Mathematical Puzzle Tales

Some of us purchase Martin Gardner's books without even looking at their titles because the author's name is a full warranty of a good investment of time and money. The warranty still works for Mathematical Puzzle Tales. In my case, the fact that I had not read details about the book was a positive coincidence. I am not a science fiction buff and my knowledge of the field is very limited. When I found out that the book was a collection of thirty-six of Martin's stories from Isaac Asimov's Science Fiction Magazine, it was too late. I had the book, and a book by such an entertaining author is always worth a try.

One might wonder if the mathematical bookshelves needed yet another puzzle collection by Martin Gardner. Maybe "needed" is too strong of a word in this case, but the book is a nice addition to what has already been published. Moreover, after working on the book (this is not a book that a mathematician just reads; pencil and paper are very helpful), my conclusion is that it targets readers who enjoy either mathematics or science fiction, with the "or" used in its inclusive meaning. Indeed, as Isaac Asimov writes in the Foreword, dated January 1981, "The puzzles that follow are woven into short fiction stories. They add amusement but are not the essence of the book. The science fiction is important, though, because it demonstrates that however times, customs and technologies alter, the essence of mathematical relationships is, was, and will be the same."

So, what is in the thirty-six stories collected in Mathematical Puzzle Tales? Some of the titles offer clues about the content (e.g.: Tanya Tackles Topology, The Toroids of Dr. Klonefake, OULIPO Wordplay, How Crock and Witson Cracked a Code, Weird Numbers from Titan, The Postage Stamps of Philo Tate, How Bagson Bagged a Board Game, The Vacancy at the Aleph-Null Inn, Tube through Earth), but some are quite mysterious (e.g.: The Doctors' Dilemma, The Shop on Bedford Street, Exploring Carter's Crater, The Explosion of Blabbage's Oracle, The Erasing of Philbert the Fudger, The Backward Banana, The Queer Story of Gardner's Magazine).

The stories include questions in geometry, topology, number theory, combinatorics, statistics, and physics. Most problems can be solved with little of what is considered "mathematics" by non-mathematicians. Logic and trial-and-error can be sufficient. It is a little difficult to talk about the problems in detail without giving away clues, but I will try with a few of them. The Doctors' Dilemma deals with doctors who are trying to avoid the spread of a contagious disease transmitted by contact. The answer to the first question can be more easily found using ideas from topology, but it is easily accessible to readers who know nothing about this subject. Later on, as the questions and the answers keep piling up, the problem reaches a nice level of generality that is better appreciated with some knowledge of mathematics. (As a side note, in its second iteration, this problem suggests that mathematicians might occasionally think about sex!) The story Machismo on Byronia is delightfully "politically incorrect." Again, obtaining the first answer requires almost no algebra. By the time the third answer is reached, there is a very interesting manipulation of a convergent series.

Some other puzzles will be much better appreciated by readers having some mathematical knowledge. For example, Exploring Carter's Crater and Lucifer at Las Vegas make use of the concept of probability and the fact that the probability of an event might change when additional information is given. At the other end of the spectrum we find some problems that do not involve mathematics, but do require some knowledge of science fiction. They include the wicked Captain Tittlebaum's Test, Professor Cracker's Antitelephone, The Backward Banana, OULIPO Word Play, and The Robots of Oz. Just remember that in any case, there is a good chance that the answer to any science fiction question is "Isaac Asimov."

Some of the short stories can be easily used for class discussion, even in classes with students with a weak mathematical background (or students who are not very interested in mathematics). My short list would include: The Third Dr. Moreau (its solution requires only the ability to multiply by 7 repeatedly), The Great Ring of Neptune (the Pythagorean Theorem is sufficient, and it includes a nice trick for remembering pi), The Explosion of Blabbage's Oracle (with a nice historical note), The Erasing of Philbert the Fudger, How Crock and Witson Cracked a Code, and Blabbage's Decision Paradox.

One nice feature of the book is the existence of repeated "extensions" of the problem presented, which are very well organized. The answers to the riddles proposed can be found in the section appropriately labeled "First Answers," and readers should read them even after solving the problem independently. Indeed the "First Answers" often contain new questions, which lead to the "Second Answers." Some of these have new questions, answered in the "Third Answers." The final answer to a puzzle sometimes includes postscripts about the history of the puzzle, some colorful related anecdote, comments, and references to relevant literature. The bibliography has a nice mix of older and newer references, including material published as recently as 1999.

Another pleasant feature of the book is the subtle humor that surfaces here and there. Martin brilliantly paraphrases the dialogues between Sherlock Holmes and Watson, disguised as Shurl and Watts in the story of The Defective Doyles, and even makes fun of himself in Pink, Blue, and Green.

I am sure that I missed a few science fiction insiders' jokes here and there, but Gardner must know about people like me, because he is very generous with explanations. Probably science fiction aficionados who do not have a strong mathematical background feel the same way about the mathematics in the book. Indeed Gardner's explanations of the mathematics/logic involved in the short stories are extremely clear. So much so that I am planning to use some of the stories from this book in one of the courses for liberal art majors I teach. They are bound to appreciate Martin's writing style and his ability to spin interesting stories from mathematical ideas. Imagine that!

Antonella Cupillari (axc5@psu.edu) is an associate professor of mathematics at Penn State Erie, The Behrend College. Her main research interests include history of mathematics and mathematics education. She is the author of the book The Nuts and Bolts of Proofs.