v1 was actually dealing with two distinct operations; in v2 I have fixed the issue. Moreover, I have fixed a few details, added some comments and discussed other possibilities
We present a definition for the sum of a sequence of combinatorial games. This sum coincides with the classical sum in the case of a converging sequence of real numbers and with the infinitary natural sum in the case of a sequence of ordinal numbers. We briefly discuss other possibilities, such as the string limit, some "magical" variants of Hackenbush, as well as "Dadaist" infinite sums, which allow transfinite runs, while still being loopfree.
In Combinatorial Game Theory, the fundamental relation of game equivalence, denoted by $=$, is introduced early on and overrides the notion of set equality. We explore what happens if set equality is given its due before game equivalence is introduced.
Combinatorial Scoring games, with the property `extra pass moves for a player does no harm', are characterized. The characterization involves an order embedding of Conway's Normal-play games. Also, we give a theorem for comparing games with scores (numbers) which extends Ettinger's work on dicot Scoring games.
Genetic programming is the practice of evolving formulas using crossover and mutation of genes representing functional operations. Motivated by genetic evolution we develop and solve two combinatorial games, and we demonstrate some advantages and pitfalls of using genetic programming to investigate Grundy values. We conclude by investigating a combinatorial game whose ruleset and starting positions are inspired by genetic structures.
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.
We define a natural equivalence relation on collections of cards from the card game SET, and enumerate some of the equivalence classes, vastly generalizing the standard game. On this basis, we describe several alternative games for the SET deck, including the game of STUN, in which the goal is to find three cards with exactly two values in each attribute.
This paper studies a single-suit version of the card game War on a finite deck of cards. There are varying methods of how players put the cards that they win back into their hands, but we primarily consider randomly putting the cards back and deterministically always putting the winning card before the losing card. The concept of a \emph{passthrough} is defined, which refers to a player playing through all cards in their hand from a particular point in the game. We consider games in which the second player wins during their first passthrough. We introduce several combinatorial objects related …
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…
The study of combinatorial games is intimately tied to the study of graphs, as any game can be realized as a directed graph in which players take turns traversing the edges until reaching a sink. However, there have heretofore been few efforts towards analyzing game graphs using graph theoretic metrics and techniques. A set $S$ of vertices in a graph $G$ resolves $G$ if every vertex in $G$ is uniquely determined by the vector of its distances from the vertices in $S$. A metric basis of $G$ is a smallest resolving set and the metric dimension is the cardinality of a metric basis. In this articl…
The $\mathscr{P}$-position sets of some combinatorial games have special combinatorial structures. For example, the $\mathscr{P}$-position set of the hexad game, first investigated by Conway and Ryba, is the block set of the Steiner system $S(5, 6, 12)$ in the shuffle numbering, $\mathcal{D}_{\text{sh}}$. There were, however, few known games related to Steiner systems like the hexad game. For a given Steiner system, we construct a game whose $\mathscr{P}$-position set is its block set. By using constructed games, we obtain the following two results. First, we characterize $\mathcal{D}_{\text{s…
In a Take-Away Game on hypergraphs, two players take turns to remove the vertices and the hyperedges of the hypergraphs. In each turn, a player must remove either a single vertex or a hyperedge. When a player chooses to remove one vertex, all of the hyperedges that contain the chosen vertex are also removed. When a player chooses to remove one hyperedge, only that chosen hyperedge is removed. Whoever removes the last vertex wins the game. Following from the winning strategy for the Take-Away Impartial Combinatorial Games on only Oddly Uniform or only Evenly Uniform Hypergraphs, this paper is a…
We develop a theory of combinatorial games that is appropriate for describing positions in Hex and other monotone set coloring games. We consider two natural conditions on such games: a game is monotone if all moves available to both players are good, and passable if in each position, at least one player has at least one good move available. The latter condition is equivalent to saying that if passing were permitted, no player would benefit from passing. Clearly every monotone game is passable, and we prove that the converse holds up to equivalence of games. We give some examples of how this t…
Combinatorial game theory (CGT), as introduced by Berlekamp, Conway and Guy, involves two players who move alternately in a perfect information, zero-sum game, and there are no chance devices. Also the games have the finite descent property (every game terminates in a finite number of moves). The two players are usually called Left and Right. The games often break up into components and the players must choose one of the components in which to play. One main aim of CGT is to analyze the components individually (rather than analyzing the sum as a whole) then use this information to analyze the …
We propose a unifying additive theory for standard conventions in Combinatorial Game Theory, including normal-, misère- and scoring-play, studied by Berlekamp, Conway, Dorbec, Ettinger, Guy, Larsson, Milley, Neto, Nowakowski, Renault, Santos, Siegel, Sopena, Stewart (1976-2019), and others. A game {\em universe} is a set of games that satisfies some standard closure properties. Here, we reveal when the fundamental game comparison problem, ``Is $G\su H$?'', simplifies to a constructive `local' solution, which generalizes Conway's foundational result in ONAG (1976) for normal-play games. This ha…
We consider three variants of a partisan combinatorial game between two players, Left and Right, played on an undirected simple graph. Left is able to delete vertices (and incident edges) while Right is able to delete edges. This natural extension of a similar impartial game gives a clear advantage to one player by allowing them the ability to play on a small subgraph which the other player can not. Our last variant removes this advantage by assuming a move is valid for one player if and only if the other player has a valid move on the same graph. In this case, we show that the ability to remo…
We introduce a one-person game that we call Padlock Solitaire which resembles the well-known clock solitaire card game. Analyzing variants of this game we obtain simple proofs of some classical results of combinatorics including ballot theorems and the enumeration of spanning trees in various graphs and hypergraphs.
We illustrate how one can use basic combinatorial theory and computer programming technique (Python) to analyze the combinatorial game: Mahjong. The results confirm some folklore concerning the game, and expose some unexpected results. Related results and possible future research in connection to artificial intelligence are mentioned. Readers interested in the subject may further develop the techniques to deepen the study of the game, or study other combinatorial games.
In 1973 Fraenkel discovered interesting sequences which split the positive integers. These sequences became famous, because of a related unsolved conjecture. Here we construct combinatorial games, with `playable' rulesets, with these sequences constituting the winning positions for the second player. Keywords: Combinatorial game, Fraenkel's conjecture, Impartial game, Normal play, Playability, Rational modulus, Splitting sequences
This paper studies 2-player impartial combinatorial games, where the outcomes correspond to updates of cellular automata (CA) which generalize Wolfram's elementary rule 60 and rule 110 (Cook 2004). The games extend the class of \emph{triangle placing games} (Larsson 2013) where at each stage of the game the previous player has the option to block certain hopeful moves of the next player. We also study fractals and partial convergence in a subclass of the CA.
We define an all-small ruleset, Bipass, within the framework of normal-play combinatorial games. A game is played on finite strips of black and white stones. Stones of different colors are swapped provided they do not bypass one of their own kind. We find a surjective function from the strips to integer atomic weights (Berlekamp, Conway and Guy 1982) that measures the number of units in all-small games. This result provides explicit winning strategies for many games, and in cases where it does not, it gives narrow bounds for the canonical form game values. We prove that the game value *2 does …
Strong placement games (SP-games) are a class of combinatorial games whose structure allows one to describe the game via simplicial complexes. A natural question is whether well-known invariants of combinatorial games, such as "game value", appear as invariants of the simplicial complexes. This paper is the first step in that direction. We show that every simplicial complex encodes a certain type of SP-game (called an "invariant SP-game") whose ruleset is independent of the board it is played on. We also show that in the class of SP-games isomorphic simplicial complexes correspond to isomorphi…
The notions of symmetry and anonymity in strategic games have been formalized in different ways in the literature. We propose a combinatorial framework to analyze these notions, using group actions. Then, the same framework is used to define partial symmetries in payoff matrices. With this purpose, we introduce the notion of the role a player plays with respect to another one, and combinatorial relations between roles are studied. Building on them, we define relations directly between players, which provide yet another characterization of structural symmetries in the payoff matrices of strateg…
Combinatorial games are two-player games of pure strategy where the players, usually called Left and Right, move alternately. In this paper, we introduce Cheating Robot games. These arise from simultaneous-play combinatorial games where one player has insider information ('cheats'). Play occurs in rounds. At the beginning of a round, both players know the moves that are available to them. Left chooses a move. Knowing Left's move, Right then chooses a move. Right's move is not constrained by Left's choice. The round is not completed until both players have made a choice. A game is finished only…
