Contents Foreword iÜ Elwyn Berlekamp and Tom Rodgers I Personal Magic ½ Martin Gardner: A “Documentary” ¿ Dana Richards Ambrose, Gardner, and Doyle ½¿ Raymond Smullyan A Truth Learned Early ½9 Carl Pomerance Martin Gardner = Mint! Grand! Rare! ¾½ Jeremiah Farrell Three Limericks: On Space, Time, and Speed ¾¿ Tim Rowett II Puzzlers ¾5 A Maze with Rules ¾7 Robert Abbott Biblical Ladders ¾9 Donald E. Knuth Card Game Trivia ¿5 Stewart Lamle Creative Puzzle Thinking ¿7 Nob Yoshigahara v vi Contents Number Play, Calculators, and Card Tricks: Mathemagical Black Holes 4½ Michael W. Ecker Puzzles from Around the World 5¿ Richard I. Hess OBeirnes Hexiamond 85 Richard K. Guy Japanese Tangram (The Sei Shonagon Pieces) 97 Shigeo Takagi How a Tangram Cat Happily Turns into the Pink Panther 99 Bernhard Wiezorke Pollys Flagstones ½¼¿ Stewart Coffin Those Peripatetic Pentominoes ½¼7 Kate Jones Self-Designing Tetraflexagons ½½7 Robert E. Neale The Odyssey of the Figure Eight Puzzle ½¾7 Stewart Coffin Metagrobolizers of Wire ½¿½ Rick Irby Beautiful but Wrong: The Floating Hourglass Puzzle ½¿5 Scot Morris Cube Puzzles ½45 Jeremiah Farrell The Nine Color Puzzle ½5½ Sivy Fahri Twice: A Sliding Block Puzzle ½6¿ Edward Hordern Planar Burrs ½65 M. Oskar van Deventer Contents vii Block-Packing Jambalaya ½69 Bill Cutler Classification of Mechanical Puzzles and Physical Objects Related to Puzzles ½75 James Dalgety and Edward Hordern III Mathemagics ½87 A Curious Paradox ½89 Raymond Smullyan A Powerful Procedure for Proving Practical Propositions ½9½ Solomon W. Golomb Misfiring Tasks ½9¿ Ken Knowlton Drawing de Bruijn Graphs ½97 Herbert Taylor Computer Analysis of Sprouts ½99 David Applegate, Guy Jacobson, and Daniel Sleator Strange New Life Forms: Update ¾¼¿ Bill Gosper Hollow Mazes ¾½¿ M. Oskar van Deventer Some Diophantine Recreations ¾½9 David Singmaster Who Wins Misère Hex? ¾¿7 Jeffrey Lagarias and Daniel Sleator An Update on Odd Neighbors and Odd Neighborhoods ¾4½ Leslie E. Shader Point Mirror Reflection ¾45 M. Oskar van Deventer How Random Are 3x + 1 Function Iterates? ¾5¿ Jeffrey C. Lagarias Forward Martin Gardner has had no formal education in mathematics, but he has had an enormous influence on the subject. His writings exhibit an extraor- dinary ability to convey the essence of many mathematically sophisticated topics to a very wide audience. In the words first uttered by mathematician John Conway, Gardner has brought “more mathematics, to more millions, than anyone else." In January 1957, Martin Gardner began writing a monthly column called “Mathematical Game” in Scientific American. He soon became the influen- tial center of a large network of research mathematicians with whom he cor- responded frequently. On browsing through Gardner’s old columns, one is struck by the large number of now-prominent names that appear therein. Some of these people wrote Gardner to suggest topics for future articles; others wrote to suggest novel twists on his previous articles. Gardner per- sonally answered all of their correspondence. Gardner’s interests extend well beyond the traditional realm of mathe- matics. His writings have featured mechanical puzzles as well as mathe- matical ones, Lewis Carroll, and Sherlock Holmes. He has had a life-long interest in magic, including tricks based on mathematics, on sleight of hand, and on ingenious props. He has played an important role in exposing char- latans who have tried to use their skills not for entertainment but to assert supernatural claims. Although he nominally retired as a regular columnist at Scientific American in 1982, Gardner’s prolific output has continued. Martin Gardner’s influence has been so broad that a large percentage of his fans have only infrequent contacts with each other. Tom Rodgers conceived the idea of hosting a weekend gathering in honor of Gardner to bring some of these people together. The first “Gathering for Gardner” (G4G1) was held in January 1993. Elwyn Berlekamp helped publicize the idea to mathematicians. Mark Setteducati took the lead in reaching the ma- gicians. Tom Rodgers contacted the puzzle community. The site chosen was Atlanta, partly because it is within driving distance of Gardner’s home. The unprecedented gathering of the world’s foremost magicians, puz- zlists, and mathematicians produced a collection of papers assembled by ix x FORWARD Scott Kim, distributed to the conference participants, and presented to Gard- ner at the meeting. G4G1 was so successful that a second gathering was held in January 1995 and a third in January 1998. As the gatherings have expanded, so many people have expressed interest in the papers presented at prior gatherings that A K Peters, Ltd., has agreed to publish this archival record. Included here are the papers from G4G1 and a few that didn’t make it into the initial collection. The success of these gatherings has depended on the generous donations of time and talents of many people. Tyler Barrett has played a key role in scheduling the talks. We would also like to acknowledge the tireless effort of Carolyn Artin and Will Klump in editing and formatting the final version of the manuscript. All of us felt honored by this opportunity to join together in this tribute to the man in whose name we gathered and to his wife, Charlotte, who has made his extraordinary career possible. Elwyn Berlekamp Tom Rodgers Berkeley, California Atlanta, Georgia Martin Gardner: A “Documentary” Dana Richards I’ve never consciously tried to keep myself out of anything I write, and I’ve always talked clearly when people interview me. I don’t think my life is too interesting. It’s lived mainly inside my brain. [21] While there is no biography of Martin Gardner, there are various interviews and articles about Gardner. Instead of a true biography, we present here a portrait in the style of a documentary. That is, we give a collection of quotes and excerpts, without narrative but arranged to tell a story. The first two times Gardner appeared in print were in 1930, while a sixteen-year-old student at Tulsa Central High. The first, quoted below, was a query to “The Oracle” in Gernsback’s magazine Science and Invention. The second was the “New Color Divination” in the magic periodical The Sphinx, a month later.Also below are two quotes showing a strong child- hood interest in puzzles. The early interest in science, magic, puzzles, and writing were to stay with him. *** “I have recently read an article on handwriting and forgeries in which it is stated that ink eradicators do not remove ink, but merely bleach it, and that ink so bleached can be easily brought out by a process of ‘fuming’ known to all handwriting experts. Can you give me a description of this process, what chemicals are used, and how it is performed?” [1] *** “Enclosed find a dollar bill for a year’s subscription to The Cryptogram.Iam deeply interested in the success of the organization, having been a fan for some time.” [2] *** An able cartoonist with an adept mind for science. [1932 yearbook caption.] *** [1934] “As a youngster of grade school age I used to collect everything from butterflies and house keys to match boxes and postage stamps — but when I grew older ... I sold my collections and chucked the whole business, and 3 4 D. RICHARDS began to look for something new to collect. Thus it was several years ago I decided to make a collection of mechanical puzzles.... “The first and only puzzle collector I ever met was a fictitious character. He was the chief detective in a series of short stories that ran many years ago in one of the popular mystery magazines.... Personally I can’t say that I have reaped from my collection the professional benefit which this man did, but at any rate I have found the hobby equally as fascinating.” [3] *** “My mother was a dedicated Methodist who treasured her Bible and, as far as I know, never missed a Sunday service unless she was ill. My father, I learned later, was a pantheist.... Throughout my first year in high school I considered myself an atheist. I can recall my satisfaction in keeping my head upright during assemblies when we were asked to lower our head in prayer. My conversion to fundamentalism was due in part to the influence of a Sunday school teacher who was also a counselor at a summer camp in Minnesota where I spent several summers. It wasn’t long until I discovered Dwight L. Moody ... [and] Seventh-Day Adventist Carlyle B. Haynes.... For about a year I actually attended an Adventist church.... Knowing little then about geology, I became convinced that evolution was a satanic myth.” [22] *** Gardner was intrigued by geometry in high school and wanted to go to Cal- tech to become a physicist. At that time, however, Caltech accepted under- graduates only after they had completed two years of college, so Gardner went to the University of Chicago for what he thought would be his first two years. That institution in the 1930s was under the influence of Robert Maynard Hutchins, who had decreed that everyone should have a broad liberal edu- cation with no specialization at first. Gardner, thus prevented from pursu- ing math and science, took courses in the philosophy of science and then in philosophy, which wound up displacing his interest in physics and Caltech. [19] *** “My fundamentalism lasted, incredibly, through the first three years at the University of Chicago, then as now a citadel of secular humanism.... I was one of the organizers of the Chicago Christian Fellowship.... There was no particular day or even year during which I decided to stop calling myself a Christian. The erosion of my beliefs was even slower than my conversion. A major influence on me at the time was a course on comparative religions taught by Albert Eustace Haydon, a lapsed Baptist who became a well- known humanist.” [22] MARTIN GARDNER: A “DOCUMENTARY” 5 “After I had graduated and spent another year at graduate work, I decided I didn’t want to teach.