Related papers: On an error in the star puzzle by Henry E. Dudeney
We consider the classical $n$-queens problem, which asks how many ways one can place $n$ mutually non-attacking queens on an $n$ x $n$ chessboard. We prove that the total number of solutions to the $n$-queens problem $Q(n)$ is divisible by…
In 1907, Henry Ernest Dudeney posed a puzzle: ``cut any equilateral triangle \dots\ into as few pieces as possible that will fit together and form a perfect square'' (without overlap, via translation and rotation). Four weeks later, Dudeney…
In 1903, noted puzzle-maker Henry Dudeney published The Spider and the Fly puzzle, which asks for the shortest path along the surfaces of a square prism between two points (source and target) located on the square faces, and surprisingly…
The $n$-queens puzzle is to place $n$ mutually non-attacking queens on an $n \times n$ chessboard. We present a simple two stage randomized algorithm to construct such configurations. In the first stage, a random greedy algorithm constructs…
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 15 puzzle is a classic reconfiguration puzzle with fifteen uniquely labeled unit squares within a $4 \times 4$ board in which the goal is to slide the squares (without ever overlapping) into a target configuration. By generalizing the…
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…
In [1] the authors studied the closed tour problem on the $8\times 8$ chessboard of a chess piece, called $k$-prince, leaving open the existence of such a tour when $k=7$. In this note we find a solution to this open case.
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…
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…
In his celebrated book "On Number and Games" (Academic Press, New-York, 1976), J.H. Conway introduced twelve versions of compound games. We analyze these twelve versions for the Node-Kayles game on paths. For usual disjunctive compound,…
The function that counts the number of ways to place nonattacking identical chess or fairy chess pieces in a rectangular strip of fixed height and variable width, as a function of the width, is a piecewise polynomial which is eventually a…
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 present an integer programming formulation of…
In 1967, Klarner proposed a problem concerning the existence of reflecting $n$-queens configurations. The problem considers the feasibility of placing $n$ mutually non-attacking queens on the reflecting chessboard, an $n\times n$ chessboard…
Inspired by a chessboard puzzle of Dudeney, the general position problem in graph theory asks for a largest set $S$ of vertices in a graph such that no three elements of $S$ lie on a common shortest path. The number of vertices in such a…
The Queen's Domination problem, studied for over 160 years, poses the following question: What is the least number of queens that can be arranged on a $m \times n$ chessboard so that they either attack or occupy every cell? We propose a…
Number the cells of a (possibly infinite) chessboard in some way with the numbers 0, 1, 2, ... Consider the cells in order, placing a queen in a cell if and only if it would not attack any earlier queen. The problem is to determine the…
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…
In his list of open problems, Martin Erickson described a certain game: "Two players alternately put queens on an n x n chess board so that each new queen is not in range of any queen already on the board (the color of the queens is…
We consider the chessboard pebbling problem analyzed by Chung, Graham, Morrison and Odlyzko [3]. We study the number of reachable configurations $G(k)$ and a related double sequence $G(k,m)$. Exact expressions for these are derived, and we…