On a generalization of the seating couples problem

TL;DR

This paper proves a conjecture that generalizes the seating couples problem to 2n seats, showing that any set of differences can partition the seats into pairs accordingly.

Contribution

It provides a proof of a conjecture extending the seating couples problem to arbitrary differences for 2n seats.

Findings

Confirmed the conjecture for all positive integers n.

Established that any set of differences can partition the seats into pairs.

Generalized the seating couples problem to a broader class of difference sets.

Abstract

We prove a conjecture of Adamaszek generalizing the seating couples problem to the case of $2n$ seats. Concretely, we prove that given a positive integer $n$ and $d_1,\ldots,d_n\in(\mathbb{Z}/2n)^*$ we can partition $\mathbb{Z}/2n$ into $n$ pairs with differences $d_1,\ldots,d_n$.