Too Many Hats

TL;DR

This paper investigates a generalized hat-guessing puzzle involving prisoners and hats, establishing the existence of perfect strategies for certain cases and providing bounds for success rates in others.

Contribution

It proves the non-existence of perfect strategies for k=2 and introduces a strategy with a success rate at least 1/O(k log k), advancing understanding of combinatorial hat puzzles.

Findings

Perfect strategies exist for k=1.

No perfect strategies for k=2.

Proposed strategy achieves success rate at least 1/O(k log k).

Abstract

A puzzle about prisoners trying to identify the color of a hat on their head leads to a version where there are k more hats than prisoners. This generalized puzzle is related to the independence number of the arrangement graph A(m, n) and to Steiner systems and other designs. A natural conjecture is that perfect hat-guessing strategies exist in all cases, where "perfect" means that the success probability is 1/(k+1). This is true when k = 1, but we show that it is false when k = 2. Further, we present a strategy with success rate at least 1/O(k log k), independent of the number of prisoners.