Euclidean Tours in Fairy Chess (2407.07903v1)

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.

• The paper establishes closed Hamiltonian Euclidean tours for the wazir, threeleaper, and zebra on k-dimensional 2×2×...×2 grids.
• The authors employ computational methods and symmetry-based combinatorial proofs to confirm results in dimensions as high as k ≥ 15.
• The research advances multi-dimensional graph theory and lays the groundwork for exploring scalable algorithms in complex chess puzzles.

An Examination of Euclidean Tours in Fairy Chess

The paper "Euclidean Tours in Fairy Chess" by Gabriele Di Pietro and Marco Ripà extends the classical knight’s tour problem to a multi-dimensional setting and introduces its application to fairy chess pieces. In this insightful work, the authors focus on the existence of Hamiltonian Euclidean tours for three specific fairy chess leapers—the wazir, the threeleaper, and the zebra—on k-dimensional chessboards of unit size, i.e., 2×2×...×2 (k times), for k ≥ 15.

Overview and Main Contributions

Primarily, the authors successfully construct closed Hamiltonian tours for the wazir, the threeleaper, and the zebra in higher dimensions. Expanding upon previous findings for knights, this paper provides a thorough investigation into the potential of other leapers under specific Euclidean distance constraints. The paper emphasizes the Euclidean interpretation of these moves, providing constructive proofs for specific dimensional cases.

Computational Approaches and Outcomes

The approach undertaken utilizes computational methods to discern these Hamiltonian paths. Due to computational power limitations, their primary focus was on achieving definitive results for dimensions up to 15 for zebras and dimensions over 10 for threeleapers. The results were corroborated by custom algorithms run on conventional computing hardware, capable of finding solutions within a reasonable time frame, as demonstrated in the appendix using Python code.

Among the central achievements:

• Theorem 3.1: Establishes the existence of Hamiltonian Euclidean tours for the wazir for each k-dimensional grid when k is a positive integer.
• Theorem 3.2: Proves the existence of Hamiltonian Euclidean threeleaper tours on C(2,k) grids for k ≥ 11, effectively utilizing symmetry and combinatorial techniques.
• Theorem 3.3: Demonstrates Hamiltonian Euclidean tours with the zebra leaper for dimensions 15 and higher.

Implications and Future Work

The results of this work help illuminate pathways in multi-dimensional graph theory and describe techniques that could be adapted to investigate higher-dimensional structures in computational mathematics and artificial intelligence. The paper also raises significant questions about the scalability of these methods to other fairy chess pieces and higher dimensions beyond those currently investigated.

Future research may involve enhancing computational efficiency to explore further dimensions and more complex leaper rules or expanding the theoretical groundwork to establish broader conditions and proofs for various fairy chess pieces. Additionally, integrating these exploration methods with broader applications in data structure traversal algorithms could provide new insights into solving complex computational problems.

This investigation marks an advancement in understanding the properties of non-traditional leapers in chess, showcasing the interplay between historical mathematical puzzles and modern computational capabilities. The clear exposition provided by Di Pietro and Ripà facilitates a stronger grasp of these mathematical constructs and lays the foundation for continued exploration in this nuanced field.