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A Gardner's Workout: Training the Mind and Entertaining the Spirit

Martin Gardner
A K Peters
Publication Date: 
Number of Pages: 
[Reviewed by
Mihaela Poplicher
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This book is a collection of some of Martin Gardner's best articles and book reviews from the past two decades, containing pieces previously unpublished in book form.

Martin Gardner is known as the "Master of Mathematical Games and Puzzles". He has enlightened, educated and delighted readers around the world during many decades as a prolific contributor to many publications such as The Washington Post, Games magazine; and especially Scientific American, in which his column "Mathematical Games" ran for 25 years. A Gardner's Workout: Training the Mind and Entertaining the Spirit will only add to the author's fans, mathematicians as well as the general public, and will be a treat for all of them.

Martin Gardner dedicates his book

To all the underpaid teachers of mathematics, everywhere, who love their subject and are able to communicate that love to their students. (p. V)

The book is divided in two parts: the first contains thirty-four articles originally published in academic journals and/or popular magazines and the second contains seven book reviews. The most controversial piece (considered as such by Martin Gardner himself (p. XI)) is the final one, in which the author criticizes the pedagogy sometimes called "new new Math." I have heard many mathematicians joining Gardner in hoping "that new new math is being abandoned almost as rapidly as the old new math faded." (p. XI) But not all will agree.

The other book reviews are about science or mathematics books. The one I enjoyed the most is about Most-Perfect Pandiagonal Magic Squares: Their Construction and Enumeration, by Dame Kathleen Ollenrenshaw, "one of England's national treasures." (p. 285) Anybody who reads this book review will probably want to get a copy of the book and read all the details of the proof of a long-standing and difficult problem involving the construction and enumeration of a certain type of magic squares.

Speaking about magic squares: this definitely is a topic much loved by Martin Gardner, and a proof of that are a few of the articles included in his book. All of them seem to have been written with the following words in mind:

Beauty is the first test: There is no permanent place in the world for ugly mathematics.
— G. H. Hardy, A Mathematician's Apology.

The other articles range over a wide range of topics, from The Square Root of 2 = 1.414 213 562 373 095..., which refers to the hundreds of books containing proofs of the irrationality of the square root of two, to Kasparov's Defeat by Deep Blue, which ends with the conclusion: "It is a long, long distance from the circuitry of Deep Blue to the mind of a mouse."

Other articles refer to directed graphs, Steiner trees, tiling, many "computer tricks" (p. 87), chess problems, "lucky numbers" (p. 149), primes in arithmetic progression. There are articles on prime magic squares, the game of domino, as well as other recreational mathematics; on "serial isogons of 90 degrees", modeling mathematics with playing cards, as well as more mathematical tricks...

I particularly enjoyed the article on Leon Bankoff and the "asymmetric propeller theorem" (p. 249) and the one on "maximum inscribed squares, rectangles, and triangles" (p.203), just because elementary geometry is so beautiful!

One article, Toroidal Currency, taught me something that I didn't know at all: the Reserve Bank of Australia issued in 1992 and 1993 two bank notes that are toroidal, with the goal of making it harder to counterfeit the bills. The article has the pictures to prove the point.

On a completely different topic, Computers Near the Threshold (p.97) is another article I liked very much, probably because I am among the ones Gardner calls "mysterians":

We do not deny that the mind emerges from a complex pattern of molecules, but we believe that the brain's complexity is so vast that at present we simply do not understand how it produces self-awareness and free will. We also believe that computers, operating with wires and switches that shift electrical currents around a network in obedience to software algorithms, will never cross the threshold at which anything resembling the human brain will emerge. (p.105)

Two other articles of particular interest to those who love Alice in Wonderland are Lewis Carroll's Pillow-Problems (p. 129) and Lewis Carroll's Word Ladders (p. 133); they remind us that Lewis Carroll (a pen name of Charles Dodgson) was a mathematician who loved mathematical games.

The book can be used as a resource for high school teachers and students (especially for a math club, or for a starting point for the discussion of interesting problems); undergraduate and graduate students of mathematics will find it to be very interesting reading; the general public will find many intriguing and challenging pieces in this gem of a book. In short: I strongly recommend Martin Gardner's book to everybody who has some interest in beautiful mathematics, science, or computers.

Mihaela Poplicher is assistant professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is

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