On the existence of reflecting $n$-queens configurations

TL;DR

This paper proves that reflecting $n$-queens configurations exist for all sufficiently large $n$, solving a longstanding problem by linking a chess puzzle to a number theory pairing problem.

Contribution

It establishes the existence of reflecting $n$-queens configurations for all large $n$, resolving Klarner's and Slater's questions for almost all integers.

Findings

Reflecting $n$-queens configurations exist for all large $n$.

The problem is equivalent to a specific number pairing problem.

The results resolve longstanding open questions in the domain.

Abstract

In 1967, Klarner proposed a problem concerning the existence of reflecting $n$-queens configurations. The problem considers the feasibility of placing $n$ mutually non-attacking queens on the reflecting chessboard, an $n\times n$ chessboard with a $1\times n$ "reflecting strip" of squares added along one side of the board. A queen placed on the reflecting chessboard can attack the squares in the same row, column, and diagonal, with the additional feature that its diagonal path can be reflected via the reflecting strip. Klarner noted the equivalence of this problem to a number theory problem proposed by Slater, which asks: for which $n$ is it possible to pair up the integers 1 through $n$ with the integers $n+1$ through $2n$ such that no two of the sums or differences of the $n$ pairs of integers are the same. We prove the existence of reflecting $n$-queens configurations for all sufficiently large $n$, thereby resolving both Slater's and Klarner's questions for all but a finite number of integers.