Jens Askgaard
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.
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We analyze the Sprague-Grundy functions for a class of almost disjoint selective compound games played on Nim heaps. Surprisingly, we find that these functions behave chaotically for smaller Sprague-Grundy values of each component game yet predictably when any one heap is sufficiently large.
We prove three conjectures of Fraenkel and Ho regarding two classes of variants of Wythoff's game. The two classes of variants of Wythoff's game feature restrictions of the diagonal moves. Each conjecture states that the Sprague-Grundy function is invariant up to a certain nim-value for a subset of that class of variant of Wythoff's game. For one class of variants of Wythoff's game, we prove that the invariance of the Sprague-Grundy function extends beyond what was conjectured by Fraenkel and Ho.
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…
We consider misère Nim as a normal-play game obtained from Nim by removing the terminal position. While explicit formulas are known for the Sprague-Grundy functions of Nim and Welter's game, no explicit formula is known for that of misère Nim. All three of these games can be considered as position restrictions of Nim. What are the differences between them? We point out that Nim and Welter's game are saturated, but misère Nim is not. Moreover, we present explicit formulas for the Sprague-Grundy functions of saturations of misère Nim, which are obtained from misère Nim by adjoining some moves.
We consider a generalization of the classical game of $NIM$ called hypergraph $NIM$. Given a hypergraph $\cH$ on the ground set $V = \{1, \ldots, n\}$ of $n$ piles of stones, two players alternate in choosing a hyperedge $H \in \cH$ and strictly decreasing all piles $i\in H$. The player who makes the last move is the winner. Recently it was shown that for many classes of hypergraphs the Sprague-Grundy function of the corresponding game is given by the formula introduced originally by Jenkyns and Mayberry (1980). In this paper we characterize symmetric hypergraphs for which the Sprague-Grundy f…
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…
Sprague-Grundy Value for Common Divisor Nim
本文针对任意正整数参数 $m$，求解 Common Divisor Nim$_m$（CDN$_m$）的 Sprague-Grundy 值。
Moore's generalization of the game of {\sc Nim} is played as follows. Let $n$ and $k$ be two integers such that $1 \leq k \leq n$. Given $n$ piles of tokens, two players move alternately, removing tokens from at least one and at most $k$ of the piles. The player who makes the last move wins. The game was solved by Moore in 1910 and an explicit formula for its Sprague-Grundy function was given by Jenkyns and Mayberry in 1980, for the case $n = k+1$ only. We introduce another generalization of {\sc Nim}, called {\sc Exact $k$-Nim}, in which each move reduces exactly $k$ piles. We give an explici…
We consider a generalization of the classical game of $NIM$ called hypergraph $NIM$. Given a hypergraph $\cH$ on the ground set $V = \{1, \ldots, n\}$ of $n$ piles of stones, two players alternate in choosing a hyperedge $H \in \cH$ and strictly decreasing all piles $i\in H$. The player who makes the last move is the winner. In this paper we give an explicit formula that describes the Sprague-Grundy function of hypergraph $NIM$ for several classes of hypergraphs. In particular we characterize all $2$-uniform hypergraphs (that is graphs) and all matroids for which the formula works. We show tha…
The classical game of {\sc Nim} can be naturally extended and played on an arbitrary hypergraph $\cH \subseteq 2^V \setminus \{\emptyset\}$ whose vertices $V = \{1, \ldots, n\}$ correspond to piles of stones. By one move a player chooses an edge $H$ of $\cH$ and reduces arbitrarily all piles $i \in H$. In 1901 Bouton solved the classical {\sc Nim} for which $\cH = \{\{1\}, \ldots, \{n\}\}$. In 1910 Moore introduced and solved a more general game $k$-{\sc Nim}, for which $\cH = \{H \subseteq V \mid |H| \leq k\}$, where $1 \leq k < n$. In 1980 Jenkyns and Mayberry obtained an explicit formula fo…
For a collection of papers in memory of Elwyn Berlekamp (1940-2019), John Conway (1937-2020), and Richard Guy (1916-2020). The Sprague-Grundy theory for finite games without cycles was extended to general finite games by Cedric Smith and by Aviezri Fraenkel and coauthors. We observe that the same framework used to classify finite games also covers the case of locally finite games (that is, games where any position has only finitely many options). In particular, any locally finite game is equivalent to some finite game. We then study cases where the directed graph of a game is chosen randomly, …
Given a hypergraph $\cH \subseteq 2^I \setminus \{\emptyset\}$ on the ground set $I = \{1, \ldots, n\}$, we assign to each $i \in I$ a nonnegative integer $x_i$, that is a pile of $x_i$ tokens, and consider the following generalization of the classical game of {\sc Nim}: Two players alternate turns. In a move a player chooses an arbitrary edge $H \in \cH$ and reduces all piles $i \in H$. The player who is out of moves loses. We call the obtained game hypergraph {\sc Nim}. Such a game is called proper {\sc Nim}, when $\cH=2^I \setminus\{I,\emptyset\}$ is the family of all proper subsets of $I$.…
Some results on LCTR, an impartial game on partitions
本文研究分拆公平博弈 LCTR，运用 Sprague-Grundy 定理证明其 SG 值不超过 2，刻画多类无限分拆族，并给出 O(n) 动态规划算法。
In this article, we investigate the Grundy values of the popular game of Fibonacci nim. The winning strategy, which amounts to understanding positions of Grundy value 0, was known since Whinihan in 1963. In this paper, we extend Whinihan's analysis by computing all the positions of Grundy value at most 3. Furthermore, we show that, when we delete the Fibonacci numbers (which have Grundy value 0), the Grundy values of the starting positions are increasing, and we give upper and lower bounds on the growth rate.
We introduce a restriction of Wythoff's game, which we call F-Wythoff, in which the integer ratio of entries must not change if an equal number of tokens are removed from both piles. We show that P-positions of F-Wythoff are exactly those positions obtained from P-positions of Wythoff's game by adding 1 to each entry. We describe the distribution of Sprague-Grundy values and, in particular, generalize two properties on the distribution of those positions which have Sprague-Grundy value k, for a given k, for variants of Wythoff's game. We analyze the misere F-Wythoff and show that the normal an…
Nim on Integer Partitions and Hyperrectangles
本文研究尼姆游戏的两个变体 PNim 和 RNim，分别以整数分拆和超矩形为操作对象，给出了 PNim 的 Sprague-Grundy 值上界及值为 1 的刻画、RNim 的完整公式，并完成二者在 Conway-Gurvich-Ho 层级中的分类。
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…
Sprague-Grundy values and complexity for LCTR
本文研究无偏组合博弈 LCTR，通过镜像策略和结构性分析，将 Sprague-Grundy 值的计算复杂度从 O(n) 改进到 O(log n)，并分析了相关复杂度度量。
Subtraction games are a class of impartial combinatorial games whose positions correspond to nonnegative integers and whose moves correspond to subtracting one of a fixed set of numbers from the current position. Though they are easy to define, sub- traction games have proven difficult to analyze. In particular, few general results about their Sprague-Grundy values are known. In this paper, we construct an example of a subtraction game whose sequence of Sprague-Grundy values is ternary and aperiodic, and we develop a theory that might lead to a generalization of our construction.
We study a variant of 3-pile Nim in which a move consists of taking tokens from one pile and, instead of removing then, topping up on a smaller pile provided that the destination pile does not have more tokens then the source pile after the move. We discover a situation in which each column of two-dimensional array of Sprague-Grundy values is a palindrome. We establish a formula for P-positions by which winning moves can be computed in quadratic time. We prove a formula for positions whose Sprague-Grundy values are 1 and estimate the distribution of those positions whose nim-values are g. We d…
The Grundy number of an impartial game G is the size of the unique Nim heap equal to G. We introduce a new variant of Nim, Restricted Nim, which restricts the number of stones a player may remove from a heap in terms of the size of the heap. Certain classes of Restricted Nim are found to produce sequences of Grundy numbers with a self-similar fractal structure. Extending work of C. Kimberling, we obtain new characterizations of these "fractal sequences" and give a bijection between these sequences and certain upper-triangular arrays. As a special case we obtain the game of Serial Nim, in which…
We present take-away games whose Sprague-Grundy functions are given by the Nim sum of heap sizes in a mixed base $β$. Let $Δ_β$ be the set of such games. We give a necessary and sufficient condition for the existence of a minimum of $Δ_β$, and a recursive construction of the maximum of $Δ_β$.
On the Sprague-Grundy values of games with a pass
本文研究带过牌的无偏博弈析取复合，建立 SG 值同态，将满足一步博弈条件的分量复合转化为带过牌的 nim 局面，从而给出必胜策略判定时间上界，并应用于巧克力博弈的 SG 值计算。
Two Games on Arithmetic Functions: SALIQUANT and NONTOTIENT
本文研究基于算术函数的两个无偏博弈 SALIQUANT 和 NONTOTIENT，获得数论引理、基本定理及关于 Sprague-Grundy 值最终密度的猜想，并计算特定值。
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…
A move in the game of nim consists of taking any positive number of tokens from a single pile. Suppose we add the class of moves of taking a nonnegative number of tokens jointly from all the piles. We give a complete answer to the question which moves in the class can be adjoined without changing the winning strategy of nim. The results apply to other combinatorial games with unbounded Sprague-Grundy function values. We formulate two weakened conditions of the notion of nim-sum 0 for proving the results.
The game of Nim as played on graphs was introduced in Nim on Graphs I and extended in Nim on Graphs II by Masahiko Fukuyama. His papers detail the calculation of Grundy numbers for graphs under specific circumstances. We extend these results and introduce the strategy for even cycles. This paper examines a more general class of graphs by restricting the edge weight to one. We provide structural conditions for which there exist a winning strategy. This yields the solution for the complete graph.
Node-Kayles on Trees
本文研究了树和由树连接形成的图的 Grundy 值序列，推导了显式公式和递推关系，并证明了这些序列的周期性。
Given $n$ piles of tokens and a positive integer $k \leq n$, we study the following two impartial combinatorial games Nim$^1_{n, \leq k}$ and Nim$^1_{n, =k}$. In the first (resp. second) game, a player, by one move, chooses at least $1$ and at most (resp. exactly) $k$ non-empty piles and removes one token from each of these piles. For the normal and misère version of each game we compute the Sprague-Grundy function for the cases $n = k = 2$ and $n = k+1 = 3$. For game Nim$^1_{n, \leq k}$ we also characterize its P-positions for the cases $n \leq k+2$ and $n = k+3 \leq 6$.
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。
This paper presents a study of restricted Nim with a pass. In the restricted Nim considered in this study, two players take turns and remove stones from the piles. In each turn, when the number of stones is m, each player is allowed to remove at least one stone and at most the ceiling of m/2 stones from a pile of m stones. The standard rules of the game are modified to allow a one-time pass, that is, a pass move that may be used at most once in the game and not from a terminal position. Once a pass has been used by either player, it is no longer available. It is well-known that in classical Ni…
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。
In a recent arXiv-manuscript Fox studies infinite subtraction games with a finite (ternary) and aperiodic Sprague-Grundy function. Here we provide an elementary example of a game with the given properties, namely the game given by the subtraction set $\{F_{2n+1}-1\}$, where $F_i$ is the $i$th Fibonacci number, and where $n$ ranges over the positive integers. Our definition of nim-dimension reflects the precise number of power-of-two-components generated by the games; the group of nim-values is of order four so the dimension is two (in the classical definition this dimension would have been one…
Let $S$ be a set of positive integers, and let $D$ be a set of integers larger than $1$. The game $i$-Mark$(S,D)$ is an impartial combinatorial game introduced by Sopena (2016), which is played with a single pile of tokens. In each turn, a player can subtract $s \in S$ from the pile, or divide the size of the pile by $d \in D$, if the pile size is divisible by $d$. Sopena partially analyzed the games with $S=[1, t-1]$ and $D=\{d\}$ for $d \not\equiv 1 \pmod t$, but left the case $d \equiv 1 \pmod t$ open. We solve this problem by calculating the Sprague-Grundy function of $i$-Mark$([1,t-1],\{d…
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…
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…
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…
Maximum Nim and Josephus Problem
本文证明了以 f(x)=floor(x/k) 为规则函数的最大尼姆的格伦迪数与每隔 k 个数移除一个的约瑟夫问题之间存在简单关系。
Tasty Chocolate Games
本文研究巧克力游戏这一尼姆游戏的推广，证明“美味条件”可作为多种形状巧克力棒的斯普莱格-格伦迪值由各维度尺寸尼姆和给出的条件。
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。
A convergence technique for the game i-Mark
本文提出一种收敛技术，为 $|D|>1$ 的 i-Mark 游戏提供 Sprague-Grundy 序列的多项式时间算法，并证明适用于所有 i-Mark$(\{1\},\{d_1,d_2\})$ 情形。
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 relatively prime numbers, and their negations.
Nim heaps in the Game of Cycles: the parity conjecture fails on trees
本文研究循环博弈中的尼姆堆，证明树上的奇偶性猜想不成立，最小反例有15个顶点，还分析了蜘蛛树的Grundy值，给出相关公式与算法，提出非负整数是否均为树的Grundy值的问题。
Partition games
本文研究了2人分割游戏，探讨了其Sprague-Grundy序列的周期性特征，并提出了一种计算测试条件以解决更多游戏。
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…
Chocolate-bar games are variants of the CHOMP game. A three-dimensional chocolate bar comprises a set of cubic boxes sized 1 X 1 X 1, with a bitter cubic box at the bottom of the column at position (0,0). For non-negative integers u,w such that u < x and w \< z, the height of the column at position (u,w) is min (F(u,w),y) +1, where F is a monotonically increasing function. We denote this chocolate bar as CB(F,x,y,z). Each player, in turn, cuts the bar on a plane that is horizontal or vertical along the grooves, and eats the broken piece. The player who manages to leave the opponent with the si…
On Remoteness Functions of Exact Slow $k$-NIM with $k+1$ Piles
本文研究具有k+1堆的精确慢速k-NIM（其中2 ≤ k = n-1），计算Smith遥远度函数以确定P-位置，并证明保留最大奇数堆或最小偶数堆并减少其余堆的简单策略对双方均最优。
Multiplicative Modular Nim (MuM)
本文提出了 Nim 的乘法模变体 MuM，通过 mumber 值和模乘法结合规则建立了完整的非加性组合博弈理论，并利用中国剩余定理处理复合模数，将其扩展至有限域以辅助 AES S-box 代数教学。
Experimental Study of the Game Exact Nim(5, 2)
本文通过理论与计算实验研究 Exact Nim(5,2) 的 P-局面，并探究其与 Moore 尼姆 nim(6,≤2) 已知 P-局面之间的密切关系。
On remoteness functions of k-NIM with k+1 piles in normal and in misère versions
本文研究了$k+1$堆$k$-NIM游戏的常规版与逆向版，针对$k=n-1$的情况提出了确定Smith remoteness函数的简单显式规则，并给出了适用于逆向版的例外情况分类及显式公式。
Removing $m$ from $m$-pile Divisor Nim
本文证明除数尼姆游戏中SG值受最小堆大小约束，改进周期界并推导公共周期，提出两个关于SG上界和周期的猜想。
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。
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。
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