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Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi

Martin Gardner
Cambridge University Press and Mathematical Association of America
Publication Date: 
Number of Pages: 
The New Martin Gardner Mathematical Library 1
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on

This book signals the beginning of The New Martin Gardner Mathematical Library, a series which will, over the next few years, reprint all of the books collecting Gardner's "Mathematical Games" columns. Fifteen volumes are planned; a list is given below. These volumes have had many titles in the course of many reprintings; the new titles are all of the form "X, Y, and Z". The volumes are presented both in paperback and hardcover, and will have a uniform look. For libraries, these famous and important books are an essential acquisition: they can have a complete set, with a uniform look, including updates.

It is hard to exaggerate the importance and influence of these books. Many mathematicians of my generation (translation: old fogeys like me) grew up reading either the columns in Scientific American or the books that collected them. These books played a role in getting us interested in mathematics, and provided solace and entertainment when mathematics turned out to be hard. As a mathematician, I am delighted to have them back in print. (Yes, a complete edition on CD-ROM was — and is — available, but while electronic form is great for searching, it is terrible for reading.) As a bibliophile, I am delighted to have these in book form, especially as books that look so good; the fifteen volumes will eventually have an honored place in my shelf. Can a Folio Society edition be far behind?

What made these columns work so well? Gardner's interest in magic clearly played a role, leading him to many mathematical themes an enabling him to show how they could be "put to work" in concrete ways. Also crucial was a wide network of mathematical collaborators. The network grew over the years as the columns became better known, but from the beginning it is clear that Gardner had good contacts in the mathematical world. He also read widely in mathematical publications: one sees references to the MAA journals, to the Mathematical Gazette, and to many books on recreational mathematics. One of the very nice things about Gardner's columns is that he is careful to give attributions (when he can). As he often points out, this has not been the tradition in the field of mathematical recreations!

In the end, however, what makes it all work is Gardner's writing. He is simple and direct, his explanations are clear, and he always includes the reader, inviting us to build models, play games, try out methods, solve problems. The readers respond, participate, and contribute, often in creative and productive ways.

Re-reading the first collection after so many years is an interesting experience. Most of the columns are still fascinating, and they are presented here with addenda, postscripts, and bibliographies that bring us up to date. In some cases, there is little to add; in others, there is so much that Gardner just points us to the references.

The column on "Hexaflexagons" turns out to have spawned the whole series; the editor of Scientific American read it and asked Gardner if he could do that every month. It turns out that he could, and did, for twenty-five years. Several other columns deal with subjects that have become standard in "recreational" books: probability paradoxes, the Towers of Hanoi, Hex, Polyominoes, Nim, mathematical card tricks. Some of these were, of course, well known long before Gardner wrote about them; others were introduced to the world in Gardner's column.

Readers will, of course, find some columns more interesting than others. I have never managed to become very interested in Polyominoes, for example; maybe it's because their name annoys me. Two of the columns are collections of problems, and these are a mixed bag: some are interesting, some are routine, and at least one I don't understand at all. (It's about poker hands, and I don't know a lot about poker. The solution is just given without explanation, and I can't understand why it is a solution, which proves that I don't understand the problem.) On the other hand, I remember making my own hexaflexagons back in the dark ages, so reading that column was a walk down memory lane. The column on probability paradoxes is particularly well done, and Hempel's Paradox, which I had forgotten, strikes me as something serious and interesting.

These books are fascinating, useful, fun, and historically significant. You must have them! Buy one for yourself, and buy many to give away. Have your students read them. Give a set to your local high school (they'll be published over five years, so it won't even be a financial burden).

Time has passed, and a generation that knoweth not Gardner has arisen. Now we can fix that.

The New Martin Gardner Mathematical Library

  1. Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi: Martin Gardner's First Book of Mathematical Puzzles and Games
  2. Origami, Eleusis, and the Soma Cube: Martin Gardner's Mathematical Diversions
  3. Sphere Packing, Lewis Carroll, and Reversi: Martin Gardner's Mathematical New Mathematical Diversions
  4. Knots and Borromean Rings, Rep-Tiles, and Eight Queens: Martin Gardner's Unexpected Hanging
  5. Klein Bottles, Op-Art, and Sliding-Block Puzzles: More of Martin Gardner's Mathematical Games
  6. Sprouts, Hypercubes, and Superellipses: Martin Gardner's Mathematical Carnival
  7. Nothing and Everything, Polyominoes, and Game Theory: Martin Gardner's Mathematical Magic Show
  8. Random Walks, Hyperspheres, and Palindromes: Martin Gardner's Mathematical Circus
  9. Words, Numbers, and Combinatorics: Martin Gardner on the Trail of Dr. Matrix
  10. Wheels, Life, and Knotted Molecules: Martin Gardner's Mathematical Amusements
  11. Knotted Doughnuts, Napier's Bones, and Gray Codes: Martin Gardner's Mathematical Entertainments
  12. Tangrams, Tilings, and Time Travel: Martin Gardner's Mathematical Bewilderments
  13. Penrose Tiles, Trapdoor Ciphers and the Oulipo: Martin Gardner's Mathematical Tour
  14. Fractal Music, Hypercards, and Chaitin's Omega: Martin Gardner's Mathematical Recreations
  15. The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications: Martin Gardner's Last Mathematical Recreations

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.

1. Hexaflexagons

2. Magic with a matrix

3. Nine problems

4. Ticktacktoe

5. Probability paradoxes

6. The icosian game and the Tower of Hanoi

7. Curious topological models

8. The game of hex

9. Sam Loyd: America’s greatest puzzlist

10. Mathematical card tricks

11. Memorizing numbers

12. Nine more problems

13. Polyominoes

14. Fallacies

15. Nim and tac tix

16. Left or right.