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Exploring, Investigating and Discovering in Mathematics

Vasile Berinde
Publication Date: 
Number of Pages: 
[Reviewed by
Bonnie Shulman
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While it may verge on the (oxy)moronic to talk of an algorithm for creativity — a process often distinguished by its spontaneity and unpredictability — it is possible to abstract certain habits of mind that do capture essential stages of the creative approach. In the spirit of George Polya's classic treatise How to Solve It: A New Aspect of Mathematical Method (Princeton University Press, latest printing in 2004), this book is Vasile Berinde's attempt to "algorithmetize" the creative process involved in problem-solving. The steps Berinde develops into a methodology are inherent in the title of his book: exploring, investigating and discovering.

The inspiration to write such a book arose naturally out of Berinde's activities as a problem proposer for mathematics journals and contests in Romania. The book was translated into English by his two daughters in 2003, two years after the Romanian edition appeared.

The author's stated goals are twofold. The most obvious aim is to provide methods for solving some difficult problems. But on a deeper level, he is trying to grow new research mathematicians by planting the seeds that will develop into the skills they will need to do original research. "In this respect, it [the book] can fill a serious gap in mathematics education: after many years of predominantly non-creative problem solving activity, graduate students are suddenly expected to do their first research work — a task for which they have not been properly prepared." (p. viii) To this end, he offers "a concrete and as exact as possible description of the act of discovery in mathematics." (p. xv)

Each of the 24 chapters begins with a source or starting problem and its solution. Then the fun begins! Remarks about the "essence" of the problem and its solution suggest new directions to explore, and these investigations lead to generalizations and the formulation of related problems, sometimes building a "factory" of new problems. Bibliographical comments end each chapter; unfortunately for most of us, the references are almost all in Romanian. In an Addendum, the author summarizes the guiding principles used in each of the preceding chapters. Some of my favorite themes among the 24 were: Chase Problems; Sequences of Integers Simultaneously Prime; Solving a Class of Nonlinear Systems (where the variables x and y appear both as exponents and as arguments of logarithmic functions in systems of two simultaneous equations); An Extension of the Leibniz-Newton Formula; A Measurement Problem; On the Convergence of Some Sequences of Real Numbers; and Difference and Differential Equations.

Many of the source problems were originally proposed in Gazeta Matematica (in fact Berinde originally hoped to publish the book in 1995, the centennial celebration of this journal). Problem 10 is from a university entrance exam in the former Soviet Union. Problem 22 (and others) are taken from the Putnam Mathematics Competition in the U.S.A. At one point the author cannot resist the opportunity to make a point about mathematics education in his country, a point which has relevance for the U.S.A. as well: "Digging around these problems — and similar ones — shows how important mathematics contests are in keeping a continuous emulation and a remarkable development process in creating mathematical problems in our country. Great efforts must be made in order to preserve these traditions, in spite of the financial restrictions which have affected the Romanian education system in the last 10-11 years." (p. 45)

The above quotation is also an example of the awkward phrasing that sometimes appears in translations. For the most part, the translation is excellent. However, I did find one place where a mathematical editor might have caught what a non-mathematician probably missed. On page 219 (Theme 23, An Application of the Integral Mean), an "increasing" function should actually have been a "non-decreasing" function, as the subsequent argument makes clear. There are also some typos, again for the most part non-intrusive, but several occurrences of the sense of inequalities being improperly reversed (see, for example, p. 47), while not serious obstacles to the professional mathematician, could present difficulties to an undergraduate student.

In one of my favorite sections, on Measurement Problems, a typo gets propagated into a mathematical error. A general algorithm is developed for moving water between three pots so that a specified amount ends up in one pot. Specifically: A pot of m + n liters is full. Remove 2(m - n) liters from an m liter pot, using an n liter pot (m, n positive integers, n < m ≤ 2n). (Problem 17b, p. 158). In the remarks that follow, it is noted that by taking particular values for n and m, the previous problems 17 and 17a are obtained. But the values for m and n are reversed, and this carries over into Problem 17c (From a pot containing 9 liters of water, remove 4 liters into a 5 liter pot, using a 7 liter pot), where the solution is given following the general algorithm, with the roles of m and n reversed. It takes 9 steps. But I found a much simpler solution taking only 4 steps. Can you find it? (Answer at end of review.)

However, this example really only reinforces the success of this book in involving the reader in the creative process of problem solving. From the first page I eagerly grabbed my pencil and a stack of scratch paper, and set to work as I happily read along, staying up much later than I should have, and neglecting a stack of papers I should have been grading! I recommend the book for all lovers of mathematics, but especially students and teachers who participate in mathematics contests and practice problem solving. Aside from the satisfaction of solving hard puzzles, there may be an even deeper fulfillment to be gained from this practice. Vasile Berinde tells us that he "wrote this book hoping that it would express one of [his] fundamental beliefs regarding the human spirit." He goes on:

I strongly believe in the existence of some latent creative resources in each of us — sparks of our creator and reflections of the divine act of creation. I would even dare to say that these latent creative resources pre-exist in the human being. An adequate methodological framework, and especially the tools specific to creation, are necessary to awaken them. No matter if he is an artist, a poet, a mathematician, an engineer, a physicist, a chemist, a tailor, a cook or a peasant, the one who seeks the way to creation, learns how to use these tools and finds the secret of creation will attain spiritual self-fulfillment. (p. xviii)

My Solution to Measurement Problem:

Step     Pot 1 (9 liters)     Pot 2 (7 liters)   Pot 3 (5 liters)   
  I            9                    0                  0
 II            4                    0                  5
 III           4                    5                  0
 IV            0                    5                  4

Bonnie Shulman is Associate Professor of Mathematics at Bates College in Lewiston, ME. She has always enjoyed solving puzzles of all kinds, loves reading mysteries, and always covered up the answers to worked out examples in her textbooks when she was a student, preferring to try and solve it on her own. She tries to inculcate this spirit of discovery and investigation in her students, likening a hard mathematics problem to a mountain, with a breathtaking view at the top.

Catching Problems.- Sequences of Numbers Simultaneously Prime.- The First Decimal of Some Sequences of Irrational Numbers .- Determinants with Alternative Entries.- Solving Some Cyclic Systems with the Fixed Point Theorem.- On a Property of Recurrent Affine Sequences.- An Extension to the Leibniz-Newton Formula.- How to Discover New Problems Using the Computer.- An Application of the Integral Mean.- Difference and Differential Equations.