AMS / MAA ANNELI LAX NEW MATHEMATICAL LIBRARY VOL 54 Hex A Playful Introduction Ryan B. Hayward Hex A Playful Introduction Ryan B. Hayward AMS / MAA ANNELI LAX NEW MATHEMATICAL LIBRARY VOL 54 Hex A Playful Introduction Ryan B. Hayward Providence, Rhode Island EDITORIAL COMMITTEE John H. McCleary, Editor Beth Malmskog Tatiana Shubin Thomas Q. Sibley Bruce F. Torrence Japheth Wood 2020 Mathematics Subject Classification. Primary 00A08, 00A09, 91A43, 91A27. For additional informationand updates on this book, visit www.ams.org/bookpages/nml-54 Library of Congress Cataloging-in-Publication Data Names: Hayward,RyanB.,1958-author. Title: Hex: aplayfulintroduction/RyanB.Hayward. Description: Providence,RhodeIsland: AmericanMathematicalSociety,2022. | Series: AnneliLaxnewmathematicallibrary,2578-6407;volume54|Includes bibliographicalreferencesandindex. Identifiers: LCCN2021049630|ISBN9781470464929(paperback)|9781470469481 (ebook) Subjects: LCSH: Game theory. | Hex (Game) | Combinatorial analysis. | AMS: General and overarching topics; collections – General and miscellaneous spe- cifictopics–Recreationalmathematics. |Generalandoverarchingtopics;col- lections–Generalandmiscellaneousspecifictopics–Popularizationofmath- ematics. | Game theory, economics, finance, and other social and behavioral sciences–Gametheory–Gamesinvolvinggraphs. Classification: LCCQA269.H3882022|DDC519.3–dc23/eng/20211201 LCrecordavailableathttps://lccn.loc.gov/2021049630 Copying and reprinting. Individual readers of this publication, and nonprofit li- braries acting for them, are permitted to make fair use of the material, such as to copyselectpagesforuseinteachingorresearch. Permissionisgrantedtoquotebrief passagesfromthispublicationinreviews,providedthecustomaryacknowledgmentof thesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthis publicationispermittedonlyunderlicensefromtheAmericanMathematicalSociety. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/ publications/pubpermissions. Send requests for translation rights and licensed reprints to reprint-permission @ams.org. (cid:2)c 2022bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttps://www.ams.org/ 10987654321 272625242322 Do you like to play games? This book is for you. In memory Fr´ed´eric Maffray (1960-2018), graph theorist and friend; Pouneh Gorji (1993-2020) and Arash Pourzarabi (1994-2020), international scholars; Audrey Jean Hayward (n´ee Banerd) (1934-2021), quirky, funny, thoughtful, ever-curious, generous, loving. contents acknowledgments xiii preface xv 1 this is Hex 1 1.1 rules 1 1.2 sample game 2 1.3 now it’s your turn 4 1.4 balancing the game 5 1.5 things to know 6 1.6 puzzles 7 1.7 solutions 8 2 basics 9 2.1 block 10 2.2 pairing 11 2.3 3×3 openings 12 2.4 safe connections 13 2.5 6×6 puzzle 14 2.6 win-threats 15 2.7 Fourth row safe connection 16 2.8 weak, safe, super connection 17 2.9 blunders 20 2.10 puzzles 22 2.11 solutions 23 3 how Hex began 25 3.1 Hein’s game 25 3.2 Lindhard’s puzzles 27 3.3 across the ocean 28 3.4 Gardner’s column and Nash’s lesson 28 3.5 puzzles 33 3.6 solutions 35 ix