Related papers: 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).…
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…
The problem of existence of closed knight's tours in $[n]^d$, where $[n]=\{0, 1, \dots, n-1\}$, was recently solved by Erde, Gol\'{e}nia, and Gol\'{e}nia. They raised the same question for a generalised, $(a, b)$ knight, which is allowed to…
The problem of existence of closed knight tours for rectangular chessboards was solved by Schwenk in 1991. Last year, in 2011, DeMaio and Mathew provide an extension of this result for 3-dimensional rectangular boards. In this article, we…
The author has constructed and enumerated tours of knight having various magic properties on 4 x n and 6 x n boards. 16 magic tours of knight have been discovered on 4 x 18 board, 88 on 4 x 20 board, 464 on 4 x 22 board, 2076 on 4 x 24…
A knight's tour is often represented as a broken line connecting the centers of successively visited squares. We say that two knight moves form a cross if the midpoints of their respective segments coincide. We show that no knight tour…
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…
Non-crossing knight's tours in 3-dimension is a new field of research. The author has shown its possibility in small cuboids and in cubes up to 8x8x8 size. It can also be extended to larger size cubes and cuboids. The author has achieved…
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…
We investigate closed knight's tours on M\"obius strip and Klein bottle chess boards. In particular, we characterize the board dimensions that admit tours that are nullhomotopic and the board dimensions that admit tours that realize…
We introduce two new metrics of "simplicity" for knight's tours: the number of turns and the number of crossings. We give a novel algorithm that produces tours with $9.25n+O(1)$ turns and $12n+O(1)$ crossings on an $n\times n$ board, and we…
We review the state of the art in the problem of counting the number open knight tours, since the publication in internet of a computation of this quantity.
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…
A well-known chessboard problem is that of placing eight queens on the chessboard so that no two queens are able to attack each other. (Recall that a queen can attack anything on the same row, column, or diagonal as itself.) This problem is…
Warnsdorffs rule for a knights tour is a heuristic, i.e., it is a rule that does not produce the desired result all the time. It is a classic example of a greedy method in that it is based on a series of locally optimal choices. This note…
A whirling knight's tour is a Hamiltonian cycle in the digraph of counter-clockwise knight steps about the centre of an $n \times n$ board; its coil count $c$ is the winding number around the centre. We prove that no such tour with $c =…
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…
How many mutually non-attacking queens can be placed on a d-dimensional chessboard of size n? The n-queens problem in higher dimensions is a generalization of the well-known n-queens problem. We provide a comprehensive overview of…
A snake path in a graph $G$ is a path in $G$ which is also an induced subgraph of $G$. For all $n$, we find the greatest length of a snake path in the $n \times n$ king graph and we give a complete description of the paths which attain this…
The aim of the paper is to enumerate all closed knight paths of length n over a square board of size n+1. The closed knight paths of length 4, 6 and 8 are classified up to equivalence. We determine that there are exactly 3 equivalence…