A Conjecture about Periods in Subtraction Games

A base-p Sprague-Grundy type theorem for p-calm subtraction games: Welter's game and representations of generalized symmetric groups

For impartial games $Γ$ and $Γ'$, the Sprague-Grundy function of the disjunctive sum $Γ+ Γ'$ is equal to the Nim-sum of their Sprague-Grundy functions. In this paper, we introduce $p$-calm subtraction games, and show that for $p$-calm subtraction games $Γ$ and $Γ'$, the Sprague-Grundy function of a $p$-saturation of $Γ+ Γ'$ is equal to the $p$-Nim-sum of the Sprague-Grundy functions of their $p$-saturations. Here a $p$-Nim-sum is the result of addition without carrying in base $p$ and a $p$-saturation of $Γ$ is an impartial game obtained from $Γ$ by adding some moves. It will turn out that Nim…

Subtraction games with FES sets of size 3

This paper extends the work done by Angela Siegel on subtraction games in which the subtraction set is N \ X for some finite set X. Siegel proves that for any finite set X, the G-sequence is ultimately arithmetic periodic, and that if |X| = 1 or 2, then it is purely arithmetic periodic. This note proves that if |X| = 3 then the G-sequence is purely arithmetic periodic. It is known that for |X| \geq 4 the sequence is not always purely arithmetic periodic.

二维减法——转移游戏

Two Dimensional Subtraction -- Transfer Games

本文推广了Tamás Lengyel的结果与猜想，证明了一类二维减法-转移游戏的nim值具有周期性，并计算了多种情况下的精确周期，同时提出了新的周期性概念。

The $\star$-operator and Invariant Subtraction Games

We study 2-player impartial games, so called \emph{invariant subtraction games}, of the type, given a set of allowed moves the players take turn in moving one single piece on a large Chess board towards the position $\boldsymbol 0$. Here, invariance means that each allowed move is available inside the whole board. Then we define a new game, $\star$ of the old game, by taking the $P$-positions, except $\boldsymbol 0$, as moves in the new game. One such game is $\W^\star=$ (Wythoff Nim)$^\star$, where the moves are defined by complementary Beatty sequences with irrational moduli. Here we give a …

Subtraction Division Games

A Subtraction-Division game is a two player combinatorial game with three parameters: a set S, a set D, and a number n. The game starts at n, and is a race to say the number 1. Each player, on their turn, can either move the total to n-s for some s in S or to \lceil\frac{n}{d}\rceil, for some d in D (i.e. either subtract s or divide by d and round up). This paper examines in detail the Sprague-Grundy sequences for the games when |S|=|D|=1. For a large class of these games, the patterns in the sequences allow for a characterization of all the winning positions. The paper also proves that, for t…

关于由随机化减法博弈产生的序列

On sequences arising from randomizing subtraction games

本文研究随机化单堆减法博弈中获胜概率序列的行为，利用线性递推关系刻画这些序列并分析其极限性质，完全解决了元素少于3个的减法集合情形并部分解决一般情形。

Invariant and dual subtraction games resolving the Duchê-Rigo conjecture

We prove a recent conjecture of Duchêne and Rigo, stating that every complementary pair of homogeneous Beatty sequences represents the solution to an \emph{invariant} impartial game. Here invariance means that each available move in a game can be played anywhere inside the game-board. In fact, we establish such a result for a wider class of pairs of complementary sequences, and in the process generalize the notion of a \emph{subtraction game}. Given a pair of complementary sequences $(a_n)$ and $(b_n)$ of positive integers, we define a game $G$ by setting $\{\{a_n, b_n\}\}$ as invariant moves.…

Comply/Constrain Subtraction

A comply/constrain game or a game with a Muller twist is a game where the next player is allowed to place constraints on opponent's next move. We develop a closed form formula for the Grundy value of the single-pile subtraction game where the next player may determine whether the previous player has to select a move from the set of some first k natural numbers or its complement. We also investigate the periodicity of Grundy values when the set of legal moves is from a set of finite arithmetic sequences.

On the expansion of three-element subtraction sets

We study the periodicity of nim-sequences for subtraction games having subtraction sets with three elements. In particular, we give solutions in several cases, and we describe how these subtraction sets can be augmented by additional numbers without changing the nim-sequences. The paper concludes with a conjecture on ultimately bipartite subtraction games.

Combinatorial Analysis of a Subtraction Game on Graphs

We define a two-player combinatorial game in which players take alternate turns; each turn consists on deleting a vertex of a graph, together with all the edges containing such vertex. If any vertex became isolated by a player's move then it would also be deleted. A player wins the game when the other player has no moves available. We study this game under various viewpoints: by finding specific strategies for certain families of graphs, through using properties of a graph's automorphism group, by writing a program to look at Sprague-Grundy numbers, and by studying the game when played on rand…

连通子图减法游戏在细分星图上的应用

Connected Subtraction Games on Subdivided Stars

研究连通子图减法游戏在细分星图上的周期性规律及特定规则下的游戏结果。

Computational Hardness of Multidimensional 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 an existence of a game in this class such that solving the game is PSPACE-hard. The results are based on the construction introduced by Larsson and Wästlund. It relates subtraction games and cellular automata.

带一次传递的减法尼姆游戏

A Subtraction Nim with a Pass

本文研究带一次传递规则的减法尼姆游戏，证明其格鲁恩迪数仍满足逆mex性质，且传递规则不影响游戏复杂性。

Cumulative subtraction games

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 moving, and get points for taking pebbles out of a joint pile. We prove that the outcome in optimal play (game value) of a CS with a finite number of possible actions is eventually periodic, with period $2s$, where $s$ is the size of the largest available action. This settles a conjecture by Stewart in his Ph.D. thesis (2011). Specifically, we find a quadratic b…

Comply subtraction games avoiding arithmetic progressions

Impartial subtraction games on the nonnegative integers have been studied by many and discussed in detail in for example the remarkable work Winning Ways by Conway, Berlekamp and Guy. We describe how comply variations of these games, similar to those introduced by Holshouser, Reiter, Smith, Stănică, can be defined as having its sets of winning positions identical to well-known sets avoiding arithmetic progressions such as $x+z=2y$, studied by Szerkeres, Erd\H os and Turán, and many others, thus exploring a new territory combining ideas from combinatorial games and combinatorial number theory. …

纯周期三步减法游戏

Purely Periodic Three-move Subtraction Games

本文研究三步减法游戏S={a,b,c}的纯周期性，给出纯周期的显式充分准则，确定满足准则的非可加集合的最小周期、P位置及nim值，还对可加情况单独处理并证明准则在特定条件下的必要性。

加法沉降减法

Additive sink subtraction

本文研究加法沉降减法游戏，证明其nim序列具有显式线性或二次周期，提出与经典墙减法的对偶性猜想。

自我利益累积减法游戏中的 tie-breaking 问题

Tie-breaking in self interest cumulative subtraction games

本文研究了自我利益累积减法游戏中，通过引入tie-breaking规则单调性，证明了在两动作减法集情况下，友好玩家优于对抗性玩家。

多维减法博弈

Subtraction games in more than one dimension

本文研究多维公平向量减法博弈，通过推广P到P原理和着色自动机求解二维三移动规则集实例，证明行列最终周期性，并提出正则性相关猜想。

加法减法游戏

Additive Subtraction Games

本文研究了加法减法游戏在原始二次情形下的全nim值结构，通过数论方法证明了每个nim值序列是经典P位置的线性移位。

Partizan Subtraction Games

Partizan subtraction games are combinatorial games where two players, say Left and Right, alternately remove a number n of tokens from a heap of tokens, with $n \in S_L$ (resp. $n \in S_R$) when it is Left's (resp. Right's) turn. The first player unable to move loses. These games were introduced by Fraenkel and Kotzig in 1987, where they introduced the notion of dominance, i.e. an asymptotic behavior of the outcome sequence where Left always wins if the heap is sufficiently large. In the current paper, we investigate the other kinds of behaviors for the outcome sequence. In addition to dominan…

全支撑与截断支撑的对抗减法游戏

Partizan Subtraction with Full and Truncated Support

本文研究全支撑与截断支撑的对抗减法游戏，明确移动集更丰富的玩家除有限情况外总能获胜，发现中间平衡截断水平会产生非平凡周期性及无限多P、N-位置。

Finding Golden Nuggets by Reduction

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 Subtraction Game has the $A$ and $B$ sequences, from Wythoff's game, as the two complementary subtraction sets. As a function of the heap size, the maximum size of the canonical forms grows quickly. However, the value of the heap is either a number or, in reduced canonical form, a switch. We find the switches by using properties of the Fibonacci word and standard…

减法游戏中获胜位置的超多项式周期长度

Superpolynomial period lengths of the winning positions in the subtraction game

本文研究减法游戏中获胜位置序列的周期长度，证明初始种子在集合大小为1或2时不能改变周期长度，但在大小为3时可使周期长度变为超多项式函数。

具有连续参数的减法尼姆游戏

Subtraction Nim with Continuous Parameters

研究减法尼姆游戏中移除数集S为有限正实数时的周期与尼姆值函数，给出了S为三元集时纯周期性的充分条件及周期公式。

论有终局偏序的减法取物游戏

On Ending Partizan Subtraction Nim

研究一种在游戏结束时根据剩余棋子奇偶性判定胜负的偏序减法取物游戏，发现大多数情况下所有足够大的n都是L位置，并给出定理解释该现象。

弧凯勒斯与非断开弧凯勒斯的复杂性及算法

Complexity and algorithms for Arc-Kayles and Non-Disconnecting Arc-Kayles

本文研究了非断开弧凯勒斯的计算复杂性，证明其在结构化图类上为PSPACE完全，并给出了多项式时间算法和二次核。

On the additive period length of the Sprague-Grundy function of certain Nim-like games

We examine the structure of the additive period of the Sprague-Grundy function of Nim-like games, among them Wythoff's Game, and deduce a bound for the length of the period and preperiod.

有向图上并行筹码点火博弈的周期与原子发射序列

Periods and atomic firing sequences of parallel chip-firing games on directed graphs

本文研究有向图上并行筹码点火博弈的周期与原子发射序列，证明无向图核心结果的有向图对应版本，并给出强连通有向图中顶点原子发射序列的完整刻画。

贪心芯片博弈中的原始周期

Primitive periods in greedy chip-firing games

本文研究贪心芯片博弈的周期性质，证明饥饿博弈及单调得分推广中所有周期轨道的最小周期均为 T，并解决了 Li 和 Propp 的两个猜想，同时揭示了与主席分配问题的联系。

On the Period Lengths of the Parallel Chip-Firing Game

The parallel chip-firing game is a periodic automaton on graphs in which vertices "fire" chips to their neighbors. In 1989, Bitar conjectured that the period of a parallel chip-firing game with n vertices is at most n. Though this conjecture was disproven in 1994 by Kiwi et. al., it has been proven for particular classes of graphs, specifically trees (Bitar and Goles, 1992) and the complete graph K_n (Levine, 2008). We prove Bitar's conjecture for complete bipartite graphs and characterize completely all possible periods for positions of the parallel chip-firing game on such graphs. Furthermor…

Cofinite Induced Subgraphs of Impartial Combinatorial Games: An Analysis of CIS-Nim

Given an impartial combinatorial game G, we create a class of related games (CIS-G) by specifying a finite set of positions in G and forbidding players from moving to those positions (leaving all other game rules unchanged). Such modifications amount to taking cofinite induced subgraphs (CIS) of the original game graph. Some recent numerical/heuristic work has suggested that the underlying structure and behavior of such "CIS-games" can shed new light on, and bears interesting relationships with, the original games from which they are derived. In this paper we present an analytical treatment of…

康威生命游戏是全周期的

Conway's Game of Life is Omniperiodic

本文宣布康威生命游戏最后两个缺失周期19和41的振荡器被发现，从而证明生命游戏是全周期的，并系统梳理了全周期问题的研究历史与解决策略。

满足$2|E|-|V|<|σ|<2|E|$的所有并行芯片点火游戏都具有周期2

All parallel chip-firing games with $2|E|-|V|<|σ|<2|E|$ have period $2$

该研究使用GPT-5.6-Sol证明了2024年Ji等人关于并行芯片点火游戏活动为$\tfrac12$时存在通用界的猜想，推广了魔鬼阶梯的中间层并统一了多类图的相关结果。

Generalized XOR non-locality games with graph description on a square lattice

We propose a family of non-locality unique games for 2 parties based on a square lattice on an arbitrary surface. We show that, due to structural similarities with error correction codes of Kitaev for fault tolerant quantum computation, the games have classical values computable in polynomial time for $d=2$ measurement outcomes. By representing games in their graph form, for arbitrary $d$ and underlying surface we provide their classification into equivalence classes with respect to relabeling of measurement outcomes, for a selected set of permutations which define the winning conditions. A ca…