Related papers: The Second Leaper Theorem

Let $p$ and $q$ be positive integers. The $(p, q)$-leaper $L$ is a generalised knight which leaps $p$ units away along one coordinate axis and $q$ units away along the other. Consider a free $L$, meaning that $p + q$ is odd and $p$ and $q$…

A leaper is a chess piece which generalises the knight. Given $n$ and a $(p, q)$-leaper $L$, we study the greatest $m$ such that the $m \times m$ grid graph can be embedded into the $n \times n$ leaper graph of $L$. We can assume that $p$…

An $\{r,s\}$-leaper is a generalized knight that can jump from $(x,y)$ to $(x\pm r,y\pm s)$ or $(x\pm s,y\pm r)$ on a rectangular grid. The graph of an $\{r,s\}$-leaper on an $m\times n$ board is the set of $mn$~vertices $(x,y)$ for $0\leq…

We give a proof of Willcocks's Conjecture, stating that if $p - q$ and $p + q$ are relatively prime, then there exists a Hamiltonian tour of a $(p, q)$-leaper on a square chessboard of side $2(p + q)$. The conjecture was formulated by T. H.…

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…

We study the $(p,q)$-Maker Breaker Crossing game introduced by Day and Falgas Ravry in 'Maker-Breaker percolation games I: crossing grids'. The game described in their paper involves two players Maker and Breaker who take turns claiming p…

Shanks's sequence of quadratic fields $\Q(\sqrt{S_{n}})$ where $S_{n}=(2^n+1)^2 + 2^{n+2}$ instances a class of quadratic fields for which the class number is large and, therefore, the continued fraction period is relatively short. Indeed,…

Motivated by problems in percolation theory, we study the following 2-player positional game. Let $\Lambda_{m \times n}$ be a rectangular grid-graph with $m$ vertices in each row and $n$ vertices in each column. Two players, Maker and…

For any two squares A and B of an m x n checkerboard, we determine whether it is possible to move a checker through a route that starts at A, ends at B, and visits each square of the board exactly once. Each step of the route moves to an…

This paper introduced a pursuit and evasion game to be played on a connected graph. One player moves invisibly around the graph, and the other player must guess his position. At each time step the second player guesses a vertex, winning if…

An alternating graph is a directed graph whose vertex set is partitioned into two classes, existential and universal. This forms the basic arena for a plethora of infinite duration two-player games where Player~$\square$ and~$\ocircle$…

The authors propose a new variation of random walks called ladder chains $L(r,s,p)$. We extend concepts such as ruin probability, hitting time, transience and recurrence of random walks to ladder chain. Take $L(2,2,p)$ for instance, we find…

We show how to construct discrete-time quantum walks on directed, Eulerian graphs. These graphs have tails on which the particle making the walk propagates freely, and this makes it possible to analyze the walks in terms of scattering…

We define a variant of the two-dimensional Silver Dollar game. Two coins are placed on a chessboard of unbounded size, and two players take turns choosing one of the coins and moving it. Coins are to be moved to the left or upward…

We study quantum walks on general graphs from the point of view of scattering theory. For a general finite graph we choose two vertices and attach one half line to each. We are interested in walks that proceed from one half line, through…

The existence of $q$-ary linear complementary pairs (LCPs) of codes with $q> 2$ has been completely characterized so far. This paper gives a characterization for the existence of binary LCPs of codes. As a result, we solve an open problem…

Lensed billiards are an extension of the notion of billiard dynamical systems obtained by adding a potential function of the form $C1_{\mathcal{A}}$, where $C$ is a real valued constant and $1_{\mathcal{A}}$ is the indicator function of an…

The $m \times n$ king graph consists of all locations on an $m \times n$ chessboard, where edges are legal moves of a chess king. %where each vertex represents a square on a chessboard and each edge is a legal move. Let $P_{m \times n}(z)$…

We apply to the $n\times n$ chessboard the counting theory from Part I for nonattacking placements of chess pieces with unbounded straight-line moves, such as the queen. Part I showed that the number of ways to place $q$ identical…

We generalize the recent results of Chaiken et al. to a rectangular $m\times n$ chessboard. An explicit formula for the number of nonattacking configurations of one-move riders on such a chessboard is calculated in two different ways, one…