The Liar Paradox and the Towers of Hanoi: The Ten Greatest Math Puzzles of All Time

We need plenty of puzzle books if we are to lure the general public into an interest in mathematics, to wheedle our way round the widespread apathy, suspicion — even fear — that many have for our subject. But with the proliferation of puzzle books, one looks for something different, and here it is! The author has the advantage over some of us in that he is not a mathematician, but a sem(e)ioticist (look that up in your Funk & Wagnalls!)

Don't be fooled by the Table of Contents, which shows ten chestnuts which have all been written about by Rouse Ball, Dudeney, Martin Gardner and a hundred others. This treatment will arouse interest, allay suspicion and banish fear. Each of the ten classics leads the reader to a great variety of puzzles, and gently slips the mathematics past the unsuspecting mathophobe by a succession of annotations, reflections, explorations (including 85 additional puzzles) and further reading.

The Riddle of the Sphinx is an example of riddles, puzzles and problems in general, and leads to deductive reasoning and insight thinking: angles of a polygon, commutativity, exponents.

Alcuin's River-Crossing Puzzle introduces similar puzzles by Tartaglia, Sam Loyd and Dudeney; the Josephus and Kirkman Schoolgirls problems, and combinatorial ideas in general.

Fibonacci's rabbits are paired with the Lucas sequence and used to develop the idea of number: integers, fractions, irrationals (esp. the golden ratio): complex and transfinite numbers are mentioned and we meet the latter again later.

Euler's Königsberg Bridges carry us though networks, graphs, topology, Euler's formula, parity and Sam Loyd's Fifteen Puzzle.

The Four-Color Problem illuminates the nature of proof: deduction, induction, contradiction: with Pythagoras's theorem and the irrationality of the square root of two as examples.

Lucas's Tower of Hanoi serves as guidepost to series, exponential growth, Mersenne and perfect numbers; Cantor and the countability of the rationals.

Sam Loyd's Get-Off-the-Earth Puzzle is an example of an optical illusion, of fallacies, impossible figures, reminding us of the dissected chessboard and a property of the Fibonacci numbers, and we are shown a less well-known example:

 

Epimenides's Liar Paradox is continued with the chicken & egg, with Russell's & Zeno's paradoxes, with Tarski & Gödel, and with Raymond Smullyan's 'knight-or-knave' statement, "You cannot prove that I am a knight." Then on to limits and functions.

The Lo Shu guides us to general magic squares, including Dürer's, Benjamin Franklin's, and Euler's knight's-tour magic square.

Finally, the Cretan Labyrinth lures us into mazes (including a hard one by Lewis Carroll) and an introduction to coordinate geometry, the Pythagoreans, triangular and square numbers, shortest paths and Dudeney's spider-&-fly problem.

The text is interspersed with minibiographies of Alcuin, Tartaglia, Pythagoras, Fibonacci, Euler, Euclid, Lucas, Cantor, Loyd, Zeno & Dürer and enlivened with quotes from Mark Twain, Jung, Aldous Huxley, Kierkegaard, Ayer, Baudelaire, Socrates, Oscar Wilde, Koestler & José Ortega y Gasset. There is a useful Glossary as well as an Index.

But a fellow has to have his quibbles: 'cube' ≠ 'cuboid' (p.234); 'digit' ≠ 'number' (p.215); Euler didn't live for 149 years (p.59); 1 is both a prime and a nonprime on p.174. Binet didn't 'elaborate' a formula — it's in Euler's work and elsewhere at least 100 years earlier; 'proven' is the pp. of an obsolete verb preve ('test' as in 'a well-proven remedy', 'Not Proven' in Scottish law, and '140-proof rum') and is inappropriate for mathematical proof. And there is loose use of 'series' where it would have been useful to compare and contrast with 'sequence'.

But get copies for your less enlightened friends.

Dick Fellow is an itinerant mathematician and free-lance writer. After leaving Cambridge University he failed to get a PhD from the University of London around the time of WW2. Since then he has wandered the world in search of permanent employment. He has written several books of his own, even better than the one under review. His motto is "If E. T. Bell could do it, so can I."