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The clairvoyant maître d'

Reed Acton, T. Kyle Petersen, Blake Shirman, Bridget Eileen Tenner

TL;DR

If the ma\^{i}tre d' learns each diner's preference for the right or left napkin before they are placed at the table, this expectation jumps to nearly $1/3$ (and converges to $1/3$ as the table size gets large) and the strategy is optimal for every sequence of diners' preferences.

Abstract

In this paper we study a variant of the Malicious Maître d' problem. This problem, attributed to computer scientist Rob Pike in Peter Winkler's book "Mathematical Puzzles: A Connoisseur's Collection", involves seating diners around a circular table with napkins placed between each pair of adjacent settings. The goal of the maître d' is to seat the diners in a way that maximizes the number of diners who arrive at the table to find the napkins on both the left and right of their place already taken by their neighbors. Previous work described a seating algorithm in which the maître d' expects to force about 18% of the diners to be napkinless. In this paper, we show that if the maître d' learns each diner's preference for the right or left napkin before they are placed at the table, this expectation jumps to nearly $1/3$ (and converges to $1/3$ as the table size gets large). Moreover, our strategy is optimal for every sequence of diners' preferences.

The clairvoyant maître d'

TL;DR

If the ma\^{i}tre d' learns each diner's preference for the right or left napkin before they are placed at the table, this expectation jumps to nearly (and converges to as the table size gets large) and the strategy is optimal for every sequence of diners' preferences.

Abstract

In this paper we study a variant of the Malicious Maître d' problem. This problem, attributed to computer scientist Rob Pike in Peter Winkler's book "Mathematical Puzzles: A Connoisseur's Collection", involves seating diners around a circular table with napkins placed between each pair of adjacent settings. The goal of the maître d' is to seat the diners in a way that maximizes the number of diners who arrive at the table to find the napkins on both the left and right of their place already taken by their neighbors. Previous work described a seating algorithm in which the maître d' expects to force about 18% of the diners to be napkinless. In this paper, we show that if the maître d' learns each diner's preference for the right or left napkin before they are placed at the table, this expectation jumps to nearly (and converges to as the table size gets large). Moreover, our strategy is optimal for every sequence of diners' preferences.
Paper Structure (11 sections, 12 theorems, 48 equations, 7 figures)

This paper contains 11 sections, 12 theorems, 48 equations, 7 figures.

Table of Contents

  1. The clairvoyant maître d'.
  2. Notation and terminology
  3. Preliminary results
  4. Maximizing balanced benches
  5. Clairvoyant trap setting
  6. Results for the distribution of maximal napkinless numbers
  7. Counting lattice paths by drift
  8. Expected number of napkinless
  9. Further questions
  10. Uneven napkin preferences
  11. Graph-theoretic generalization

Key Result

Theorem 1

Let $n\geq 3$ and $q=\lfloor n/3 \rfloor$. Under the clairvoyant trap setting strategy, we have the following results.

Figures (7)

  • Figure 1: The expected proportion of napkinless diners under the trap setting strategy (diamonds), the napkin shunning strategy (open circles), and the clairvoyant trap setting strategy (filled circles). The dashed lines are at heights $1/6$ and $1/3$.
  • Figure 2: The seating arrangement $(1,\Circled{5}, -2, -8, 4, -6, \Circled{7}, -3)$, with preferences that lead to two (circled) napkinless diners (and two unclaimed napkins). The thick lines indicate the napkins claimed by Diners $1$, $2$, $3$, $4$, $6$, and $8$, while the dotted line indicates that Diner $6$ had been a negative diner (wanting the napkin on their left), but was forced to take the napkin on their right.
  • Figure 3: A portion of a seating arrangement in which Diner 9 is napkinless and Diners 2 and 5 are happy. Diners 3, 8, and 7 are frustrated. The minimal napkinless block for Diner 9 is $B(9) = \{2,9,5\}$.
  • Figure 4: The lattice path $p(\sigma)$ corresponding to the preference order $\sigma = (1,1,-1,1,-1,-1,1,1,1,1,1,1,1,-1)$. The dotted lines $y=x$ and $y=x+7$ have been drawn to show that this path has drift $7$.
  • Figure 5: The $\{N,E\}$ lattice path grid for set $L_{n,h}$ with (a) $n=18$, $h=2$, and (b) $n=18$, $h=5$.
  • ...and 2 more figures

Theorems & Definitions (30)

  • Theorem 1
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • ...and 20 more