Impartial Combinatorial Game Research Articles

Auction in Impartial Games

Combinatorial game theory (CGT) is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. We prove an importance of a tiebreaking marker related with N-position and P-position in the bidding variant of Impartial combinatorial games.

On the linearity of the periods of subtraction games

A subtraction game is an impartial combinatorial game involving a finite set S of positive integers. The nim-sequence GS associated with this game is ultimately periodic. In this paper, we study the nim-sequence GS∪{c} where S is fixed and c varies. We conjecture that there is a multiple q of the period of GS, such that for sufficiently large c, the pre-period and period of GS∪{c} are linear in c if c modulo q is fixed. We prove it in several cases.We also give new examples with period 2 inspired by this conjecture.

The combinatorial game Nofil played on Steiner triple systems

AbstractWe introduce an impartial combinatorial game on Steiner triple systems called Next One to Fill Is the Loser (N ofil) . Players move alternately, choosing points of the triple system. If a player is forced to fill a block on their turn, they lose. By computing nim‐values, we determine optimal strategies for N ofil on all Steiner triple systems up to order 15 and a sampling for orders 19, 21 and 25. The game N ofil can be thought of in terms of play on a corresponding hypergraph which will become a graph during play. At that point N ofil is equivalent to playing the game N ode K ayles on the graph. We prove necessary conditions and sufficient conditions for a graph to reached playing N ofil. We conclude that the complexity of determining the outcome of the game N ofil on Steiner triple systems is PSPACE‐complete for randomized reductions.

Some i-Mark games

Let S be a set of positive integers, and let D be a set of integers larger than 1. The game ▪ 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∈S from the pile, or divide the size of the pile by d∈D, if the pile size is divisible by d. Sopena partially analyzed the games with S=[1,t−1] and D={d} for d≢1(modt), but left the case d≡1(modt) open.We solve this problem by calculating the Sprague–Grundy function of ▪ for d≡1(modt), for all t,d≥2. We also calculate the Sprague–Grundy function of ▪ for all k, and show that it exhibits similar behavior. Finally, following Sopena's suggestion to look at games with |D|>1, we derive some partial results for the game ▪, whose Sprague–Grundy function seems to behave erratically and does not show any clear pattern. We prove that each value 0,1,2 occurs infinitely often in its SG sequence, with a maximum gap length between consecutive appearances.

Functional take-away

Combinatorial game theory [1] affords a rich collection of interesting problems for students to sharpen their coding skills. Impartial combinatorial games with complete information live on acyclic ...

Two impartial games on finite p-groups

This paper is about two impartial combinatorial games, namely achievement and avoidance games. These games can be played on groups by two persons. If G is a finite group, then both players chose elements of G alternately. The player who builds a generating set for G is called the loser in avoidance game whereas the builder of generating set is called the winner in achievement game. Dana and Sieben have been provided concept of nim-numbers by plotting their structure diagrams for both games. Since, the nim-numbers provide a more easier analysis of impartial combinatorial games and provide final outcome of these impartial games. But it is hard to calculate the nim-numbers of an impartial game. In this paper, the nim-numbers for achievement and avoidance games on some non-abelian groups have been calculated. In particular, the simplified structure diagrams and nim-values of these two games for finite non-abelian p-groups, dicyclic type groups, and groups of the form ℂ3 × ℂp has been determined.

Multi-player Last Nim with Passes

We introduce a class of impartial combinatorial games, Multi-player Last Nim with Passes, denoted by MLNim\(^{(s)}(N,n)\): there are N piles of counters which are linearly ordered. In turn, each of n players either removes any positive integer of counters from the last pile, or makes a choice ‘pass’. Once a ‘pass’ option is used, the total number s of passes decreases by 1. When all s passes are used, no player may ever ‘pass’ again. A pass option can be used at any time, up to the penultimate move, but cannot be used at the end of the game. The player who cannot make a move wins the game. The aim is to determine the game values of the positions of MLNim\(^{(s)}(N,n)\) for all integers \(N\ge 1\) and \(n\ge 3\) and \(s\ge 1\). For \(n>N+1\) or \(n=N+1\ge 3\), the game values are completely determined for any \(s\ge 1\). For \(3\le n\le N\), the game values are determined for infinitely many triplets (N, n, s). We also present a possible explanation why determining the game values becomes more complicated if \(n\le N\).

Multi-player Wythoff’s game and its variants

Wythoff’s game is a well-known 2-player impartial combinatorial game, introduced by W.A. Wythoff in 1907. In recent years, many scholars studied the variants of Wythoff’s game, including mainly extensions and restrictions, with fruitful results achieved. One way of solving n-player impartial games was presented by W.O. Krawec in 2012. We employ Krawec’s function in this paper to analyze n-player Wythoff’s game and its nine restricted versions. The game values are completely determined for all ten n-player impartial games.

Algebraic games—playing with groups and rings

Two players alternate moves in the following impartial combinatorial game: Given a finitely generated abelian group A, a move consists of picking some $$0 \ne a \in A$$ . The game then continues with the quotient group $$A/\langle a \rangle $$ . We prove that under the normal play rule, the second player has a winning strategy if and only if A is a square, i.e. $$A \cong B \times B$$ for some abelian group B. Under the misere play rule, only minor modifications concerning elementary abelian groups are necessary to describe the winning situations. We also compute the nimbers, i.e. Sprague–Grundy values of 2-generated abelian groups. An analogous game can be played with arbitrary algebraic structures. We study some examples of non-abelian groups and commutative rings such as R[X], where R is a principal ideal domain.

A class of extensions of Restricted (s, t)-Wythoff’s game

Abstract Restricted (s, t)-Wythoff’s game, introduced by Liu et al. in 2014, is an impartial combinatorial game. We define and solve a class of games obtained from Restricted (s, t)-Wythoff’s game by adjoining to it some subsets of its P-positions as additional moves. The results show that under certain conditions they are equivalent to one case in which only one P-position is adjoined as an additional move. Furthermore, two winning strategies of exponential and polynomial are provided for the games.

Global Fibonacci nim

Fibonacci nim is a popular impartial combinatorial game, usually played with a single pile of stones: two players alternate in removing no more than twice the previous player’s removal. 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.

A-Wythoff’s game with n players

Letting a = 1 in a- Wythoff’s game introduced by Fraenkel yields Wythoff’s game which is a well-known 2-player impartial combinatorial game introduced by Wythoff in 1907. A method of solving n-player impartial games was presented by Krawec in 2012. In this paper, we employ Krawec’s function to analyze n > 2 players a-Wythoff’s game and obtain game values for all a 1. The results obtained cover n-player Wythoff’s game, a special case 1-Wythoff’s game.

I-Mark: A new subtraction division game

Given two finite sets of integers S⊆N∖{0} and D⊆N∖{0,1}, the impartial combinatorial game i-Mark(S,D) is played on a heap of tokens. From a heap of n tokens, each player can move either to a heap of n−s tokens for some s∈S, s≤n, or to a heap of n/d tokens for some d∈D if d divides n. Such games can be considered as an integral variant of Mark-type games, introduced by Elwyn Berlekamp and Joe Buhler and studied by Aviezri Fraenkel and Alan Guo, for which it is allowed to move from a heap of n tokens to a heap of ⌊n/d⌋ tokens for any d∈D.Under normal convention, it is observed that the Sprague–Grundy sequence of the game i-Mark(S,D) is aperiodic for any sets S and D. However, we prove that in many cases this sequence is almost periodic and that the sequence of winning positions is periodic. Moreover, in all these cases, the Sprague–Grundy value of a heap of n tokens can be computed in time O(log⁡n).We also prove that under misère convention the outcome sequence of these games is purely periodic.

N-Player impartial combinatorial games with random players

In this paper we develop a method of analyzing n-player impartial combinatorial games where n−1 players behave optimally while one of the players plays randomly. More specifically, we develop a function gr that maps impartial combinatorial games to n-tuples called game values. A game value of some impartial combinatorial game G describes the probability that any player can win assuming player r plays randomly (i.e. player r chooses games at random without strategy).After devising our game value function gr, we apply it to the analysis of three-player Chomp whereby the second player plays randomly. Our analysis gives game values for single rowed Chomp positions and two rowed Chomp positions where the bottom row has 1 or 2 elements.We next discuss three alternative methods of analysis. First we extend our game value function to support more than just a single random player. Our second extension allows the random player to choose from an arbitrary distribution over games (as opposed to uniform which is assumed in our analysis of Chomp). This second extension also allows us to model a player who makes correct decisions for “simple” games (those that are, for instance, close to an end game) while playing randomly for more complex games. Finally, the third method considers the case where the random player only makes a mistake with probability ϵ and otherwise plays strategically.We conclude with many open problems and some interesting observations on the behavior of games containing a random player.

Circular Nim Games

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.

Nimbers are inevitable

This article concerns the resolution of impartial combinatorial games, in particular games that can be split in sums of independent positions. We prove that in order to compute the outcome of a sum of independent positions, it is always more efficient to compute separately the nimber of at least one of the independent positions, rather than to develop directly the game tree of the sum. The concept of the nimber is therefore inevitable to accelerate the computation of impartial games, even when we only try to determine the winning or losing outcome of a starting position. We also describe algorithms to use nimbers efficiently and to conclude, we give a review of the results obtained on two impartial games: Sprouts and Cram.

A RESTRICTION OF EUCLID

AbstractEuclid is a well-known two-player impartial combinatorial game. A position in Euclid is a pair of positive integers and the players move alternately by subtracting a positive integer multiple of one of the integers from the other integer without making the result negative. The player who makes the last move wins. There is a variation of Euclid due to Grossman in which the game stops when the two entries are equal. We examine a further variation which we called M-Euclid where the game stops when one of the entries is a positive integer multiple of the other. We solve the Sprague–Grundy function for M-Euclid and compare the Sprague–Grundy functions of the three games.

Analyzing Two-Color Babylon

AbstractWe examine the impartial combinatorial game Babylon. We abstract the game so that it is suitable to combinatorial analysis, and present a full characterization and strategies for the cases with odd number of tokens and only two colors. We also demonstrate partial results on games with even tokens with two colors, an initial extension to three colors, and offer directions for future work.

Lattice point methods for combinatorial games

We encode arbitrary finite impartial combinatorial games in terms of lattice points in rational convex polyhedra. Encodings provided by these lattice games can be made particularly efficient for octal games, which we generalize to squarefree games. These encompass all heap games in a natural setting where the Sprague–Grundy theorem for normal play manifests itself geometrically. We provide an algorithm to compute normal play strategies.The setting of lattice games naturally allows for misère play, where 0 is declared a losing position. Lattice games also allow situations where larger finite sets of positions are declared losing. Generating functions for sets of winning positions provide data structures for strategies of lattice games. We conjecture that every lattice game has a rational strategy: a rational generating function for its winning positions. Additionally, we conjecture that every lattice game has an affine stratification: a partition of its set of winning positions into a finite disjoint union of finitely generated modules for affine semigroups. This conjecture is true for normal-play squarefree games and every lattice game with finite misère quotient.

Taming the Wild in Impartial Combinatorial Games

Taming the Wild in Impartial Combinatorial Games