Martin Gardner's 100th Anniversary

by Frederic Friedel
10/21/2014 – "With a career spanning most of a century, Martin Gardner was the last of the polymaths," writes Neil deGrasse Tyson, and Douglas Hofstradter calls him "a major shaping force in my life" – as does Frederic Friedel, who has written a long tribute to this giant of recreational mathematics, micromagic, literature, philosophy, scientific skepticism, religion – and not to forget: chess.

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Remembering Martin Gardner

By Frederic Friedel

Martin Gardner, born on October 21, 1914, was an American popular mathematics and science writer specializing in recreational mathematics, but with interests encompassing micro-magic, literature (especially the writings of Lewis Carroll and G.K. Chesterton), philosophy, scientific skepticism, religion – and chess. He published more than 100 books and almost all of his columns have been collected in book form. Martin Gardner died in 2010 at age 95, razor sharp until the end. Today he would have been 100 years old.

The caption to the above pictures by Colm Mulcahy, Spelman College, reads: "Martin standing by every word he ever wrote: the six shelves consist entirely of his own publications, dating back to 1930."

I corresponded with Martin Gardner in my teen years, and I can safely say that he played a greater role in my intellectual development than almost anyone else (and I am in good company here). I corresponded with him during my school and college days and followed his columns and writings for decades. Our readers must forgive this rather lengthy celebration of his 100th anniversary – it has personal reasons.

The skeptic

In 1979 I visited Martin Gardner in his home in Hastings-on-Hudson. He and many others (including myself) had recently founded the skeptical Committee for Investigation of Claims of the Paranormal, which mutated into the Committee of Skeptical Inquiry and became the publisher of the Skeptical Inquirer. It was the first time I met him in person.

Gardner's uncompromising attitude toward pseudoscience made him one of the foremost anti-pseudoscience polemicists of the 20th century. His book Fads and Fallacies in the Name of Science (1952, revised 1957) is a classic and seminal work of the skeptical movement. It explored myriad dubious outlooks and projects including Fletcherism, creationism, food faddism, Charles Fort, Rudolf Steiner, Scientology, Dianetics, UFOs, dowsing, extra-sensory perception, the Bates method, and psychokinesis. This book and his subsequent efforts earned him a wealth of detractors and antagonists in the fields of "fringe science" and New Age philosophy, with many of whom he kept up running dialogs (both public and private) for decades.

The puzzlist

Martin Gardner's column on mathematical puzzles and diversions in Scientific American were my first contact with the work of this extraordinary man. I was still an early teen in school, but I sent letters (handwritten, on paper, with envelopes and stamps) with comments and solutions. He replied to every single one of them – in my boundless stupidity I did not keep his type-written letters, which are lost forever. I have added some comments to the following puzzles from memory.

If you want to get an impression of the vintage Martin Gardner get hold, if you can, of a copy of his book "Hexaflexagons and other mathematical diversions". It is also available as a PDF file here. You can spend a week or two, as I did in my college days, building hexahexaflexagons (or read the article on them here in Scientific American). Or you can try some of the puzzles that have become legendary. Here a sample:

  • The props for this problem are a chessboard and 32 dominoes. Each domino is of such size that it exactly covers two adjacent squares on the board. The 32 dominoes therefore can cover all 64 of the chessboard squares. But now suppose we cut off two squares at diagonally opposite corners of the board and discard one of the dominoes. Is it possible to place the 31 dominoes on the board so that all the remaining 62 squares are covered? If so, show how it can be done. If not, prove it impossible.

  • A logician vacationing in the South Seas finds himself on an island inhabited by the two proverbial tribes of liars and truth-tellers. Members of one tribe always tell the truth, members of the other always lie. He comes to a fork in a road and has to ask a native bystander which branch he should take to reach a village. He has no way of telling whether the native is a truth-teller or a liar. The logician thinks a moment, then asks one yes-no question only. Form the reply he knows which road to take. What question does he ask? In a delightful additional twist two readers noted that if the logician knows that 'pish' and 'tush' are the native words for 'yes' and 'no' but has forgotten which is which (though otherwise he can speak the native language), he can still determine which road leads to the village. It is worth reading the discussion to the solutions, which is included in the book.

I participated vigorously the the discussion given in the book. Some ChessBase readers will recall that I used this problem at the end of one of our Christmas Puzzle weeks.

  • This is an incredible problem – incredible because it seems to lack sufficient data for a solution. A cylindrical hole six inches long has been drilled straight through the center of a solid sphere. What is the volume remaining in the sphere?

My most memorable exchange with Martin Gardner was over this problem. At the time I sent him the following solution: "I do not (yet) have the mathematical skills to solve this problem. But assuming that the text does not contain a misprint it would appear that it is not necessary to know the diameter of the sphere or the drill bit. Just the length of the cylindrical hole through the middle is enough, the remaining volume will be constant. If that is the case then I can drill a hole of zero diameter through a sphere with a diameter of six inches. So the remaining volume will be 4/3 Pi r³ = 113.09733 cubic inches (I can do volume of a sphere)." Martin wrote back commending my ingenuity, and many years later, when I finally met him, the remembered this exchange and even the fairly silly pseudonym I had used at the time.

The answers to the puzzles are given in the above book – print or PDF. A more recent puzzle involved a desk block perpetual calendar (like this one). The day is indicated simply by arranging two cubes so that their front faces gave the date. The face of each cube has a single digit, 0 through 9, and one can arrange the cubes so that their front faces indicated any date from 01, 02, 03 … to 31. On the left cube of the calendar shown on the right you can see two faces whose digits are 1 and 2. On the right cube you can see three faces, whose digits are 1, 3 and 4. What are the four digits that cannot not be seen on the left cube and the three that cannot be seen on the right cube?

In case you get hooked, there are many other Gardner books with mathematical and logical puzzles. Here I would like to add one (of the many) he gave me when I visited him in in Hastings-on-Hudson in 1979: There are three on/off switches on the ground floor of a house. One is connected to a light bulb on the second floor, the other two are unconnected. The switches are all in the off position. You can switch any position and then climb the stairs to the second floor and examine the bulb. Can you tell which switch is connected in a single try?

The chess enthusiast

In May 2010, shortly after the death of Martin Gardner at the age of 95, Tom Braunlich wrote a eulogy for Chess Life. A link to the article is given below – here are some excerpts:

Martin Gardner was an enduring fan of chess and once played two tournament games with Sammy Reshevsky! Gardner was born in Tulsa, Oklahoma, in 1914 and played chess there in high school until he was graduated in 1931. Later at the University of Chicago he continued as an enthusiastic amateur player. There he met the famous grandmaster. Gardner told me the story.

In those days [the early 1930s] Reshevsky was having difficulty making a living as a professional chess player during the Great Depression, and he had decided to give up the game and take up accounting. He enrolled at the University of Chicago to study for a degree. We had a small chess club going at the university then where I would play, and of course we became aware that Reshevsky was at the school. But we were disappointed that he never came by the chess club to play. Of course, he had no reason to come, as we were all far below his level, and he was trying to give up the game anyway. But those of us in the club hatched a plan to get him. We took up a collection and raised $50 for a guaranteed first prize for a tournament. We put up flyers around the school advertising a double round robin event, making sure to put plenty of flyers around the accounting department. Sure enough, Reshevsky showed up to register for the tournament! $50 was a lot of money in those days and, as we expected, the temptation was too much for him.

Since it was a double-round robin, I got to play him twice. In the first game, I made an error in the opening leaving myself open to a direct attack. But I was surprised he didn’t take that opportunity, continuing instead to just make strong building moves and playing for position, eventually overwhelming me.

Gardner never really lost his love for chess, often using unusual chess puzzles in his columns, but only if they related to mathematical principles he was discussing. An example of this would be the “Eight Queens” puzzle (i.e., how do you place eight queens on a chessboard so that no two queens attack each other?), which could then be generalized to n number of queens on an n by n board and underlying principles revealed. – Read the full article in Chess Life Online here.

The Eight Queens problem has 92 destinct solutions, twelve if you discount those that differ only by symmetry operations (rotations and reflections) of the board. Gardner took it further: place three white queens and five black queens on a 5 x 5 chessboard so that no queen of one colour is attacking one of the other colour. There is only one solution to this problem, excluding reflections and rotations.

The 5 x 5 board comes from a minichess version invented by Gardner in 1969. He proposed a chess variant played on a 5×5 board in which all chess moves, including pawn double-move, en-passant capture as well as castling can be made. The game was largely played in Italy (including by correspondence) and opening theory was developed. The statistics of the finished games is the following: White won 40% of games, Black won 28%, 32% were draws (Gardner's minichess was solved in 2013, and the game-theoretic value turned out to be a draw). In 1980 HP shipped HP-41C programmable calculator, which could play this game at a fairly decent level.

In 1989, Gardner proposed another setup, which he called Baby chess. In difference from Gardner minichess, kings are placed into opposite corners here. – For more information on minichess see this Wikipedia article.

Chess and cheating

In January 1948 (!) Martin Gardner wrote a short story for the magazine Esquire, Nora Says 'Check', which was subsequently reprinted in a volume of Gardner’s early fiction, The No-Sided Professor. In this story Sierpinsky, the world champion, alleviates his chess boredom by guiding a not very bright waitress named Nora to chess fame with the help of a confederate. During each of Nora’s tournaments the confederate watches from the audience, communicating with a hidden Sierpinsky via a toe-interfaced shoe radio and with the waitress through a language of gestures. The story also anticipates the so-called hippopotamus chess opening – further Sierpinsky boredom alleviation – by nine years. A third Gardnerian anticipation of future chess history flows out of the success of Sierpinski’s hijinks. A world championship chess match transcends all previous bounds of public attention, inflaming the masses from the front page of Pravda to the cover of Time. Of course, this particular world championship match is especially amusing: puppet master and puppet on stage together, still mediated by cigar chomping confederate. Even more amusing is the Frankensteinian catastrophe that befalls Sierpinski at the very climax of his carefully constructed finale. A fourth manifestation of chess prescience – this one almost spooky – appears in the last sentences of the story in the form of an eight-year-old boy from the Bronx. Summary by Danny Purvis – read the entire story here.

Scientific American

Martin Gardner was best known for sustaining and nurturing interest in recreational mathematics for a large part of the 20th century, particularly through his "Mathematical Games" column (from 1956 to 1981) in Scientific American. The October issue has a six-page article on him.

Like a good magic trick, a clever puzzle can inspire awe, reveal mathematical truths and prompt important questions. At least that is what Martin Gardner thought. His name is synonymous with the legendary Mathematical Games column he wrote for a quarter of a century in Scientific American. Thanks to his own mathemagical skills, Gardner presented noteworthy mathematics every month with all the wonder of legerdemain and, in so doing, captivated a huge readership worldwide. Many people—obscure, famous and in between—have cited Mathematical Games as informing their decisions to pursue mathematics or a related field professionally. – From Scientific American October 2014, pp 78-83.

Communications Award

The Mathematical Association of America has just shown 40 minutes of interview footage recorded with the 79-year-old Gardner in 1994 at his Hendersonville, North Carolina, home. Until recently it was stored only on a pair of video cassettes formatted for broadcast television. They were found in a cupboard at MAA headquarters and transferred to DVD. A 14-minute segment of the interview was uploaded to Youtube on October 15, 2014.

Joint Policy Board for Mathematics Communications Award Presentation to Martin Gardner


There are countless articles and columns on the Internet, dealing with Martin Gardner and his activities. You can spend days and weeks – or perhaps an entire lifetime – exploring them. Or you can buy some of his 100 or so books he published. Here are a few links we have selected:

Editor-in-Chief emeritus of the ChessBase News page. Studied Philosophy and Linguistics at the University of Hamburg and Oxford, graduating with a thesis on speech act theory and moral language. He started a university career but switched to science journalism, producing documentaries for German TV. In 1986 he co-founded ChessBase.


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RJS RJS 10/23/2014 08:48
The chessboard and dominoes problem as posed here misses the point entirely. The original version of the puzzle entails simply an eight by eight grid of squares all the same colour. Colouring them in chessboard fashion is the conceptual leap required to find the solution. Unfortunately the chessboard version has now become almost universal.
RJS RJS 10/23/2014 08:47
The chessboard and dominoes problem as posed here misses the point. The original version of the puzzle entails simply an eight by eight grid of squares all the same colour. Colouring them in chessboard fashion is the conceptual leap required to find the solution. Unfortunately the chessboard version has now become almost universal.
Adam P Adam P 10/23/2014 02:38
What an ignorant, mean-spirited attack, vincero. 'Gardner was a quack'! Lol. Saying it doesn't make it so. You offer exaggeration and irrelevancies. "Wasn't interested in the truth if it got in the way of his personal attacks." - look and the mirror and tell me if this isn't you. Shameful. Do you have anything more specific than vague exaggerated accusations and outright falsehoods to throw? Still, I guess sceptics must appear this way to people who know nothing of how science works, and don't care to learn. Or of the studies of the many ways humans can fool themselves, get things wrong, reason badly, misestimate probabilities etc. Or of how a healing 'modality' can cure or benefit people, while the benefit evidently comes from something other than the medicine/procedure. (e.g. homeopathy, acupuncture for starters)

Nice article, FF, thanks. :-) Gardner's recreational maths and sceptical books have been a huge influence on me too. I read Hofstadter's GEB aged 14 (being already a fan of Escher, a musician and programmer) so I should give Gardner partial credit for that as well! It takes a special kind of patience, I imagine, to be willing to delve so deeply, fairly and accurately into pseudoscientific investigations - it is foul air in that ugly and all-too-human world. And, it seems to me, he was very fair, never claiming more than the evidence warranted, never mocking, in his lively and admirable prose style. What a magnificent service for all of us. Although, it seems, there are more who are fans of pseudoscience than are fans of science. The work - the battle against ignorance - continues!

Also lastly, I came across recently in one of Gardner's math books, a chapter on a large rectangular piece of metal that had been installed in a New York street (i.e. as modern sculpture), and another well-known artist who puts bricks together in squares or paths. And writing of it as he did pseudoscience, drolly examining their claims to being good art, giving some ridiculous laudatory reviews these works had received, and a history of their public and critical reception. I showed this to my partner, whose job is in the contemporary arts, she thought it hilarious too. So this was, it seems, another aspect of his oeuvre, as early critic of that Emperor-without clothes, the 'pile of bricks' school of contemporary sculpture. A magnificent essay, I thought. :-) Thank you MG.
Gametheory Gametheory 10/22/2014 02:30
A good article - I had never heard of Mr. Gardner; he seems to have been a very interesting fellow.

I was surprised that there was no mention of the "Magic Square" (I read the piece rather quickly, sorry if it was in there), a great intro to the topics discussed and the more elaborate mathematical puzzles that were posed; The first time one examines a Magic Square, the explanation that it is a coincidence feels inadequate, even somewhat annoying but then, what is the explanation?

The "Fork In The Road" riddle that was mentioned is an all-time classic, it can be very confusing in its total simplicity - sort of like an endgame in Chess.
billybudd billybudd 10/21/2014 09:56
Vincero: Do you know what they call alternative medicine that's been proved to work? Medicine.
Morphée06 Morphée06 10/21/2014 05:50
With M. Mahlla we have proved the theoretical value of Gardner's chess : it is draw.

Details can be found in the article : Mehdi Mhalla, Frédéric Prost, Gardner’s Minichess Variant is Solved. ICGA Journal, Vol. 36, No. 4
vincero vincero 10/21/2014 04:22
martin gardner may have done things that you admire but not everyone admires him.
gardner was a quack who pretended nothing but main stream views held any authority.
he attacked people like Oprah Winfrey and doctor oz....for even suggesting that main stream medicine didn't have all the answers.

there are literally 100s of thousands of studies and doctors all over the world speaking to the need of vitamins and alternative healing ,methods yet gardner wasn't interested in the truth if it got in the way of his personal attacks.

admire him if you want to but enlightened people will have nothing but disdain for someone who sought to harm as many people as he could.
billybudd billybudd 10/21/2014 03:59
Thanks Frederic, these kind of stories are the reason I keep coming back to your site.

I had never heard about Martin Gardner before, but he seems to have a lot in common with Raymond Smullyan. Did they know each other well?