Too Many Hats
Combinatorics
2019-03-25 v5
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.
Cite
@article{arxiv.1810.08263,
title = {Too Many Hats},
author = {Rob Pratt and Stan Wagon and Michael Wiener and Piotr Zielinski},
journal= {arXiv preprint arXiv:1810.08263},
year = {2019}
}