A New Approach on the Seating Couples Problem

TL;DR

This paper investigates the seating arrangement problem for couples around a round table with specific distance constraints, establishing that solutions exist for all configurations if and only if the number of seats is prime, using Combinatorial Nullstellensatz.

Contribution

It provides a complete characterization of when seating arrangements with prescribed distances are possible, linking the problem to prime numbers and applying algebraic combinatorics.

Findings

Solutions exist for all distance configurations if and only if 2n+1 is prime.

The problem is solved using the Combinatorial Nullstellensatz theorem.

The result connects seating arrangements with number theory and algebraic methods.

Abstract

A king invites n couples to sit around a round table with 2n+1 seats. For each couple, the king decides a prescribed distance d between 1 and n which the two spouses have to be seated from each other (distance d means that they are separated by exactly d-1 chairs). We will show that there is a solution for every choice of the distances if and only if 2n+1 is a prime number, using a theorem known as Combinatorial Nullstellensatz.