The closed knight tour problem in higher dimensions

TL;DR

This paper extends the known results on closed knight tours from 2D and 3D rectangular boards to n-dimensional rectangular boards for all dimensions n ≥ 4, solving a long-standing problem.

Contribution

It provides a general solution for the existence of closed knight tours on n-dimensional rectangular boards, expanding prior work limited to lower dimensions.

Findings

Established the existence of closed knight tours in n-dimensional rectangular boards for all n ≥ 4.

Extended previous results from 2D and 3D to higher dimensions.

Unified the understanding of knight tours across multiple dimensions.

Abstract

The problem of existence of closed knight tours for rectangular chessboards was solved by Schwenk in 1991. Last year, in 2011, DeMaio and Mathew provide an extension of this result for 3-dimensional rectangular boards. In this article, we give the solution for $n$-dimensional rectangular boards, for $n\geq 4$.