Euclidean Tours in Fairy Chess

TL;DR

This paper extends the knight's tour problem to various fairy chess leapers on high-dimensional hypercubes, demonstrating the existence of closed tours for specific leapers in dimensions 15 and above.

Contribution

It constructs explicit closed tours for wazir, threeleaper, and zebra leapers on hypercubes, generalizing recent Euclidean knight's tour results to other fairy chess pieces.

Findings

Closed tours exist for wazir, threeleaper, and zebra in high-dimensional hypercubes.

Tours are constructed for all dimensions k ≥ 15.

The work opens new research directions in fairy chess by exploring Euclidean-length jumps.

Abstract

The present paper aims to extend the knight's tour problem for $k$-dimensional grids of the form $\{0,1\}^k$ to other fairy chess leapers. Accordingly, we constructively show the existence of closed tours in $2 \times 2 \times \cdots \times 2$ ($k$ times) chessboards concerning the wazir, the threeleaper, and the zebra, for all $k \geq 15$. Our result considers the three above-mentioned leapers and replicates for each of them the recent discovery of Euclidean knight's tours for the same set of $2 \times 2 \times \cdots \times 2$ grids, opening a new research path on the topic by studying different fairy chess leapers that perform jumps of fixed Euclidean length on given regular grids, visiting all their vertices exactly once before coming back to the starting one.