Related papers: Board games, random boards and long boards
We analyze the computational complexity of the popular computer games Threes!, 1024!, 2048 and many of their variants. For most known versions expanded to an m x n board, we show that it is NP-hard to decide whether a given starting…
Consider an n by n array of cards shuffled in the following manner. An element x of the array is chosen uniformly at random; Then with probability 1/2 the rectangle of cards above and to the left of x is rotated 180 degrees, and with…
We consider a card guessing game with complete feedback. An ordered deck of $n$ cards labeled $1$ up to $n$ is riffle-shuffled exactly one time. Given a value $p\in(0{,}1)\setminus\{\frac12\}$, the riffle shuffle is assumed to be…
We continue our studies of burn-off chip-firing games from [Discrete Math. Theor. Comput. Sci. 15 (2013), no. 1, 121-132; MR3040546] and [Australas. J. Combin. 68 (2017), no. 3, 330-345; MR3656659]. The latter article introduced randomness…
We analyze what happens to the average duration of a game of Chutes and Ladders as the probability of rolling $\delta \in \{ 1,2,3,4,5,6\}$ approaches 100%. We utilize Markov models, and Monte Carlo simulations in Python. We also introduce…
The billiard table is modeled as an $n$-dimensional box $[0,a_1]\times [0,a_2]\times \ldots \times [0,a_n] \subset \mathbb{R}^n$, with each side having real-valued lengths $a_i$ that are pairwise commensurable. A ball is launched from the…
We consider the one-person game of peg solitaire played on a computer. Two popular board shapes are the 33-hole cross-shaped board, and the 15-hole triangle board---we use them as examples throughout. The basic game begins from a full board…
In this paper, we consider a game played on a rectangular $m \times n$ gridded chocolate bar. Each move, a player breaks the bar along a grid line. Each move after that consists of taking any piece of chocolate and breaking it again along…
We consider the generalized game Lights Out played on a graph and investigate the following question: for a given positive integer $n$, what is the probability that a graph chosen uniformly at random from the set of graphs with $n$ vertices…
Tetravex is a widely played one person computer game in which you are given $n^2$ unit tiles, each edge of which is labelled with a number. The objective is to place each tile within a $n$ by $n$ square such that all neighbouring edges are…
We consider the tiling of an $n$-board (a $1\times n$ array of square cells of unit width) with half-squares ($\frac12\times1$ tiles) and $(\frac12,\frac12)$-fence tiles. A $(\frac12,\frac12)$-fence tile is composed of two half-squares…
A 5x5 board is the smallest board on which one can set up all kind of chess pieces as a start position. We consider Gardner's minichess variant in which all pieces are set as in a standard chessboard (from Rook to King). This game has…
In this paper, we address a natural question at the intersection of combinatorial game theory and computational complexity: "Can a sum of simple tepid games in canonical form be intractable?" To resolve this fundamental question, we…
The worst-case expected length f(n) of the path taken by the simplex algorithm with the Random Edge pivot rule on a 3-dimensional linear program with n constraints is shown to be bounded by 1.3445 n <= f(n) <= 1.4943 n for large enough n.
This paper intends to solve the Transversal achievement on an n times n grid problem proposed by Dr. Martin Erickson. We approach the problem using mathematical induction and case analysis and prove the hypothesis that for all n greater…
Conway Checkers is a game played with a checker placed in each square of the lower half of an infinite checkerboard. Pieces move by jumping over an adjacent checker, removing the checker jumped over. Conway showed that it is not possible to…
A formal n-square is the set of positions in an square matrix of size n. A shuffle of a formal n-square consists of independent rotations of each row and of each column. A key result turns out to be valid at least for n <= 34 and n = 37:…
Random walks are powerful tools to analyze spatial-temporal patterns produced by living organisms ranging from cells to humans. At the same time, it is evident that these patterns are not completely random but are results of a convolution…
The classic Rock-Paper-Scissors game of size 3 and its extension, Rock-Paper-Scissors-Lizard-Spock, are modeled by directed graphs called tournaments. They can be further extended to any odd size. The extended games are regular tournaments…
Game theory is the study of tractable games which may be used to model more complex systems. Board games, video games and sports, however, are intractable by design, so "ludological" theories about these games as complex phenomena should be…