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game_partI - GAME THEORY Thomas S Ferguson Part I Impartial...

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GAME THEORY Thomas S. Ferguson Part I. Impartial Combinatorial Games 1. Take-Away Games. 1.1 A Simple Take-Away Game. 1.2 What is a Combinatorial Game? 1.3 P-positions, N-positions. 1.4 Subtraction Games. 1.5 Exercises. 2. The Game of Nim. 2.1 Preliminary Analysis. 2.2 Nim-Sum. 2.3 Nim With a Larger Number of Piles. 2.4 Proof of Bouton’s Theorem. 2.5 Mis` ere Nim. 2.6 Exercises. 3. Graph Games. 3.1 Games Played on Directed Graphs. 3.2 The Sprague-Grundy Function. 3.3 Examples. 3.4 The Sprague-Grundy Function on More General Graphs. 3.5 Exercises. 4. Sums of Combinatorial Games. 4.1 The Sum of n Graph Games. 4.2 The Sprague Grundy Theorem. 4.3 Applications. I – 1

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4.4 Take-and-Break Games. 4.5 Exercises. 5. Coin Turning Games. 5.1 Examples. 5.2 Two-dimensional Coin Turning Games. 5.3 Nim Multiplication. 5.4 Tartan Games. 5.5 Exercises. 6. Green Hackenbush. 6.1 Bamboo Stalks. 6.2 Green Hackenbush on Trees. 6.3 Green Hackenbush on General Rooted Graphs. 6.4 Exercises. References. I – 2
Part I. Impartial Combinatorial Games 1. Take-Away Games. Combinatorial games are two-person games with perfect information and no chance moves, and with a win-or-lose outcome. Such a game is determined by a set of positions, including an initial position, and the player whose turn it is to move. Play moves from one position to another, with the players usually alternating moves, until a terminal position is reached. A terminal position is one from which no moves are possible. Then one of the players is declared the winner and the other the loser. There are two main references for the material on combinatorial games. One is the research book, On Numbers and Games by J. H. Conway, Academic Press, 1976. This book introduced many of the basic ideas of the subject and led to a rapid growth of the area that continues today. The other reference, more appropriate for this class, is the two-volume book, Winning Ways for your mathematical plays by Berlekamp, Conway and Guy, Academic Press, 1982, in paperback. There are many interesting games described in this book and much of it is accessible to the undergraduate mathematics student. This theory may be divided into two parts, impartial games in which the set of moves available from any given position is the same for both players, and partizan games in which each player has a different set of possible moves from a given position. Games like chess or checkers in which one player moves the white pieces and the other moves the black pieces are partizan. In Part I, we treat only the theory of impartial games. An elementary introduction to impartial combinatorial games is given in the book Fair Game by Richard K. Guy, published in the COMAP Mathematical Exploration Series, 1989. We start with a simple example. 1.1 A Simple Take-Away Game. Here are the rules of a very simple impartial combinatorial game of removing chips from a pile of chips.

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