1. Introduction

A natural enumeration question for combinatorial games is: “How many legal positions are possible in a game?” Surprisingly, few have actually considered this problem. Farr [7; 8], and Tromp and Farnebäck [20] consider the problem of “counting the number of end positions in GO.” Similar enumeration questions are addressed by Hetyei [12], who analyses a game where the number of P-positions (second player win positions) of length n is related to the n-th Bernoulli number of the second kind, and in [17], where it is shown that for the game of TIMBER, on paths, the number of P-positions of length n is related to the Catalan and Fine numbers.

In Section 3, we enumerate the positions of several well-known games. A natural subset of combinatorial games, which we call placement games, are those that consist of placing pieces on a board until the board “fills” and there are no further moves. For each game, except NOGO, we find an auxiliary graph for which a position in the game corresponds to an independent set in the auxiliary graph.

In Section 4, we exhibit bijections between the games with identical generating functions.

2. Background

A placement game can be abstractly represented as a game on a graph, with the following properties.

• • The game begins on a graph that contains no pieces.

• • A move is to place a piece on one (or more) vertices subject to the rules of the particular game.

• • The rules must imply that if a piece can be placed in a certain position on the board then it was legal to place it in that position at any time earlier in the game.

• • Once played, a piece remains on the graph; it is never moved or removed from the graph.

Placement games were first identified during the seminar which led to this paper and have become of interest because of their properties; see [13; 6; 5; 16].

1. Introduction

The games STRATEGO, OVID's GAME, THREE-, SIX-, and NINE-MEN's MORRIS [3] and also BUILDING NIM [7] are examples of combinatorial games that have two phases. Specifically, in Phase 1 the board is set up and in Phase 2 the game is played. There are many other “math games” (which mathematicians want to analyze regardless of whether people actually want to play them) in which Phase 1 is not even defined, but instances of Phase 2 are analysed. For example, BOXCARS [2], END-NIM [1], PUSH [2], TOPPLING DOMINOES [8] and their variants.

In this paper, we consider playing Phase 1 as a combinatorial game as well as Phase 2 and analyze two specific games. We were introduced to this concept by Kyle Burke and Urban Larsson (personal communication).

To avoid confusion with the multiple meanings of the word “game”, we refer to the ruleset, which describes the legal moves, and a position, which is an instance of the game after several (including zero) legal moves. By necessity, the position also describes the board upon which play takes place.

In an impartial game, both players have the same moves available.

Definition 1. Let F and H be two impartial rulesets. The conjoined ruleset (F ▸ H) is to play Phase 1 under the F ruleset and when play is no longer possible to start Phase 2 which is played under the ruleset of H.

Forming a conjoined game allows for an interesting Phase 1 battle before the “real” game begins. Since play in Phase 1 sets up the board, it is convenient to have the corresponding game be a placement game [5], i.e., pieces are placed but not moved or removed. The positions at the beginning of Phase 2 will have structure reflecting the Phase 1 rules, and this allows for some partial analysis.

1. Introduction

Our goal in this paper is to study complexes of placement games (Definition 1.1). In [3] we demonstrated that to a placement game G played on a board B one can associate a simplicial complex ΔG,B, where G can be considered as a game played on ΔG,B.

The main question addressed in this paper is: What complexes can be legal complexes of a placement game?

We give partial answers to this question in specific cases: when the board is a path, a cycle, or a complete graph (also see [5]).

We begin by introducing some of the concepts needed. A complete introduction is given in [3].

Definition 1.1. A strong placement game is a combinatorial game played on a graph which satisfies the following:

• (i) The starting position is the empty board.

• (ii) Players place pieces on empty spaces of the board according to the rules.

• (iii) Pieces are not moved or removed once placed.

• (iv) If it is possible to reach a position through a sequence of legal moves, then any sequence of moves leading to this position consists only of legal moves.

The TRIVIAL placement game on a board is the placement game that has no additional rules.

Throughout this paper “placement game” refers to a strong placement game. Since placement games are played on a graph, we use the terms board and graph, and space and vertex interchangeably.

A basic position is a board with only one piece placed. Any position, whether legal or illegal, in a placement game can be decomposed into basic positions.

Definition 1.2. A simplicial complex Δ on a finite vertex set V is a set of subsets (called faces) of V with the conditions that if A ϵ Δ and B ⊆ A, then B ϵ Δ. The f -vector ( f0, f1, … , fk) of a simplicial complex Δ enumerates the number of faces fi with i vertices. Note that if Δ ≠ ∅, then f0 = 1.

1. Introduction

We will demonstrate a relationship between a subclass of combinatorial games, such as DOMINEERING and COL, and algebraic structures defined on simplicial complexes. There are two relationships, one via the maximal legal positions and the other through the minimal illegal positions. We will begin by giving the necessary background, first from combinatorial game theory, then from combinatorial commutative algebra.

For a game, perfect information means that both players know which game they are playing, on which board, and the current position. No chance means that no dice can be rolled or cards can be dealt, or any other item involving probability can be used.

Definition 1.1. A combinatorial game is a 2-player game with perfect information and no chance, where the two players are Left and Right (denoted by L and R respectively) and they do not move simultaneously. Then a game is a set P of positions with a specified starting position. Rules determine from which position to which position the players are allowed to move. A legal position is a position that can be reached by playing the game from the starting position (which is legal) according to the rules. Moving from position P to position Q is called a legal move if both P and Q are legal positions and the move is allowed according to the rules. Q is usually called an option of P.

In this paper, a combinatorial game will be denoted by its name in SMALL CAPS. Well-known examples of combinatorial games are CHESS, CHECKERS, TIC-TAC-TOE, GO, and CONNECT FOUR. Examples of games that are not combinatorial games include bridge, backgammon, poker, and snakes and ladders.

Although games usually have a “winning condition” associated to them, i.e., rules as to which player wins, for the purposes of this paper games do not need to have a notion of winning identified.

We will assume that the board on which games are played is a graph (or can be represented as a graph). A space on a board is then equivalent to a vertex and we use the two terms interchangeably.

1. Introduction

Recent progress in scoring-play combinatorial game theory motivates a survey on the subject. There are similarities with classical settings in normal- and misere-play, but the subject is richer than both those combined. This survey has a three-fold purpose: first to survey the combinatorial game theory (CGT) work in the area (such as disjunctive sum, game comparison, game reduction and game values), secondly to point at some important ideas about scoring rulesets (in relation with normal- and misère-play), and at last we show that existing literature includes many scoring combinatorial games which have not yet been studied in the broader CGT context. Although CGT was first developed in positional (scoring) games by Milnor (inspired by game decomposition in the game of Go), the field took off only with the advances in normal-play during the 1970-80s, and recently via successes in understanding misère games.

1.1. Normal- and misère-play.The family of combinatorial games consists of two-player games with perfect information (no hidden information as in some card games), no chance moves (no dice), and where the two players move alternately. We primarily consider games in which the positions decompose into independent subpositions. This class of games has been called additive to distinguish it from maker-maker and maker-breaker positional games [5], such as HEX.

When a player has no more moves (in any game component, which is called the “long” disjunctive sum) the game ends and some convention is required to be able to determine the outcome. There are two natural conventions that tie together the last move and the outcome:

• (1) Normal-play convention: last player wins;

• (2) Misere-play convention: last player loses.

The obtained body of results considering these conventions is what we call classical combinatorial game theory. See [16; 7; 1; 47] for background, [25] for a survey, and [39] for a list of open problems.

The theory of normal-play was developed first. The analysis of NIM, by Charles Bouton, was published in the early twentieth century [10].

During the recent development of combinatorial game theory many more problems have been suggested than solved. Here is a collection of problems that many people have found interesting. Included is a discussion of recent developments in these areas. This collection was started by Richard Guy in 1991, and has been updated in each Games of No Chance volume. The problems are divided into five sections:

• A. Taking and breaking

• B. Pushing and placing pieces

• C. Playing with pencil and paper

• D. Disturbing and destroying

• E. Theory of games

In the sections, the problems are identified by letters and numbers. Any number in parenthesis is the old number used in each of the lists of unsolved problems given on pp. 183–189 of AMS Proc. Sympos. Appl. Math. 43 (1991), called PSAM 43 below; on pp. 475–491 of Games of No Chance, hereafter referred to as GONC.

Previous versions of the unsolved problems can also be found on pp. 457–473 of More Games of No Chance (MGONC); pp. 475–500 of Games of No Chance 3 (GONC3); and pp. 279–308 of Games of No Chance 4 (GONC4).

Some problems have little more than the statement of the problem if there is nothing new to be added. References [#] are at the end of this article. Useful references for the rules and an introduction to many of the specific games mentioned below are:

• • M. Albert, R. J. Nowakowski, and D. Wolfe, Lessons in Play: An Introduction to the Combinatorial Theory of Games, A K Peters, 2007 (LIP);

• • Berlekamp, Conway, and Guy, Winning Ways for your Mathematical Plays, vol. 1–4, A K Peters, 2000–2004 (WW);

• • Siegel, Combinatorial Game Theory, American Math. Society, 2013 (CGT).

The term “game” is ambiguous and needs to be clarified. If a board is not specified then the question pertains to all possible boards. Berlekamp noted that many games have a standard opening position, an empty board for example. However, some results include positions, such as Garden-of-Eden positions, that could never occur by any sequence of moves from a standard opening position, and ones which would never occur in decent play. Indeed, Plambeck has asked at several meetings whether there is any difference in the complexity results for all positions and those that would occur in decent play.

A taking-and-breaking game [Albert et al. 2007; Berlekamp et al. 2001] is an impartial combinatorial game, played with heaps of beans on a table. A move for either player consists of choosing a heap, removing a certain number of beans from the heap, and then possibly splitting the remainder into several heaps; the winner is the player making the last move. For example, both Grundy’s Game (choose a heap and split it into two unequal heaps) and Couples-Are-Forever (choose a heap with at least three beans and split it into two) are taking-and-breaking games with very simple rules, however neither has been solved.

We present an overview of the required theory of impartial games. The reader can consult the references above for a more in-depth grounding in the theory of, and for more details about, subtraction and octal games.

The numbers in parentheses are the old numbers used in each of the lists of unsolved problems given on pp. 183–189 of AMS Proc. Sympos. Appl. Math. 43 (1991), called PSAM 43 below; on pp. 475–491 of Games of No Chance, hereafter referred to as GONC; on pp. 457–473 of More Games of No Chance (MGONC); and on pp. 475–500 of Games of No Chance 3 (GONC3). Some numbers have little more than the statement of the problem if there is nothing new to be added. References [year] may be found in Fraenkel’s bibliography at the end of this volume. References [#] are at the end of this article. A useful reference for the rules and an introduction to many of the specific games mentioned below is M. Albert, R. J. Nowakowski and D. Wolfe, Lessons in Play: An Introduction to the Combinatorial Theory of Games, A. K. Peters, 2007 (LIP) or Berlekamp, Conway and Guy, Winning Ways for your Mathematical Plays, vol. 1–4, A. K. Peters, 2000–2004 (WW).

Subtraction games with finite subtraction sets are known to have periodic nim-sequences. Investigate the relationship between the subtraction set and the length and structure of the period. The same question can be asked about partizan subtraction games, in which each player is assigned an individual subtraction set. See Fraenkel and Kotzig [1987].

TOPPLING DOMINOES, introduced by Albert, Nowakowski and Wolfe [1], is a combinatorial game played with a row of dominoes, such as the one pictured in Figure 1. Here each domino is colored blue or red (black or white, respectively, when color printing is unavailable). On his turn, Left selects any bLue (black) domino and topples it either east or west (his choice). This removes the toppled domino from the game, together with all other dominoes in the chosen direction. Likewise, Right’s options are to topple Red (white) dominoes east or west. For example, the Left options of are

Here A and B result from toppling the westmost domino respectively west or east, while C and D result from toppling the eastern black domino respectively west or east.