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A New Variation of Hat Guessing Games

  • Tengyu Ma
  • Xiaoming Sun
  • Huacheng Yu
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

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

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