Related papers: Non-crossing Knight's Tour in 3-Dimension
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…
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…
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…
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…
There are 216 magic tours of knight in 4x4x4 cube.
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…
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…
Three-dimensional arrays of silicon transistors increase the density of bits. Solid-state qubits are much larger so could benefit even more from using the third dimension given that useful fault-tolerant quantum computing will require at…
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…
To visualize a higher dimensional object it is convenient to consider its two-dimensional cross-sections. The set of quantum states for a three level system has eight dimensions. We supplement a recent paper by Goyal et al by considering…
We explore a continuous-time quantum walk starting at a single vertex on the discrete path and cycle with a cubic nonlinearity. Such nonlinearities arise in Bose-Einstein condensates described by the Gross-Pitaevskii equation or by…
A kinematic chain in three-dimensional Euclidean space consists of $n$ links that are connected by spherical joints. Such a chain is said to be within a closed configuration when its link lengths form a closed polygonal chain in three…
On a convex polygonal chessboard, the number of combinatorial types of nonattacking configuration of three identical chess riders with $r$ moves, such as queens, bishops, or nightriders, equals $r(r^2+3r-1)/3$, as conjectured by Chaiken,…
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 =…
This article describes sixteen different ways to traverse d-dimensional space recursively in a way that is well-defined for any number of dimensions. Each of these traversals has distinct properties that may be beneficial for certain…
We study the enumeration of different classes of grand knight's paths in the plane. In particular, we focus on the subsets of zigzag knight's paths that are subject to constraints. These constraints include ending at $y$-coordinate 0,…
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.
Evidence is presented to suggest that, in three dimensions, spherical 6-designs with N points exist for N=24, 26, >= 28; 7-designs for N=24, 30, 32, 34, >= 36; 8-designs for N=36, 40, 42, >= 44; 9-designs for N=48, 50, 52, >= 54; 10-designs…