We play a variation of Nim on stacks of tokens. Take your favorite increasing sequence of positive integers and color the tokens according to the following rule. Each token on a level that corresponds to a number in the sequence is colored red; if the level does not correspond to a number in the sequence, color it green. Now play Nim on a arbitrary number of stacks with the extra rule: if all top tokens are green, then you can make any move you like. On two stacks, we give explicit characterizations for winning the normal play version for some popular sequences, such as Beatty sequences and the evil numbers corresponding to the 0s in the famous Thue-Morse sequence. We also propose a more general solution which depends only on which of the colors `dominates' the sequence. Our construction resolves a problem posed by Fraenkel at the BIRS 2011 workshop in combinatorial 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.
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…
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.
A circular Nim game is a two player impartial combinatorial game consisting of n stacks of tokens placed in a circle. A move consists of choosing k consecutive stacks, and taking at least one token from one or more of the k stacks. The last player able to make a move wins. We prove results on the structure of the losing positions for small n and k and pose some open questions for further investigations.
The ordinary game of Nim has a long history and is well-known in the area of combinatorial game theory. The solution to the ordinary game of Nim has been known for many years and lends itself to numerous other solutions to combinatorial games. Nim was extended to graphs by taking a fixed graph with a playing piece on a given vertex and assigning positive integer weight to the edges that correspond to a pile of stones in the ordinary game of Nim. Players move alternately from the playing piece across incident edges, removing weight from edges as they move. This paper solves Nim on hypercubes in…
We present a new family of Nim games where the rules depend on a given `coloring' of the tokens, each token being either black or white. The rules are as in Nim with the restriction that a white token on top of each heap is not allowed. We resolve the winning strategies of two disjoint game families played on two heaps. The heap-sizes with black tokens correspond to the numbers $\lfloor βn \rfloor$, where $β>2$ is an integer for one of the families and irrational for the other, and where $n$ ranges over the positive integers. In the process, new notions of \emph{invariant games} are introduced…
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…
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…
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 $n = k+3 \leq 6$.
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.
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.
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.
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…
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…
Aim: Present a systematic development of part of the theory of combinatorial games from the ground up. Approach: Computational complexity. Combinatorial games are completely determined; the questions of interest are efficiencies of strategies. Methodology: Divide and conquer. Ascend from Nim to chess in small strides at a gradient that's not too steep. Presentation: Informal; examples of games sampled from various strategic viewing points along scenic mountain trails, which illustrate the theory.
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…
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 $Δ_β$.
Candy Nim is a variant of Nim in which both players aim to take the last candy in a game of Nim, with the added simultaneous secondary goal of taking as many candies as possible. We give bounds on the number of candies the first and second players obtain in 3-pile $\mathcal{P}$ positions as well as strategies that are provably optimal for some families of such games. We also show how to construct a game with $N$ candies such that the loser takes the largest possible number of candies and bound the number of candies the winner can take in an arbitrary $\mathcal{P}$ position with $N$ total candi…
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 $I$.…
Let A be a finite subset of the naturals and let n be a natural. Let NIM(A;n) be the two player game in which players alternate removing $a\in A$ stones from a pile with $n$ stones; the first player who cannot move loses. This game has been researched thoroughly. We discuss a variant of NIM in which Player 1 and Player 2 start with d and e dollars, respectively. When a player removes a stones from the pile, he loses a dollars. The first player who cannot move loses, but this can now happen for two reasons: (1) The number of stones remaining is less than min(A), (2) The player has less than min…
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.
Fibonacci nim is a popular impartial combinatorial game, usually played with a single pile of stones. The game is appealing due to its surprising connections with the Fibonacci numbers and the Zeckendorf representation. In this article, we investigate some properties of a variant played with multiple piles of stones, and solve the 2-pile case. A player chooses one of the piles and plays as in Fibonacci nim, but here the move-size restriction is a global parameter, valid for any pile.
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…
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…
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 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…
In this paper we study a family of 2-pile Take Away games, that we denote by Generalized Diagonal Wythoff Nim (GDWN). The story begins with 2-pile Nim whose sets of options and $P$-positions are $\{\{0,t\}\mid t\in \N\}$ and $\{(t,t)\mid t\in \M \}$ respectively. If we to 2-pile Nim adjoin the main-\emph{diagonal} $\{(t,t)\mid t\in \N\}$ as options, the new game is Wythoff Nim. It is well-known that the $P$-positions of this game lie on two 'beams' originating at the origin with slopes $Φ= \frac{1+\sqrt{5}}{2}>1$ and $\frac{1}Φ < 1$. Hence one may think of this as if, in the process of going f…
We study a variation of the combinatorial game of 2-pile Nim. Move as in 2-pile Nim but with the following constraint: Suppose the previous player has just removed say $x>0$ tokens from the shorter pile (either pile in case they have the same height). If the next player now removes $x$ tokens from the larger pile, then he imitates his opponent. For a predetermined natural number $p$, by the rules of the game, neither player is allowed to imitate his opponent on more than $p-1$ consecutive moves. We prove that the strategy of this game resembles closely that of a variant of Wythoff Nim--a varia…
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…
Let A be a finite subset of $\nat$. Then NIM(A;n) is the following 2-player game: initially there are $n$ stones on the board and the players alternate removing $a\in A$ stones. The first player who cannot move loses. This game has been well studied. We investigate an extension of the game where Player I starts out with d dollars, Player II starts out with e dollars, and when a player removes a\in A he loses a dollars. The first player who cannot move loses; however, note this can happen for two different reasons: (1) the number of stones is less than min(A), (2) the player has less than $\min…
We enumerate P-positions in the game of Nim in two different ways. In one series of sequences we enumerate them by the maximum number of counters in a pile. In another series of sequences we enumerate them by the total number of counters. We show that the game of Nim can be viewed as a cellular automaton, where the total number of counters divided by 2 can be considered as a generation in which P-positions are born. We prove that the three-pile Nim sequence enumerated by the total number of counters is a famous toothpick sequence based on the Ulam-Warburton cellular automaton. We introduce 10 …
We build off the game, NimG to create a version named Neighboring Nim. By reducing from Geography, we show that this game is PSPACE-hard. The games created by the reduction share strong similarities with Undirected (Vertex) Geography and regular Nim, both of which are in P. We show how to construct PSPACE-complete versions with nim heaps *1 and *2. This application of graphs can be used as a form of game sum with any games, not only Nim.
We relate the Sierpinski triangle and the game of Nim. We begin by defining both a new high-dimensional analog of the Sierpinski triangle and a natural geometric interpretation of the losing positions in Nim, and then, in a new result, show that these are equivalent in each finite dimension.
Given $n$ piles of tokens and a positive integer $k \leq n$, the game Nim$^1_{n, =k}$ of exact slow $k$-Nim is played as follows. Two players move alternately. In each move, a player chooses exactly $k$ non-empty piles and removes one token from each of them. A player whose turn it is to move but has no move loses (if the normal version of the game is played, and wins if it is the misére version). In Integers 20 (2020) 1-19, Gurvich et al gave an explicit formula for the Sprague-Grundy function of Nim$^1_{4, =2}$, for both its normal and misére version. Here we extend this result and obtain an…
The game of nim, with its simple rules, its elegant solution and its historical importance is the quintessence of a combinatorial game, which is why it led to so many generalizations and modifications. We present a modification with a new spin: building nim. With given finite numbers of tokens and stacks, this two-player game is played in two stages (thus belonging to the same family of games as e.g. nine-men's morris): first building, where players alternate to put one token on one of the, initially empty, stacks until all tokens have been used. Then, the players play nim. Of course, because …
We give short rules for two-pile take-away games satisfying that a pair of complementary homogeneous Beatty sequences together with $(0,0)$ constitute a complete set of $P$-positions.
In this paper, we consider a modular extension to the game of Nim, which we call $m$-Modular Nim, and explore its optimal strategy. In $m$-Modular Nim, a player can either make a standard Nim move or remove a multiple of $m$ tokens in total. We develop a winning strategy for all $m$ with $2$ heaps and for odd $m$ with any number of heaps.
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…
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…
New combinatorial games are introduced, of which the most pertinent is Maharaja Nim. The rules extend those of the well-known impartial game of Wythoff Nim in which two players take turn in moving a single Queen of Chess on a large board, attempting to be the first to put her in the lower left corner. Here, in addition to the classical rules a player may also move the Queen as the Knight of Chess moves. We prove that the second player's winning positions of Maharaja Nim are close to the ones of Wythoff Nim, namely they are within a bounded distance to the lines with slope $\frac{\sqrt{5}+1}{2}…
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…
Poset games are a class of combinatorial game that remain unsolved. Soltys and Wilson proved that computing wining strategies is in \textbf{PSPACE} and aside from special cases such as Nim and N-Free games, \textbf{P} time algorithms for finding ideal play are unknown. This paper presents methods calculate the nimber of posets games allowing for the classification of winning or losing positions. The results present an equivalence of ideal strategies on posets that are seemingly unrelated.
While the game chromatic number of a forest is known to be at most 4, no simple criteria are known for determining the game chromatic number of a forest. We first state necessary and sufficient conditions for forests with game chromatic number 2 and then investigate the differences between forests with game chromatic number 3 and 4. In doing so, we present a minimal example of a forest with game chromatic number 4, criteria for determining the game chromatic number of a forest without vertices of degree 3, and an example of a forest with maximum degree 3 and game chromatic number 4.
We modify Wythoff's game by allowing an additional move, which we call a "split", and show how the $P$-positions are coded by the Tribonacci word. We analyze the table of letter positions of arbitrary $k$-bonacci words and find a $\mathrm{mex}$-rule that generates the Quadribonacci table.
