In Pólya's Footsteps

If the first seven numbers in a sequence are 1, 2, 3, 4, 9, 10, and 17, then what is the next number? Answer: 19. The book under review is volume 19 of the twenty books published so far in the MAA's Dolciani Mathematical Expositions series, and it is Ross Honsberger's eighth contribution to that series. This volume contains well over a hundred problems, the majority of them from the Olympiad Corner columns of Crux Mathematicorum. Lest it be thought that this is an easy way to put a book together, note that Honsberger has taken the trouble, in most cases, of constructing his own solutions to the problems. Although the source journal is Canadian, the problems themselves have a strong international flavor. Within the first 10 pages alone, we find entries originating from the Asian Pacific, Bulgarian, Ibero-American, International, Spanish, and U. S. A. Olympiads.

Honsberger comments in the Preface that the problems "are offered solely for [the reader's] enjoyment," and this is indeed a book meant for browsing. The problems are not organized by subject, and may be read and enjoyed in any order. There are geometry problems, number theory problems, combinatoric and coloring problems, maximum/minimum and inequality problems, and so on, each followed by a solution that in many cases discusses how the solution was arrived at, or ruminates on problem-solving in general. Here is a sampler of problems that will give some sense of the material to be found in the book:

• Page 5: There are arbitrarily long strings of consecutive positive integers, none of which is a prime power.

• Page 23: Suppose twelve identical disks (each tangent to its two neighbors) cover the unit circle, and express their total area in the form , where sqrt(c) is irrational. Find a + b + c.

• Page 42: If each point of the plane is colored either red or blue, then some equilateral triangle has all vertices the same color.

• Page 48: Determine the number of towers of pennies having n pennies in the bottom row.

  (A tower of pennies consists of rows of pennies on top of other rows, with no gaps or overhang. The problem was proposed by Richard Guy, and its solution - which involves Fibonacci numbers - is due to University of Waterloo student Colin Springer.)

• Page 127: Start with any positive integer, and form a sequence in which succeeding terms are obtained by adding to a term the product of its digits. For example:
  127, 141, 145, 165, 195, 240, 240 . . . or perhaps 1997, 2564, 2804, 2804, . . . .

  Clearly, whenever a 0 appears, the sequence becomes constant. Does this always happen?

• Page 140: On an island where chameleons can adopt any of three colors, it happens that any two different-colored chameleons change to the third available color when they meet. Is it possible for all chameleons on the island to be the same color at some point?

  Editor's Note: this one's reminiscent of the problems discussed in Alex Bogomolny's Tribute to Invariance. Check it out!

• Page 145: Color the integers from 1 through 1987 with four colors so that no arithmetic progression with 10 terms is monochromatic. (This was an unused problem from the 1987 International Olympiad, and like others in the book manages to slip the date into the problem.)

• Page 157: A plane arc of length 1, lying entirely on one side of the line through its endpoints, can be covered by an isosceles right triangle of hypotenuse 1.

A change of pace occurs on pages 251 - 266, where Honsberger presents some identities (due to Liouville) involving the arithmetic functions sigma, tau, phi, etc. Then, after a few more Olympiad-style problems, the book concludes with 25 exercises that are not as difficult as those encountered earlier. These are stated as a group, followed by the solutions similarly grouped, making it somewhat easier to resist the temptation to proceed directly to a solution.

As with many of Pólya's works, Honsberger's latest entertaining volume ranges widely over several fields, but consists of problems that are reasonably accessible to a capable undergraduate mathematics student (or even, in many cases, a high school student). Honsberger shares with Pólya a concern for general methods of discovery and problem-solving, methods that go beyond the particular problem at hand. He is also unabashedly interested in clever problems as objects of pleasure and entertainment. Many colleges offer courses in Music Appreciation, and perhaps this book (together with his previous Morsels, Gems, and Plums) could help point the way to more emphasis on the concept of Mathematics Appreciation.

David Graves (dgraves@elmira.edu) is Associate Professor of Mathematics at Elmira College, where he is active as a pianist, and has taught courses in opera and history of astronomy as