COCOON 2011: Computing and Combinatorics pp 616-626

# A New Variation of Hat Guessing Games

• Tengyu Ma
• Xiaoming Sun
• Huacheng Yu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6842)

## 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 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 a perfect strategy for (n,k) is open for only a few values of 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.

## Keywords

Hat guessing game perfect strategy hypercube k- dominating set perfect code

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## Authors and Affiliations

• Tengyu Ma
• 1
• Xiaoming Sun
• 2
• Huacheng Yu
• 1
1. 1.Institute for Theoretical Computer ScienceTsinghua UniversityChina
2. 2.Institute for Advanced StudyTsinghua UniversityChina