Knight's Tours in Higher Dimensions

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.