1. Introduction

The main result of this paper states that all the short combinatorial games are game values of particular positions of KONANE (Theorem 14 in Section 4). To prove this result constructively, two instrumental lemmas giving the needed building “pieces” are used (Lemmas 12 and 13 in Section 3). Some fundamental results of combinatorial game theory, like reduction concepts and the largest game value of the day n, are also used. Readers can find these results in Section 3. The next paragraphs of this introduction and Section 2 have some basics: the rules of KONANE, the state of the art and motivation for this research, and the definitions of habitat and universality of a ruleset (Definitions 3 and 7). Readers who know the rules of KONANE and are fluent in combinatorial game theory may wish to proceed to Lemma 12, Lemma 13 and Theorem 14 (Sections 3 and 4).

One of the principal goals of combinatorial game theory (CGT) is the study of combinatorial rulesets with the following properties ([1; 2; 5; 15] are fundamental references, [7] is a complete survey):

• • There are two players who take turns moving alternately.

• • No chance devices such as dice, spinners, or card deals are involved, and each player is aware of all the details at all times.

• • Even ignoring the alternating condition, the play must end in a finite number of moves, and the winner is often determined on the basis of who made the last move. Under normal play, the last player wins, while in misere play, the last player loses.

We distinguish between multiple meanings of the word game by using the words ruleset and game. The word ruleset has a concrete meaning related to some particular set of rules (KONANE, AMAZONS, NIM are examples of rulesets).

1. Introduction

Officers is an impartial game played with beans arranged into heaps. On each move, a player selects a heap with at least two beans and removes exactly one bean from the heap. If at least two beans remain in the heap, then the player may also optionally split the remaining heap into two. Officers is an example of an octal game [Berlekamp, Conway and Guy 2001]: a take-and-break game where each legal move can be described as removing k beans from a heap, then possibly splitting any remaining beans, leaving behind exactly 0, 1 or 2 nonempty heaps. The name “octal game” comes from the fact that such games can conveniently be described as a string of octal digits (traditionally preceded by a decimal point with trailing zeros omitted), where the k-th digit after the decimal point encodes the legal moves involving the removal of k beans from a heap. The binary representation of a digit has three bits, with bit m indicating whether or not it is legal to leave exactly m nonempty heaps. Thus, if the k-th digit is zero then it is never legal to remove exactly k beans. Under this encoding, Officers is the game .6: only the first digit after the decimal point is nonzero (only one bean can be removed), and this digit has bits 1 and 2 set (exactly 1 or 2 nonempty heaps can remain).

Leibniz's binary arithmetics

In the seventeenth century, Francis Bacon used binary 5-tuples as a code, but binary arithmetic as we currently understand it—doing actual arithmetic with binary numbers rather than just using binary representations—starts with Leibniz about 1679, though he didn't publicize it until the late 1600s. He heard about the Fu-Hsi ordering of the I-Ching hexagrams from Jesuit missionaries in China in 1701 and wrote a good deal about it thereafter (see Figures 1 and 2 and p. 219).

Figure 1. The title page of Leibniz's booklet [Leibniz 1734] explaining binary notation to a nobleman shows a medallion he created, later borrowed by the Stadtsparkasse of Hanover to honor Leibniz himself.

Figure 2. Leibniz's first writing on binary arithmetic, dated 11 March 1679.

However, Leibniz was anticipated by Thomas Harriot, 1604, who did not publish, and by John Napier, whose Rabdologiæ of 1617 gave binary arithmetic as far as computing square roots, but this seems to have been ignored.

But binary ideas go much further back. Some simple counting systems are more or less base 2 and there are many instances of duality in nature—hands, sexes, etc. But we are interested in material that is somewhat more mathematical.

Binary multiplication

The earliest implicit use of binary representations occurs in ancient Egyptian mathematics. Figure 3 is Problem 30 of the Rhind Mathematical Papyrus, ca. 1700 B.C.E., computing (⅔ + ⅒) ×13. The problem is to solve (⅔ + ⅒)x =10, which is being done by false position, using x = 13 as a trial. Because of their complicated notation for numbers, especially fractions, they multiplied by repeatedly doubling, then adding the appropriate terms. For instance, to multiply a number by 13, they computed successively the double, the quadruple and the octuple of the number, then added the number to its quadruple and its octuple.

1. Introduction

Combinatorial games where the winner is determined by a “score”, rather than who moves last, have been largely ignored by combinatorial game theorists. As far as this author is aware, there have been four previous studies of scoring play combinatorial games, all of which focused on the universe of “well-tempered” scoring games.

There are the works of Milnor [9], Hanner [6], Ettinger [3; 4] and most recently, Johnson [7]. The definition of a scoring game that all of them used is the following.

Definition 1. A scoring game is defined as

The authors would say that a game G, where GL = GR = ∅, is atomic and all other games are not atomic. Using this terminology, they were able to define concepts such as the disjunctive sum, and the “left outcome” and “right outcome”, which are the score at the end of a game under optimal play, when Left and Right move first respectively. Their mathematical definitions are given here.

Definition 2. The disjunctive sum is defined as follows:

Definition 3. The left outcome L(G) and right outcome R(G) are defined as follows:

Effectively they showed that under the disjunctive sum, this class of games forms a nontrivial monoid, and that with certain restrictions, it is equivalent to the set of all small normal play combinatorial games.

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.

Overview

In its broadest sense, combinatorial game theory (CGT) is the study of twoperson, perfect information games of no chance. For each position in such a game, the theory defines a temperature, which is a measure of the importance of the next move. CGT differs from economic game theory, which emphasizes multiplayer games including elements of chance and imperfect information. Most economic games focus on maximizing some payoff or score; CGT was originally more concerned with getting the last move, but it now also applies to games whose outcomes are determined by scores.

CGT is a branch of mathematics. It seeks to find and understand strategies which can provably succeed against any opposition. This differs from the primary goal of human or computer competitors, who are more focused on making fewer serious mistakes than their opponents. CGT seeks to understand every position, including composed problems. It assigns no special importance to any official “opening” position, nor to who gets the first move. Each position is treated as its own game, and both possibilities for who moves next are given appropriate consideration. Most CGT results employ the “divide and conquer” methodology:

• (1) partition the board into disjoint regions;

• (2) analyze each region, condensing it into an appropriate data structure;

• (3) analyze the entire board position as the (disjunctive) sum of these disjoint regions.

The results are so interesting that many combinatorial game theorists now also play and analyze hybrid games, which are sums of positions in different games. Such a hybrid sum is called a gallimaufry.

The most successful application of CGT to Anglo-American checkers has been to composed problems (e.g., [Berlekamp 2002]). Elkies [1996] has successfully applied CGT to composed chess problems, and even to at least one position which occurred in a world championship chess match. But in most historical games of chess and checkers, every position that occurs is already as well understood by players who know no CGT as by those who do.

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.

1. Introduction

Most research in combinatorial game theory assumes normal play, where the first player unable to move loses, as opposed to misere play, where the first player unable to move wins. It is rather remarkable how much changes when we simply switch the goal from getting the last move to avoiding the last move. At first glance one might think misere play is merely the “opposite” of normal play, but this is not at all the case. There is actually no relationship between normal outcome and misère outcome: for every pair of (not necessarily distinct) outcomes , there is a game with normal outcomeand misère outcome [11]. Likewise, strategies from normal play are in general neither the same nor reversed for misère play. For example, in normal play, Left would always choose a move to 1 = ﹛0|·﹜ over a move to 0 = ﹛·|·﹜, but in misère play there are situations in which Left should choose 1 over 0 and others where Left prefers 0 over 1. This means that 0 and 1 are incomparable in misere play, which goes against our intuition that Left is trying to run out of moves before Right.

So we really are in a fog in misere play. We look to the elegant algebra of normal-play games and hope for some semblance of structure, but we are dismayed at every turn:

Zero is trivial. In normal play we have the wonderful property that every previous-win game is equal to zero. In misere, the zero game is next-win, but our hopes that perhaps every next-win game is equal to zero are more than dashed; in fact, only the game ﹛·|·﹜ is equal to zero [11].

1. Introduction

At the workshop in combinatorial games in BIRS 2011, Aviezri Fraenkel posed the following intriguing problem: find nice (short/simple) rules for a 2-player combinatorial game for which the P-positions are obtained from a pair of complementary

Beatty sequences [Be]. We begin by solving this problem, by defining a class of heap games, dubbed bichromatic nim, or just chromatic nim, and then later in Section 4, we explain some background to the problem. In Section 3, we solve a similar game on arithmetic progressions. In Section 5, we discuss the general environment for chromatic nim on two heaps. At last, in Section 6, we study the famous evil numbers, also known as the indexes of the 0's in the Thue–Morse sequence.

2. Bichromatic nim finds a game for your complementary Beatty solution

Let S denote a subset of the positive integers. We let the i-th token in a stack be red if i ϵ S, and otherwise the i-th token is green. We play a take-away game on k ≥ 0 copies of such stacks of various finite sizes. Classical nim rules are always allowed; any number of tokens can be removed from precisely one of the stacks.

Philosophers’ Phutball, invented in the towers of Cambridge by Conway and company, and named after a much-beloved Monty Python sketch of a similar name, is a two-player game played on a m×n board. The official Phutball pitch, as described by Berlekamp–Conway–Guy [BCG03], is 19×15 (that is, there are 19 rows and 15 columns); however, Phutball is usually played on a 19×19 Go board.

The rules of the game are fairly simple. One of the grid points is occupied by the ball, usually a black Go stone. Alfred, the first player, wins if the ball crosses the topmost row or ends up in the topmost row at the end of a turn. Betty, going second, wins if the ball crosses the bottommost row or ends up in the bottommost row at the end of a turn. On their turn, the players may either place a chap, usually a white Go stone, or move the ball. The ball is moved by jumping over a line of chaps in one of the eight possible directions, and those chaps are immediately removed; and multiple jumps are allowed, although not required. However, in all our diagrams, we will represent the ball by a gray stone and the chaps by black stones (Phutball played with reversed colors is called floodlit Phutball).

Despite the simplicity of the rules, the game is fairly complicated. Phutball on an n×n board is PSPACE-hard [Der10] and to even check if one has a win in one is NP-complete [DDE02]. Several variants of Phutball have been analyzed in great detail, such as “directional Phutball” [Loo08] when the players are constrained to jump in certain directions, or “one-dimensional Phutball” [GN02] played on an m ×1 board. Even the one-dimensional game is surprisingly complicated: It has only been analyzed fully in [GN02] when the players are forbidden to place “off-parity” chaps.