Related papers: A solution to the 7-prince's tour problem

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,…

This paper solves a pursuit-evasion problem in which a prince must find a princess who is constrained to move on each day from one vertex of a finite graph to another. Unlike the related and much studied `Cops and Robbers Game', the prince…

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…

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…

In this note we show that every tournament on $n$ vertices contains the $k$-th power of a directed path of length $n/2^{6k+7}$, which improves upon the recent bound of Scott and Kor\'{a}ndi of $n/2^{2^{3k}}$. By doing so, we get an inverse…

Building on the results of our previous work on Euclidean leaper tours, considering all integers $k>1$ and $h>0$, we study the existence of Hamiltonian cycles in the vertex set $C(2,k):=\{0,1\}^k$ of the $k$-dimensional hypercube when the…

A linear algorithm is described for solving the n-Queens Completion problem for an arbitrary composition of k queens, consistently distributed on a chessboard of size n x n. Two important rules are used in the algorithm: a) the rule of…

Let c be an integer. A c-partite tournament is an orientation of a complete c-partite graph. A c-partite tournament is rich if it is strong, and each partite set has at least two vertices. In 1996, Guo and Volkmann characterized the…

If $T$ is an $n$-vertex tournament with a given number of $3$-cycles, what can be said about the number of its $4$-cycles? The most interesting range of this problem is where $T$ is assumed to have $c\cdot n^3$ cyclic triples for some $c>0$…

We introduce a two player game on an n x n chessboard where queens are placed by alternating turns on a chessboard square whose availability is determined by the number of queens already on the board which can attack that square modulo two.…

The $n$ queens problem considers the maximum number of safe squares on an $n \times n$ chess board when placing $n$ queens; the answer is only known for small $n$. Miller, Sheng and Turek considered instead $n$ randomly placed rooks,…

In how many ways can $n$ queens be placed on an $n \times n$ chessboard so that no two queens attack each other? This is the famous $n$-queens problem. Let $Q(n)$ denote the number of such configurations, and let $T(n)$ be the number of…

We present a new problem called the incomplete Traveling Tournament problem, which introduces the well known Traveling Tournament Problem into the realm of incomplete round-robin tournaments. We focus on the case where teams can face each…

We introduce two new metrics of "simplicity" for knight's tours: the number of turns and the number of crossings. We give a novel algorithm that produces tours with $9.25n+O(1)$ turns and $12n+O(1)$ crossings on an $n\times n$ board, and we…

In Parts I-III we showed that the number of ways to place $q$ nonattacking queens or similar chess pieces on an $n\times n$ chessboard is a quasipolynomial function of $n$ whose coefficients are essentially polynomials in $q$. In this part…

The aim of this paper is to solve the "gift exchange" problem: you are one of n players, and there are n wrapped gifts on display; when your turn comes, you can either choose any of the remaining wrapped gifts, or you can "steal" a gift…

We show that for each non-negative integer k, every bipartite tournament either contains k arc-disjoint cycles or has a feedback arc set of size at most 7(k - 1).

For an integer $k\geq 1$, a graph is called a $k$-circulant if its automorphism group contains a cyclic semiregular subgroup with $k$ orbits on the vertices. We show that, if $k$ is even, there exist infinitely many cubic arc-transitive…

We prove that king chasing problem in Chinese Chess is NP-hard when generalized to $n\times n$ boards. `King chasing' is a frequently-used strategy in Chinese Chess, which means that the player has to continuously check the opponent in…

We consider a problem of shuffling a deck of cards with ordered labels. Namely we split the deck of N=k^tq cards (where t>=1 is maximal) into k equally sized stacks and then take the top card off of each stack and sort them by the order of…