Related papers: Depth in Bingo Closure
Retrograde analysis is used in game-playing programs to solve states at the end of a game, working backwards toward the start of the game. The algorithm iterates through and computes the perfect-play value for as many states as resources…
We revisit the classic 'guess my number' game and extend it from its familiar binary form to representations in any integer base. For each base we derive formulas for the number of cards needed to identify a given integer and, conversely,…
At some places (see the references) Martin Erickson describes a certain game: "Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an n x n grid. The first player (if any) to occupy four cells…
Jigsaw puzzles are typically labeled with their finished area and number of pieces. With this information, is it possible to estimate the area required to lay each piece flat before assembly? We derive a simple formula based on…
We present an algorithm for computing the integral closure of a reduced ring that is finitely generated over a finite field.
We determine the maximal number of steps required to sort $n$ labeled points on a circle by adjacent swaps. Lower bounds for sorting by all swaps, not necessarily adjacent, are given as well.
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…
It is proved that in card games similar to 'Beggar-my-neighbour' the mathematical expectation of the playing time is finite, provided that the player who starts the round is determined randomly and the deck is shuffled when the trick is…
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…
So Long Sucker is a strategy board game that requires 4 players, each with $c$ chips of their designated color, and a board made of $k$ empty piles. With a clear set-up comes intricate rules, such as: players taking turns but not in a fixed…
We consider systems of "pinned balls," i.e., balls that have fixed positions and pseudo-velocities. Pseudo-velocities change according to the same rules as those for velocities of totally elastic collisions between moving balls. The times…
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 prove an explicit upper bound on the amount of entanglement required by any strategy in a two-player cooperative game with classical questions and quantum answers. Specifically, we show that every strategy for a game with n-bit questions…
In this paper we give a mathematical model for a game that we call picture cube puzzle and investigate its properties. The central question is the number of moves required to solve the puzzle. A mathematical discussion is followed by the…
We introduce a 2-player game played on an infinite grid, initially empty, where each player in turn chooses a vertex and colours it. The first player aims to create some pattern from a target set, while the second player aims to prevent it.…
The orbit closure of the unfolding of every rational right and isosceles triangle is computed and the asymptotic number of periodic billiard trajectories in these triangles is deduced. This follows by classifying all orbit closures of rank…
The winning rule of billiards is to drive the billiard ball on the table into the designated holes. We try to study the trajectory of the billiard ball, so that we can predict the direction of the ball. For rational slopes, we got cutting…
We present a Spades bidding algorithm that is superior to recreational human players and to publicly available bots. Like in Bridge, the game of Spades is composed of two independent phases, \textit{bidding} and \textit{playing}. This paper…
This article is dedicated to domino tilings of square grids. In each of these grids domino tilings are represented using linear-recurrent sequences. For different grids are determined new dependencies.
We establish a relationship between the word complexity and the number of generalized diagonals for a polygonal billiard. We conclude that in the rational case the complexity function has cubic upper and lower bounds. In the tiling case the…