Hats: all or nothing

TL;DR

This paper investigates optimal strategies for a hat-guessing game with multiple players and colors, providing solutions for equal and unequal probability scenarios, and identifying maximum winning probabilities.

Contribution

It introduces new optimal strategies using Hamming Complete Sets and Optimal Hamming Sets for different numbers of colors and players, expanding understanding of the game.

Findings

Optimal strategies constructed for up to 4 colors and any number of players.

Maximum winning probability determined for two-color case with unequal probabilities.

Strategies for 5 colors and up to 5 players using Optimal Hamming Sets.

Abstract

N players are randomly fitted with a colored hat (q different colors). All players guess simultaneously the color of their own hat observing only the hat colors of the other N-1 players. The team wins if all players guess right. No communication of any sort is allowed, except for an initial strategy session before the game begins. In the first part of our investigation we have q different colors with equal probability. Up to 4 colors we construct optimal strategies for any number of players using Hamming Complete Sets. For 5 colors we find optimal strategies up to 5 players using Optimal Hamming Sets. In the second part we have two colors where the probabilities may differ. We construct optimal strategies and maximal probability of winning the game for any number of players.