Martin Gardner. (Courtesy of Colm Mulcahy)

 

The "Mathematical Games" column in Scientific American that began in January 1957 is a legend in publishing, even though it's been almost 30 years since the last one appeared. The columns are still considered models of clarity and elegance for introducing fresh and engaging ideas in mathematics in non-technical ways.

As we pause to celebrate the centennial of the man who wrote them, the ever-prolific Martin Gardner (1914–2010), we note that while many of his articles fell under the umbrella of “recreational mathematics,” others touched on cutting-edge concepts involving contributions from some of the world's most creative minds. Even the articles that seemed to be purely for entertainment sometimes inspired important research, some of which led to developments with real impact on science, technology and society. This success is all the more remarkable considering that Gardner had no formal training in mathematics.

In his memoirs Undiluted Hocus-Pocus (Princeton, 2013), Gardner recalls:

 

"One of the pleasures in writing the column was that it introduced me to so many top mathematicians, which of course I was not. Their contributions to my column were far superior to anything I could write, and were a major reason for the column’s growing popularity. The secret of its success was a direct result of my ignorance. Even today my knowledge of math extends only through calculus, and even calculus I only dimly comprehend. As a result, I had to struggle to understand what I wrote, and this helped me write in ways that others could understand.”

 

How does one choose the top ten Gardner articles from the roughly 300 that he wrote for Scientific American? The vast majority of those were "Mathematical Games" columns, which appeared monthly from January 1957 to December 1980, and then sporadically until June 1986. Should we focus on the ones that are most talked about today, those that generated the biggest volume of correspondence upon publication, or those that were the scientifically influential?

The following list takes all of these points into consideration, so without further ado, I present, in order of publication, an annotated list of what are, in my opinion, Gardner’s top ten articles for Scientific American:

1. “Flexagons, in which strips of paper are used to make hexagonal figures with unusual properties” (December 1956)

 

Including Gardner’s “first” Scientific American article, “Flexagons,” is a no-brainer. It was actually the second piece he wrote for the magazine (“Logic Machines” had appeared as a one-off back in March 1952) but it was such a hit that editor Gerry Piel promptly invited Gardner to write a monthly column. Thus, in January 1957 "Mathematical Games" proper was born.

Hexaflexagons, as they’re known today, are hexagonal folded paper objects which can be transformed repeatedly by “turning them inside out” to reveal new faces. In his memoirs, Gardner recounts how he was introduced to them by Royal V. Heath, the man credited with popularizing the term “mathemagic” from 1951 onwards. An English graduate student named Arthur Stone had accidentally discovered flexagons at Princeton in 1939, and he and fellow students John Tukey, Bryant Tuckerman and Richard Feynman then explored them mathematically. The war intervened and the paper curiosities were forgotten. It fell to Gardner to revive them 15 years later, little dreaming that it would launch him on the most successful phase of his career.

As he noted shortly before his death, “Today there are some fifty websites devoted to flexagon theory and variants of the original forms.” Here are two from more recent times that will guide you through making your own: Hexaflexagon Templates and Make Your Own Hexaflexagons…and Snap Pictures of Them

2. “More about complex dominoes” (December 1957)

 

All 18 pentominoes. (Credit: R. A. Nonenmacher via Wikimedia Commons)

 

This column is remembered today for introducing readers to mathematician Sol Golomb's five-square versions of “polyominoes” – pieces formed by fitting several unit squares together along their edges. As Gardner notes in his memoirs:

 

“A single square is the monomino, two squares are the dominoes, three the trominoes, four the tetrominoes, and five the pentominoes. The problem of finding a formula for the number of n-ominoes, given n, is still a deep unsolved combinatorial problem. My first column on Golomb’s twelve pentominoes was an instant hit. I returned to polyominoes in several later columns.”

 

3. “A third collection of 'brain-teasers’” (August 1958)

Gardner’s regular compendiums of short head-scratchers (often simply called “Nine Problems”) forced readers either to buckle down and conquer the problems therein or wait a full month to see what solutions and comments the next column offered. The only “websearch” option available back in the Sputnik and Apollo eras was to write to the man himself, which many people did, but those were usually readers with new solutions or fresh material to offer.

Surveying those special columns today, one is struck by how many gems they contain: The Returning Explorer (aka one mile south, one mile east, one mile north), The Mutilated Chessboard, The Fork in the Road (aka truth-tellers and liars), the mind-boggling Hole in the Sphere, and the obvious-once-you-get-it Touching Cigarettes. (There's an exciting new development in the last problem, Gardner's bibliographer and biographer Dana Richards recently reported.)

 

 

In the face of such tough competition it's not easy to select an outright winner here, but we selected the head-scratchers from August 1958. It opens with the iconic Twiddled Bolts, depicted above, and continues with The Cork Plug, and The Sliding Pennies. Pride of place in this particular collection goes to The Colliding Missiles, however. Here’s the problem:

 

Two missiles speed directly towards each other, one at 9,000 mph, the other at 21,000 mph. They start 1,317 miles apart. Without using pencil and paper, calculate how far apart they are one minute before they crash.

 

Needless to say, one should not use a calculator, slide rule or abacus either! (Gardner's son Jim reports that the last-named device was his dad's weapon of choice when balancing his check book.) A tiny knowledge of physics helps, and we believe there is an important lesson for teachers hiding within this question's presentation. See the solution and comment at the very end below.

4. “Concerning the diversions in a new book on geometry” (April 1961)

This column, and its associated Scientific American cover, introduced many people to the pre-psychedelic creations of Dutch artist M. C. Escher. It reviewed the book An Introduction to Geometry (Wiley, 1961) by Toronto-based geometer H. S. M. Coxeter.

It's hard to believe today, but by the early 1960s geometry and visualization in mathematics had fallen out of favor in deference to more abstract branches of the subject and more formal reasoning. Indeed, it was not unheard of for mathematics books to contain few if any pictures. Coxeter’s book, as Gardner gleefully reported with wonderful accompanying images, revealed many delicious surprises, including Morley's Theorem and the difficulty of proving the Internal Bisector Theorem. Gardner then quotes "The Kiss Precise"—a verse immortalizing a curious result about any three mutually touching circles—before moving on to tessellations.

Escher's knights on horseback from Coxeter's book make an appearance, but the cover is a Scientific American exclusive, the now famous flying geese (which the art department colored without consulting the artist). As it happens, Escher was already a fan of Gardner’s and in particular of his recent book, Annotated Alice (Potter, 1960).

5. “The fantastic combinations of John Conway's new solitaire game ‘life’” (October 1970)

Better known today as “Life” or “The Game of Life,” this column ventured into very new territory as it explored a cellular automaton creation of English mathematician John Horton Conway. To quote Gardner at the time:

 

“Because of its analogies with the rise, fall and alterations of a society of living organisms, it belongs to a growing class of what are called ‘simulation games’ – games that resemble real-life processes. To play Life without a computer you need a fairly large checkerboard and a plentiful supply of flat counters of two colors.”

 

 

 

 

 

 

 

 

Gosper's Glider Gun. (Credit: Kieff via Wikimedia Commons)

 

 

 

As it turned out, many people with then-rare access to mainframes seized the opportunity to program Life. But there was also great theoretical interest in the new game. Referring to the animation above, which was reported in a later column, Gardner relayed in his memoirs that, “Conway was the first to prove that Gosper’s glider gun turned Life into a Turing machine that in principle can do everything the most powerful computers can do.” He continued:

 

“All over the world mathematicians with computers were writing Life programs. I heard about one mathematician who worked for a large corporation. He had a button concealed under his desk. If he was exploring Life, and someone from management entered the room, he would press the button and the machine would go back to working on some problem related to the company!”

 

Gardner also noted that his first column on Life “made Conway an instant celebrity. The game was written up in Time.”

6. "Free will revisited, with a mind-bending prediction paradox by William Newcomb" (July 1973)

Better known today as “Newcomb's Paradox,” this column concerns a free will paradox devised by physicist William Newcomb in 1960, and then featured in a paper by philosopher Robert Nozick in 1970. Imagine two closed boxes on a table. Box 1 is known for sure to contain $1,000, whereas Box 2 contains either nothing or $1,000,000, but you don't know which. Two courses of action are open to you: either take what is in both boxes or take only what is in Box 2.

Here's the catch: we are asked to believe that a superior being has predicted in advance which choice you will make, and if the being predicts you will choose both boxes, the being has left Box 2 empty, otherwise the being has put $1,000,000 in it. Also, if the being expects you to flip a coin to decide on your course of action, the being has definitely left Box 2 empty.

 

 

 

Gardner goes on to present very strong arguments for why each course of action is superior to the other using expected payoff value computations, and he discusses both sides of the argument at length. He concludes, “Can it be that Newcomb's paradox validates free will by invalidating the possibility, in principle, of a predictor capable of guessing a person's choice between two equally rational actions with better than 50 percent accuracy?”

 

7. "Six sensational discoveries that somehow or another have escaped public attention" (April 1975)

This unprecedented April Fools prank column went over the heads of many readers. At a time when “new results” of any type could not be researched on the Internet, calculators only displayed eight digits and very few people had access to computers, Gardner got away with the astonishing claim that eπ√163 = 262,537,412,640,768,744 simply because nobody could really check, and it seemed close to true to anyone who did a ballpark calculation! He attributed it to Indian mystic mathematician Ramanujan, which was a total red herring, but this “near miss” turns out to have mathematical significance.

Gardner also revealed that Leonardo da Vinci invented the valve flush toilet, and Gardner produced convincing looking drawings (“courtesy of N.Y. Public Library”) to prove it. Then there was the announcement of a computer proof that in chess, the move “pawn to king's rook 4” is a win for white, “with a high degree of probability.”

Most dramatically, Gardner unveiled a 110-region map that he said could not be colored with fewer than five colors. If true, this would have provided a counterexample to the then long-standing four-color-map conjecture. It certainly wasn't easy to color this particular map with four colors. Gardner’s timing was impeccable: a year later Appel and Haken announced a “computer assisted” proof that all maps can indeed be four-colored. From Gardner’s memoirs:

 

“I received hundreds of letters showing how to color my map with four colors. Many readers, including a few scientists, thanked me for alerting them to such important discoveries but chided me for being totally mistaken about one of them.”

 

8. "In which 'monster' curves force redefinition of the word `curve'" (December 1976)

Later known as “Mandelbrot's Fractals,” this column begins by discussing the nature of what's been understood by the word “curve” throughout history, from ancient Greece to 17th century notions based on analytic geometry, and the assumptions resulting from the age of calculus that soon followed. Early examples of so-called pathological or monster curves, such as Koch snowflakes, are next, along with their unavoidable paradoxes. How can a curve fill the plane, or space, and how can the distance between two points on a curve be infinite?

Mandelbrot's formalization of a new type of dimension, which he named fractal dimension only a year or two before this column was published, is explained in terms of self-similarity at various scales. Examples covered include square snowflakes (shown above) and Cantor dust.

Mandelbrot lived not far from Gardner when this piece was written, and around this time, at his own home, Gardner introduced Mandelbrot to Conway. As Gardner relays in his memoirs, “Conway had been making new discoveries about Penrose tiling, and Mandelbrot was interested because Penrose tiling patterns are fractals. You can keep enlarging or diminishing them, always to obtain similar patterns.”

9. "Extraordinary nonperiodic tiling that enriches the theory of tiles" (January 1977)

Gardner’s column on Penrose tiling gave him another cover story, the cover artwork having been sketched by John Conway (and later colored by a Scientific American staff artist).

 

 

It starts with traditional tilings, such as those done with squares, dominoes and hexagons, which are generally periodic. Gardner then explains how many—but not all—of these are also associated with nonperiodic tilings. He draws on images of M. C. Escher and Sol Golomb (the latter's “reptiles”) before asking, “Are there any sets of tiles that tile only nonperiodically?” This leads to the fascinating story of Penrose's mid 1970s discovery of what are now known as Penrose tiles (or, darts and kites, following a suggestion of Conway's).

In his memoirs, Gardner comments:

 

“To Penrose’s vast surprise, it turned out that three-dimensional forms of his tiles would tile space only aperiodically! Not only that, but such shapes could actually be fabricated in laboratories. They became known as quasicrystals. Hundred of papers have since been published about them. They are a marvelous example of how a mathematical discovery, made with no inkling of its application to reality, may turn out to have been anticipated by Mother Nature!”

 

In 2011, chemist Dan Shechtman was awarded a Nobel Prize for “the discovery of quasicrystals.

10. "A new kind of cipher that would take millions of years to break" (August 1977)

This trapdoor ciphers column introduced RSA cryptography, a new “public key” method of secret communication previously not believed possible. It was based on an MIT memo by Ron Rivest, Adi Shamir and Leonard Adleman from April 1977, which they sent to Gardner. He was so impressed that he broke his usual rule of planning his column several months in advance, and quickly wrote it up for publication.

The basic idea is to secretly take two very large prime numbers (p, q) at least 40 digits long each, and form their product r=pq, assuming that it would be an insurmountable task for an outsider to factor r. It’s considered safe to reveal r, as well as a related odd number s, to all and sundry; that's the public key. Anyone wishing to send a secret numerical “word” w to the person who selected p and q does the following: find the remainder e when ws is divided by r, and communicate e openly. An easy mathematical trick allows a person who knows e to reconstruct w from it provided they know the factors p and q of n, but it seems unlikely that somebody not knowing p and q would have a chance.

To prove the point, the RSA team provided Gardner with a 128-digit coded message e, computed using a specified 129-digit n, which was the product of mysterious top-secret, 64-digit and 65-digit primes p and q, respectively. They also indicated that s = 9007. A prize of $100 was offered for anyone who could recover the original message w from which e had been computed. Given the title of the column, it was assumed that no one would crack it anytime soon. In fact, Gardner hedged his bets and prefaced the piece with an Edgar Allan Poe quote: “Yet it may be roundly asserted that human ingenuity cannot concoct a cipher which human ingenuity cannot resolve.”

RSA cryptography became an industry standard and variations of it are still in use today, though in recent times the question of how secure it is has been revisited. Despite the groundbreaking nature of Gardner’s column, it didn't quite live up to its title. The challenge message posed in it was successfully decoded as early as April 1994.

Apologies and Addendum

Opinions will of course vary on the items to include in any top ten list, and it's likely that many reader favorites are missing above. Apologies. I also regret any offense caused to fans of the late great Dr. Irving Joshua Matrix. None of his adventures made the final cut either. Maybe if the list went to 11...

There is no shortage of Gardner memorabilia to explore. Check out Scientific American’s In-Depth Report, “A Centennial Celebration of Martin Gardner,” the Scientific American e-book, Martin Gardner: The Magic and Mystery of Numbers, and the official Martin Gardner site.

Solution and comment for The Colliding Missiles

We may as well assume that one missile is stationary and the other is barreling towards it at 30,000 mph (their combined speed). Since distance traveled in this case is found by multiplying speed by time taken, it follows that in one minute (1/60 of an hour) the moving missile travels (30,000 mph) x (1/60 hour) = 500 miles. That's how far apart the missiles must be a minute before they collide.

Most readers are surprised—and some feel disappointed or cheated—that we don't need the fact that the missiles started 1,317 miles apart. Does this make it a “trick” question? That depends on one's perspective. Consider this (and there’s a lesson here for those of us in the teaching profession): in the real world, unlike in many textbook situations, we're bombarded with information and data. Learning to distinguish the essential from the irrelevant is a key skill worth acquiring. Once more, with this puzzler, Gardner the rationality champion subtly points us in the right direction.