Applications of recursive operators to randomness and complexity
Applications of recursive operators to randomness and complexity
SIAM Journal on Discrete Mathematics
On Optimal Strategies for a Hat Game on Graphs
SIAM Journal on Discrete Mathematics
Hypercube orientations with only two in-degrees
Journal of Combinatorial Theory Series A
On a conjecture of Butler and Graham
Designs, Codes and Cryptography
| Hi-index | 0.00 |
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 a perfect strategy for (n, k) is open for only a few values of 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 k-dominating 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.