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].

In 2001, Albert, Grossman, Nowakowski and Wolfe investigated a new 2-player partisan game called Clobber; they developed the first results in [Albert et al. 2005]. In terms of game values, it turns out that Clobber is difficult even when played on basic positions. This complexity explains the author’s motivation for studying Clobber. The description of the game follows below.

Black and white stones are placed on the vertices of an undirected graph, at most one per vertex. The first player moves only black stones and the second player the white ones. A player moves by picking up one of his stones and “clobbering” an adjacent stone of the opposite color (vertically or horizontally). The clobbered stone is deleted and replaced by the one that was moved. The last player to move wins.

Clobber is usually played on a grid where the initial position is the one of a checkerboard, as depicted by Figure 1.

In the last few years, several events were organized around Clobber like the first international Clobber tournament. It was held at the 2002 Dagstuhl seminar on algorithmic and combinatorial game theory (see the report in [Grossman 2004]). Since 2005 the game has been one of the events of the Computer Olympiad.

An impartial combinatorial game Γ is a two-player game with no hidden information and no chance elements, in which both players have exactly the same moves available at all times. When Γ is played under the misère play convention, the player who makes the last move loses.

Thirty years ago, Conway [] showed that the misère-play combinatorics of such games are often frighteningly complicated. However, new techniques recently pioneered by Plambeck [2005] have reinvigorated the subject. At the core of these techniques is the misère quotient, a commutative monoid that encodes the additive structure of an impartial combinatorial game (or a set of such games). See [Siegel 2015] for a gentle introduction to misère quotients, and [Plambeck and Siegel 2008] for a more rigorous one; see [Plambeck 2009] for a survey of the theory.

The introduction of misère quotients opens up a fascinating new area of study: the investigation of their algebraic properties. Such investigations are intrinsically interesting, and also have the potential to reveal new insights into the misère-play structure of combinatorial games. In this paper, we introduce several new results that expose quite a bit of structure in misère quotients.

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.

This paper gives an algorithm for computing certain misère indistinguishability quotient monoids. The approach employed here is not the genus theory of [Berlekamp et al. 2003, Chapter 13], but rather the quotient monoid approach introduced by Thane Plambeck [2005].

The algorithm described here was initially designed to analyze octal games, but is also valid for a broader class of games which will be called “heap rulesets”.

The notion of a heap ruleset comes from Nim, which is played with heaps of beans. The rules of Nim, and its variations, specify how a player may remove beans from a heap. The terminology below is influenced greatly by play of Nim, and readers may wish to keep games like Nim in mind when reading this paper. However, the collection of heap rulesets includes many other impartial games whose standard descriptions do not involve heaps. Chomp and Cram are examples.

Recall the classic children’s game Dots-and-Boxes [Berlekamp et al. 2003]. We start with an m x n square grid of dots. Players alternate drawing individual edges of the grid. If a player completes a box of the grid, s/he gets a point and must draw another edge; this process can repeat several times within a single turn. The game ends when all edges have been drawn, i.e., when all mn boxes have been completed. In normal Dots-and-Boxes, the player to receive the most points wins. In misère Dots-and-Boxes, the player to receive the fewest points wins. A draw (tie) occurs when mn is even and the players complete the same number of boxes.

Normal Dots-and-Boxes endgames are known to be NP-hard; see [Demaine and Hearn 2009]. In addition, no winning strategies are known when m or n is sufficiently large. To our knowledge, even the 1xn case is open for arbitrary n. On the other hand, misère Dots-and-Boxes may be easier to analyze.

Retrograde analysis has been applied to many problems. It enables to generate databases of positions or databases of patterns. For each possible position or pattern it enables to find the status of the position and other information such as the minimal number of moves required to win in the position. Once generated, databases enable to control, reduce or even replace search.

Retrograde analysis was first used to solve chess endgames [van den Herik and Herschberg 1985; Thompson 1986; Stiller 1996; Thompson 1996] containing up to six pieces. Chess endgame databases enable to play endgames perfectly and even discovered new chess knowledge about endgames [Nunn 1993].

Another successful application of retrograde analysis is the computation of Checkers endgames by Chinook [Lake et al. 1994; Schaeffer 1997] which is an important part of the program that solved Checkers [Schaeffer 2007]. Retrograde analysis has also been used in single player games such as the 16 puzzle. It consisted in computing an admissible heuristics involving only some of the pieces [Culberson and Schaeffer 1998]. Pattern database can also be combined and improve on single pattern databases [Korf and Felner 2002]. Another application of pattern databases is Rubik’s cube [Korf 1997] where separate databases for corner and side cubes can be computed and improve much the admissible heuristic. Pattern databases can also be used for the game of Go, computing for example databases on eyes or on life [Cazenave 1993; Cazenave 1996b; Cazenave 1996a]. Improvements include associating patterns to abstract conditions such as external liberties [Cazenave 2001] and reducing memory requirements using metarules [Cazenave 2003].

Some complex games such as Awari have been completely solved with retrograde analysis [Romein and Bal 2003].

In his thesis [Fraser 2002], Bill Fraser describes the BruteForce program that searches an endgame region in Go to calculate thermographs for every position. It enables his program to find means, temperatures, and orthodox lines of play. Our work is related since we use a brute force approach that takes ko into account, however we simply compute the values of positions and not the associated thermograph. Moreover we deal with long loops in the game graph, long loops only very rarely occur in Go positions.

The combinatorial game of Wythoff Nim [Wythoff 1907] is a so-called (2-player) impartial game played on two piles of tokens. As an addition to the rules of the game of Nim [Bouton 1901/02], where the players alternate in removing any finite number of tokens from precisely one of the piles (at most the whole pile), Wythoff Nim also allows removal of the same number of tokens from both piles. The player who removes the last token wins.

Monte-Carlo Go has recently improved to compete with the best Go programs [Coulom 2007; Gelly et al. 2006; Gelly and Silver 2007]. We are interested in the use of Monte-Carlo methods when there are independent games. In such cases it might be interesting to analyze the games independently instead of considering them as a unified game.

Section 2 describes related works Section 3 presents the Monte-Carlo algorithms we have tested. Section 4 details experimental results. Section 5 concludes.

In this section we expose related works on Monte-Carlo Go. We first explain basic Monte-Carlo Go as implemented in Gobble in 1993. Then we address the combination of search and Monte-Carlo Go, followed by the UCT algorithm, and previous works on the approximation of temperature.

The first Monte-Carlo Go program is Gobble [Brueg-mann 1993]. It uses simulated annealing on a list of moves. The list is sorted by the mean score of the games where the move has been played. Moves in the list are switched with their neighbor with a probability dependent on the temperature. The moves are tried in the games in the order of the list. At the end, the temperature is set to zero for a small number of games. After all games have been played, the value of a move is the average score of the games it has been played in first. Gobble-like programs have a good global sense but lack of tactical knowledge. For example, they often play useless Ataris, or try to save captured strings.

In this instalment of the Games Bibliography, Richard Nowakowski joined as a coauthor. Unlike previous instalments, this one is restricted, mainly, to new entries of the last few years that did not appear in previous versions, such as in “Games of No Chance 3” and the Electronic J. of Combinatorics (Surveys). We apologize profusely that due to lack of time, this version does not include all the new game papers we are aware of, and also lacks some unifying editing and polishing.

Roughly speaking, 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 outcome restricted to (lose, win), (tie, tie) and (draw, draw) for the two players who move alternately. Tie is an end position such as in tic-tac-toe, where no player wins, whereas draw is a dynamic tie: any position from which a player has a nonlosing move, but cannot force a win. Both the easy game of nim and the seemingly difficult chess are examples of combinatorial games. And so is go. The shorter terminology game, games is used below to designate combinatorial games.

Before reading this article one can consider the following Amazons position:

In Amazons there are a lot of hot positions: the players want to play to gain some territory. However, the position shown is not of this kind. If we use [Siegel 2011] to analyze it, this position proves to have value *3 + *2 = *. We will attempt to analyze the options of such a position and construct nimbers in partizan games. For instance, *3 = {0, *, *2 | 0, *, *2}, however, when we study the players options with [Siegel 2011] we see that the options are not {0,*,*2}. In a partizan game we have a bigger number of possible options to construct a nimber than in an impartial game. For instance, we know that a game like { ↑ | 0} has value * too. When we think about higher stars the number of possibilities is just gigantic. So it’s important to make some mathematical considerations to classify the games that “can act as nimbers”. In this article, we prove some useful results about the construction of nimbers and show some interesting examples in Amazons.

Combinatorial game theory has been applied to many kinds of existing games and has produced many excellent results. In the case of the game of Go, applications of CGT have been focused on endgames [Berlekamp and Wolfe1994; Berlekamp 1996; Müller et al. 1996; Nakamura and Berlekamp 2003; Spight 2003] and eyespace values [Landman 1996] so far. But it can be applied to any situations that involve counting. Recently, we developed a new genre of application of CGT to Go, that is, to count liberties in capturing races [Nakamura 2003; Nakamura 2009; Nakamura 2006].

Capturing races, or semeai is a particular kind of life and death problem in which two adjacent opposing groups are each fighting to capture the opponent’s group. A player’s strength in Go depends on their skills in winning capturing races as well as opening and endgame skills. In order to win a complicated capturing race, various techniques in counting liberties, taking away the opponent’s liberties, and extending self-liberties, are required in addition to broad and deep reading. Human expert players usually count liberties for each part of the blocks involved in semeai, sum them, and decide the outcome. A position of capturing races can also be decomposed into independent subpositions, as in the cases of endgames and eyespaces, and we can apply CGT to analyse the capturing races. We propose a method of analysing capturing races that have no shared liberty or have only simple shared liberties, and then, using combinatorial game values of external liberties, give an evaluation formula to find the outcome of the capturing races.

MAZE was introduced in [Albert et al. 2007], but apart from a few scattered observations, nothing was known about the values of the game. In the original article, MAZE is played on a rectangular grid oriented 45◦ to the horizontal.

The token starts at the top of the board and highlighted edges are walls that may not be crossed. Left is allowed to move a token any number of cells in a southwesterly direction and Right is allowed to move similarly in a southeasterly direction. However, for ease of referring to specific places in the position, we re-orient the sides parallel to the page so that Left moves downward and Right moves to the right; see Figure 1. One interesting feature is that any number of consecutive Left (Right) moves also can be accomplished in one move. This feature had been noted in several games, including HACKENBUSH strings [Berlekamp et al. 2001], and given the name of option-closed in [Nowakowski and Ottaway 2011], a reference we henceforth abbreviate as [NO]. Siegel [2011] notes that the partial order of option-closed games born on day n forms a planar lattice.

In two-player combinatorial games, the last player to move either wins (normal play) or loses (misère play). Traditionally, normal play games have garnered more attention due to the group structure which arises on such games. Less work has been done with games played under the misère play convention, Just as in normal play, misère games can be placed in equivalence classes, where two games G and H are equivalent if the outcome class of G + K is the same as the outcome class of H+K for all games K. However, Conway showed that, unlike in normal play, these misère equivalence classes are sparsely populated, making the analysis of misère games under such equivalence classes far less useful than their normal play counterparts [ONAG]. Even though these equivalence classes are sparse, Conway developed a method, called genus theory, for analyzing impartial games played under the misère play convention [Allen 2006; WW; ONAG]. For years, this was the only universal tool available for those studying misère games.

In [Plambeck 2009; 2005; Plambeck and Siegel 2008; Siegel 2006; 2015b], many results regarding impartial misère games have been achieved. These results were obtained by taking a game, restricting the universe in which that game was played, and obtaining its misère quotient. However, while, as Siegel [2015a] says “a partizan generalization exists”, few results have been obtained regardingthe structure of the misère quotients which arise from partizan games.

For a game G ={GL|GR}, we define Ḡ = {GR|GL}. Those familiar with normal play will notice that under the normal play convention rather than Ḡ, we would generally write̶G. In normal play, this nomenclature is quite sensible as G+(-G)=0 [Albert et al. 2007], giving us the Tweedledum–Tweedledee principle; the second player can always win the game G+(-G) by mimicking the move of the first player, but in the other component. However, in misère play, not only does the Tweedledum–Tweedledee strategy often fail, G + Ḡ is not necessarily equivalent to 0. For example, *2+*2=*2+*2 is not equivalent to 0 [Allen 2006; WW]. However, having the property that G+Ḡ is equivalent to 0 is much desired, as it gives a link to which partizan misère games may behave like their normal counterparts.