相关论文: The Period of the subtraction games

We study algorithmic complexity of solving subtraction games in a~fixed dimension with a finite difference set. We prove that there exists a game in this class such that any algorithm solving the game runs in exponential time. Also we prove…

Node-Kayles is a well-known impartial combinatorial game played on graphs, where players alternately select a vertex and remove it along with its neighbors. By the Sprague-Grundy theorem, every position of an impartial game corresponds to a…

Combinatorial Game Theory has also been called `additive game theory', whenever the analysis involves sums of independent game components. Such {\em disjunctive sums} invoke comparison between games, which allows abstract values to be…

We study zero-sum games, a variant of the classical combinatorial Subtraction games (studied for example in the monumental work "Winning Ways", by Berlekamp, Conway and Guy), called Cumulative Subtraction (CS). Two players alternate in…

The concept of nimbers--a.k.a. Grundy-values or nim-values--is fundamental to combinatorial game theory. Nimbers provide a complete characterization of strategic interactions among impartial games in their disjunctive sums as well as the…

This paper concerns two-player alternating play combinatorial games (Conway 1976) in the normal-play convention, i.e. last move wins. Specifically, we study impartial vector subtraction games on tuples of nonnegative integers (Golomb 1966),…

This paper introduces a variant of the impartial combinatorial game nim, called tree nim, as well as a particular case of tree nim called tripod nim. A certain existence-uniqueness result and a periodicity result are proven about the…

Arithmetic functions in Number Theory meet the Sprague-Grundy function from Combinatorial Game Theory. We study a variety of 2-player games induced by standard arithmetic functions, such as Euclidian division, divisors, remainders and…

Given two finite sets of integers $S\subseteq\NNN\setminus\{0\}$ and $D\subseteq\NNN\setminus\{0,1\}$,the impartial combinatorial game $\IMARK(S,D)$ is played on a heap of tokens. From a heap of $n$ tokens, each player can moveeither to a…

Subtraction games are played with one or more heaps of tokens, with players taking turns removing from a single heap a number of tokens belonging to a specified subtraction set; the last player to move wins. We describe how to compute the…

The Games-Chan algorithm finds the minimal period of a periodic binary sequence of period $2^n$, in $n$ iterations. We generalise this to periodic $q$-ary sequences (where $q$ is a prime power) using generating functions and polynomials and…

In this survey we revisit {\sc finite subtraction}, one-heap subtraction games on finite rulesets. The main purpose is to give a general overview of the development, and specifically to draw attention to Flammenkamp's thesis (1997), where…

This thesis will be discussing scoring play combinatorial games and looking at the general structure of these games under different operators. I will also be looking at the Sprague-Grundy values for scoring play impartial games, and…

We introduce a class of normal play partizan games, called Complementary Subtraction. Let $A$ denote your favorite set of positive integers. This is Left's subtraction set, whereas Right subtracts numbers not in $A$. The Golden Nugget…

For impartial games $\Gamma$ and $\Gamma'$, the Sprague-Grundy function of the disjunctive sum $\Gamma + \Gamma'$ is equal to the Nim-sum of their Sprague-Grundy functions. In this paper, we introduce $p$-calm subtraction games, and show…

A finite impartial game is a two-player game in which the players take turns making moves and the game ends after finitely many moves. In this paper, we study a class of finite impartial games introduced by H.~Lenstra, which we call coin…

We determine the full nim-value structure of additive subtraction games in the {\em primitive quadratic} regime. The problem appears in Winning Ways by Berlekamp et al. in 1982; it includes a closed formula, involving Beatty-type {\em…

In this article, we study the behavior of a broad family of real sequences derived from randomized one-pile subtraction games. For any subtraction set $S$, we allow any valid number of chips $s\in S$ to be removed at equal probability at…

The Game of Cycles is a combinatorial game introduced by Francis Su in 2020 in which players take turns marking arrows on the edges of a simple plane graph, avoiding the creation of sinks and sources and seeking to complete a "cycle cell."…

We encode arbitrary finite impartial combinatorial games in terms of lattice points in rational convex polyhedra. Encodings provided by these \emph{lattice games} can be made particularly efficient for octal games, which we generalize to…