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Proving the existence of Euclidean knight's tours on $n \times n \times \cdots \times n$ chessboards for $n < 4$

Marco Ripà

Abstract

The Knight's Tour problem consists of finding a Hamiltonian path for the knight on a given set of points so that the knight can visit exactly once every vertex of the mentioned set. In the present paper, we provide a $5$-dimensional alternative to the well-known statement that it is not ever possible for a knight to visit once every vertex of $C(3,k) := \{0,1,2\}^k$ by performing a sequence of $3^k-1$ jumps of standard length, since the most accurate answer to the original question actually depends on which mathematical assumptions we are making at the beginning of the game, when we decide to extend a planar chess piece to the third dimension and above. Our counterintuitive outcome follows from the observation that we can alternatively define a $2$D knight as a piece that moves from one square to another on the chessboard by covering a fixed Euclidean distance of $\sqrt{5}$ so that also the statement of Theorem~3 in [Erde, J., Gol{é}nia, B., \& Gol{é}nia, S. (2012), The closed knight tour problem in higher dimensions, The Electronic Journal of Combinatorics, 19(4), \#P9] does not hold anymore for such a Euclidean knight, as long as a $2 \times 2 \times \cdots \times 2$ chessboard with at least $2^6$ cells is given. Moreover, we show a classical closed knight's tour on $C(3,4)-\{(1,1,1,1)\}$ whose arrival is at a distance of $2$ from $(1,1,1,1)$, and we finally construct closed Euclidean knight's tours on $\{0,1\}^k$ for each integer $k \geq 6$.

Proving the existence of Euclidean knight's tours on $n \times n \times \cdots \times n$ chessboards for $n < 4$

Abstract

The Knight's Tour problem consists of finding a Hamiltonian path for the knight on a given set of points so that the knight can visit exactly once every vertex of the mentioned set. In the present paper, we provide a -dimensional alternative to the well-known statement that it is not ever possible for a knight to visit once every vertex of by performing a sequence of jumps of standard length, since the most accurate answer to the original question actually depends on which mathematical assumptions we are making at the beginning of the game, when we decide to extend a planar chess piece to the third dimension and above. Our counterintuitive outcome follows from the observation that we can alternatively define a D knight as a piece that moves from one square to another on the chessboard by covering a fixed Euclidean distance of so that also the statement of Theorem~3 in [Erde, J., Gol{é}nia, B., \& Gol{é}nia, S. (2012), The closed knight tour problem in higher dimensions, The Electronic Journal of Combinatorics, 19(4), \#P9] does not hold anymore for such a Euclidean knight, as long as a chessboard with at least cells is given. Moreover, we show a classical closed knight's tour on whose arrival is at a distance of from , and we finally construct closed Euclidean knight's tours on for each integer .
Paper Structure (5 sections, 4 theorems, 2 equations, 4 figures)

This paper contains 5 sections, 4 theorems, 2 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Euclidean knight's tours on a 3 x 3 x 3 x 3 x 3 chessboard
  3. Closed knight's tours on C(3,k)-1,1,...,1
  4. Closed Euclidean knight's tours on any C(2,k) : k >= 6
  5. Conclusion

Key Result

Theorem 2.1

Let $h \in \{0,1,2,\ldots,3^k-1\}$ and assume that the knight move rule from the vertex ${\rm{V}}_h \equiv (x_1,x_2, \ldots, x_k)$ to the next vertex, ${\rm{V}}_{h+1} \equiv (y_1,y_2, \ldots, y_k)$, of $C(3,k) \coloneqq \{0,1,2\}^k$ is given by $d({{\rm{V}}}_h, {\rm{V}}_{h+1}) \coloneqq \sqrt{\sum_{

Figures (4)

  • Figure 1: A graphical representation of an open Euclidean knight's tour on $C(3,5) \coloneqq \{\{0,1,2\} \times \{0,1,2\}\times \{0,1,2\}\times \{0,1,2\} \times \{0,1,2\}\}$.
  • Figure 2: Coloring the $3 \times 3 \times 3 \times 3 \times 3$ chessboard in a proper way (thanks to the parity argument).
  • Figure 3: The polygonal chains $P_c(3,2;\{(1,1)\})$ and $\overline{P}(3,3;\{(1,1,1),(2,0,0)\})$ visit all the vertices of $C(3,2)-\{(1,1)\}$ and $C(3,3)-\{(1,1,1), (2,0,0)\}$, respectively. Both of them follow the regular (and thus also the Euclidean) knight move rule and end at the corners of $C$.
  • Figure 4: A graphical representation of the closed (regular) knight's tour $P_c(3,4;\{(1,1,1,1)\})$ on $C(3,4)-\{(1,1,1,1)\}$.

Theorems & Definitions (8)

  • Theorem 2.1
  • proof
  • Corollary 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 4.1
  • proof