Conversational Problem Solving

Although this book is based on material accumulated while teaching the M.I.T. freshman problem solving seminar focused on the Putnam competition, it is not the usual collection of topics and tricks that constitute many “competition math” books. This book consists of a series of dialogues between a teacher (“Professor Mortimer Ignatius Blakely”) and the participants of “Problem Solving Camp”—eight fictitious (and very talented) students from the US and abroad.  The professor in this situation serves mainly as the guide on the side, rather than as the sage on the stage. The conversations are divided into sixteen sessions with titles such as Polynomials, Two Triangles, and Self-referential Mathematics, with a total of over 200 problems discussed.

 

At the beginning of each chapter there is a List of Problems that highlights the mathematical content of the session. The topics range widely over classical geometry, set theory, algebra, analysis, combinatorics, number theory, and probability. The level of these “conversations” is high, but the material is leavened by a sprinkling of anecdotes and mathematical dad jokes. Each chapter is supported by notes which provide historical background and suggestions for further reading. The book ends with a Bibliography of 193 items, including references to online material.

 

As an example of the content and style of this book, I give a brief outline of Chapter 4 (A Mysterious Visitor). The professor starts by posing a warmup problem, the “elementary inequality”

 

\( \sum_{d|n} d \leq H_{n} + (\log H_{n}) e^{H_{n}} \)

 

where H n denotes the nth harmonic number. After initial unsuccessful attempts, the professor reveals that this inequality is equivalent to the Riemann Hypothesis and adds “…if anyone can solve this problem before the next class, I will exempt them from the next problem set.” Blakely then introduces \( \mathfrak{G}_{n} \), the symmetric group of all permutations of \( 1 \), \( 2\) , \( \ldots \), \( n\), digressing briefly to recommend the Fraktur alphabet as a way to confound colleagues. By using generating functions, a student finds \( f_{n} \) , the expected number of cycles in the disjoint cycle decomposition of a random \( w \in \mathfrak{G}_{n}: f_{n} =H_{n} \). The signless Stirling numbers of the first kind arise in this discussion, and the students are challenged to come up with a conceptual way of seeing the answer, eschewing induction, generating functions, and other such tools. However, this

discussion bogs down and further work on this challenge is postponed to Chapter 7.

 

The dialogue in this chapter continues with a discussion of the expected number of cycles of length \( k \), and the expected length \( L(n) \) of the longest cycle in a random permutation. There is a brief anecdote about the combinatorialist William Tutte, and then the professor reveals (after some consideration by the students) that the limit of \( L(n) n \) exists and is given by 

 

\( \int_{0}^{\infty} \mathop{exp} \left( -x - \int_{x}^{\infty} \frac{e^{-y}}{y} dy \right) dx \)

 

(The notes for this chapter cite a 1966 paper of Shepp and Lloyd.) The session ends with the entry of a “stately male cat” and ensuing silliness as the class tests the cat’s mathematical knowledge, only to have the animal run out of the room when asked about The Twin Prime Conjecture.

 

In the Preface, the author states that “the primary purpose of the problems in this book is not didactic, but rather to entertain.” I recommend this collection of mathematical colloquies highly, both for its mathematical content and for the interesting way in which it is presented. I believe the book will be both challenging and stimulating to its intended audience of “mathematicians at all levels…undergraduates…and even high school students, who are adept at solving challenging problems.” If, as Paul Halmos believed, the heart of mathematics consists of problems and problem solving, then Stanley has shown himself to be a skilled cardiologist.

---

Henry Ricardo (odedude@yahoo.com) retired from Medgar Evers College (CUNY) as Professor of Mathematics and is currently affiliated with the Westchester Area Math Circle. He is the author of A Modern Introduction to Differential Equations (Third Edition) and A Modern Introduction to Linear Algebra. He has a special interest in mathematical problem solving.