The Most Malicious Ma\^{i}tre D'

The problem of the malicious ma\^{i}tre d' is introduced and solved by Peter Winkler in his book Mathematical Puzzles: A Connoisseur's Collection [1]. This problem is about a ma\^{i}tre d' seating diners around a table, trying to maximize the number of diners who don't get napkins. Along with this problem, Winkler introduces a variation called the adaptive ma\^{i}tre d' and presents a strategy. This problem was later investigated and a better strategy was discovered by Acton et al. [2]. We describe an even better strategy which we call ``long trap setting" and prove that it is optimal. We also derive a formula for the expected number of napkinless diners under our optimal strategy.