Non-crossing Knight's Tour in 3-Dimension

Magic Knight's Tours in Higher Dimensions

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 to…

Tourneys and the Fast Generation and Obfuscation of Closed Knight's Tours

New algorithms for generating closed knight's tours are obtained by generating a vertex-disjoint cycle cover of the knight's graph and joining the resulting cycles. It is shown experimentally that these algorithms are significantly faster in practice than previous methods. A fast obfuscation algorithm for closed knight's tours that obscures obvious artifacts created by their method of generation is also given, along with visual and statistical evidence of its efficacy.

The Topology of Knight's Tours on Surfaces

We investigate the homotopy classes of closed knight's tours on cylinders and tori. Specifically, we characterize the dimensions of cylindrical chessboards that admit closed knight's tours realizing the identity of the fundamental group and those that admit closed tours realizing a generator of the fundamental group. We also produce analogous results for toroidal chessboards.

Knight's Tours in Higher Dimensions

In this paper we are concerned with knight's tours on high-dimensional boards. Our main aim is to show that on the $d$-dimensional board $[n]^d$, with $n$ even, there is always a knight's tour provided that $n$ is sufficiently large. In fact, we give an exact classification of the grids $[n_1] \times ... \times [n_d]$ in which there is a knight's tour. This answers questions of DeMaio, DeMaio and Mathew, and Watkins.

非可定向曲面上的零伦与生成元骑士巡游

Nullhomotopic and Generating Knight's Tours on Non-Orientable Surfaces

本文研究莫比乌斯带与克莱因瓶棋盘上的闭合骑士巡游，刻画了允许零伦巡游及实现基本群生成元巡游的棋盘维度。

证明 $n \times n \times \cdots \times n$ 棋盘上 $n < 4$ 时欧几里得骑士巡游的存在性

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

本文针对低维棋盘上的骑士巡游问题，通过重新定义欧几里得骑士，证明了在 $n<4$ 时存在闭欧几里得骑士巡游，并给出了 $k \geq 6$ 时 $\{0,1\}^k$ 上的构造。

Counting Knight's Tours through the Randomized Warnsdorff Rule

We give an estimate of the number of geometrically distinct open tours $\G$ for a knight on a chessboard. We use a randomization of Warnsdorff rule to implement importance sampling in a backtracking scheme, correcting the observed bias of the original rule, according to the proposed principle that ``most solutions follow Warnsdorff rule most of the time''. After some experiments in order to test this principle, and to calibrate a parameter, interpreted as a distance of a general solution from a Warnsdorff solution, we conjecture that $\G=1.22\times 10^{15}$.