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The Inquisitive Problem Solver

Paul Vaderlind, Richard Guy and Loren Larson
Mathematical Assciation of America
Publication Date: 
Number of Pages: 
Problem Books
[Reviewed by
Dick Fellow
, on

Here's a problem book with a difference. Problem solvers and mathematicians are of two kinds: those who solve problems and those who ask questions. Erdős used to call Szemerédi "the Mad Prover", but one day he said, "a miracle has occurred; Szemerédi has made a conjecture." As a well-known mathematician has said "Mathematics often owes more to those who ask questions than to those who answer them.

The authors' aim is to encourage the reader to ask questions. The book starts in the traditional way, with Problems, Hints and Solutions. But among the Solutions are several Queries: generalizations and extensions that The Inquisitive Problem Solver might be expected to make. Then follow Responses, more complete solutions, many tantalizing tidbits which are left to Rikki-Tikki-Tavi (read Kipling's The Jungle Book if you don't know the reference), and frequent references to the Treasury.

The aptly-named Treasury is a valuable collection of brief essays on recurring mathematical themes: e.g., error-correcting codes, the Fano configuration (due to Woolhouse), graphs, modular arithmetic, nim-addition, parity, partitions, Pascal's triangle (due to Omar Khayyam), round robin tournaments, strategy stealing, triangular numbers, vectors, and 101 other headings. It serves as Index, Glossary, and as a list of Terms, Techniques and Tricks of the Trade.

When Elwyn Berlekamp read the publisher's blurb he learned that one of Ron Graham's favorite problems was packing tennis balls in a rectangular box. The authors show how to pack 477 balls in a 2 x 238 box, but Berlekamp promptly improved this to 333 balls in a 2 x 166 box — just the kind of reaction that the authors hope to stimulate.

Apart from Disquisitiones Arithmeticae, all mathematics books contain mistakes and TIPS is no exception. In Problem 127, Barry Cipra (The Sol LeWitt Puzzle, in Puzzlers' Tribute: A Feast for the Mind, Tom Rodgers, David Wolfe, editors, AKPeters, Natick MA, 2001) asks if the 16 "tiles" of Sol LeWitt's etching can be rearranged, without rotation, so that the segments form continuous lines across the square.

In the Treasury, under Configurations, we find the limerick:

Those squares of LeWitt, with the lines,
Will make a great many designs.
    But can you bestow,
    In each column and row,
Two each of four kinds of inclines?

Under Magic squares we find Cipra's observation that assigning binary digits 1 or 0 to indicate presence or absence of vertical, horizontal, up diagonal, down diagonal segments gives each tile a different number from 0 to 15. A solution to Loren's limerick gives a semi-magic square with row and column sums, but not necessarily diagonals, each equal to 30.

On p.306 the authors rashly state that the converse is true, but Cipra hastens to provide numerous counterexamples.

15 5 3 7 15 5 9 1 15 5 10 0
10 14 6 0 10 14 0 6 3 14 6 7
4 2 13 11 2 4 13 11 4 2 13 11
1 9 8 12 3 7 8 12 8 9 1 12

Is there a magic square counterexample? Even a pandiagonal one? TIPS may have spawned even more problems than it intended.

There's a mystery surrounding the final, 13-coin, problem. Many will recall the notorious 12-coin problem, suggested for dropping over enemy territory during WW2, and Freeman Dyson's elegant ternary label solution. The authors collected the 13-coin problem from Cedric Smith, not long before he died. It is attributed to Herbert Wright around 1960, but if you look in Math. Gaz., 31(1947), p.39, at the end of an article by Cedric, you will find the same problem! C. A. B. Smith was a most modest and generous man: the best interpretation I can give is that he received a solution from Herbert Wright and wanted to give him some acknowledgement. The solution in the book (Wright's is said to have been mislaid) is pure Cedric Smith, a delight that deserves to be as widely known as the undoubted success of the book will make it.

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, but few are expected to achieve the success of the volume under review. His motto is "If E. T. Bell could do it, so can I."

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