The Most Malicious Maître D'

Abstract

We revisit the family of "napkin problems" first discussed by Peter Winkler in his 2004 puzzle book and discussed subsequently in several papers. This family of problems involves diners being seated around a circular table in which napkins are placed between the seats, so it is ambiguous which napkin belongs to which seat. The problems seek to quantify the proportion of diners who end up with no napkin under a variety of assumptions, one of which is that the diners have an adaptive adversary known as "the malicious maitre d'," who seeks to maximize the number of napkinless diners. Winkler's original strategy for the adaptive maitre d' was shown to be suboptimal in 2023 when Acton, Petersen, Shirman, and Toal presented a better (yet also suboptimal) strategy. In this paper we demonstrate an optimal strategy for the malicious maitre d' and compute the expected proportion of napkinless diners as a function of the probability of a diner taking the left napkin. Interestingly, the strategy does not depend on the exact probability; rather, it only depends on which napkin (left or right) the diners tend to prefer.