AMS / MAA SPECTRUM VOL 75

Martin Gardner in the Twenty-First Century

Michael Henle and Brian Hopkins, Editors

i

i

“master” — 2013/8/14 — 15:08 — page i — #1i

i

i

i

i

i

Martin Gardner

in the

Twenty-First Century

10.1090/spec/075

i

i

“master” — 2013/8/14 — 15:08 — page ii — #2i

i

i

i

i

i

c 2012 by the Mathematical Association of America, Inc.

Library of Congress Catalog Card Number 2012954077

Print edition ISBN: 978-0-88385-913-1

Electronic edition ISBN: 978-1-61444-801-3

Printed in the United States of America

Current Printing (last digit):

10 9 8 7 6 5 4 3 2 1

i

i

“master” — 2013/8/14 — 15:08 — page iii — #3i

i

i

i

i

i

Martin Gardner

in the

Twenty-First Century

edited by

Michael Henle

Oberlin College

and

Brian Hopkins

Saint Peter’s University

Published and Distributed by

The Mathematical Association of America

i

i

“master” — 2013/8/14 — 15:08 — page iv — #4i

i

i

i

i

i

i

i

“master” — 2013/8/14 — 15:08 — page v — #5i

i

i

i

i

i

Preface

Martin Gardner and the MAA share a long history. In 1958, around the time he started his

famous “Mathematical Games” column for Scientific American, he submitted the first of

many problems to The American Mathematical Monthly. In 1982, as his column wound

down, Gardner’s first MAA article was published in The Two Year College Mathematics

Journal. He wrote for MAA journals the rest of his life, particularly The College Mathe-

matics Journal and Math Horizons. Gardner contributed to the latter almost annually from

its founding in 1993 until 2005.

Gardner’s prodigious writing activity continued right until his death in 2010. Articles,

stories, problems, solutions, Quickies, and other kinds of contributions continued to flow.

His last mathematical article to appear in an MAA journal, “L-tromino Tiling of Mutilated

Chessboards,” was the centerpiece of a special puzzle issue of The College Mathematics

Journal in 2009, and it is included here.

Early in 2010, Math Horizons editors Steve Abbott and Bruce Torrence were surprised

to receive a typescript manuscript. Gardner used a typewriter his whole life, never email.

The submission was accompanied by a note, “Is this short story something you can use? I

wrote the math column in Scientific American for 25 years. If my piece is not right for Math

Horizons, there is no need to send it back. All best, Martin.” There was not enough time

for the editors to thank Martin for his submission [6]. Fittingly, this story, “Superstrings

and Thelma,” is the last piece in this collection.

Apart from his own work, Martin Gardner, by enormously expanding the field of recre-

ational mathematics, opened up vast mathematical tracts for exploration by others. This

was quite deliberate. In an interview in The Two-Year College Mathematics Journal [1],

Gardner said, “I’m defining [recreational mathematics] in the very broad sense to include

anything that has a spirit of play about it.” Gardner, of course, had a refined and very

well-developed sense of play, one quality that made his pieces so enjoyable to read. In

almost everything he wrote, Gardner posed problems to challenge his readers, and they

responded. He maintained an extensive correspondence with mathematicians, both profes-

sional and amateur. Their work fueled his own pieces, but then his correspondents turned

around and wrote their own articles.

One consequence was the expansion of recreational mathematics into a major research

area (also helped by the development of the computer and the corresponding expanded in-

terest in discrete mathematics) that is such a feature of the current mathematical landscape.

The CMJ devoted the January 2012 issue to papers on topics that Gardner introduced to the

mathematical public. There were so many articles to include that half of the March 2012

issue continued the theme.

v

i

i

“master” — 2013/8/14 — 15:08 — page vi — #6i

i

i

i

i

i

vi Preface

Another consequence of the flowering of recreational mathematics is this volume. We

have collected MAA journal articles starting from 1999 on topics that Gardner developed.

Some are written by Gardner, but most are by others. The tribute articles from the January

and March 2012 CMJ issues are all here, but they constitute less than half of this collection.

All the MAA journals are represented, Mathematics Magazine and the Monthly, as well as

CMJ and Math Horizons. The limitation to pieces published roughly in the twenty-first

century is a practical one. Even so, some puzzle collections, longer articles, and pieces less

directly linked to Martin Gardner have been omitted.

The 41 pieces collected here are grouped around common themes, such as geometry,

number theory and graph theory, and cards and probability. Flexagons, the topic of Gard-

ner’s first Scientific American column, are seen to be associated with Catalan numbers and

together merit their own section. Geometric tiling and various “magic” number puzzles

are all about “Making Things Fit,” and there are enough other puzzles and games to fill

another section.

Gardner’s interests ranged far beyond mathematics. A fan of magicians and magic

tricks from childhood (“I waste a lot of time on it” [2]), he wrote several books on magic.

He annotated Lewis Carroll’s Alice books and other classics, and produced two novels of

his own. Other topics included philosophy, religion, literature and pseudo-science, leading

to some 70 books.

The last section of this volume highlights some of these other facets of Martin Gard-

ner’s wide-ranging interests. It includes two short stories by Gardner and several other

pieces that demonstrate his support for amateur mathematicians, his love of play (about an

April Fool’s joke he played on his Scientific American readers), and his interest in debunk-

ing false science. Also included is Gardner’s review of a 2004 novel in which an important

character seems to be based on him.

Our hope is that this volume will play a role in perpetuating the memory of Martin

Gardner, modest celebrity, larger-than-life character, self-confessed amateur as a mathe-

matician, popularizer of recreational mathematics in the broadest sense, prolific and bril-

liant writer. Given the lasting impression he made on several generations of mathematics

enthusiasts of all backgrounds, we are confident that the MAA and others will be publish-

ing articles inspired by Gardner’s work for a long time.

Gardner, like the readers of this book, loved mathematics. We close this preface with

Gardner’s own words on his background, the community, and why he enjoyed the field so

much (from [5], [4], and [3], respectively).

I took no math in college. I’m like a person who loves music but can’t hum

a tune or play an instrument. My understanding of math does not go beyond

a minimal understanding of calculus. I hasten to add that I consider this one

reason for the success of my Scientific American column. I had to work hard

to understand whatever I wrote about, and this made it much easier for me to

write in a way that readers could understand.

When I first started the column, I was not in touch with any mathemati-

cians, and gradually mathematicians who were creative in the field found out

about the column and began corresponding with me. So my most interesting

columns were columns based on material that I got from them, so I owe them

a big debt of gratitude.

i

i

“master” — 2013/8/14 — 15:08 — page vii — #7i

i

i

i

i

i

Preface vii

[I enjoy mathematics] because it has a strange kind of unearthly beauty.

There is a strong feeling of pleasure, hard to describe, in thinking through an

elegant proof, and even greater pleasure in discovering a proof not previously

known.

Acknowledgments

The editors are grateful for the encouragement and hard work of the MAA publications

staff: Director of Publications Ivars Peterson, who conceived the book, Production Man-

ager Carol Baxter, who designed it, and especially Electronic Publication Manager Beverly

Joy Ruedi, who set the book in type.

Bibliography

[1] Anthony Barcellos and Martin Gardner, A Conversation with Martin Gardner, Two-Year College

Math. J. 10 (1979) 233–244.

[2] Don Albers and Martin Gardner, On the Way to “Mathematical Games”: Part I of an Interview

with Martin Gardner, College Math. J. 36, (2005) 178–190.

[3] Don Albers and Martin Gardner, “Mathematical Games” and Beyond: Part II of an Interview

with Martin Gardner, College Math. J. 36 (2005) 301–314.

[4] Colm Mulcahy and Martin Gardner, An Interview with Martin Gardner, Card Colm, October

2006, available at www.maa.org/columns/colm/cardcolm200610.html.

[5] Michael Henle and Martin Gardner, Interview with Martin Gardner, College Math. J. 40 (2009)

158–161.

[6] Bruce Torrence and Stephen Abbott, To Our Readers, Math Horizons 18 (2010) 2–4.

www.maa.org/columns/colm/cardcolm200610.htmlhttp://www.maa.org/columns/colm/cardcolm200610.html

i

i

“master” — 2013/8/14 — 15:08 — page viii — #8i

i

i

i

i

i

i

i

“master” — 2013/8/14 — 15:08 — page ix — #9i

i

i

i

i

i

Contents

Preface v

I Geometry 1

1 The Asymmetric Propeller 3

Martin GardnerGardner, paying tribute to dentist and geometer Leon Bankoff, discusses some of his unpub-

lished results and concludes with an open question.

2 The Asymmetric Propeller Revisited 7

Gillian Saenz, Christopher Jackson, and Ryan CrumleyThree University of Texas students use dynamic geometry software to confirm Bankoff’s re-

sults and resolve Gardner’s question.

3 Bracing Regular Polygons As We Race into the Future 11

Greg W. FredericksonA problem Gardner published in 1963 continues to spur generalizations and improved solu-

tions around the world.

4 A Platonic Sextet for Strings 19

Karl SchafferThe professor and dance company co-director details string polyhedra constructions for ten

participants.

5 Prince Rupert’s Rectangles 25

Richard P. Jerrard and John E. WetzelA 17th century puzzle that Gardner posed in higher dimensions is here solved in the case of

three-dimensional boxes.

II Number Theory and Graph Theory 35

6 Transcendentals and Early Birds 37

Martin GardnerGardner moves from Liouville to an “innocent but totally useless amusement” that nonetheless

captured the attention of Solomon Golomb.

7 Squaring, Cubing, and Cube Rooting 39

Arthur T. BenjaminThe professor and “mathemagician,” inspired as a high school student by Gardner’s tricks for

mental calculations, extends some of them here.

ix

i

i

“master” — 2013/8/14 — 15:08 — page x — #10i

i

i

i

i

i

x Contents

8 Carryless Arithmetic Mod 10 45

David Applegate, Marc LeBrun, and N. J. A. SloaneInspired by the carryless arithmetic of the game Nim, this trio of authors explores the number

theory of a South Pacific island.

9 Mad Tea Party Cyclic Partitions 53

Robert Bekes, Jean Pedersen, and Bin ShaAnother playful trio analyzes cyclic arrangements that build from integer partitions in a Lewis

Carroll setting.

10 The Continuing Saga of Snarks 65

sarah-marie belcastroA type of graph, given a fanciful name by Gardner from Lewis Carroll, was the subject of a

Branko Grünbaum conjecture for 39 years.

11 The Map-Coloring Game 73

Tomasz Bartnicki, Jaroslaw Grytczuk, H. A. Kierstead, and Xuding ZhuDaltonism and half-dollar coins are used in this exploration of a Steven Brams game theory

approach to the Four Color Theorem.

III Flexagons and Catalan Numbers 85

12 It’s Okay to Be Square If You’re a Flexagon 87

Ethan J. Berkove and Jeffrey P. DumontThis article details the 1939 origin of flexagons at Princeton University and focuses on the

neglected tetraflexagons.

13 The V-flex, Triangle Orientation, and Catalan Numbers in Hexaflexagons 103

Ionut E. Iacob, T. Bruce McLean, and Hua WangThis trio of Georgia Southern University authors examines a once-illegal variety of flex and

makes a connection between “pat classes” and Catalan numbers.

14 From Hexaflexagons to Edge Flexagons to Point Flexagons 109

Les PookAn engineer and author of two books on flexagons considers the more general edge flexagons

and recently discovered point flexagons.

15 Flexagons Lead to a Catalan Number Identity 113

David CallanExamining the descent permutation statistic on flexagon pats leads the author to full binary

trees and a combinatorial proof.

16 Convergence of a Catalan Series 119

Thomas Koshy and Z. GaoCalculus is brought to bear on the infinite sum of Catalan number reciprocals and related

series; � and the golden ratio make appearances.

IV Making Things Fit 125

17 L-Tromino Tiling of Mutilated Chessboards 127

Martin GardnerIn his last MAA mathematics article, Gardner moves from classic chessboard domino tiling

problems to new results.

i

i

“master” — 2013/8/14 — 15:08 — page xi — #11i

i

i

i

i

i

Contents xi

18 Polyomino Dissections 135

Tiina Hohn and Andy LiuThe authors introduce a new technique for solving dissection problems, often presented in the

context of quilts, leaving several puzzles for the reader.

19 Squaring the Plane 143

Frederick V. Henle and James M. HenleA father and son team resolve Golomb’s “heterogenous tiling conjecture” and discuss another

dozen open questions.

20 Magic Knight’s Tours 153

John BeasleyThe author surveys results combining a knight’s tour on the chessboard with magic squares,

including a computer-aided solution to a Gardner question.

21 Some New Results on Magic Hexagrams 159

Martin GardnerHere Gardner focuses on three types of puzzles about placing numbers on six-pointed stars,

mentioning a “rare mistake” of the British puzzlist Henry Dudeney.

22 Finding All Solutions to the Magic Hexagram 167

Alexander Karabegov and Jason HollandThe authors relate magic hexagrams to magic edge labelings of cubes, using card shuffling to

enumerate distinct solutions.

23 Triangular Numbers, Gaussian Integers, and KenKen 173

John J. WatkinsMiyamoto’s contemporary puzzle is expanded to complex numbers where a different unique

factorization adds to the challenge.

V Further Puzzles and Games 179

24 Cups and Downs 181

Ian StewartOne of Gardner’s mathematical successors at Scientific American uses graph theory and linear

algebra on two related puzzles.

25 30 Years of Bulgarian Solitaire 187

Brian HopkinsSome recent math history explains this oddly-named puzzle on integer partitions, visualized

with state diagrams and generalized to a new two-player game.

26 Congo Bongo 195

Hsin-Po WangA high school student uses state diagrams and Dennis Shasha’s detectives to open a tricky

treasure chest.

27 Sam Loyd’s Courier Problem with Diophantus, Pythagoras,

and Martin Gardner 201

Owen O’SheaA Classroom Capsule extends Gardner’s solution of related Sam Loyd puzzles to other army

formations.

i

i

“master” — 2013/8/14 — 15:08 — page xii — #12i

i

i

i

i

i

xii Contents

28 Retrolife and The Pawns Neighbors 207

Yossi ElranAn inverse version of Conway’s game Life, famously popularized by Gardner, is examined

using chessboards.

29 RATWYT 213

Aviezri FraenkelThe combinatorial game theorist uses the Calkin Wilf tree to devise a rational number version

of Wythoff’s Nim.

VI Cards and Probability 219

30 Modeling Mathematics with Playing Cards 221

Martin GardnerIn addition to probability applications, Gardner uses a deck of cards for a discrete version of a

fluid mixing puzzle and mentions a correction to W. W. Rouse Ball.

31 The Probability an Amazing Card Trick Is Dull 227

Christopher N. SwansonRook polynomials and the principle of inclusion-exclusion help determine the likelihood that

the author’s students were underwhelmed.

32 The Monty Hall Problem, Reconsidered 231

Stephen Lucas, Jason Rosenhouse, and Andrew ScheplerThese authors remind us of Gardner’s early role in this infamous problem that still “arouses

the passions” and examine variations.

33 The Secretary Problem from the Applicant’s Point of View 243

Darren GlassChanging perspective, the author reconsiders a classic strategy in order to help job seekers

choose the best interview slot.

34 Lake Wobegon Dice 249

Jorge Moraleda and David G. StorkLake Wobegon Dice, named after Garrison Keillor’s Minnesota town, have the property that

each is “better than the set average.”

35 Martin Gardner’s Mistake 257

Tanya KhovanovaAnother controversial problem about probability and information is carefully discussed,putting

Gardner in the company of Dudeney and Ball.

VII Other Aspects of Martin Gardner 263

36 Against the Odds 265

Martin GardnerIn this short story, a principal recognizes the potential in a student whose unconventional

thinking irritates his teacher.

37 A Modular Miracle 271

John StillwellGardner used an obscure result of Hermite and the limitations of 1970’s calculators for an

April Fool’s Day prank.

i

i

“master” — 2013/8/14 — 15:08 — page xiii — #13i

i

i

i

i

i

Contents xiii

38 The Golden Ratio—A Contrary Viewpoint 273

Clement FalboBuilding on a Gardner article in The Skeptical Inquirer, the author argues that � “is not entirely

astonishing.”

39 Review of The Mysterious Mr. Ammann by Marjorie Senechal 285

Philip StraffinThis Media Highlight discusses an example of Gardner’s support of an amateur mathematician

who independently discovered Penrose tiles.

40 Review of PopCo by Scarlett Thomas 287

Martin GardnerThis popular 2004 novel includes a character based on Gardner, so he was a natural choice to

review the book.

41 Superstrings and Thelma 289

Martin GardnerGardner’s last MAA submission, a short story about a physics graduate student and a waitress

who quips, “How are strings?”

Index 293

About the Editors 297

i

i

“master” — 2013/8/14 — 15:08 — page xiv — #14i

i

i

i

i

i

i

i

“master” — 2013/8/14 — 15:08 — page 293 — #307i

i

i

i

i

i

IndexAbe, Gakuho, 163

Adamatzky, Andrew, 208

Aitken, A. C., 39

Almkvist, Gert, 188

Ammann, Robert, 285

Ascher, Marcia, 45

Ball, W. W. Rouse, 223

Bankoff, Leon, 3, 7

Bernhart, Frank, 164

Beverley, William, 154

Bodlaender, Hans, 73

Bojanov, Borislav, 188

Bolt, Brian, 159

Bottomley, Henry, 46

Brams, Steven, 73

Brandt, Jørgen, 188

Brooks, R. L., 143

Carroll, Lewis, vi, 58, 66

Carver, W. B., 26

Catalan, Eugene Charles, 119

Christ, Henry, 227

Chu, I. Ping, 131

Cipra, Barry, 176

Coeter, H. S. M., 3

Conway, John, 207, 285

Coxeter, H. S. M., 224, 285

da Vinci, Leonardo, 273

de Vasa, H. E., 155

Denef, Yann, 154

diophantine, 203, 269

Dodgson, Charles, 58

Dudeney, Henry, 162, 167

Efron, Bradley, 249

Eggleton, Roger, 159

Erdős, Paul, 4, 288

Eriksson, Henrik, 188

Euler, Leonhard, 20

Feynman, Richard, 87, 110

Fibonacci numbers, 51, 143, 274

Fields Medal, 269

Ford, L. R., 26

Foshee, Gary, 258

Four Color Theorem, 67, 73

Frederickson, Greg, 135

Friedman, Erich, 16

Fulves, Karl, 181, 224

Gale, David, 221

games

Go, 78, 207

KenKen, 173

Life, 207, 287

map coloring, 73

Nim, 45

sudoku, 173

ticktacktoe, 266

two-player Bulgarian solitaire, 191

two-player Monty, 236

Wythoff, 213

Garnett, F. M., 26

Gauss, Carl Friederich, 271

Gessel, Ira, 114

Gilbreath, Norman, 221

Gilks, Joe, 159

golden ratio, 123, 273

Golomb, Solomon, 38, 128, 136, 140, 150

Gomory, Ralph, 127

Gosper, Bill, 208

Grünbaum, Branko, 66, 150

Grabarchuk, Serhiy, 139

Graham, Ron, 188

Gutenmacher, Victor, 188

293

i

i

“master” — 2013/8/14 — 15:08 — page 294 — #308i

i

i

i

i

i

294 Index

Guthrie, Francis, 65

Hall, Monty, 223, 231, 287

Hamilton circuit, 68, 191

Harary, Frank, 268

Hardy, G. H., 54

Hein, Piet, 288

Hermite, Charles, 271

Hickerson, Dean, 208

High, Robert, 74

Hirayama, Akira, 163

Hoffman, Paul, 288

Hoggatt, Verner, Jr., 150

Huber, Greg, 33

Jaenisch, C. F., 154

Jefferson, Thomas, 287

Jelliss, George, 154

Jensen, Christopher, 132

Johnsonbaugh, Richard, 131

Jones, Kate, 132

Karatsuba, Anatolii Alexeevich, 188

Karp, Richard, 222

Keillor, Garrison, 250

Khodulev, Andrei, 16

Kim, Scott, 19

Kirstead, Friend, Jr., 225

Klamkin, Murray, 4

Knuth, Donald, 188, 222

Kronecker, Leopold, 271

Kumar, Awani, 155

Ligocki, Terry, 33

Lindgren, Harry, 135

Liouville, Joseph, 37

Loyd, Sam, 201

Mackay, Hughues, 154

magic square, 153, 160, 225

Mahler, Kurt, 38

Marlow, T. W., 154

Maxwell, Brian, 226

Mayrignac, Jean-Charles, 154

McIntosh, Harold, 99

McLean, Bruce, 103

Milgram, Stanely, 288

Miyamoto, Tetsuya, 173

Moseteller, Fred, 231

Murray, H. J. R., 154

Nelsen, Roger, 128

Nieuwland, Pieter, 25

Niven, Ivan, 38

Nobel Prize, 290

O’Beirne, T. H., 13

Ollerenshaw, Kathleen, 224

Online Encyclopedia of Integer Sequences,

46, 107, 217

Oskolkov, Konstantin, 188

Pólya Award, 3

Palmatelli-Palmarini, Massimo, 232

partitions, 51, 53, 188

Penrose, Roger, 285

Petkov, Milko, 188

Pomerance, Carl, 150

Poniachik, Jaime, 38

Putnam Competition, 3

Rademacher, Hans, 54

Ramanujan, Srinivasa, 54, 272

Ransom, William, 202

Reiter, Harold, 163

Rendell, Paul, 208

Ritchie, David, 164

Roberts, T. S., 154

Robinson, Raphael, 11

Schattschneider, Doris, 11

Scherer, Karl, 150

Schreck, D. J. E., 25

Selvin, Steve, 231

Senechal, Marjorie, 285

Shapley, Lloyd, 74

Shasha, Dennis, 199

Sherman, Scott, 111

shuffle, 170, 221

Silver, Stephen, 208

Siu, Man-Keung, 184

Smith, C. A. B., 143

i

i

“master” — 2013/8/14 — 15:08 — page 295 — #309i

i

i

i

i

i

Index 295

Smith, John Maynard, 231

Starr, Norton, 133

state diagram, 182, 189, 197

Stertenbrink, Guenter, 154

Stone, Arthur, 87, 103, 109, 113, 143

Stork, David, 249

Susco, Barbara, 19

Theobald, Gavin, 14

Thiel, Von J. Christian, 162

Thomas, Scarlett, 287

Toom, Andrei, 188

tree

Calkin Wilf, 215

Monty Hall, 233

pat, 115

Stern Brocot, 217

ticktacktoe, 268

Tucker, Bryant, 87

Tukey, John, 87, 110

Tutte, William, 143

vos Savant, Marilyn, 232

Wainwright, Robert, 208

Wallis, John, 25

Wenzelides, Karl, 154

Willcocks, T. H., 155

Witten, Ed, 290

Wythoff, Willem Abraham, 213

i

i

“master” — 2013/8/14 — 15:08 — page 296 — #310i

i

i

i

i

i

i

i

“master” — 2013/8/14 — 15:08 — page 297 — #311i

i

i

i

i

i

About the Editors

Michael Henle is a professor of mathematics at Oberlin College in Oberlin, Ohio. He is

the author of several previous books including Which Numbers are Real? which was just

published by the MAA in 2012. Trained as a functional analyst, he has written as well on

combinatorial subjects and geometry. He is serving as editor of The College Mathematics

Journal through 2013.

Brian Hopkins is a professor of mathematics at Saint Peter’s University in Jersey City,

New Jersey. He won, with Robin Wilson, the 2005 George Pólya Award, edited the 2008

MAA Notes volume Resources for Teaching Discrete Mathematics, and was given the

2011 MAA New Jersey Section Award for Distinguished College or University Teaching

of Mathematics. Much of his research stems from Bulgarian Solitaire, a topic popularized

by Martin Gardner. Hopkins will be the editor of The College Mathematics Journal from

2014 to 2018.

297

AMS / MAA SPECTRUM

Martin Gardner enormously expanded the field of recreational mathematics with the Mathematical Games columns he wrote for Scientific American for over 25 years and the more than 70 books he published. He also had a long relationship with the Mathematical Association of America, pub-lishing articles in MAA journals right up to his death in 2010. This book collects the articles Gard-ner wrote for the MAA in the twenty-first century, together with other articles the MAA published from 1999 to 2012 that spring from and comment on his work.

Martin Gardner in the Twenty-First Century

Michael Henle and Brian Hopkins, Editors