The N-queens Problem on a symmetric Toeplitz matrix

TL;DR

This paper investigates the placement of nonattacking queens on symmetric Toeplitz matrices, revealing specific size conditions for such arrangements based on modular arithmetic.

Contribution

It characterizes the exact sizes of symmetric Toeplitz matrices that allow nonattacking queen placements, extending the classical N-queens problem to a new matrix class.

Findings

Nonattacking queens can be placed if and only if n ≡ 0 or 1 mod 4.

Provides a complete characterization of feasible matrix sizes.

Extends the classical N-queens problem to symmetric Toeplitz matrices.

Abstract

We consider the problem of placing $n$ nonattacking queens on a symmetric $n \times n$ Toeplitz matrix. As in the $N$-queens Problem on a chessboard, two queens may attack each other if they share a row or a column in the matrix. However, the usual diagonal restriction is replaced by specifying that queens may attack other queens that occupy squares with the same number value in the matrix. We will show that $n$ nonattacking queens can be placed on such a matrix if and only if $n\equiv 0,1 \mod 4$.