Euclidean Tours in Fairy Chess

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 $k\ge 15$. 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.

9 pages