Combinatorial Games, Theory and Applications -. [ Tim Wylie
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On Variations of Nim and Chomp Arxiv:1705.06774V1 [Math.CO] 18
On Variations of Nim and Chomp June Ahn Benjamin Chen Richard Chen Ezra Erives Jeremy Fleming Michael Gerovitch Tejas Gopalakrishna Tanya Khovanova Neil Malur Nastia Polina Poonam Sahoo Abstract We study two variations of Nim and Chomp which we call Mono- tonic Nim and Diet Chomp. In Monotonic Nim the moves are the same as in Nim, but the positions are non-decreasing numbers as in Chomp. Diet-Chomp is a variation of Chomp, where the total number of squares removed is limited. 1 Introduction We study finite impartial games with two players where the same moves are available to both players. Players alternate moves. In a normal play, the person who does not have a move loses. In a misère play, the person who makes the last move loses. A P-position is a position from which the previous player wins, assuming perfect play. We can observe that all terminal positions are P-positions. An N-position is a position from which the next player wins given perfect play. When we play we want to end our move with a P-position and want to see arXiv:1705.06774v1 [math.CO] 18 May 2017 an N-position before our move. Every impartial game is equivalent to a Nim heap of a certain size. Thus, every game can be assigned a non-negative integer, called a nimber, nim- value, or a Grundy number. The game of Nim is played on several heaps of tokens. A move consists of taking some tokens from one of the heaps. The game of Chomp is played on a rectangular m by n chocolate bar with grid lines dividing the bar into mn squares. -
An Introduction to Conway's Games and Numbers
AN INTRODUCTION TO CONWAY’S GAMES AND NUMBERS DIERK SCHLEICHER AND MICHAEL STOLL 1. Combinatorial Game Theory Combinatorial Game Theory is a fascinating and rich theory, based on a simple and intuitive recursive definition of games, which yields a very rich algebraic struc- ture: games can be added and subtracted in a very natural way, forming an abelian GROUP (§ 2). There is a distinguished sub-GROUP of games called numbers which can also be multiplied and which form a FIELD (§ 3): this field contains both the real numbers (§ 3.2) and the ordinal numbers (§ 4) (in fact, Conway’s definition gen- eralizes both Dedekind sections and von Neumann’s definition of ordinal numbers). All Conway numbers can be interpreted as games which can actually be played in a natural way; in some sense, if a game is identified as a number, then it is under- stood well enough so that it would be boring to actually play it (§ 5). Conway’s theory is deeply satisfying from a theoretical point of view, and at the same time it has useful applications to specific games such as Go [Go]. There is a beautiful microcosmos of numbers and games which are infinitesimally close to zero (§ 6), and the theory contains the classical and complete Sprague-Grundy theory on impartial games (§ 7). The theory was founded by John H. Conway in the 1970’s. Classical references are the wonderful books On Numbers and Games [ONAG] by Conway, and Win- ning Ways by Berlekamp, Conway and Guy [WW]; they have recently appeared in their second editions. -
Impartial Games
Combinatorial Games MSRI Publications Volume 29, 1995 Impartial Games RICHARD K. GUY In memory of Jack Kenyon, 1935-08-26 to 1994-09-19 Abstract. We give examples and some general results about impartial games, those in which both players are allowed the same moves at any given time. 1. Introduction We continue our introduction to combinatorial games with a survey of im- partial games. Most of this material can also be found in WW [Berlekamp et al. 1982], particularly pp. 81{116, and in ONAG [Conway 1976], particu- larly pp. 112{130. An elementary introduction is given in [Guy 1989]; see also [Fraenkel 1996], pp. ??{?? in this volume. An impartial game is one in which the set of Left options is the same as the set of Right options. We've noticed in the preceding article the impartial games = 0=0; 0 0 = 1= and 0; 0; = 2: {|} ∗ { | } ∗ ∗ { ∗| ∗} ∗ that were born on days 0, 1, and 2, respectively, so it should come as no surprise that on day n the game n = 0; 1; 2;:::; (n 1) 0; 1; 2;:::; (n 1) ∗ {∗ ∗ ∗ ∗ − |∗ ∗ ∗ ∗ − } is born. In fact any game of the type a; b; c;::: a; b; c;::: {∗ ∗ ∗ |∗ ∗ ∗ } has value m,wherem =mex a;b;c;::: , the least nonnegative integer not in ∗ { } the set a;b;c;::: . To see this, notice that any option, a say, for which a>m, { } ∗ This is a slightly revised reprint of the article of the same name that appeared in Combi- natorial Games, Proceedings of Symposia in Applied Mathematics, Vol. 43, 1991. Permission for use courtesy of the American Mathematical Society. -
Algorithmic Combinatorial Game Theory∗
Playing Games with Algorithms: Algorithmic Combinatorial Game Theory∗ Erik D. Demaine† Robert A. Hearn‡ Abstract Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in Combinatorial Game Theory, which analyzes ideal play in perfect-information games, and Constraint Logic, which provides a framework for showing hardness. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer. 1 Introduction Many classic games are known to be computationally intractable (assuming P 6= NP): one-player puzzles are often NP-complete (as in Minesweeper) or PSPACE-complete (as in Rush Hour), and two-player games are often PSPACE-complete (as in Othello) or EXPTIME-complete (as in Check- ers, Chess, and Go). Surprisingly, many seemingly simple puzzles and games are also hard. Other results are positive, proving that some games can be played optimally in polynomial time. In some cases, particularly with one-player puzzles, the computationally tractable games are still interesting for humans to play. We begin by reviewing some basics of Combinatorial Game Theory in Section 2, which gives tools for designing algorithms, followed by reviewing the relatively new theory of Constraint Logic in Section 3, which gives tools for proving hardness. -
COMPSCI 575/MATH 513 Combinatorics and Graph Theory
COMPSCI 575/MATH 513 Combinatorics and Graph Theory Lecture #34: Partisan Games (from Conway, On Numbers and Games and Berlekamp, Conway, and Guy, Winning Ways) David Mix Barrington 9 December 2016 Partisan Games • Conway's Game Theory • Hackenbush and Domineering • Four Types of Games and an Order • Some Games are Numbers • Values of Numbers • Single-Stalk Hackenbush • Some Domineering Examples Conway’s Game Theory • J. H. Conway introduced his combinatorial game theory in his 1976 book On Numbers and Games or ONAG. Researchers in the area are sometimes called onagers. • Another resource is the book Winning Ways by Berlekamp, Conway, and Guy. Conway’s Game Theory • Games, like everything else in the theory, are defined recursively. A game consists of a set of left options, each a game, and a set of right options, each a game. • The base of the recursion is the zero game, with no options for either player. • Last time we saw non-partisan games, where each player had the same options from each position. Today we look at partisan games. Hackenbush • Hackenbush is a game where the position is a diagram with red and blue edges, connected in at least one place to the “ground”. • A move is to delete an edge, a blue one for Left and a red one for Right. • Edges disconnected from the ground disappear. As usual, a player who cannot move loses. Hackenbush • From this first position, Right is going to win, because Left cannot prevent him from killing both the ground supports. It doesn’t matter who moves first. -
~Umbers the BOO K O F Umbers
TH E BOOK OF ~umbers THE BOO K o F umbers John H. Conway • Richard K. Guy c COPERNICUS AN IMPRINT OF SPRINGER-VERLAG © 1996 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1996 All rights reserved. No part of this publication may be reproduced, stored in a re trieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Published in the United States by Copernicus, an imprint of Springer-Verlag New York, Inc. Copernicus Springer-Verlag New York, Inc. 175 Fifth Avenue New York, NY lOOlO Library of Congress Cataloging in Publication Data Conway, John Horton. The book of numbers / John Horton Conway, Richard K. Guy. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4612-8488-8 e-ISBN-13: 978-1-4612-4072-3 DOl: 10.l007/978-1-4612-4072-3 1. Number theory-Popular works. I. Guy, Richard K. II. Title. QA241.C6897 1995 512'.7-dc20 95-32588 Manufactured in the United States of America. Printed on acid-free paper. 9 8 765 4 Preface he Book ofNumbers seems an obvious choice for our title, since T its undoubted success can be followed by Deuteronomy,Joshua, and so on; indeed the only risk is that there may be a demand for the earlier books in the series. More seriously, our aim is to bring to the inquisitive reader without particular mathematical background an ex planation of the multitudinous ways in which the word "number" is used. -
ES.268 Dynamic Programming with Impartial Games, Course Notes 3
ES.268 , Lecture 3 , Feb 16, 2010 http://erikdemaine.org/papers/AlgGameTheory_GONC3 Playing Games with Algorithms: { most games are hard to play well: { Chess is EXPTIME-complete: { n × n board, arbitrary position { need exponential (cn) time to find a winning move (if there is one) { also: as hard as all games (problems) that need exponential time { Checkers is EXPTIME-complete: ) Chess & Checkers are the \same" computationally: solving one solves the other (PSPACE-complete if draw after poly. moves) { Shogi (Japanese chess) is EXPTIME-complete { Japanese Go is EXPTIME-complete { U. S. Go might be harder { Othello is PSPACE-complete: { conjecture requires exponential time, but not sure (implied by P 6= NP) { can solve some games fast: in \polynomial time" (mostly 1D) Kayles: [Dudeney 1908] (n bowling pins) { move = hit one or two adjacent pins { last player to move wins (normal play) Let's play! 1 First-player win: SYMMETRY STRATEGY { move to split into two equal halves (1 pin if odd, 2 if even) { whatever opponent does, do same in other half (Kn + Kn = 0 ::: just like Nim) Impartial game, so Sprague-Grundy Theory says Kayles ≡ Nim somehow { followers(Kn) = fKi + Kn−i−1;Ki + Kn−i−2 j i = 0; 1; :::;n − 2g ) nimber(Kn) = mexfnimber(Ki + Kn−i−1); nimber(Ki + Kn−i−2) j i = 0; 1; :::;n − 2g { nimber(x + y) = nimber(x) ⊕ nimber(y) ) nimber(Kn) = mexfnimber(Ki) ⊕ nimber(Kn−i−1); nimber(Ki) ⊕ nimber(Kn−i−2) j i = 0; 1; :::n − 2g RECURRENCE! | write what you want in terms of smaller things Howe do w compute it? nimber(K0) = 0 (BASE CASE) nimber(K1) = mexfnimber(K0) ⊕ nimber(K0)g 0 ⊕ 0 = 0 = 1 nimber(K2) = mexfnimber(K0) ⊕ nimber(K1); 0 ⊕ 1 = 1 nimber(K0) ⊕ nimber(K0)g 0 ⊕ 0 = 0 = 2 so e.g. -
Blue-Red Hackenbush
Basic rules Two players: Blue and Red. Perfect information. Players move alternately. First player unable to move loses. The game must terminate. Mathematical Games – p. 1 Outcomes (assuming perfect play) Blue wins (whoever moves first): G > 0 Red wins (whoever moves first): G < 0 Mover loses: G = 0 Mover wins: G0 Mathematical Games – p. 2 Two elegant classes of games number game: always disadvantageous to move (so never G0) impartial game: same moves always available to each player Mathematical Games – p. 3 Blue-Red Hackenbush ground prototypical number game: Blue-Red Hackenbush: A player removes one edge of his or her color. Any edges not connected to the ground are also removed. First person unable to move loses. Mathematical Games – p. 4 An example Mathematical Games – p. 5 A Hackenbush sum Let G be a Blue-Red Hackenbush position (or any game). Recall: Blue wins: G > 0 Red wins: G < 0 Mover loses: G = 0 G H G + H Mathematical Games – p. 6 A Hackenbush value value (to Blue): 3 1 3 −2 −3 sum: 2 (Blue is two moves ahead), G> 0 3 2 −2 −2 −1 sum: 0 (mover loses), G= 0 Mathematical Games – p. 7 1/2 value = ? G clearly >0: Blue wins mover loses! x + x - 1 = 0, so x = 1/2 Blue is 1/2 move ahead in G. Mathematical Games – p. 8 Another position What about ? Mathematical Games – p. 9 Another position What about ? Clearly G< 0. Mathematical Games – p. 9 −13/8 8x + 13 = 0 (mover loses!) x = -13/8 Mathematical Games – p. -
Combinatorial Game Theory, Well-Tempered Scoring Games, and a Knot Game
Combinatorial Game Theory, Well-Tempered Scoring Games, and a Knot Game Will Johnson June 9, 2011 Contents 1 To Knot or Not to Knot 4 1.1 Some facts from Knot Theory . 7 1.2 Sums of Knots . 18 I Combinatorial Game Theory 23 2 Introduction 24 2.1 Combinatorial Game Theory in general . 24 2.1.1 Bibliography . 27 2.2 Additive CGT specifically . 28 2.3 Counting moves in Hackenbush . 34 3 Games 40 3.1 Nonstandard Definitions . 40 3.2 The conventional formalism . 45 3.3 Relations on Games . 54 3.4 Simplifying Games . 62 3.5 Some Examples . 66 4 Surreal Numbers 68 4.1 Surreal Numbers . 68 4.2 Short Surreal Numbers . 70 4.3 Numbers and Hackenbush . 77 4.4 The interplay of numbers and non-numbers . 79 4.5 Mean Value . 83 1 5 Games near 0 85 5.1 Infinitesimal and all-small games . 85 5.2 Nimbers and Sprague-Grundy Theory . 90 6 Norton Multiplication and Overheating 97 6.1 Even, Odd, and Well-Tempered Games . 97 6.2 Norton Multiplication . 105 6.3 Even and Odd revisited . 114 7 Bending the Rules 119 7.1 Adapting the theory . 119 7.2 Dots-and-Boxes . 121 7.3 Go . 132 7.4 Changing the theory . 139 7.5 Highlights from Winning Ways Part 2 . 144 7.5.1 Unions of partizan games . 144 7.5.2 Loopy games . 145 7.5.3 Mis`eregames . 146 7.6 Mis`ereIndistinguishability Quotients . 147 7.7 Indistinguishability in General . 148 II Well-tempered Scoring Games 155 8 Introduction 156 8.1 Boolean games . -
Combinatorial Game Theory
Combinatorial Game Theory Aaron N. Siegel Graduate Studies MR1EXLIQEXMGW Volume 146 %QIVMGER1EXLIQEXMGEP7SGMIX] Combinatorial Game Theory https://doi.org/10.1090//gsm/146 Combinatorial Game Theory Aaron N. Siegel Graduate Studies in Mathematics Volume 146 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Daniel S. Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 91A46. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-146 Library of Congress Cataloging-in-Publication Data Siegel, Aaron N., 1977– Combinatorial game theory / Aaron N. Siegel. pages cm. — (Graduate studies in mathematics ; volume 146) Includes bibliographical references and index. ISBN 978-0-8218-5190-6 (alk. paper) 1. Game theory. 2. Combinatorial analysis. I. Title. QA269.S5735 2013 519.3—dc23 2012043675 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. -
Crash Course on Combinatorial Game Theory
CRASH COURSE ON COMBINATORIAL GAME THEORY ALFONSO GRACIA{SAZ Disclaimer: These notes are a sketch of the lectures delivered at Mathcamp 2009 containing only the definitions and main results. The lectures also included moti- vation, exercises, additional explanations, and an opportunity for the students to discover the patterns by themselves, which is always more enjoyable than merely reading these notes. 1. The games we will analyze We will study combinatorial games. These are games: • with two players, who take turns moving, • with complete information (for example, there are no cards hidden in your hand that only you can see), • deterministic (for example, no dice or flipped coins), • guaranteed to finish in a finite amount of time, • without loops, ties, or draws, • where the first player who cannot move loses. In this course, we will specifically concentrate on impartial combinatorial games. Those are combinatorial games in which the legal moves for both players are the same. 2. Nim Rules of nim: • We play with various piles of beans. • On their turn, a player removes one or more beans from the same pile. • Final position: no beans left. Notation: • A P-position is a \previous player win", e.g. (0,0,0), (1,1,0). • An N-position is a \next player win", e.g. (1,0,0), (2,1,0). We win by always moving onto P-positions. Definition. Given two natural numbers a and b, their nim-sum, written a ⊕ b, is calculated as follows: • write a and b in binary, • \add them without carrying", i.e., for each position digit, 0 ⊕ 0 = 1 ⊕ 1 = 0, 0 ⊕ 1 = 1 ⊕ 0 = 1. -
Combinatorial Game Theory: an Introduction to Tree Topplers
Georgia Southern University Digital Commons@Georgia Southern Electronic Theses and Dissertations Graduate Studies, Jack N. Averitt College of Fall 2015 Combinatorial Game Theory: An Introduction to Tree Topplers John S. Ryals Jr. Follow this and additional works at: https://digitalcommons.georgiasouthern.edu/etd Part of the Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation Ryals, John S. Jr., "Combinatorial Game Theory: An Introduction to Tree Topplers" (2015). Electronic Theses and Dissertations. 1331. https://digitalcommons.georgiasouthern.edu/etd/1331 This thesis (open access) is brought to you for free and open access by the Graduate Studies, Jack N. Averitt College of at Digital Commons@Georgia Southern. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of Digital Commons@Georgia Southern. For more information, please contact [email protected]. COMBINATORIAL GAME THEORY: AN INTRODUCTION TO TREE TOPPLERS by JOHN S. RYALS, JR. (Under the Direction of Hua Wang) ABSTRACT The purpose of this thesis is to introduce a new game, Tree Topplers, into the field of Combinatorial Game Theory. Before covering the actual material, a brief background of Combinatorial Game Theory is presented, including how to assign advantage values to combinatorial games, as well as information on another, related game known as Domineering. Please note that this document contains color images so please keep that in mind when printing. Key Words: combinatorial game theory, tree topplers, domineering, hackenbush 2009 Mathematics Subject Classification: 91A46 COMBINATORIAL GAME THEORY: AN INTRODUCTION TO TREE TOPPLERS by JOHN S. RYALS, JR. B.S. in Applied Mathematics A Thesis Submitted to the Graduate Faculty of Georgia Southern University in Partial Fulfillment of the Requirement for the Degree MASTER OF SCIENCE STATESBORO, GEORGIA 2015 c 2015 JOHN S.