Impartial Games Research Articles

Nimber-preserving reduction: Game secrets and homomorphic Sprague-Grundy theorem

The concept of nimbers—a.k.a. Grundy-values or nim-values—is fundamental to combinatorial game theory. Beyond the winnability, nimbers provide a complete characterization of strategic interactions among impartial games in disjunctive sums. In this paper, we consider nimber-preserving reductions among impartial games, which enhance the winnability-preserving reductions in traditional computational characterizations of combinatorial games. We prove that Generalized Geography is complete for the natural class, IP, of polynomially-short impartial rulesets, under polynomial-time nimber-preserving reductions. We refer to this notion of completeness as Sprague-Grundy-completeness. In contrast, we also show that not every PSPACE-complete ruleset in IP is Sprague-Grundy-complete for IP.By viewing every impartial game as an encoding of its nimber—a succinct game secret richer than its winnability alone—our technical result establishes the following striking cryptography-inspired homomorphic theorem: Despite the PSPACE-completeness of nimber computation for IP, there exists a polynomial-time algorithm to construct, for any pair of games G1,G2 in IP, a Generalized Geography game G satisfying:nimber(G)=nimber(G1)⊕nimber(G2).

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.

Graph convexity impartial games: Complexity and winning strategies

Accordingly to Duchet (1987), the first paper of convexity on general graphs, in english, is the 1981 paper “Convexity in graphs”. One of its authors, Frank Harary, introduced in 1984 the first graph convexity games, focused on the geodesic convexity, which were investigated in a sequence of five papers that ended in 2003. In this paper, we continue this research line, extend these games to other graph convexities, and obtain winning strategies and complexity results. Among them, we obtain winning strategies for general convex geometries and winning strategies for trees from the Sprague-Grundy theory on impartial games. We also obtain the first PSPACE-hardness results on convexity games, by proving that the normal play and the misère play of the impartial hull game on the geodesic convexity is PSPACE-complete even in graphs with diameter two.

Some remarks on the Game of Cycles

Abstract The Game of Cycles is an impartial game on a planar graph that was introduced by Francis Su. In this short note we address some questions that have been raised on the game, and raise some further questions. We assume that the reader is familiar with basic notions from combinatorial game theory.

Feedback game on [formula omitted]-chromatic Eulerian triangulations of surfaces

A new impartial game on a connected graph was introduced, called a feedback game, which is a variant of generalized geography. In this paper, we study the feedback game on 3-chromatic Eulerian triangulations of surfaces. We prove that the winner of the game on every 3-chromatic Eulerian triangulation of a surface all of whose vertices have degree 0 modulo 4 is always fixed. Moreover, we also study the case of 3-chromatic Eulerian triangulations of surfaces which have at least two vertices whose degrees are 2 modulo 4. In addition, as a concrete class of such graphs, we consider the octahedral path, which is obtained from an octahedron by adding octahedra in the same face, and completely determine the winner of the game on those graphs.

Some results on LCTR, an impartial game on partitions

Some results on LCTR, an impartial game on partitions

Impartial games with decreasing Sprague–Grundy function and their hypergraph compound

AbstractThe Sprague–Grundy (SG) theory reduces the disjunctive compound of impartial games to the classical game of NIM. We generalize this concept by introducing hypergraph compounds of impartial games. An impartial game is called SG-decreasing if its SG value is decreased by every move. Extending the SG theory, we reduce hypergraph compounds of SG-decreasing games to hypergraph compounds of single-pile NIM games. We show that this reduction works only if all games involved in the compound are SG-decreasing. A hypergraph is called SG-decreasing if the corresponding hypergraph compound of single-pile NIM games is an SG-decreasing game. We provide some necessary and some sufficient conditions for a hypergraph to be SG-decreasing. In particular, for hypergraphs with hyperedges of size at most 3 we obtain a necessary and sufficient condition verifiable in polynomial time.

Some notes on disjunctive short sum: polychromatic nim

polychromatic nim is a version of the classic game nim , played with colored stones, in which each pile has stones of a single color, and the player who successfully extinguishes a color wins the game. This game is closely related to the concept of short rule, the ending condition that states that a disjunctive sum ends as soon as any one of the components ends. Here, we discuss that rule, namely when applied to impartial games, and prove that the Grundy-values of polychromatic nim present an arithmetic-periodic behavior.

Emperor nim and emperor sum: a new sum of impartial games

Emperor nim and emperor sum: a new sum of impartial games

Developing an Impartial Game by Mathematical Approach

In this study, we developed an impartial game namely Nim by using a mathematical approach. The game consists of multiple ways of movement for player winning the game. A total of five players participating in the game testing. The experiment shows that the proposed impartial game for learning process is easy to use.

On pruning search trees of impartial games

In this paper we study computing Sprague-Grundy values for short impartial games under the normal play convention. We put forward new game-agnostic methods for effective pruning search trees of impartial games. These algorithms are inspired by the α-β, a well-known pruning method for minimax trees. However, our algorithms prune trees whose node values are assigned by the mex function instead of min/max.We have verified the effectiveness of our algorithms experimentally on instances of some standard impartial games (that is Nim, Chomp, and Cram). Based on the results of our experiments we have concluded that: (1) our methods generally perform well, especially when transposition table can store only a small fraction of all game positions (which is typical when larger games are concerned); (2) one of our algorithms constitutes a more universal alternative to the state-of-the-art algorithm proposed by Lemoine and Viennot.

Construction and investigation of Cram endgame databases

In this paper we investigate the game of Cram, which is the impartial version of Domineering. We have built Cram endgame databases for all board sizes < 30 squares. We developed a program that fills the databases with their Combinatorial Game Theory (CGT) values. Since Cram is an impartial game, all CGT values for Cram positions are so-called nimbers, indicated by ∗ n. The nimber value of a position not only directly determines the game-theoretic value (first- or second-player win), but also provides an optimal playing strategy. When analyzing the resulting databases we observed the following facts. Firstly we confirmed that the CGT values of all investigated empty boards are in agreement with results published in the literature. Since the value of an empty board depends completely on the values of many partially filled positions in the database, this is a strong indication that our process of filling the database with CGT values is correct. Secondly, although the series of values for 2 × n boards is completely regular, namely a value ∗ 0 for n being even and ∗ 1 for n being odd, this was not proven formally so far. We were able to provide such a proof. We also investigated the databases for their contents. So far we encountered nimber values up to ∗ 11 among single-fragment Cram positions. It appears that for a ∗ n value to occur a board size with ≈ 3 n squares is needed, some more for very tall boards ( 1 × n), some less for wider boards ( 4 × n). So far no single-fragment Cram positions were encountered with nimber values ⩾ ∗ 12, although we show a construction of larger multi-fragment positions with nimber values from ∗ 12 up to ∗ 15. In a preliminary experiment we incorporated the CGT endgame databases constructed into a simple alpha-beta solver for the game. Results revealed a large improvement in solving power.

Multiplayer games as extension of misère games

We introduce i-misere play to multiplayer impartial games, which is an extension of Li’s rank-based play and, at the same time, which contains the normal play and the misere play as special cases when the number of players is two. We characterize losing positions of such plays for both NIM and Moore’s game. i-misere play is defined as a special case of a more general notion of preference-based play, and we also study properties of some other preference-based plays.

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.

Impartial achievement games for generating nilpotent groups

Abstract We study an impartial game introduced by Anderson and Harary. The 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 finite groups of the form T × H {T\times H} , where T is a 2-group and H is a group of odd order. This includes all nilpotent and hence abelian groups.

PRIME STEP Plays Games

A group of students in 7-9 grades are inventing combinatorial impartial games. The games are played on graphs, piles, and grids. We found winning positions, optimal strategies, and other interesting facts about the games.

Multi-player End-Nim games

W.O. Krawec (2012) [17] introduced a method of analyzing multi-player impartial games, and derived a recursive function capable of determining which of the n players has a winning strategy. The present paper is devoted to “Multi-player End-Nim games”, abbreviated by ENim(N,n), assuming that the standard alliance matrix is adopted. The game values of ENim(N,n) are completely determined for three cases n>N+1, n=N+1 and n=N. For the case n<N, by letting d=N−n≥1, the game values of ENim(n+d,n) are completely determined if n≥d+4. The case 3≤n≤d+3 is more complicated, we present the game values for d=1.

The Sprague–Grundy function for some nearly disjunctive sums of Nim and Silver Dollar games

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 in each strip. We give an algorithm that, for each Sprague–Grundy value g, computes the positions of two-token Star Nim whose Sprague–Grundy values are g. We establish that the sequence of differences of coordinates of g-positions is ultimately periodic.

On tame, pet, domestic, and miserable impartial games

Playing impartial games under the normal and misère conventions may differ a lot. However, there are also many “exceptions” for which the normal and misère Sprague–Grundyfunctions are very similar. The first such example, the game of Nim, was considered by Bouton as early as in 1901. 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 efficiently verifiable characterizations. These games are closely related to another important subclass of the tame games introduced in 2007 by the first author and called miserable games. We show that tame, pet, and domestic games turn into miserable games by “slight modifications” of the definitions. We also show that the sum of miserable games is miserable and find several other classes that respect summation. The developed techniques allow us to prove that very many well-known impartial games fall into classes mentioned above. Such examples include all subtraction games, which are pet; game Euclid, which is miserable (and, hence, tame), as well as many versions of the Wythoff game and Nim, which may be miserable, pet, or domestic.

Ordinal sums of impartial games

In an ordinal sum of two combinatorial games G and H, denoted by G:H, a player may move in either G (base) or H (subordinate), with the additional constraint that any move on G completely annihilates the component H. It is well-known that the ordinal sum does not depend on the form of its subordinate, but depends on the form of its base. In this work, we analyze G(G:H) where G and H are impartial forms, observing that the G-values are related to the concept of minimum excluded value of orderk. As a case study, we introduce the ruleset oak, a generalization of green hackenbush. By defining the operation gin sum, it is possible to determine the literal forms of the bases in polynomial time.