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AMS / MAA SPECTRUM VOL 100

Barrycades and Septoku Papers in Honor of and Tom Rodgers

Thane Plambeck Tomas Rokicki Editors 10.1090/spec/100 Barrycades and Septoku Papers in Honor of Martin Gardner and Tom Rodgers

Editors AMS/MAA SPECTRUM

VOL 100

Barrycades and Septoku Papers in Honor of Martin Gardner and Tom Rodgers

Thane Plambeck Tomas Rokicki Editors 2019 Editorial Committee James J. Tattersall, Editor

Michael Barany Andrew Beveridge Virginia M. Buchanan Thomas L. Drucker Evan D. Fisher Donna L. Flint Richard K. Guy Dominic Klyve John Lorch Cayla Danielle McBee 2010 Mathematics Subject Classification. Primary 00A08, 00A09, 00B10, 97A20.

For additional information and updates on this book, visit www.ams.org/bookpages/spec-100

Library of Congress Cataloging-in-Publication Data Names: Plambeck, Thane E., editor. | Rokicki, Tomas, editor. | Gathering 4 Gardner Foundation. Title: Barrycades and septoku : papers in honor of Martin Gardner and Tom Rodgers / Thane Plam- beck, Tomas Rokicki, editors. Description: Providence, Rhode Island : American Mathematical Society, [2020] | : Spectrum ; volume 100 | Copyrighted by Gathering 4 Gardner, Inc. | Includes bibliographical references. Identifiers: LCCN 2019041150 | ISBN 9781470448707 (paperback) | ISBN 9781470455187 (ebook) Subjects: LCSH: Mathematical recreations. | . | Gardner, Martin, 1914–2010. | Rodgers, Tom, 1943–2012 | AMS: General – General and miscellaneous specific topics – Recreational mathe- matics [See also 97A20]. | General – General and miscellaneous specific topics – Popularization of mathematics. | General – Conference proceedings and collections of papers – Collections of articles of general interest. | Mathematics education – General, mathematics and education – , games [See also 00A08]. Classification: LCC QA95 .B3585 2020 | DDC 793.74–dc23 LC record available at https://lccn.loc.gov/2019041150

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© 2020 by Gathering 4 Gardner, Inc. All rights reserved. Printed in the United States of America. ⃝∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 25 24 23 22 21 20 Contents

Preface ix

Remembrances xi xi Nancy Blachman xii Rob Jones xiii Dick Esterle xv

Part 1 Sequences, Tiling, and Packing 1

1 Eight Hateful Sequences 3 N. J. A. Sloane

2 Building Barrycades and Constructing Corrals 11 Richard K. Guy

3 Limited Placements of Polyominoes on Rectangles 19 Solomon W. Golomb

4 Polyominoes on a Multicolored Infinite Grid 29 Hans Hung-Hsun Yu

Part 2 Fun and Games 37

5 A Chess Tribute to 39 Carlos Pereira dos Santos

6 Some New Combinatorial Games 51 Aaron N. Siegel

7 Factor Subtractor 59 Barry Cipra

v vi Contents

8 The Mathematics of Septoku 63 George I. Bell

Part 3 History 77

9 Thirty Years of Computer Cubing: The Search for ’s Number 79 Tomas Rokicki

10 Nim-like Games: An Ancestry 99 Lisa Rougetet

Part 4 Puzzles 113

11 Triangles of Absolute Differences 115 Brian Chen, YunHao Fu, Andy Liu, George Sicherman, Herbert Taylor, and Po-Sheng Wu

12 Generalization of a Involving Set Partitions 125 Michael L. Fredman, Daniel A. Kleitman, and Peter Winkler

13 A Generalization of Retrolife 131 Yossi Elran

14 Coins of Three Different Weights 139 Tanya Khovanova and Konstantin Knop

15 Rubber Bandzzles: Three -Art Challenges 155 George Hart

Part 5 Art, Sculpture, and Design 161

16 Comet! 163 George Hart

17 Developing Topsy Turvy and Number Planet 173 M. and Igor Kriz Contents vii

Part 6 Magic and Miscellany 181

18 Coin-Flipping Magic 183 Nadia Benbernou, Erik D. Demaine, Martin L. Demaine, and Benjamin Rossman

19 Three Puzzle Fonts 203 Erik D. Demaine, Martin L. Demaine, Belén Palop, and Jason Ku

Preface

This volume continues a series of books (Pegg et al., 2009; Wolfe and Rodgers, 2002; Gardner et al., 2008, 2009, 1999) containing papers contributed to the Gath- ering 4 Gardner (G4G) meetings that have been taking place in , Geor- gia, every two years since the early 1990s. In 1991, Thomas Malin Rodgers, Jr., an Atlanta-based entrepreneur, philan- thropist, investor, and puzzle and book collector, conceived the idea of bringing together fans of Martin Gardner for a meeting. Contacting many people directly via phone calls, he received an enthusiastic response, and the first Gathering 4 Gardner (G4G1) took place in the spring of 1993. Martin Gardner attended this first Gathering, and attendees exchanged papers and gifts amongst themselves on many topics, principally including recreational mathematics, but also magic, mathematical art, and puzzle-related content. Papers contributed to the initial Gathering and subsequent ones (most re- cently G4G11, in the spring of 2014) have appeared in several previous volumes (see “Further Reading”, below), and this volume continues the series with contri- butions both from original G4G participants over 20 years ago as well as students as young as fifteen years old when they wrote their included papers. Rodgers indefatigably continued organizing these meetings, and advising the nonprofit G4G Foundation organized in 2006 to support them, almost untilthe very moment of his death in the spring of 2012. We’ve chosen to dedicate this volume to his memory. Thank you, Tom!

Thane Plambeck and Tomas Rokicki

Acknowledgments The editors are grateful for the encouragement of Stephen Kennedy and the MAA publications staff.

ix x Preface

Further Reading M. Gardner, E. R. Berlekamp, and T. Rodgers (1999). The mathemagician and pied puzzler: a collection in tribute to Martin Gardner. A. K. Peters, Natick, Mass. M. Gardner, E. D. Demaine, M. L. Demaine, and T. Rodgers (2008). A lifetime of puzzles: A collection of puzzles in honor of Martin Gardner’s 90th birthday. A. K. Peters, Ltd., Wellesley, Mass. M. Gardner, E. Pegg, A. H. Schoen, and T. Rodgers (2009). Mathematical Wizardry for a Gardner. A. K. Peters, Wellesley, Mass. E. Pegg, A. H. Schoen, and T. Rodgers (2009). Homage to a pied puzzler. A. K. Peters, Wellesley, Mass. D. Wolfe and T. Rodgers (2002). Puzzlers’ tribute: a feast for the mind. A. K. Peters, Natick, Mass. http://site.ebrary.com/id/10159715. Remembrances

I I met Tom Rodgers in the late 1980s at a puzzle party in Livermore, CA. Soon thereafter, he recruited me to help him organize the first Gathering 4 Gardner. He concurrently invited Mark Setteducati to invite some magicians. I invited some of the whom I knew to be fans of Martin Gardner, and we all came to G4G1 in Atlanta. My wife and our son came too. Tom showed us his collections of puzzles and books, and we met his family. Buck Rodgers was going to spend part of his summer at one of the language camps in northern Minnesota, which inspired David Berlekamp to go to another one there. From the beginning, Tom was always the driving force behind the Gather- ings. Under his leadership, they provided the forum for many wonderful con- tributions to recreational mathematics. Some of them are documented in the Tribute volumes, the first six of which Tom coedited. Others were distributed in the puzzle exchanges. G4G also provided a gathering point and erudite audience for some world-class magicians and mathematical artists, many of whom were personally recruited by Tom. He was also the Gatherings’ most generous finan- cial supporter. In later years, he hosted visits by all attendees at each Gathering to a big celebration at his Japanese-style house and grounds. These events became one of the major attractions of the Gatherings.

Elwyn Berlekamp

xi xii Remembrances

II At my first Gathering 4 Gardner1—G4G9 at the Ritz hotel in Atlanta—I arrived during dinner, after registration was closed, and soon found myself being inter- rogated by Tom Rodgers to make sure I wasn’t crashing the conference, since I didn’t have a badge. For the rest of the Gathering, Tom checked in with me sev- eral times each day to make sure I was settling in comfortably and connecting with people I found interesting. The thought-provoking presentations, the magic shows, the art exhibit, and the attendees made G4G among the most intriguing conferences I’ve attended. After G4G9, Tom invited me to join the Gathering 4 Gardner Celebration of Mind2 host committee and encouraged me to invite women, minorities, and young people to G4G conferences. Practically daily, Tom sent me fascinating emails and he also sent me the book Complexities: Women in Mathematics. I put together a list of over 20 people with short bios of each one and, at Tom’s request, invited them all. In December of 2011, Tom asked me to join the G4G board of directors; while I was deeply honored at the invitation, I told Tom I wanted to attend a board meet- ing before deciding. At the board meeting at Tom’s house on Saturday, March 31, 2012, which coincided with the G4GX (the 10th G4G) conference, Tom was ob- viously in poor health, a fraction of his former self, sitting in a wheelchair. Tom told me that he hadn’t been sleeping and that he wasn’t able to eat. Nevertheless, he was fired up when he began the meeting and told the board that weneeded to get the Gathering 4 Gardner Foundation on a solid financial footing. When he spoke, he sounded just like he did before cancer ravaged his body. His passion and commitment persuaded me to join the board. Thank you, Tom, for enriching my life.

Nancy Blachman Founder, Julia Robinson Mathematics Festival

1Taking place every two years in Atlanta and honoring the legacy of the late Martin Gardner, the Gathering 4 Gardner (G4G) brings people together from around the world to explore, create, and experience that Aha! moment when disparate bits of knowledge converge to create a new discovery. 2In the same spirit, Celebration of Mind (celebrationofmind.org) takes Gathering 4 Gardner activities to local and online communities, so everyone can share the fun. Remembrances xiii

III Devoted to helping people, Tom Rodgers was a man of vision, passion, and action. Tom cherished civilization and surrounded himself with objects (and people) that evinced and propelled it. Known to enjoy life’s finer things, Tom respected those who had attained them as he had, through education and work. He also loved to figure things out and appreciated journeys of discovery. Like Martin Gardner, Tom valued curiosity and derived great satisfaction from getting people to use their minds—especially those typically not so inclined. He inspired students, educators, and artists. His life made the world better. Tom and I met at one of Jerry Slocum’s puzzle parties. Neighbors during the all-day puzzle exchange, we spent much of it getting to know each other. He enjoyed sharing his knowledge and I eagerly soaked it up. The next year, I was honored to receive an invitation to a G4G conference. In G4G Tom created an oasis where especially curious people could engage, collaborate, and celebrate their delightful idiosyncrasies. What a thrill for me, as a first-timer, to find myself among so many people eager to share discoveries that had piqued or rewarded their curiosity. Spirited exchanges abounded: questions, strategies, and wonder were everywhere. Numerous insights and partnerships trace their origins to these curiosity-fueled interactions. Tom fashioned G4G as a progress-dedicated utopia which valued respect, fairness, openness, candor, beauty, ingenuity, virtuosity, and generosity. Dur- ing conferences, we gather in a giant ballroom for hours on end, with speaker after speaker attempting to arouse our enthusiasm. Many do and a few really do. To those that don’t, we show more than enough appreciation to keep them trying. G4G presentations emphasize math, puzzles, and games but attendees especially enjoy (as Tom did) artists whose work strongly engages the mind. I vividly recall John Edmark receiving thunderous applause while demonstrating his kinetic sculptures (one is now in the Museum of Modern Art Design Store). Crucially, the evenings offer the most accessible and irresistible brain-teasers of all: the magicians, challenging us to figure out how they do it. Leading by example, Tom upheld a tradition of welcoming the several hun- dred G4G attendees to his home for an afternoon of loosely structured fun. Situ- ated on a glorious property, finely landscaped to evoke the great palaces of Kyoto, Tom’s house was a masterpiece of hand-rendered Japanese esthetic. His hospi- tality included opportunities to explore his vast collections of puzzles, books, and art. He could not have been more welcoming, and his standard influenced every- one lucky enough to experience it. Tom’s medical diagnosis forced him to work fast. He had much to preserve and many duties to transfer. G4G was at the top of his list. It was his personal gift to humanity, his legacy, and it was in danger. For 20 years, he had operated it in a tightly controlled, personal way. Suddenly, he needed a succession plan. Two xiv Remembrances weeks before he died, Tom called six of us to his home to form a leadership group and receive his instructions. It was intensely emotional and extremely powerful to be with Tom as he branded us with his searing devotion to all that G4G is and means. Sparked by a compelling vision—individual intellectual engagement through benevolent collaboration, enhancing community prosperity—Tom Rodgers suc- cessfully launched a movement whose vast potential remains largely untapped. G4G’s value, while obvious to those it has touched, defies trite categorization. This makes it challenging for conventional grant-makers to embrace. Conse- quently, the G4G community itself must step into Tom’s shoes and accept re- sponsibility for G4G’s future. The G4G Foundation, the organization that administers the G4G programs, relies on the G4G community’s financial support and welcomes G4G community- members’ involvement in the discourse shaping G4G’s trajectory. For the first 20 years of its existence, Tom sustained G4G, largely with seemingly unfettered access to his personal time and resources. This high calling has now shifted to the community he created.

Rob Jones Remembrances xv

IV Can you imagine? That is something I heard Tom say more than a few times. I can still hear it in his special inflection, and enjoy the sound. And though it was often said in response to something thought over the top, as I think about him, it’s a fitting phrase, for he truly did imagine, and asked anyone near him to do so as well. He kept his eye on the Gathering as a special place to foster that kind of curiosity and camaraderie. He loved it, with all the attendant problematics of love. My first introduction was at G4G7, and I didn’t really get to know Tom until G4G9. After that, we began working toward the AHA retrospective exhibit for G4G10. My first impression of G4G was what a special event. Like Martin Gardner’s book titles, G4G contained all the elements of surprises, insights, and curiosity one wouldn’t expect. It was a blast. Tom had a knack for enlisting one to volunteer beyond one’s expectations of what they might like, and could deliver, and surprisingly do it and feel good about it. Tom was like the ringmaster at a carnival and the host of a party almost out of control. Sarah was part of that behind-the-scenes support as well as his children, along with many other volunteers. A few times I saw Scott juggling for his life as he tried to fulfill a last minute almost impossible request of Tom. Andlike something family run, G4G has the flavor and hospitality that includes coming out to the house and gardens which is still one of the most memorable aspects for the many who attend. Through this I got to know Tom and how close a friend he was to Martin Gardner. Whenever I stayed at the house, his big table was covered in newly arrived books, puzzles, and curiosities from anywhere for him to review and ex- amine. He did it with a clear delight. He welcomed visitors and was happy to talk. With his eye on inclusiveness, it was evident that he saw the role of G4G as a special place to foster and enrich the qualities he and Martin Gardner so loved, and by extension, the world.

Dick Esterle

AMS / MAA SPECTRUM

The Gathering 4 Gardner is a biannual conference founded—and for many years organized—by Tom Rodgers to celebrate the spirit of Martin Gardner. While primarily concerned with recreational mathematics, most of Gardner’s intellectual interests are featured, including magic, , , puzzles, art, and rationality. Gardner’s writing inspired several generations of mathematicians by introducing us to the joy of discovery and exploration, and the Gathering’s aim is to continue that tradition of inspiration. This volume, a tribute to Rodgers and Gardner, consists of papers originally presented at the Gathering 4 Gardner meetings. Recreational mathematics is strongly prominent with contributions from , Richard Guy, Solomon Golomb, Barry Cipra, , and many others. There are games and puzzles, including new Nim-like games, chess puzzles, coin weighings, coin fl ippings, and contributions that combine art and puzzles or magic and puzzles. Two historical articles present the stories of combi- natorial game theory and the search for God’s number for Rubik’s Cube. Anyone who fi nds pleasure in clever and intriguing intellectual puzzles will fi nd much to enjoy in Barrycades and Septoku.

For additional information and updates on this book, visit www.ams.org/bookpages/spec-100

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