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Mathematical Explorations

Alan F. Beardon
Cambridge University Press
Publication Date: 
Number of Pages: 
AIMS Library Series
Problem Book
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on

One way in which the education of undergraduate mathematics majors has changed from the way it was when I was an undergraduate, in the late 60s and early 70s, is that there is now more of an effort to get undergraduates interested in, and actually doing, research. A number of events and organizations both reflect and encourage this trend: some universities offer REU (Research Experience for Undergraduates) summer programs; the mathematics institutes AIM and ICERM administer a REUF (Research Experiences for Undergraduate Faculty) program “for undergraduate faculty who are interested in mentoring undergraduate research”; Brigham Young University hosts the Center for Undergraduate Research in Mathematics; and there are several journals devoted to undergraduate research.

The book under review is intended to help smooth the transition from undergraduate classroom study to research. It is a sequel to the author’s previous (unseen by me) book Creative Mathematics, which, according to the author’s description, constitutes an introduction to solving problems and writing proofs. Explorations is a more sophisticated book than the earlier one and assumes a greater degree of mathematical maturity. The author describes the purpose and format of the new book as follows:

This book is written to help the reader learn how to do research mathematics. Each chapter contains a project that has been chosen not because of its mathematical importance but because (in the view of the author) it provides a good illustration of how arguments develop, and how new questions arise once one some progress is made. These projects have also been chosen because they do not require a deep mathematical background in order to understand the problem and start investigation. Nevertheless, the reader will probably have to learn some more mathematics in order to solve the problems. Some of the problems do not have easy answers, and some are not yet completely solved.

One important point, implicit in the last sentence above, should be noted at the outset. This book is intended to try and teach what mathematical research is like, but it is not intended as a book of projects whose solutions will lead to a published paper. The problems in this book teach important lessons about research — the importance of precise definitions and statements of problems, how one problem can lead to others, how areas of mathematics that are not obviously related to a problem can find unexpected use, etc. — but the problems themselves are not generally unsolved ones. The author points this out in the preface, where he states that a major goal of the book is to “give the reader experience in working on (as far as the reader is concerned) unsolved problems.” (The emphasis is mine.)

This italicized caveat is a potential problem in trying to teach research out of this textbook. After all, when a student tackles a standard problem in, say, an abstract algebra textbook, he or she is also looking at a problem that is, “as far as the reader is concerned”, unsolved. A book that strives to prepare a student to do mathematical research should be more than just a book of problems. I’ll return to this theme shortly, but first let’s look at the structure of the book in more detail.

There are 14 chapters, focusing on topics in discrete mathematics (number theory, combinatorics, geometry, etc.), but with an occasional glimpse of analysis (such as the gamma function in the last chapter). Some of these chapters cover material that is fairly closely related to topics that the student might have seen in undergraduate courses, others go a bit beyond this. While considerations of space make it impossible to survey all the chapters here (for a list, click on the “table of contents” link above), we can discuss a few as representative samples.

Chapter 10, for example, is a good illustration of a topic that is related very strongly to undergraduate coursework. Titled “Primes and Irreducible Elements”, it is concerned with fairly standard topics in factorization theory — primes and irreducibles, unique factorization, Gaussian integers, and so on. Some of the problems posed here (e.g., prove that the set of units of a commutative ring with identity constitutes a group with respect to the operation of ring multiplication) are just trivial results that typically appear as exercises in such a course. Problems like this will hardly prepare a student for research. In the last few pages, however, the chapter does go on to address other issues, such as irreducible matrices, that are not typically taught in such a course. There is also a section called “irreducible polynomials” that refers to irreducibility under function composition rather than polynomial multiplication; this should be a new concept for many undergraduates.

Chapter 11, on symmetries of a quadrilateral, begins with a discussion of planar quadrilaterals and their symmetry groups and does not, it seems to me, go much beyond what a student would likely have encountered in a good first semester of abstract algebra. The chapter ends with a discussion of quadrilaterals in three-space and here may go beyond a typical course, but on the whole the questions in this chapter of the text seemed to me to be quite similar to exercises in a textbook rather than questions pointing towards research.

Other chapters, however, do address material that a typical undergraduate will not generally see as part of a standard curriculum. For example, in chapter 8 (“Sums of Powers of Digits”) the author begins by defining, for a positive integer \(n\), the number \(f(n)\) to be the sum of the squares of the digits of \(n\). (So, for example, \(f(17) = 50\).) Questions are then posed concerning the sequence obtained by repeatedly applying the function \(f\). These kinds of questions can lead to undergraduate research topics. Numbers, such as \(19\) and \(230\), for which this sequence eventually reach \(1\) are called happy numbers. Although this term is not used in the chapter, these numbers have been studied at the undergraduate research level: in fact, I first heard the term “happy number” in a REUF talk by Helen Grundman at the American Institute of Mathematics.

The various project chapters contain not just problems but also some mathematical development and proofs of theorems. However, the exposition is reasonably succinct and it is an implied exercise for each chapter that the reader fill in the details of the arguments.

None of the projects come with solutions, which is as it should be: no student can seriously learn how to do mathematical research if all he or she has to do is flip to the back of the book for answers. What might have been helpful (for faculty members teaching this material) is a short, password-protected, manual linked from the publisher’s webpage, giving a survey of results and the relevant literature, and perhaps suggestions for additional research, perhaps even to unsolved but accessible questions in the area, if there are any.

In fact, the lack of discussion of the relevant literature is, I think, my biggest criticism of the book. Only 12 references are given; I would have liked to have seen at least twice that many. Many of the topics discussed in the book must surely be the subject of relatively accessible textbooks or journal articles, and, for obvious reasons, it would be helpful to have a list of these. Learning how to read journal articles is an important part of learning how to do research. Of course, learning how to find them is also an important part; the text also does not provide any discussion of how to find existing literature.

There is another problem with having students use this text as a way of learning how to do research, one that was alluded to earlier. What we have here, essentially, is a problem book, and I harbor serious doubts as to whether a problem book can teach a student how to do research. One difficulty that some students have in transitioning to research is coming to grips with its more “open-ended” aspects, and to the extent that this book resembles a collection of textbook exercises, it may not assist with that difficulty.

These concerns aside, I think this book is an interesting and useful one. Instructors teaching a course doing undergraduate research, or mentoring students doing so, might get some useful ideas from this book, even if they don’t assign it to students. Another possible use that the author does not explicitly mention is as a text for a senior seminar or capstone course for mathematics majors that does not focus on undergraduate research. The broad array of mathematics that is covered in this text might provide a thought-provoking summary of topics and give students an opportunity to expand their horizons.

Mark Hunacek ( teaches mathematics at Iowa State University.

How to use this book
1. Paying for parking
2. Lengths and angles
3. Magic squares
4. Intersecting chords
5. Crossing squares
6. Repeated vector products
7. A rolling disc
8. Sums of powers of digits
9. The metric dimension
10. Primes and irreducible elements
11. The symmetries of a quadrilateral
12. Removing a vertex
13. Squares within squares
14. Catalan numbers