Too Many Hats
Rob Pratt, Stan Wagon, Michael Wiener, Piotr Zielinski

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.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Computational Geometry and Mesh Generation
