Related papers: The $n$-Queens Problem in Higher Dimensions
The classical Maker-Breaker positional game is played on a board which is a hypergraph $\mathcal{H}$, with two players, Maker and Breaker, alternately claiming vertices of $\mathcal{H}$ until all the vertices are claimed. When the game…
Some of the most important open problems for linear layouts of graphs ask for the relation between a graph's queue number and its stack number or mixed number. In such, we seek a vertex order and edge partition of $G$ into parts with…
In this paper, we show that the number of points that can be placed in the grid $n\times n\times \cdots \times n~(d~times)=n^d$ for all $d\in \mathbb{N}$ with $d\geq 2$ so that no three points are collinear satisfies the lower bound…
We study the discrete variation of the classical cake-cutting problem where n players divide a 1-dimensional cake with exactly (n-1) cuts, replacing the continuous, infinitely divisible "cake" with a necklace of discrete, indivisible…
We say that a chessboard filled with integer entries satisfies the neighbour-sum property if the number appearing on each cell is the sum of entries in its neighbouring cells, where neighbours are cells sharing a common edge or vertex. We…
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…
Following on the King Chicken Theorems originally proved by Maurer, we examine the idea of multiple flocks of chickens by bringing the chickens from tournaments to multipartite tournaments. As Kings have already been studied in multipartite…
We study the disproportionate version of the classical cake-cutting problem: how efficiently can we divide a cake, here $[0,1]$, among $n$ agents with different demands $\alpha_1, \alpha_2, \dots, \alpha_n$ summing to $1$? When all the…
We prove that chess played on the infinite chessboard $\mathbb{Z}^2$ with infinitely many pieces is as powerful as it could possibly be, by showing that every open Gale-Stewart game with draws is strategically equivalent to some infinite…
A permutation $\sigma=[\sigma_1,\dots,\sigma_n] \in S_n$ is called a {\em cylindrical king permutation} if $ |\sigma_{i+1}-\sigma_{i}|>1$ for each $1\leq i \leq n-1$ and $|\sigma_1-\sigma_n|>1$. The name comes from the the way one can see…
In 1983 the chess periodical EG published a summary of a letter from Julius Telesin outlining how a king, a bishop and a knight can checkmate a lonely king on an arbitrarily large chessboard. The Telesin checkmating procedure doesn't seem…
Graph neural networks (GNNs) have attracted increasing interests. With broad deployments of GNNs in real-world applications, there is an urgent need for understanding the robustness of GNNs under adversarial attacks, especially in realistic…
A king in a directed graph is a node from which each node in the graph can be reached via paths of length at most two. There is a broad literature on tournaments (completely oriented digraphs), and it has been known for more than half a…
Erd\H{o}s asked what is the maximum number $\alpha(n)$ such that every set of $n$ points in the plane with no four on a line contains $\alpha(n)$ points in general position. We consider variants of this question for $d$-dimensional point…
New combinatorial games are introduced, of which the most pertinent is Maharaja Nim. The rules extend those of the well-known impartial game of Wythoff Nim in which two players take turn in moving a single Queen of Chess on a large board,…
The Parks Puzzle is a paper-and-pencil puzzle game that is classically played on a square grid with different colored regions (the parks). The player needs to place a certain number of "trees" in each row, column, and park such that none…
We study a generalization of the Mermin-Peres magic square game to arbitrary rectangular dimensions. After exhibiting some general properties, these rectangular games are fully characterized in terms of their optimal win probabilities for…
Cops and robbers is a pursuit-evasion game played on graphs. We completely classify the cop numbers for $n \times n$ knight graphs and queen graphs. This completes the classification of the cop numbers for all $n \times n$ classical chess…
We study the proportional chore division problem where a protocol wants to divide an undesirable object, called chore, among $n$ different players. The goal is to find an allocation such that the cost of the chore assigned to each player be…
We consider the well-studied cake cutting problem in which the goal is to find an envy-free allocation based on queries from $n$ agents. The problem has received attention in computer science, mathematics, and economics. It has been a major…