The three-colour hat guessing game on the cycle graphs

TL;DR

This paper analyzes a cooperative three-colour hat guessing game on cycle graphs, establishing conditions under which a guaranteed winning strategy exists based on the number of players.

Contribution

It characterizes the exact cycle lengths for which a guaranteed winning strategy exists in the three-colour hat guessing game.

Findings

A winning strategy exists if and only if N is divisible by 3 or N=4.

The problem exemplifies relational systems with incomplete information.

Provides a complete characterization of when guaranteed success is possible.

Abstract

We study a cooperative game in which each member of a team of $N$ players, wearing coloured hats and situated at the vertices of a cycle graph $C_N$, is guessing their own hat colour merely on the basis of observing the hats worn by their two neighbours without exchanging the information. Each hat can have one of three colours. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colours. We prove that a winning strategy exists if and only if $N$ is divisible by $3$ or $N=4$. This problem represents an example of a relational system using incomplete information about an unpredictable situation, where at least one participant has to act properly.