The Mathematical Legacy of Martin Gardner

By Elwyn Berlekamp

As a high school student in the late 1950s, I became an ardent follower of Martin Gardner’s monthly “Mathematical Games” column in Scientific American. I continued to read it for many years. It wasn’t until the 1970s when I was a professor that I came to realize how many other mathematicians had also been influenced by him. 

Gardner’s thought-provoking columns required almost no prerequisites. He often posed a problem, a puzzle, a trick, or an effect that seemed paradoxical. Yet sufficient thought usually led the reader to a joyous “Aha!” moment when the resolution became clear. There was a bit of an addictive quality to these intellectual “Eureka!” exclamations. Success bred the quest for more success, which then led to a productive and joyous cycle of continual progress.

Gardner answered postal correspondence from his many readers. In 1960, I learned that a fellow MIT student named William L. Black had invented a Gardner-like game, which we called the Game of Black. I solved it and sent a note about it to Martin Gardner. To my delight and surprise, he published it. It was the first time I had seen my name in print in such a prestigious setting.¹

It was in Gardner’s columns that I first learned of the popular early-20th-century writings of Henry Dudeney² and Sam Loyd³, which publicized the impartial game of Kayles⁴ and the solution to it written by Richard Guy and Cedric A.B. Smith.⁵ 

Partly because of what I had read about them in Martin Gardner’s columns, I was appropriately awestruck in the 1960s when I first met Sol Golomb and then Richard Guy, each of whom had a large influence on my subsequent work. Richard invited me to Calgary to give a colloquium on the game of Dots-and-Boxes and its relationship to Kayles. The talk got an enthusiastic reception, after which I pursued research on other mathematical games with increased vigor.

In 1969 Richard introduced me to John Horton Conway, and the three of us immediately began collaborating on a book that eventually became Winning Ways for Your Mathematical Plays (WW). In the 1970s, I joined Conway in some of his many visits to Gardner’s home on Euclid Avenue, in Hastings-on-Hudson, New York. Gardner soon became an enthusiastic advocate of our book project, and he previewed various snippets of it in his Scientific American columns. Some of his readers became students in courses taught by each of us, and/or thesis students under the supervision of one of us.

Both the breadth and the depth of combinatorial game theory grew, and we began to branch out in somewhat different directions. In his book On Numbers and Games, Conway introduced “surreal numbers,” combining and superseding classical works of Dedekind and Cantor. Time marched on, and by 1982, when the first edition of our WW was published, Gardner was retiring from his Scientific American column. Readers viewed his retirement as a big setback. In 1993, in an effort to rekindle their enthusiasm, Tom Rodgers organized the first “Gathering for Gardner.” At this event, many of Martin’s fans––mathematicians, magicians, puzzlers, skeptics, and others––gathered in his honor for several days in Atlanta.  A second such gathering was held three years later, after which it became a biennial event. The eleventh G4G was held March 19–23, 2014. 

Research on mathematical games thrived after the publication of WW. Led by Aviezri Fraenkel, studies of various mathematical games resulted in refinements of the algorithmic complexity classes used in theoretical computer science. Conferences on combinatorial game theory led to the publication of several volumes of mathematical results. The first of these, Games of No Chance, included Fraenkel’s survey of the subject and an early version of his selected bibliography, which already contained 666 entries. Many of these new results were included in the second edition of Winning Ways, published by A.K. Peters in four volumes conveniently dated 2001, 2002, 2003, and 2004. Games of No Chance was followed by More Games of No Chance and Games of No Chance III. In 2007, Michael Albert, Richard Nowakowski, and David Wolfe published Lessons in Play, a wonderful undergraduate textbook in which they introduced and analyzed scores of new games. In 2013, Aaron Siegel’s book Combinatorial Game Theory introduced several additional major new results, and also provided a clear and concise mathematical summary of nearly all that was then known. 

On the more “applied” side, combinatorial game theory found applications to several classical board games, including the ancient Hawaiian game of Konane. In our 1994 book Mathematical Go, David Wolfe and I applied combinatorial game theory to late-stage endgames in what for several millennia has been the foremost intellectual board game in East Asia (and arguably in the world). More recently, a dozen of the world’s best professional Go players in Korea and China have competed in tournaments of  “Coupon Go,” a mathematically based version of the game that sheds more light both on real Go endgames and on the related mathematics. 

The first three volumes of Winning Ways dealt with traditional two-player games of no chance, but the fourth considered one-person games, such as sliding block puzzles. Martin Gardner’s “Mathematical Games” columns covered an even wider range of topics, including magic. These helped some students to become professional magicians, others to become professional mathematicians, and some (e.g., Persi Diaconis) to become both. 

Some of us view our own mathematical work as both recreational mathematics and “serious” mathematics, but many others try to distinguish between the two. I think that whatever distinctions exist are to be found not in the subject matter, but in the researchers’ motivations. Many mathematicians, as well as scientists and historians, agree that much of the best research is primarily driven by curiosity. The distinguishing feature of Martin Gardner’s writings is that they encouraged curiosity-driven investigations by people of many ages and at many educational levels. To continue this tradition, following Gardner’s death in 2010 the G4G foundation began to sponsor annual “Celebration of Mind” events all over the world on or near October 21, Martin’s birthday. In 2013, these events occurred at more than 150 locations on all seven continents. Events like these not only encourage intellectual pursuits, they also increase popular appreciation of mathematics.  

¹Scientific American, October 1963.
²Scientific American, June 1958. 
³Scientific American, August 1957.
⁴Scientific American, February 1967.
⁵R.K. Guy and C.A.B. Smith, in Proceedings of the Cambridge Philosophical Society, 52: 1956, 516–526.

Elwyn Berlekamp is professor emeritus at the University of California, Berkeley.