Koki Suetsugu, Tomoaki Abuku
In this paper, we study an impartial game called Delete Nim. In this game, there are two heaps of stones. The player chooses one of the heaps and delete the other heap. Next, she takes away one stone from the chosen heap and optionally splits it into two heaps. We show the way to calculate the G-value of this game by using the OR operation and 2-adic valuation.
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The Sprague-Grundy (SG) theory reduces the sum of impartial games to the classical game of $NIM$. We generalize the concept of sum and introduce $\cH$-combinations of impartial games for any hypergraph $\cH$. In particular, we introduce the game $NIM_\cH$ which is the $\cH$-combination of single pile $NIM$ games. An impartial game is called SG decreasing if its SG value is decreased by every move. Extending the SG theory, we reduce the $\cH$-combination of SG decreasing games to $NIM_\cH$. We call $\cH$ a Tetris hypergraph if $NIM_\cH$ is SG decreasing. We provide some necessary and some suffi…
Cofinite Induced Subgraphs of Impartial Combinatorial Games: An Analysis of CIS-Nim
给定一个公平的组合博弈G，我们通过在G中指定一个有限集的位置，并禁止参与者移动到这些位置来创建一类相关博弈（CIS-G）（保留所有其他博弈规则不变）。这样的修改相当于取原博弈图的余有限诱导子图（CIS）。最近的一些数值/启发式的工作表明，这种“CIS游戏”的基本结构和行为可以揭示新的光，并承担有趣的关系，从原来的游戏，他们是派生。本文给出了（三堆）Nim博弈的上有限诱导子图的一个解析处理。这是CIS博弈中最简单的非平凡情形之一。我们的主要发现是，尽管CIS-Nim博弈中获胜策略的结构与Nim博弈中获胜策略的结构有很大不同，但CIS-Nim博弈继承了Nim博弈中的周期二尺度不变性。
Nim is a well-known combinatorial game with several variants, e.g., Delete Nim and Variant Delete Nim. In Variant Delete Nim, the player deletes one of the two heaps of stones and splits the other heap on his/her turn. In this paper, we discuss generalized Variant Delete Nim, which generalizes the number of stone heaps to three or more, All-but-one-delete Nim, Half-delete Nim, No-more-than-half-delete Nim, and Single-delete Nim. We study the win-loss conditions for each of these games.
The study of the combinatorial game Nim and its variants is rich and varied, but little is known of the game Nim with a Pass. It is Nim, but once per game a player is permitted to skip their turn but this can only be done if a nonempty pile remains. In this paper we define a new binary operation on games which we use to prove novel properties of Nim with a Pass, as well as games which are generalizations of it. Most importantly, we make small progress on finding the $\mathcal{P}$-positions and optimal strategies of the game.
We determine the misère equivalence classes of Nim positions under two equivalence relations: one based on playing disjunctive sums with other impartial games, and one allowing sums with partizan games. In the impartial context, the only identifications we can make are those stemming from the known fact about adding a heap of size 1. In the partizan context, distinct Nim positions are inequivalent.
Nimbers are inevitable
这篇文章关注公正的组合游戏的决议，特别是游戏，可以在独立的立场和分裂。我们证明，为了计算的结果，一个独立的位置，它总是更有效地分别计算每个独立的位置比直接开发的对策树的总和的号码。因此，敏捷度的概念对于解决公平博弈是不可避免的，即使我们只是试图确定一个起始位置的输赢结果。我们还描述了算法，以有效地使用nimbers，最后，我们给出了审查的结果，两个公正的游戏：发芽和克拉姆。
Some results on split sum of impartial games
本文针对带通过的Nim游戏这一组合游戏难题，扩展拆分和的应用，明确其与经典析取和的等价情形，推广Grundy值并揭示其在拆分和下的不同性质，为公平游戏研究提供新结果。
In this paper, we study a colorful, impartial combinatorial game played on a two-dimensional grid, Transverse Wave. We are drawn to this game because of its apparent simplicity, contrasting intractability, and intrinsic connection to two other combinatorial games, one inspired by social influence and another inspired by quantum superpositions. More precisely, we show that Transverse Wave is at the intersection of social-influence-inspired Friend Circle and superposition-based Demi-Quantum Nim. Transverse Wave is also connected with Schaefer's logic game Avoid True. In addition to analyzing the…
Chocolate bar games are variants of the game of Nim in which the goal is to leave your opponent with the single bitter part of the chocolate bar. The rectangular chocolate bar game is a thinly disguised form of classical multi-heap Nim. In this work, we investigate the mathematical structure of triangular chocolate bar games in which the triangular chocolate bar can be cut in three directions. In the triangular chocolate bar game, a position is a $\mathcal{P}$-position if and only if $x \oplus y \oplus z = 0$, where the numbers $x,y,z$ stand for the maximum number of times that the chocolate b…
Single-delete Nim
本文将 Delete Nim 游戏推广到 n 堆石子场景，每回合移除一堆并拆分另一堆，通过分析 n=4 的情况推导获胜策略条件，拓展了经典 Nim 游戏的变体研究。
Impartial Scoring Play Games
在本文中，我们将研究公平得分游戏。我们首先给出了什么是公平得分博弈的基本定义，并在析取和下研究了它们的一般结构。然后，我们将研究nim和所有八进制博弈，并定义一个函数来帮助我们分析这些博弈。最后，我们将查看该函数具有的属性，并对该函数表现出的行为做出许多猜测。
Isomorphism Theorems for Impartial Combinatorial Games
该研究引入选项图与选项保持映射的范畴作为无偏组合博弈研究模型，证明泛代数四同构定理在此范畴成立，提出商选项图可简化博弈分析。
Circular Nim Games
圆形尼姆游戏是一个两人公平的组合游戏，由放置在一个圆圈中的n堆代币组成。移动包括选择k个连续的堆栈，并从k个堆栈中的一个或多个堆栈中取出至少一个令牌。最后一个能移动的玩家获胜。我们证明了结果的结构上的损失的位置为小n和k，并提出了一些有待进一步研究的问题。
Global Fibonacci Nim
斐波那契尼姆是一种流行的公平组合游戏，通常用一堆石头玩。这个游戏很吸引人，因为它与斐波那契数和泽肯多夫表示法有着惊人的联系。在这篇文章中，我们研究了多堆石头的变体的一些性质，并解决了2堆的情况。玩家选择其中一个牌堆，然后像斐波那契尼姆那样进行游戏，但这里的移动大小限制是一个全局参数，对任何牌堆都有效。
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 winnability. In this paper, we initiate a study of nimber-preserving reductions among impartial games. These reductions enhance the winnability-preserving reductions in traditional computational characterizations of combinatorial games. We prove that Generalized Geography is complete for the natural class, $\cal{I}^P$ , of polynomially-short impartial rulesets…
Tree and Tripod Nim
本文提出尼姆游戏的新变体树尼姆及其特例三脚架尼姆，证明其P位置与Grundy值的分布性质，并借助序列复合游戏中的数组给出部分分析及周期猜想。
We study a game where two players take turns selecting points of a convex geometry until the convex closure of the jointly selected points contains all the points of a given winning set. The winner of the game is the last player able to move. We develop a structure theory for these games and use it to determine the nim number for several classes of convex geometries, including one-dimensional affine geometries, vertex geometries of trees, and games with a winning set consisting of extreme points.
Playing impartial games under the normal and misere conventions may differ a lot. However, there are also many "exceptions" for which the normal and misere plays are very similar. As early as in 1901 Bouton noticed that this is the case with the game of Nim. In 1976 Conway introduced a large class of such games that he called tame games. Here we introduce a proper subclass, pet games, and a proper superclass, domestic games. For each of these three classes we provide an efficiently verifiable characterization based on the following property. These games are closely related to another important…
Fractal Sequences and Restricted Nim
公平博弈G的格伦迪数是唯一尼姆堆的大小等于G。我们介绍了一个新的变种尼姆，限制尼姆，它限制了数量的石头，球员可以从一个堆的大小方面删除堆。某些类的限制尼姆被发现产生序列的Grundy数与自相似的分形结构。C. Kimberling，我们得到了这些“分形序列”的新的特征，并给出了这些序列和某些上三角阵列之间的双射。作为一个特殊情况，我们得到了串行尼姆游戏，其中尼姆堆是从左到右排列的，玩家只能在最左边的非空堆中移动。
Slow $k$-Nim
给定$n$牌堆和一个正整数$k \leq n$，我们研究了以下两个公平组合博弈Nim $^1_{n, \leq k}$和Nim $^1_{n, =k}$ .在第一个（resp。第二）游戏，玩家通过一步棋选择至少$1$和至多（分别地）$1$和$1$。精确地）$k$非空堆，并从这些堆中的每一个中移除一个令牌。对于每个游戏的正常和misère版本，我们计算$n = k = 2$和$n = k+1 = 3$情况下的Sprague-Grundy函数。对于博弈Nim $^1_{n, \leq k}$，我们也刻画了它在$n \leq k+2$和$n = k+3 \leq 6$情形下的P-位置。
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…
Impartial Avoidance Games on Convex Geometries
本文研究了凸几何中的无偏回避游戏，确定了游戏的nim数，并探讨了树结构和极点集合等不同场景下的游戏规则与结果。
Row Impartial Terminus
提出整数分拆上的公平组合游戏Row Impartial Terminus，证明其局面可分解为核心与残部，Conway对等价于残部对应的Nim游戏局面，据此给出完整获胜策略并完成层级分类。
On Aperiodic Subtraction Games with Bounded Nim Sequence
减法游戏是一类公平的组合游戏，其位置对应于非负整数，其移动对应于从当前位置减去一组固定数字中的一个。尽管减法游戏很容易定义，但事实证明它很难分析.特别是，很少有一般的结果，他们的Sprague-Grundy值是已知的。在本文中，我们构建了一个减法游戏的例子，其序列的Sprague-Grundy值是三元和非周期性的，我们开发了一个理论，可能会导致我们的建设推广。
Maharaja Nim, Wythoff's Queen meets the Knight
新的组合游戏介绍，其中最相关的是王公尼姆。规则延伸了著名的Wythoff Nim的公平游戏，其中两名球员轮流在一个大棋盘上移动一个国际象棋皇后，试图成为第一个把她放在左下角的人。在这里，除了经典的规则，玩家还可以像国际象棋骑士移动一样移动皇后。我们证明了Maharaja Nim的第二个玩家的获胜位置与Wythoff Nim的获胜位置接近，即它们分别在斜率为$\frac{\sqrt{5}+1}{2}$和$\frac{\sqrt{5}-1}{2}$的直线的有界距离内。对于一个近亲王公尼姆，骑士的跳跃的形式是$(2,3)$和$(3,2)$（而不是$(1,2)$和$(2,1)$），我们也证明了多项式时间复杂性的判定问题的结果，一个给定的位置。
On the Nature and Complexity of an Impartial Two-Player Variant of the Game Lights-Out
本文研究了 Lights-Out 的双人公平博弈变体，目标是在简单图上做出最后一步合法移动，并利用递归算法计算了该游戏在 2 x n 网格图和部分广义 Petersen 图上的 Nimbers。
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 assigned to them. However, there are rulesets with {\em entailing moves} that break the alternating play axiom and/or restrict the other player's options within the disjunctive sum components. These situations are exemplified in the literature by a ruleset such as {\sc nimstring}, a normal play variation of the classical children's game {\sc dots \& boxes}, and {…
We introduce and analyse an extension of the disjunctive sum operation on some classical impartial games. Whereas the disjunctive sum describes positions formed from independent subpositions, our operation combines positions that are not completely independent but interact only in a very restricted way. We extend the games Nim and Silver Dollar, played by moving counters along one-dimensional strips of cells, by joining several strips at their initial cell. We prove that, in certain cases, computing the Sprague-Grundy function can be simplified to that of a simpler game with at most two tokens…
Circular Nim CN(7,4)
本文研究圆形 Nim 游戏 CN(7,4)，刻画其 $\mathcal{P}$-局面结构，并发现其分裂为多个子集，区别于同类已知游戏。
Impartial achievement and avoidance games for generating finite groups
研究生成有限群的两个无偏博弈，利用结构图工具确定阿贝尔群和二面体群的nim数，并提出基于计算的猜想。
We study an impartial game introduced by Anderson and Harary. This game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for generalized dihedral groups, which are of the form $\operatorname{Dih}(A)= \mathbb{Z}_2 \ltimes A$ for a finite abelian group $A$.
We examine short combinatorial games for three or more players under a new play convention in which a player who cannot move on their turn is the unique loser. We show that many theorems of impartial and partizan two-player games under normal play have natural analogues in this setting. For impartial games with three players, we investigate the possible outcomes of a sum in detail, and determine the outcomes and structure of three-player Nim.
Impartial geodetic building games on graphs
研究Buckley和Harary提出的图上测地凸构建博弈变体，两名玩家轮流选顶点直到凸包过大，确定了若干图族此类成就与避免博弈的nim值。
Impartial avoidance games for generating finite groups
研究有限群上的公正回避博弈，通过极大子群判定nim数，并应用于幂零、零星及对称群。
Impartial achievement games for generating nilpotent groups
研究Anderson和Harary提出的公平博弈，确定形如$T \ imes H$（$T$为$2$-群，$H$为奇数阶群）的有限群的nim数，涵盖所有幂零群和阿贝尔群。
Impartial Combinatorial Games and the Nuclear Escalation Ladder
将赫尔曼·卡恩的升级阶梯建模为无偏组合博弈，通过重新索引阶梯为减法博弈并采用反常规则，证明了单阶梯稳定性由同余条件决定，多战区联合稳定性在正常规则下为Nim和，在反常规则下需用反常商群刻画。
Impartial removing games on grid graphs
本文研究网格图上的两种无偏移除博弈（达成与回避），利用延迟博弈图等工具确定了其尼姆值，并给出格图的部分结果。
The spectrum of nim-values for achievement games for generating finite groups
本文研究Anderson和Harary提出的有限群生成成就博弈，证明其nim值谱为{0,1,2,3,4}，肯定回答了两个猜想。
Impartial avoidance and achievement games for generating symmetric and alternating groups
本研究分析对称群与交错群上的两种公正博弈，通过计算尼姆数确定其胜负结果，为有限群生成集博弈提供了理论判定。
The authors introduce the impartial game of the generalized Ryūō Nim, a variant of the classical game of Wythoff Nim. In the latter game, two players take turns in moving a single queen on a large chessboard, attempting to be the first to put her in the upper left corner, position $(0,0)$. Instead of the queen used in Wythoff Nim, we use the generalized Ryūō for a given natural number $p$. The generalized Ryūō for $p$ can be moved horizontally and vertically, as far as one wants. It also can be moved diagonally from $(x,y)$ to $(x-s,y-t)$, where $s,t$ are non-negative integers such that $1 \le…
Impartial geodetic removing games on graphs
研究两位玩家轮流选择图中顶点，直到联合未选顶点的凸包过小，最后移动者获胜的两种游戏，并计算了多种图族的nim值。
Blocking Wythoff Nim
Wythoff Nim的2人公平游戏在两堆代币上进行。移动包括从其中一堆中删除任何数量的代币，或者从两堆中删除相同数量的代币。获胜者是移除最后一个代币的玩家。我们研究这个游戏的封锁策略，也就是说，对于每一个移动，下一个球员移动之前，前一个球员可以宣布最多一个预定的数量，$k - 1 \ge 0$，禁止的选项。当下一个玩家移动时，任何阻挡动作都会被遗忘，对游戏没有任何进一步的影响。我们解决了这个游戏的$k = 2$和$k = 3$的获胜策略，并支持计算机模拟，状态的渐近“行为”的$P$ -位置为各自的游戏时，$4 \le k \le 20$。
The $\star$-operator and Invariant Subtraction Games
我们研究2人公平游戏，即所谓的\emph{invariant subtraction games}，给定一组允许的移动，玩家轮流将大棋盘上的一颗棋子移向位置$\boldsymbol 0$。在这里，不变性意味着每个允许的移动在整个棋盘中都是可用的。然后我们定义一个新的博弈，旧博弈的$\star$，通过把除了$\boldsymbol 0$之外的$P$ -位置作为新博弈中的移动。一个这样的游戏是$\W^\star=$（Wythoff Nim）$^\star$，其中移动由具有无理模的互补贝蒂序列定义。本文给出了$\W^\star$的无穷多个$P$ -位置的多项式时间算法. $\star$的重复应用结果是为不变减法游戏的某个子类提供了特别好的性质，\emph{permutation games}，我们在这里介绍。我们还介绍了\emph{ornament games}家族，其$P$ -位置定义了具有有理模的互补Beatty序列-因此与A有关。S. Fraenkel的'变种'老鼠和老鼠游戏-并给出了封闭的形式为这样的游戏的举动。我们还证明了（$k$ -pile Nim）$^{\star\star}$ = $k$ -pile Nim.
A complete solution for the partisan chocolate game
本文提出一种受chomp启发的党派巧克力博弈，通过证明其等价于blue-red hackenbush strings，证明了博弈值为数字并给出了多巧克力棒时的最优算法。
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。
Games as recursive coalgebras: A categorical view on the Nim-sum
本文从范畴论视角利用递归余代数重释组合博弈与尼姆和，分别为博弈论、范畴论研究者提供对应框架与性质良好的博弈范畴，文末列出7个开放问题。
We settle two long-standing complexity-theoretical questions-open since 1981 and 1993-in combinatorial game theory (CGT). We prove that the Grundy value (a.k.a. nim-value, or nimber) of Undirected Geography is PSPACE-complete to compute. This exhibits a stark contrast with a result from 1993 that Undirected Geography is polynomial-time solvable. By distilling to a simple reduction, our proof further establishes a dichotomy theorem, providing a "phase transition to intractability" in Grundy-value computation, sharply characterized by a maximum degree of four: The Grundy value of Undirected Geog…
$(1,2)$-GDWN splits
我们研究了2个无序堆的令牌$(x,y)$的有限非负数的公平拿走游戏。两个玩家根据一定的规则交替从各自的堆中取出至少一个和最多所有的代币，并且当玩家反过来无法移动时游戏终止。我们遵循正常的游戏规则，这意味着不能移动的玩家输了。在Wythoff Nim游戏中，玩家可以从其中一堆中取出任意数量的代币，或者从两堆中取出相同数量的代币。令$ϕ= \frac{1+\sqrt{5}}{2}$和对于所有非负整数$n$，$A_n=\lfloorϕn \rfloor$和$B_n=A_n+n$。Wythoff Nim的P-位置是分别具有$A_n$和$B_n$令牌的所有桩对。我们研究了一个概括的这个游戏称为PDMATHAMT，其中，除了Wythoff尼姆的规则，玩家可以选择从其中一个桩中删除一个积极的令牌数量，并从另一个桩中删除该数量的两倍。我们证明了对于给定的实数$α>1$和$ε> 0$，存在一个无穷扇区$α\le y/x \le α+ε$，其中每个$(x,y)$是一个N-位置，但是对于$1\le y/x <α$和$α+ε< y/x $，存在无穷多个P-位置.这证明了最近一篇论文中的一个猜想。也就是说，邻接的集合在$(1,2)\G$ \emph{splits}中移动Wythoff Nim的斜率$ϕ$ P-位置的梁。我们还提供了一个下界上的较低的渐近密度较低的堆高度的P-位置的Wythoff尼姆的扩展。设…
Ordinal Sums with Substitution of Impartial Games
本文提出一种新的序数和游戏，称为带有替代的序数和，研究其基本性质并推导出其变体集合的简单公式，同时给出链式结构的Grundy数表达式及例子。
GM-rule and its applications to impartial games
本文定义了适用于满足特定条件整数向量的GM-rule，证明了当向量极差不超过ℓ时的三个关键性质，实现了线性时间的向量序列计算，并介绍了该规则在ℓ=2时的变体及在无偏博弈中的应用背景。
Computing Remoteness Functions of Moore, Wythoff, and Euclid's games
该研究针对无偏博弈的距离函数$\u211b$，给出了Euclid游戏、广义Wythoff游戏的$\u211b$多项式算法，以及Moore's NIM偶数值的显式公式，并证明其奇数值计算为NP难问题。
Enforce and selective operators of combinatorial games
本文研究了公平规则集上的“强制算子”与“选择算子”，给出了支配关系的充要条件，探讨了类分配格结构，并证明了强制规则集下Sprague-Grundy理论依然成立。
One-Dimensional Peg Solitaire, and Duotaire
我们解决的问题，一维钉接龙。特别是，我们表明，一组配置，可以减少到一个单一的挂钩形式的正规语言，并存在一个线性时间算法减少任何配置的最小数量的挂钩。然后我们来看看拉维库马尔提出的公平的两人游戏，其中两名玩家轮流做出挂钩动作，哪个玩家没有动作就输了。我们计算了一些简单的尼姆值，并讨论了当游戏分离成一个小游戏的析取和。在一系列的跳跃可以在一个单一的移动的版本中，我们表明，无论是P-位置，也不是N-位置（即前一个或下一个玩家的胜利）是由一个定期或上下文无关的语言描述。
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