Knight's Tours in Higher Dimensions

TL;DR

This paper proves that for sufficiently large even-sized high-dimensional boards, a knight's tour always exists, and it classifies all grid sizes where such tours are possible, answering longstanding open questions.

Contribution

It provides a complete classification of high-dimensional grids that admit knight's tours and establishes existence results for large even-sized boards.

Findings

Knight's tours exist on all sufficiently large even-sized d-dimensional boards.

Complete classification of grid sizes allowing knight's tours.

Answers to open questions by DeMaio, DeMaio, Mathew, and Watkins.

Abstract

In this paper we are concerned with knight's tours on high-dimensional boards. Our main aim is to show that on the $d$-dimensional board $[n]^d$, with $n$ even, there is always a knight's tour provided that $n$ is sufficiently large. In fact, we give an exact classification of the grids $[n_1] \times ... \times [n_d]$ in which there is a knight's tour. This answers questions of DeMaio, DeMaio and Mathew, and Watkins.