean hypercube, which is worth to study in its own right. For example, it is related to the $k$-dominating set of the hypercube. It also might be interesting in coding theory. The existence of $(d_1,d_2)$-regular partition is explored in the paper and the existence of perfect $k$-dominating set follows as a corollary.