Magic Knight's Tours in Higher Dimensions

TL;DR

This paper explores the existence of magic and closed knight's tours in higher-dimensional hypercubes, proving impossibility in odd dimensions and identifying specific minimal configurations where such tours are possible.

Contribution

It establishes new theoretical results on the existence and non-existence of magic knight's tours in higher dimensions, including minimal configurations for tours in hypercubes.

Findings

No magic or closed tours in odd-dimensional hypercubes.

Smallest cuboid for n-dimensional tours: 3×4×(2n-2).

Magic tours exist in 4^4 and 4^5 hypercubes.

Abstract

A knight's tour on a board is a sequence of knight moves that visits each square exactly once. A knight's tour on a square board is called magic knight's tour if the sum of the numbers in each row and column is the same (magic constant). Knight's tour in higher dimensions (n > 3) is a new topic in the age-old world of knight's tours. In this paper, it has been proved that there can't be magic knight's tour or closed knight's tour in an odd order n-dimensional hypercube. 3 \times 4 \times 2n-2 is the smallest cuboid (n \geq 2) and 4 \times 4 \times 4n-2 is the smallest cube in which knight's tour is possible in n-dimensions (n \geq 3). Magic knight's tours are possible in 4 \times 4 \times 4 \times 4 and 4 \times 4 \times 4 \times 4 \times 4 hypercube.