The Three Hat Problem

TL;DR

This paper analyzes the Three Hat Problem, providing an optimal strategy for players to deduce their hat numbers based on logical reasoning and known constraints, with proofs of correctness and optimality.

Contribution

It introduces a novel, proven optimal strategy for solving the Three Hat Problem, expanding understanding of logical deduction in constrained scenarios.

Findings

The strategy guarantees a solution for all configurations.

The strategy is proven to be optimal.

Players can deduce their numbers through turn-wise reasoning.

Abstract

In this paper we study the Three Hat Problem which appeared in Puzzle Corner of the Technology Review magazine. This puzzle gives a scenario in which three players wearing hats are sitting together and each hat can be seen by everyone except the player that is wearing that hat. Each player is told that all of the hats contain a positive integer and that two of the integers add to the third. In an ordered, turn-wise, modular fashion, each player truthfully states whether or not he knows his integer. We give a strategy which allows for one of the players to solve for his integer for all possible integer configurations of the puzzle and prove it is the optimal such strategy.