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].

Throughout this paper, the natural numbers = ﹛0, 1, 2, …﹜ are denoted , the set of sequences of natural numbers (integers) are denoted N∞ (Z∞) and the origin is denoted O.

For an introduction to combinatorial game theory, including impartial games, see [WW]. For a detailed description of misère quotient monoids, see [PS]. An algorithm for computing quotient monoids is found in [W].

1. Introduction

Traditionally, normal play impartial games and misère play impartial games are thought to differ by their respective winning conditions, that is, the goal of normal play is to make the final move to the empty game, whereas the goal of misère play is to force the opponent to make the final move. Both versions end when the last bean is removed. Here we present a different perspective, that normal play and misère play differ in the set of positions which are legal, so that both end with the same winning condition: having your opponent unable to move to a legal position.

Also, the traditional description of impartial games relies on the notion of the sum of two games and the theory is typically presented in terms of arbitrary sums of games. Here we adopt a notation that includes arbitrary sums of games under a particular ruleset as a set of lattice points ⊆ N∞. Hence we need not talk about sums of positions; all discussion will be localized to legal moves from a particular lattice point, since the notion of sum is inherent in the lattice point definition of a position.

Our basic framework is to play on several (but a finite number of) heaps of beans, where the rules allow a heap to be replaced by specified finite multisets of heaps of smaller sizes.

2. Heap games

Our definitions are motivated by the game of Nim, in which the set of all legal positions is generated by the set of heaps of various sizes.

1. Background

1.1. Algorithmic combinatorial game theory. Most of the results here revolve around the computational complexity of determining which player has a winning strategy from a given game position. There exist faster algorithms to solve this problem for some rulesets than for others. For each ruleset, we consider the computational problem that could be solved by such an algorithm. We will refer to both the ruleset and problem by the same name. We strongly encourage readers unfamiliar with these topics to refer to [1].

1.2. Terminology. A small amount of nonstandard terminology is used:

• • We use the word sticks to refer to the objects in nim heaps. Thus, a nim heap of size six contains six sticks.

• • An optimal sequence set is a set of sequences of plays for both players such that any move deviating from all of the sequences results in an _-position (meaning, the _ext player has a winning move). No move in that sequence should be nonoptimal for either player. Thus, if a player does not know whether they have a winning strategy, adhering to an optimal sequence is at least as good as any other move.

1.3. NIM. NIM is an impartial game played on a collection of heaps, each with a nonnegative number of sticks. On a player's turn, they choose a nonempty pile and remove as many sticks as desired (at least one) from that pile. A player loses when they cannot remove sticks (all piles are empty).

NIM is a classic impartial game, being the basis of Nimbers and Sprague– Grundy theory [8; 5]. NIM has lots of nice properties, from easy evaluation of games to obvious composition of two NIM games (the sum is just a new NIM game).

Introduction

This course is concerned with impartial combinatorial games, and in particular with misere play of such games. Loosely speaking, a combinatorial game is a two-player game with no hidden information and no chance elements. We usually impose one of two winning conditions on a combinatorial game: under normal play, the player who makes the last move wins, and under misère play, the player who makes the last move loses. We will shortly give more precise definitions.

The study of combinatorial games began in 1902, with C. L. Bouton's published solution to the game of NIM [2]. Further progress was sporadic until the 1930s, when R. P. Sprague [17; 18] and P. M. Grundy [6] independently generalized Bouton's result to obtain a complete theory for normal-play impartial games.

In a seminal 1956 paper [8], R. K. Guy and C. A. B. Smith introduced a wide class of impartial games known as octal games, together with some general techniques for analyzing them in normal play. Guy and Smith's techniques proved to be enormously powerful in finding normal-play solutions for such games, and they are still in active use today [4].

At exactly the same time (and, in fact, in exactly the same issue of the Proceedings of the Cambridge Philosophical Society), Grundy and Smith published a paper on misere games [7]. They noted that misère play appears to be quite difficult, in sharp contrast to the great success of the Guy–Smith techniques.

Despite these complications, Grundy remained optimistic that the Sprague– Grundy theory could be generalized in a meaningful way to misère play. These hopes were dashed in the 1970s, when Conway [3] showed that the Grundy–Smith complications are intrinsic. Conway's result shows that the most natural misereplay generalization of the Sprague–Grundy theory is hopelessly complicated, and is therefore essentially useless in all but a few simple cases.

1. Introduction

This paper uses ideas from combinatorial game theory, Beatty sequences, and Sturmian words. We have in many cases given the pertinent information in this paper, but have chosen to omit some material on these subjects. The reader who wishes to have more background information on certain topics is directed to the following references.

For standard terminology of impartial removal games on heaps of tokens, see [WW]; for Beatty sequences, see [B]; for k-Wythoff Nim, see [W; F]; for Sturmian words, see [L]; and for continued fractions, see [K].

Our problem is an inverse to that of the main field of research, which for a given an impartial ruleset Γ, (for example, Γ = k-Wythoff Nim) is to determine the P-positions of Γ (within reasonable time-complexity). Here we rather start with a particular (candidate) set of P-positions and search for “simple” game rules. Let us explain the setting.

Throughout this paper, we will denote the position consisting of two heaps of x ≥ 0 and y ≥ 0 tokens as (x, y). When the values of x and y are known, we adopt the convention that x ≤ y, though in general we regard such a position as an unordered multiset, so we identify (y, x) with (x, y).

Similarly, we let the move (u, v) denote a removal of v >0 tokens from one of the heaps and u from the other, where 0≤u ≤v; thus from the position (x, y), the move (u, v) is ambiguous, being either (x, y)→(x −u, y −v), provided both x−u ≥0 and y−v ≥0, or (x, y)→(x−v, y−u), provided both x−v ≥0 and y −u ≥ 0.

Combinatorial 2-player games can be studied from different perspectives. Traditionally the goal has been to acquire a perfect strategy, and to this purpose an efficient procedure (polynomial in succinct input size) is required. However, most combinatorial games are intrinsically hard to analyze; success is limited to a small number of games with predominant “mathematical structure”. The classical games of Nim (Bouton 1901) and Wythoff Nim (Wythoff 1907) are easy to analyze rigorously, but already seemingly modest variants, like (p, q)-GDWN (Larsson 2012, 2014), appear to withstand log-polynomial descriptions. Therefore, development of new methods is highly desirable.

Here, we use methods from physics, such as renormalization, in an attempt to understand the larger geometry of a game's P-positions (safe positions for the Previous player), rather than their exact configurations (Friedman et al. 2007, 2009). By studying evolution diagrams of a general class of linear extensions of Nim, Wythoff Nim and GDWN, we observe that P-positions often distribute uniformly along lines (a.k.a. P-beams), visually separated from the move lines. Given a fundamental hypothesis, a filling property which generalizes directly from Wythoff Nim, we formulate natural equations on the slopes and densities of P-positions along these lines; here, a key innovation, a reorganization model, guides us in selecting the relevant rules (move lines). The exceptional case of the symmetric (p, q)-GDWN is interesting, because of observed quasi-log repetitive fluctuations, and these games have defied all previous analysis.

1. Introduction to the class Linear Nimhoff

In this paper we study a class of combinatorial games using renormalizationbased techniques from physics in combination with computer simulations. This approach leads to a probabilistic geometric analysis of the underlying structure and behavior of a game. A number of interesting features are revealed, including observations of quasi log-periodic fluctuations. Our class of games, dubbed Linear Nimhoff, is a generalization of the classical Wythoff's game [Wythoff 1907] and the more recent GDWN [Larsson 2012a].

The renormalization approach to games involves:

• 1) Identifying broad, overall patterns in games (on a course grained level) by focusing on their scaling, asymptotic, and/or global, probabilistic behavior;

• 2) Analyzing these patterns using scaling/course-graining techniques derived from self-consistency conditions;

• 3) A novel method for this paper, a reorganization model, which filters out game rules that do not contribute to the broad overall patterns.

1. Bidding Chess

In chess, positions with only three pieces (the two kings and one more piece) are perfectly understood. The only such endgame requiring some finesse is king and pawn versus king, but even that endgame is played flawlessly by amateur players. In this article we investigate a chess variant where already positions with three pieces exhibit a complexity far beyond what can be embraced by a human.

Bidding Chess is a chess variant where instead of the two players alternating turns, the move order is determined by a bidding process. Each player has a stack of chips and at every turn, the players bid for the right to make the next move. The highest bidding player then pays what they bid to the opponent, and makes a move. The goal is to capture the opponent's king, and therefore there are no concepts of checkmate or stalemate.

As the total number of chips tends to infinity, there is in each position a limit proportion of chips that a player needs in order to force a win. We have computed these limits for all positions with three pieces, and the results (see for example Figure 7) show that already with such limited material, the game displays a remarkable intricacy.

Similar bidding games were introduced by David Richman in the 1980s, and presented in [4; 5] only after his tragic death. Bidding chess has been discussed in [1; 2; 3].

There are various reasonable protocols for making bids and handling situations of equal bids [3].

1. Introduction

The first author posed an intriguing problem at the GONC-workshop: “Describe nice/short rulesets for games with so-called complementary Beatty sequences as sets of P-positions.” The problem is the opposite of the main field of research in this area, which is to, given a game, search for its set of P-positions. Here we describe three such rule sets, which resolves the question for any pair of complementary homogenous Beatty sequences.

Let us recall the rules of d-Wythoff [10], d a fixed positive integer. The available positions are (x, y), x and y nonnegative integers. The legal moves are

• (I) Nim-type: (x, y) → (x − t, y), if x − t ≥ 0 and (x, y) → (x, y − t), if y −t ≥ 0; t > 0.

• (II) Extended diagonal type: (x, y)→(x −s, y−t) if |t −s|<d and x −s ≥ 0, y −t ≥ 0; s > 0, t > 0.

This game is a so-called impartial take-away game [2], vol. 1. We restrict attention to normal play; that is, the player first unable to move loses. For our games it means that the player called upon to move from (0, 0) loses.

Rules (I) and (II) imply that d-Wythoff is a so-called invariant [5; 16] (takeaway) game; that is, each available move is legal from any position as long as the resulting position has nonnegative coordinates. Every move in any invariant game is an invariant move. In this note we study another type of take-away game, where certain positions have some local restrictions on the set of otherwise invariant moves. Such games are called variant [5; 16]. We define these notions in Section 4.

1. A modification of the game of NIM

The game of NIM only preceded Wythoff's modification by a few years. By the famous theory of Sprague and Grundy some decades later, NIM drew a lot of attention. WYTHOFF NIM (here also called Wythoff's game), on the other hand, only became regularly revisited towards the end of the 20th century, but its winning strategy is related to Fibonacci's old discovery of the evolution of a rabbit population. The subject has been receiving more attention in recent decades thanks in part to new studies of WYTHOFF NIM and its variants by Fraenkel, and investigations into related sequences and arrays by Kimberling. In this paper we provide surveys of six different aspects of this subject by six of its current masters. Let us recall Wythoff's original definition of the game, given in item 1 in his paper [158]:

Wythoff proceeds by designating the safe positions (P-positions in current jargon) of his game, first by noting that the heaps are unordered, which implies

Table 1. The first few terms of the A and B sequences.

that (x, y) is safe if and only if (y, x) is, where x and y denote the respective number of counters in each pile. Then he proceeds to the nowadays celebrated minimal exclusive algorithm (mex), but without giving it a name. Let U be a finite subset of the nonnegative integers. Then the minimal excludant of U, mexU, is the smallest nonnegative integer not in U.

1. Introduction

At the 2011 Banff International Research Station Workshop on Combinatorial Game Theory, I asked professional 9-dan Ziang Zhujiu (“Jujo”) about the Go proverb your opponent's good move is your good move [?]. His instant response was “it's not always true”, and of course he is right. For example, in Figure 1 each player's winning move (in the opponent's territory) is the opponent's losing move, so no move is good for both a player and their opponent.

This paper is about good moves in no-draw HyperHex, a game that generalizes Hex. We consider the Go proverb above, and the general proverb it's never too late, which when applied to games could be expanded as it's never too late for a good move.

2. No-draw HyperHex

In 1942 Piet Hein invented Polygon, the classic alternate-turn two-player connection game now known as Hex [?]. In 1949 John Nash described the game to David Gale, who built a board that was soon in frequent use in Princeton's Fine Hall [?; ?]. Later John Milnor and independently Claude Shannon [?] and independently Charles Titus (personal communication with Craige Schensted) [?] invented Y, and Shannon created his eponymous (switching) game [?].

This book consists of 23 invited, original peer-reviewed papers in Combinatorial Game Theory (CGT) [5; 11; 46]—seven surveys and sixteen research papers. This is the fifth volume in the subseries Games Of No Chance (GONC) of the Mathematical Sciences Research Institute Publications. The name emphasizes these volumes’ focus on play with no dice and no hidden cards, situating them in the landscape of game theory at large, where incomplete and/or imperfect information is common. Considering our class of games, perfect play can in theory be computed, and thus we include games such as CHESS, GO and CHECKERS, but not YAHTZEE, BACKGAMMON and POKER.

Another characterizing feature is that combinatorial games are usually zerosum, typically win-loss situations, although in some games draws are also possible. Players alternate in moving, so for any game description, we include a move flag of who starts. Game positions can be very sensitive to this move flag, and a common question is, given a combinatorial game, if I offer you to start, should you accept?

Often it is better to start, but not always. In the popular game of GO, the second player is rewarded a “komi” advantage of about 6.5 points before the game begins. In CHESS it is also regarded that White has a slight advantage. In neither of these games there is a mathematical proof, of this believed advantage, but since the games have been played for thousands of years, the belief seems well founded through overwhelming play-evidence.

There are play-games which are also math-games. The first player loses the game TWENTYONE. The rules are as follows: start with the number 21. The players alternate in subtracting 1 or 2 from the current number. If you start, then (in perfect play) the other player will “complement your move modulo 3”, and win after 7 such rounds. Here, the game is specially designed to be a second player win.

We include three papers (13 on p. 313, 14 on p. 333 and 19 on p. 403) in the spirit of mechanism design in game theory; here, given a candidate set of P-positions4, related to Beatty's classical theorem [2; 3], these contributions construct three classes of game rules with this set as the set of P-positions.