Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi

by Martin Gardner

Martin Gardner's First Book of Mathematical Puzzles and Games

Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi is the inaugural volume in The New Martin Gardner Mathematical Library series. Based off of Gardener's enormously popular Scientific American columns, his puzzles and challenges can now fascinate a whole new generation! Paradoxes and paper-folding, Moebius variations and mnemonics, fallacies, magic square, topological curiosities, parlor tricks, and games ancient and modern, from Polyminoes, Nim, Hex, and the Tower of Hanoi to four-dimensional ticktacktoe. These mathematical recreations, clearly and cleverly presented by Martin Gardner, delight and perplex while demonstrating principles of logic, probability, geometry, and other fields of mathematics. Now the author, in consultation with experts, has added updates to all the chapters, including new game variations, mathematical proofs, and other developments and discoveries.

Genres: Mathematics, Nonfiction, Science, Puzzles, Stem, Reference, Games

208 pages, Paperback

First published January 1, 1959

About the author

Martin Gardner was an American mathematics and science writer specializing in recreational mathematics, but with interests encompassing micromagic, stage magic, literature (especially the writings of Lewis Carroll), philosophy, scientific skepticism, and religion. He wrote the Mathematical Games column in Scientific American from 1956 to 1981, and published over 70 books.

Reviews

[Nandakishore Mridula · Rating 5 out of 5]

This is the first book by Martin Gardner I read - and I have been a fan ever since. Just thinking of this book fills me with nostalgia.

A lazy summer afternoon: the youth section of our city library, housed in an ancient mammoth of a colonial building: the musty smell of old books: the summer vacation stretching in front of me... and the pretty girl who sat across me at the table, at whom I stole glances now and then, but never got up the courage to speak to...

Ah, the halcyon days of youth!...

[Tara · Rating 4 out of 5]

I made the dumb mistake of starting to read this book on the train on my way to work.
Rule number one: Do not open this book unless you have access to paper, pencils, a ruler and a flat surface. This book requires three-dimensional aides.

[Mir]

This author was recommended to me when I was 21, by a high school friend who is now an artificial intelligence expert. I'm probably too dumb by now to understand math, but I'll give it a try.

[Nick Davies · Rating 3 out of 5]

Mildly interesting, my appreciation of this was stilted mainly by the fact that it was short and sixty years old, hence the vast majority of the mathematical puzzles covered I was already very aware of, or they felt a bit basic in terms with the modern maths trivia of this age of powerful computers. It did remind me pleasantly of a book of Jonny Ball's maths puzzles which I had as a child, and was reasonably enjoyable.

[Laura L. Van Dam · Rating 4 out of 5]

En algunas otras reseñas ya conté lo importante y definitoria que fue en mi vida la revista Juegos (la que editaba en los 80s la Editorial DeMente). Gracias a ella no sólo me hice aficionada a los crucigramas si no a los juegos matemáticos, puzzles lógicos y mecánicos, e incluso conocí varios libros y autores que tuvieron un profundo impacto en mi vida. Uno de ellos fue Martin Gardner, cuyo libro The Ambidextrous Universe me decidió a estudiar mi carrera universitaria.

En este libro, Gardner recopila muchos temas que ya tocó en ese libro y en otros, como por ejemplo la simetría y la distinción izquierda-derecha, los acertijos de Sam Loyd, los tetrominos y el juego del Nim (varios de estos también recuerdo haberlos leído en Juegos). Así que no todos los capítulos me parecieron novedosos, pero fueron como un flashback a mi preadolescencia.

Recomiendo leer el libro con papel y lápiz a mano, cartulina, un mazo de cartas, algunas monedas, para poder recrear las actividades propuestas; en algunos casos lo que se explica sólo me pareció tener sentido llevándolo a la práctica.

[Liedzeit Liedzeit · Rating 3 out of 5]

Die klassischen Puzzler. Aber nichts wirklich aufregendes.

[G · Rating 3 out of 5]

Hexaflexagons is the first of Gardner's series of Scientific American compilations, all now available as PDFs on a single DVD. It hearkens back to the halcyon days of nerdery before the Internet, when instead of watching youtube videos and writing python code, math geeks sat around folding strips of paper in certain ways.

The chapters aren't terribly even; some are much more interesting and thought provoking than others. My favorites are the nine puzzles chapters, which are collections of easy but thought provoking brain teasers. Some of the game analysis is fascinating, but Gardner assumes that everyone is deeply familiar with and enjoys chess throughout the book (I'm not and don't), yet goes into detail explaining how one might create different shapes out of four attached squares; surely the assumption that people know the rules of chess but have never seen tetris is a sign that the columns are a bit dated.

[David Miller · Rating 3 out of 5]

An interesting collection of logic puzzles, game strategies, and interesting physical/mathematical curiosities. I say interesting, despite the fact that there are plenty of sections wherein I definitely lost interest, for while I find math interesting in the abstract I am not accustomed to the kind of thinking it requires. I did not force myself to understand it all, nor beat myself up when I failed to tease out the answer to a posed problem, though. I was just along for the ride, and it was pleasantly diverting at that.

[Alison M · Rating 5 out of 5]

This book was given to me at school and I absolutely adored it. It was the start of my interest in actual mathematics - not the humdrum stuff you learn from people who just want you to pass exams, but the subject itself. And it has been a lifelong fascination.

[Brok3n · Rating 5 out of 5]

Twenty-five years of recreational mathematics

In 1967 my Aunt Althea, the very best of all possible aunts, gave me a subscription to Scientific American for my twelfth birthday. I remined a subscriber until the 1990s. Among the best features of SA were the monthly columns "The Amateur Scientist", where you could learn how to build a laser in your garage -- you think I'm joking, but I'm serious -- and Martin Gardner's Recreational Mathematics column "Mathematical Games". Yes, I know that to many of you the phrase "recreational mathematics" makes about as much sense as "recreational colonoscopy", but there are enough people who were willing to entertain the idea that math could be fun to sustain Gardner's column for 26 years. I was one, and Gardner was brilliant.

These columns were collected and published in fifteen books by SA. The best way to get them now is in electronic form. There is a searchable CD-ROM version, and also the American Mathematical Society will sell you PDFs of all 15 as a package deal. The fifteen books, as numbered by AMS, are

1. Hexaflexagons and Other Mathematical Diversions
2. The Second Scientific American Book of Mathematical Puzzles and Diversions
3. New Mathematical Diversions
4. The Unexpected Hanging and Other Mathematical Diversions
5. Sixth Book of Mathematical Games from Scientific American
6. Mathematical Carnival
7. Mathematical Magic Show
8. Mathematical Circus
9. The Magic Numbers of Dr. Matrix
10. Wheels, Life, and Other Mathematical Amusements
11. Knotted Doughnuts and Other Mathematical Entertainments
12. Time Travel and Other Mathematical Bewilderments
13. Penrose Tiles to Trapdoor Ciphers and the Return of Dr. Matrix
14. Fractal Music, Hypercards & More Mathematical Recreations from Scientific American Magazine
15. The Last Recreations

(Most of The Magic Numbers of Dr. Matrix was originally published as number four in the series, which was later reprinted and expanded, explaining why Sixth Book of Mathematical Games from Scientific American is number five in the AMS numbering.)

There's a lot here. A "Greatest Hits" compilation would be nice. The closest thing to that that I have found is this SA blog post, listing ten of Gardner's best articles. Unfortunately, the links take you to paywalled SA pages. Still, if you have bought the books, it's useful.

Gardner sometimes made the silly claim that he was not a professional mathematician. There is no sensible understanding of the term "professional mathematician" under which this claim is true. I was tempted to write "uncharacteristically silly" in the above sentence, but then I remembered that Gardner was often very silly, for instance in his annual April Fools column. This was a man who was always up for a little fun and knew how to find it.

Gardner's work was revered by professional mathematicians, and Mathematical Games was great in part because they talked to him and told him about their latest work. Some important new math was thus first published in Mathematical Games.

A regular feature of the column was puzzles. I remember this one in particular:

EVE/DID = .TALKTALK...

The puzzle is to assign a distinct digit to each letter (E, V, ..., K) such that the equation is true. The fraction is in lowest terms. The unique solution is

242/303 = .79867986...

As a high-school student I had fun solving that. Gardner published a collection of his best puzzles My Best Mathematical and Logic Puzzles (Dover Recreational Math).

The AMS collection is a great deal for anyone who likes math.

Blog review.

[Andrea Nicora · Rating 4 out of 5]

Il libro è indubbiamente datato, soprattutto quando deve trattare dell'approccio "a calcolatore" dei problemi, ma rimane assolutamente interessantissimo. Carino il capitolo 11 sulla memorizzazione dei numeri - vorrei dargli una chance.

[Елена Louis · Rating 5 out of 5]

Любимая книга детства. Играла в игры описанные в книге, мастерила тригексафлексагоны, разбиралась в топологии и получала пятерки по математике, алгебре и геометрии.
Одна из лучших детских книг.

[Isaac · Rating 3 out of 5]

An interesting bunch of problems. Some of them are a little dated, now being more common knowledge in both puzzle and answer. Still very interesting with lots of content in very few pages.

[Fiona · Rating 5 out of 5]

Martin Gardner was a columnist for Scientific American, and notably described himself as a recreational mathematician. When I found this out, I already loved him, it was just a matter of negotiating the degree. I sat down with this book, a pad of paper, some colouring pencils and a cup of tea. Two hours later I was grinning broadly and surrounded by hexaflexagons. Best few quid on Amazon I have spent in ages, and recommended to anyone who thinks maths is pretty but you wouldn't want to do it for a living. This book is accessible, broad, clever and playful. It is full of good anecdotes, card tricks, elegant puzzles, and clear explanations. There is a piece of commentary on knight/liar puzzles that, if it weren't two pages long, I would copy out right here, because it is the most ingenious and off-the-wall solution I have ever come across.

This may come across as a little condescending, but part of me pities people who insist they wouldn't like a book called 'Mathematical Puzzles and Diversions'. They are quite possibly lingering under a misconception about how difficult numbers are or how opaque (or rubbish) the humour is, and either way they are missing out on an awful lot. Get a copy. It's really sad that this book might be considered 'niche'.

[Lyndon Hardy]

When I was a sophomore in high school in 1956, I remember going to the library once and seeing a magazine I had not noticed before— the Scientific American. I thumbed through it, and in the back was a column titled ‘Mathematical Games’ by Martin Gardner.

Unlike the other articles in the issue which were hard to understand fully, Gardner was very lucid. He talked about folding strips of paper into fascinating shapes called ‘Flexagons’. I did not find out to much later that it was his very first column in the magazine and that I had climbed aboard his bus at the very first stop.

What amazing and so well presented topics he covered: Tic Tac Toe, Probability Paradoxes, The Tower of Hanoi, Memorizing Numbers, … For the next twenty five years he kept turning them out, and they were anthologized into thirteen volumes. I have a copy of them all.

Don’t let the word ‘Mathematical’ scare you. Everyone can enjoy what is called ‘recreational’ mathematics—especially the way that Gardner presents their concepts.

His writing instilled in me, among other things, a delight in puzzles—especially ones that can be stated simply but sometimes had solutions of surprising beauty.

[Ste · Rating 5 out of 5]

Martin Gardner's "Mathematical Games" column in Scientific American is considered one of the most important collections of recreational mathematics - computer scientist Donald Knuth called it "the Canon". Reading it 65 years later, the main thing you notice is just how influential it was - every book on recreational mathematics since clearly borrows from Gardner (which unfortunately means you'll already know the solutions to some of the puzzles), but you get a bit more depth here - plus later revisions explaining the latest progress and variations on the subjects. My only complaint is that sometimes - possibly to do with editing to fit a magazine column - the questions aren't always clearly defined. One puzzle in particular - about a face-up variant of poker - baffles me because I can't understand what is meant to be complex about it or why the naive solution wouldn't work. Consulting the answer didn't help either.

[Justin · Rating 5 out of 5]

What an incredible collection of mathematical brain candy. I discovered hexaflexagons from YouTube user ViHart this past school year. I showed the videos to my math classes, and they were hooked. Getting to read the original essay that introduced hexaflexagons to the general public was a joy. There is so much material in this little volume (quite a bit of it genuinely challenging for me, and my degree is in mathematics!) that I'm sure I will return to it again and again.

[Mark · Rating 5 out of 5]

My dad gave me his copy of this book. I think I was in high school. My interest in his work hasn't waned one iota since. He truly is the king of recreational mathematics. And, yes, that really is a thing.

[Ryan Work · Rating 4 out of 5]

I just recently discovered this author and his huge library of books. I plan to read others. they are great distractions although I plan to spread them out. Too many puzzles at a time takes away some of the fun.

[Noah Meyer]

Uptill now I really like the book and its puzzles. I think that it explains the maths behind it quite well and the book has got a quick pace making it a fun read. It also makes every chapter independent from the next which means if you are very interested in only one chapter of it you can read it.

[Lost Planet Airman · Rating 5 out of 5]

Great stuff... guess I'm a geek

[Rachel · Rating 5 out of 5]

Love it. Love Martin Gardner. An excellent adventure in recreational mathematics!

[Jim Razinha · Rating 4 out of 5]

I love these books, though I don't share a Gardner's enthusiasm for topography. Nearly timeless puzzles.