Introduction

An enthusiastic group of puzzlers, magicians, and mathematical game buffs held weekend gatherings in Atlanta in January or February of 1993, 1996, 1998, and 2000. These meetings, which honor Martin Gardner, the well-known author and former Scientific American games columnist, are now called “Gatherings for Gardner”. A collection of papers presented at the earlier gatherings was published by [?]. The problems shown in Figure 1 were presented at the fourth such gathering in February 2000. The problem statement appears in [?]. The present paper presents solutions to the problems that appeared in “Four Games for Gardner” for Gathering for Gardner IV. The solutions require some background in combinatorial game theory and thermography, topics with which the readers of this volume are assumed to be familiar.

Superficially, in Figure 1 there appears to be one problem in each of four different well-known games: Go, Domineering, checkers and chess. In each of the four games, the reader is to play white (horizontal in domineering). But lurking below the surface is a more interesting and much deeper problem which occurs if we play the sum of all four games added together.

There may be disputes about how to interpret the rules. Do they matter? In Go, purists might quibble over whether to use the North American, Chinese, or Japanese version of the superko rule. (As in nearly all Go positions, it makes no difference here.)

1. Introduction

In [Elkies 1996] we showed that certain chess endgames can be analyzed by the techniques of combinatorial game theory (CGT). We exhibited such endgames whose components show a variety of CGT values, including integers, fractions, and some infinite and infinitesimal values. Conspicuously absent were the values *k of Nim-heaps of size k > 1. Towards the end of [Elkies 1996] we asked whether any chess endgames, whether on the standard 8 x 8 chessboard or on larger rectangular boards, have components equivalent to *2, *4 and higher Nimbers. In the present paper we answer this question affirmatively by constructing a new class of pawn endgames on large boards that include *k for many large k, and conjecture — though we cannot yet prove — that all *k arise in this class.

Our construction begins with a variation of a pawns game called “Dawson's Chess” in [Berlekamp et al. 1982]. In §3 we introduce this variation and show that, perhaps surprisingly, all quiescent (non-entailing) components of the modified game are equivalent to Nim-heaps (Theorem 1). We then determine the value of each such component, and characterize non-loony moves (Theorem 2). In § 4 we construct pawn endgames on large chessboards that incorporate those components. These endgames do not yet attain our aim, because the values determined in Theorem 2 are all 0 or *1. In §5 we modify one of our components to obtain *2. In §6 we further study components modified in this way, showing that they, too, are equivalent to Nim-heaps (Theorem 3). We conclude with the numerical evidence suggesting that all Nim-heaps can be simulated by components of pawn endgames on large rectangular chessboards.

1. Introduction

John Conway's game Phutball [1-3,13], also known as Philosopher's Football, starts with a single black stone (the ball) placed at the center intersection of a rectangular grid such as a Go board. Two players sit on opposite sides of the board and take turns. On each turn, a player may either place a single white stone (a man) on any vacant intersection, or perform a sequence of jumps. To jump, the ball must be adjacent to one or more men. It is moved in a straight line (orthogonal or diagonal) to the first vacant intersection beyond the men, and the men so jumped are immediately removed (Figure 1). If a jump is performed, the same player may continue jumping as long as the ball continues to be adjacent to at least one man, or may end the turn at any point. Jumps are not obligatory: one can choose to place a man instead of jumping. The game is over when a jump sequence ends on or over the edge of the board closest to the opponent (the opponent's goal line) at which point the player who performed the jumps wins. It is legal for a jump sequence to step onto but not over one's own goal line. One of the interesting properties of Phutball is that any move could be played by either player, the only partiality in the game being the rule for determining the winner.

It is theoretically possible for a Phutball game to return to a previous position, so it may be necessary to add a loop-avoidance rule such as the one in Chess (three repetitions allow a player to claim a draw) or Go (certain repeated positions are disallowed). However, repetitions do not seem to come up much in actual practice.

1. Introduction

Consider a configuration of coins such as the one on the left of Figure 1. The player is allowed to move any coin to a position that is determined rigidly by incidences to other coins. In other words, a coin can be moved to any position adjacent to at least two other coins. The puzzle or 1-player game is to reach the configuration on the right of Figure 1 by a sequence of such moves. This particular puzzle is most interesting when each move is restricted to slide a coin in the plane without overlapping other coins.

This puzzle is described in Gardner's Mathematical Games article on Penny Puzzles [7], in Winning Ways [1], in Tokyo Puzzles [6], in Moscow Puzzles [8], and in The Penguin Book of Curious and Interesting Puzzles [11]. Langman [9] shows all 24 ways to solve the puzzle in three moves. Another classic puzzle of this sort [2; 6; 7; 11] is shown in Figure 2. A final classic puzzle that originally motivated our work is shown in Figure 3; its source is unknown. Other related puzzles are presented by Dudeney [5], Fujimura [6], and Brooke [4].

The preceding puzzles always move the centers of coins to vertices of the equilateral-triangle grid. Another type of puzzle is to move coins on the square grid, which appears less often in the literature but has significantly more structure and can be more difficult.

Introduction and Notation

In the game of Chomp, cookies are laid out at the lattice points where denotes the natural numbers and play is in dimensions. The cookie at the origin is poisonous. Two players alternate biting into the configuration, each bite eating the cookies in an infinite box from some lattice point outward in all directions, until one player loses by eating the poison cookie. The game can start from a position with finitely many bites already taken from rather than from all of. Chomp was invented by David Gale in [Ga74] and christened by Martin Gardner. When Chomp begins from a finite rectangle it is isomophic to an earlier game, Divisors, due to Schuh [Sch52]. See also [BCG82b], pp.598-606.

This paper considers Chomp on where denotes the ordinals, a subject the first author began studying in the early 1990s. This transfinite version of Chomp has been mentioned in Mathematical Intelligencer columns [Ga93; Ga96]; these are anthologized in [Ga98].

Identifying each ordinal a with the set, the sets

are the boxes at the origin and the Chomp bites in one and two dimensions. Similarly in d dimensions, the Chomp boxes and bites are the corresponding for. Every Chomp position X is a finite union of boxes, and conversely. As we will see, every Chomp game must terminate after finitely many bites.

1. Introduction

Alpha-beta (α-β) pruning is widely used to reduce the amount of search needed to analyze game trees. However, almost all discussion of α-β in the literature is restricted to game trees with real or integer valued positions. It may be useful to consider game trees with other valuation schema, such as vectors, sets, or constraints. In this paper, we attempt to find the most general conditions on a value set under which α-β may be used.

The intuition underlying α-β is that it is possible to eliminate from consideration portions of the game tree that can be shown not to be on the “main line.” Thus if one player P has a move leading to a position of value v, any alternative or future move that would let the opponent produce a value v’ worse for P than v need not be considered, since P can (and should) always make choices in a way that avoid the value v’.

The assumption in the literature has been that terms such as “better” and “worse” refer to comparisons made using a total order; there has been almost no consideration of games where payoffs may be incomparable. As an example, imagine a game involving a card selected at random from a standard 52-card deck. If I make move m1, I will win the game if the card is an ace.

As in [2], let Ug be the group of all partizan combinatorial games, let No be the field of surreal numbers, and for G in Ug, let L(G) and R(G) be the Left and Right sections of G, respectively. If L(G) is the section just to the left or right of some number z, we say that z is the Left stop of G, and similarly for R(G) and the Right stop. Let ShUg be the group of all short games in Ug; that is, ShUg is the set of all games born before day a;, or of all games which can be expressed in a form with only finitely many positions. For games U and integers n, we write

Also, recall from [1, Chapter 8] the definition of Norton multiplication, for a game G and a game U > 0:

Here, GL, GR, UL, and UR range independently over the left options of G, right options of G, left options of U, and right options of U, respectively. To define G.U, we must fix a form of G and sets of Left and Right options for U.

We will say that a subgroup X of Ug has the integer translation property if it contains the integers and, whenever either Left or Right has a winning move in a sum A1+…+An of games from X, not all integers, he also has a winning move in an Aj which is not equal to an integer.

Lemma 1. The real numbers have the integer translation property.

1. Solitaire

Peg Solitaire is a game for one player. Each move consists of hopping a peg over another one, which is removed. The goal is to reduce the board to a single peg. The best-known forms of the game take place on cross-shaped or triangular boards, and it has been marketed as “Puzzle Pegs” and “Hi-Q.” Discussions and various solutions can be found in [1; 2; 3; 4; 5].

In [6], Guy proposes one-dimensional Peg Solitaire as an open problem in the field of combinatorial games. Here we show that the set of solvable configurations forms a regular language, i.e. it can be recognized by a finite-state automaton. In fact, this was already shown in 1991 by Plambeck ([7], Introduction and Ch.5) and appeared as an exercise in a 1974 book of Manna [8]. More generally, B. Ravikumar showed that the set of solvable configurations on rectangular boards of any finite width is regular [9], although finding an explicit grammar seems to be difficult on boards of width greater than 2.

Thus there is little new about this result. However, it seems not to have appeared in print, so here it is.

1. Introduction and Background

We assume the reader is familiar with the first volume of Winning Ways [Berlekamp et al. 1982], including Conway's axiomatization of G, the group of partesan games under addition, which can also be found in [Conway 1976]. I now call the elements of this group traditional games. Each of the traditional games considered in this paper has only a finite number of positions. The identity of G is the game called 0, which is an immediate win for the second player. We investigate some interesting extensions of the group G to a bigger semi-group, S, generated by some new elements which are idempotents in the sense that each of them satisfies the equation G + G = G. We also present an addition table for these idempotents.

Some of these idempotents have long been well-known in other contexts. The newer ones all fall into a class I have been calling enriched environments. A companion paper [Berlekamp 2002] shows how these idempotents prove useful in solving a particular hard problem involving a gallimaufry of checkers, chess, domineering and Go.

We begin with a review of definitions, modified slightly to fit our present purposes.

Moves. In Go, a move is the change on the board resulting from the act of a single player. In chess, this is commonly called a ply, and the term move is used to describe a consecutive pair of plys, one by White and one by Black.