Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Mathematical Puzzles and Games by Martin Gardner

Math games and puzzles mostly for the Mensa set

For much of an authorial career that spanned over three-quarters of a century, Martin Gardner (1914–2010) was America’s most popular writer on matters mathematical. He is best known for “Mathematical Games,” a column he wrote in Scientific American magazine for 25 years. The articles from that column were collected into a series of at least 15 books, the first volume being The Scientific American Book of Mathematical Puzzles and Diversions, published in 1959. Many of the books in this series were later republished under different titles. This first book was rereleased in 1988 as Hexaflexagons and Other Mathematical Diversions. The 1988 edition includes additional new content from Gardner in the form of addenda to each chapter and an afterward at the end of the book.

Gardner’s “Mathematical Games” was more than just a brain-teaser-of-the-month. In fact, very few of the chapters in this book are the sort of puzzles you would find in a typical brain teaser book. A couple chapters entitled “Nine Problems” feature such riddles. For the most part, however, a typical Gardner column was more of a recreational exploration into a topic of mathematical interest. The word “Games” in the title is no casually chosen word. Gardner does spend many pages talking about games, those in the abstract strategy category that includes chess, tic tac toe, and everything in between. In this volume he discusses chess, tic tac toe (2D and 3D), Hex, Nim, and Tac Tix. Beyond going over the rules for such games, Gardner might discuss the number of possible moves, play on different board sizes, the winning probabilities of certain opening moves, whether the first or second player has a mathematical advantage, and sure-fire strategies to ensure victory. Other favorite Gardner topics include magic tricks, geometry puzzles, and the branch of mathematics known as topology. Even when he’s discussing card tricks, magic squares, or hand-held sliding-square puzzles, the emphasis is on permutations and probabilities.

It’s amazing how many of the games and puzzles that Gardner features here were created by scientists and mathematicians in some of the most prestigious research institutions in the country. Apparently physicists, when they’re not splitting the atom at MIT or Fermilabs, spent their leisure time on such mathematical diversions. Hexaflexagons are strips of paper that are divided into equilateral triangles and folded into a hexagon-shaped object somewhat like a kid’s paper fortune teller. Yet Gardner tells us that one of the pioneers of hexaflexagons was Richard P. Feynman, the theoretical physicist who won the Nobel in 1965.

Nowadays, Scientific American magazine is aimed at a broad popular audience, but back in Gardner’s day, it was written for a high-IQ clientele. Those same mathematicians and scientists who were fooling around with flexagons, polyominoes, and board games would have been reading the magazine and tuning into Gardner’s column every month. For that reason, this ain’t always easy stuff for the average Joe to wrap his head around. In some cases, that’s because the math may be beyond your level. In others, it’s because Gardner’s explanations are confusing. For the most part, he’s great at explaining higher mathematical concepts to laymen, but sometimes the content is just too complex to be dumbed down for mere mortal mathematical minds like mine.

By its very nature, this book is a grab bag, and as such, some topics will fascinate particular readers more than others. I, for example found the geometry stuff more interesting than the magic tricks. Since a good portion of the book is about abstract strategy games, it helps to have an interest in those, which I do. I can’t say this book was fun and fascinating form start to finish, but it hooked me enough to know that I’ll likely delve into the second volume of Gardner’s Scientific American writings.

Mathematical Puzzles by Martin Gardner

An early collection of brain teasing quickies

Martin Gardner was a popular and prolific nonfiction author of the latter half of the twentieth century. Just as Carl Sagan was the great popularizer of science during that time period, Gardner was the great popularizer of mathematics. For over 25 years, he wrote a column in Scientific American magazine entitled Mathematical Games, in which he presented all sorts of mathematical curiosities and puzzles and educated readers on the mathematical laws and theories behind them. These columns were collected into over a dozen books that enlarged his audience well beyond the subscription list of Scientific American.

Mathematical Puzzles, an early collection of Gardner’s mathematical writings, was published in 1961 and later rereleased by Dover Publications in 1986 as Entertaining Mathematical Puzzles. The puzzles in this volume are not taken from Gardner’s column in Scientific American. Some of them previously appeared in a magazine called Science World, for which Gardner also wrote a column entitled On the Light Side from 1957 to 1959. These are not the chapter-length mathematical explorations from Mathematical Games but rather what one typically finds in a collection of brain teasers: items of one to two pages, essentially a word problem followed by an explanation of its solution. Altogether, there are about 60 puzzles in this book. They are arranged into ten categories such as money puzzles, speed puzzles, plane and solid geometry puzzles, and probability puzzles.

Compared to his writings for Scientific American, these little brain teasers are not Gardner’s best work. It’s amazing how many of these puzzles conclude with “Nope. It can’t be solved.” You read the first page, wrack your brain to try to come up with a solution, and then turn the page to find out there’s no viable answer. Where’s the fun in that? If it were one or two puzzles, that would be reasonable, but it seems like nearly half the book is unsolvable. Another frustrating aspect of brain teasers in general is that they are essentially mathematics written with words, and if the author isn’t exactly precise and clear in the wording of the problem, then you end up feeling like you’ve been cheated or you wasted your time. Gardner’s guilty of that as well in a number of these riddles. Such occasions call to mind the questions on your old school math tests that no one in the class got right because the teacher worded them poorly.

Nevertheless, there’s still fun to be had here and math to be learned. Every puzzle has its little twist that forces you to think outside the box of straightforward solutions. As the aforementioned list of categories indicates, there are little lessons on geometry, probability, and arithmetic. These are not easy math puzzles, but you don’t need to know algebra or calculus to figure them out. The most unfamiliar area for most will probably be the little section on topology. Gardner also describes some games of the abstract strategy variety (think checkers or tic tac toe, for example) that can be played with pencil and paper, coins, or other everyday items.

The material here may not be Gardner’s most ingenious writings, but this little book still does its job adequately. The next time I download a volume of Gardner’s mathematical puzzles, however, I’ll make sure it’s one of the 15 volumes from his Scientific American series. 

Logic Machines and Diagrams by Martin Gardner

Early graphic representations of mathematical reasoning

For almost a quarter of a century, Martin Gardner (1914–2010) wrote a column in Scientific American magazine entitled Mathematical Games, in which he would present all sorts of math curiosities, puzzles, and brain teasers. These columns were assembled into many collected volumes that were popular with general math-curious readers. He also wrote plenty of books outside of that series on topics concerning math, science, and magic (card tricks and other forms of prestidigitation). Logic Machines and Diagrams, published in 1958, is one of Gardner’s earlier works, written when he had just started with Scientific American. In regards to its particular subject, the book has a sort of master’s-thesis feel to it, but it is still written very accessibly for curious non-mathematicians.

A logic diagram is a graph or chart that helps illustrate or prove the veracity and validity of a series of logical propositions. The most obvious and commonly known example of a logic diagram is the Venn diagram composed of interlocking circles, introduced by English mathematician John Venn in the 1880s. Gardner covers these as well as more complicated forms of logic diagrams. If one takes such diagrams and converts them into a three-dimensional analog—with levers and gears, for instance—then you have a logic machine. The structure of logical propositions can also be translated into electronic circuitry, thus creating an electric logic machine. From there, one naturally progresses to digital logic machines like the precursors to today’s computers. (In 1958 personal computers as we know them today did not exist, but there were giant industrial and military computers like ENIAC.) To solve problems and learn, the “AI” systems that we use today employ the same laws of logic that were explicated by Aristotle, George Boole, and others, just as human brains constantly use these same logical laws without consciously thinking about them.

Perhaps because I’m a graphic designer, the diagrams interested me far more than the machines. I found the first few chapters of this book really fascinating. First, Gardner introduces us to Ramón Lull (or Llull), a 13th century Spanish philosopher who devised a series of complicated diagrams with moving parts in the form of rotating paper wheels. These weren’t so useful for logic, as Gardner points out, but rather they amounted to a sort of Western European I Ching used to inspire philosophical contemplation and intellectual creativity. In Chapter 2, Gardner discusses Venn and other logical diagrammists like him, including the author/mathematician Lewis Carroll of Alice in Wonderland fame. In Chapter 3, Gardner presents his own system of logical diagrams designed to move beyond the computational capabilities of Venn diagrams.

At about the halfway point of the book, Gardner turns his attention to logic machines—mechanical, electronic, and digital. He ends up admitting, however, that none of these machines were as effective as a decent diagramming system, and they amounted to little more than experimental novelties perhaps useful as classroom learning tools. In his final chapter, Gardner speculates about the future of logic machines. Much of what he predicts about computing are things that we now take for granted every day with our desktop computers, tablets, smart phones, and other electronic devices.

For over half a century, Gardner was mathematics’ most charming and prolific ambassador to the general public. The subject matter of Logic Machines and Diagrams isn’t as deliberately fun as Gardner’s Mathematical Games columns, but it will still appeal to the same audience—anyone interested in mathematical “mental floss.”