相关论文: Rook Endgame Problems in $m$ by $n$ Chess

The famous $n$-queens problem asks how many ways there are to place $n$ queens on an $n \times n$ chessboard so that no two queens can attack one another. The toroidal $n$-queens problem asks the same question where the board is considered…

We study a game on a graph $G$ played by $r$ {\it revolutionaries} and $s$ {\it spies}. Initially, revolutionaries and then spies occupy vertices. In each subsequent round, each revolutionary may move to a neighboring vertex or not move,…

We study the Torus Puzzle, a solitaire game in which the elements of an input $m \times n$ matrix need to be rearranged into a target configuration via a sequence of unit rotations (i.e., circular shifts) of rows and/or columns. Amano et…

Consider the following variant of Rock, Paper, Scissors (RPS) played by two players Rei and Norman. The game consists of $3n$ rounds of RPS, with the twist being that Rei (the restricted player) must use each of Rock, Paper, and Scissors…

Consider n cards that are labeled 1 through n with n an even integer. The cards are put face down and their ordering starts with card labeled 1 on top through card labeled n at the bottom. The cards are top to random shuffled m times and…

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…

We consider infinite duration alternating move games. These games were previously studied by Roth, Balcan, Kalai and Mansour. They presented an FPTAS for computing an approximated equilibrium, and conjectured that there is a polynomial…

For two graphs $B$ and $H$ the strong Ramsey game $\mathcal{R}(B,H)$ on the board $B$ and with target $H$ is played as follows. Two players alternately claim edges of $B$. The first player to build a copy of $H$ wins. If none of the players…

This paper analyzes a simple game with $n$ players. We fix a mean, $\mu$, in the interval $[0, 1]$ and let each player choose any random variable distributed on that interval with the given mean. The winner of the zero-sum game is the…

Two standard algorithms for approximately solving two-player zero-sum concurrent reachability games are value iteration and strategy iteration. We prove upper and lower bounds of 2^(m^(Theta(N))) on the worst case number of iterations…

Recent work in reinforcement learning demonstrated that learning solely through self-play is not only possible, but could also result in novel strategies that humans never would have thought of. However, optimization methods cast as a game…

A tournament is an orientation of a complete graph. A vertex that can reach every other vertex within two steps is called a \emph{king}. We study the complexity of finding $k$ kings in a tournament graph. We show that the randomized query…

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 consider the one-round Voronoi game, where player one (``White'', called ``Wilma'') places a set of n points in a rectangular area of aspect ratio r <=1, followed by the second player (``Black'', called ``Barney''), who places the same…

We study a two-person game played on graphs based on the widely studied chip-firing game. Players Max and Min alternately place chips on the vertices of a graph. When a vertex accumulates as many chips as its degree, it fires, sending one…

We consider the following game. A deck with $m$ copies of each of $n$ distinct cards is shuffled in a perfectly random way. The Guesser sequentially guesses the card from top to bottom. After each guess, the Guesser is informed whether the…

The triangle game introduced by Chv\'{a}tal and Erd\H{o}s (1978) is one of the most famous combinatorial games. For $n,q\in\mathbb{N}$, the $(n,q)$-triangle game is played by two players, called Maker and Breaker, on the complete graph…

We extend the study of the 2-Solo Chess problem which was first introduced by Aravind, Misra, and Mittal in 2022. 2-Solo Chess is a single-player variant of chess in which the player must clear the board via captures such that only one…

The game of memory is played with a deck of n pairs of cards. The cards in each pair are identical. The deck is shuffled and the cards laid face down. A move consists of flipping over first one card then another. The cards are removed from…

We look at the unbiased Maker-Breaker Hamiltonicity game played on the edge set of a complete graph $K_n$, where Maker's goal is to claim a Hamiltonian cycle. First, we prove that, independent of who starts, Maker can win the game for $n =…