Mathematical Problems and Proofs: Combinatorics, Number Theory, and Geometry

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MATHEMATICAL PROBLEMS AND PROOFS: COMBINATORICS, NUMBER THEORY, AND GEOMETRY

by Branislav Kisacanin

Plenum Press, 1998

My first reading of the book's title piqued my curiosity about the glue that the author might use to bind together combinatorics, number theory, and geometry into a single volume, and about the book's intended purpose. Although the dust jacket bills the book as a "gentle introduction to the highly sophisticated world of discrete mathematics" (front flap), and "an excellent entree to discrete mathematics for advanced students interested in mathematics, engineering, and science" (back flap), the author does not himself describe his work in these terms, but rather as a work "for those who enjoy seeing mathematical formulas and ideas, interesting problems, and elegant solutions" (p. vii). This seeming disconnect between the way the book's dust jacket markets the work and the author's probable intentions frames some important questions about the purpose of the work and the niche it might fill in a reader's mathematical education.

The book is an interesting collection of excursions into the topical areas of combinatorics, number theory, and geometry, which comprise Chapters 2, 3, and 4, respectively. Chapter 1 introduces set theory terminology and concepts. The book also contains four appendices on mathematical induction, important mathematical constants, and great mathematicians, plus a listing of characters in the Greek alphabet. The discussions of induction and the mathematical constants pi, epsilon, delta, and phi are especially nice, with many interesting details and historic facts.

While each topical excursion slices sufficiently into the mathematical landscape to give the reader an appreciation for the topic, the rationale for the author's choices - which problems and proofs are or aren't included - isn't always immediately apparent. In the chapter on combinatorics, for example, one encounters many of the expected basics for solving enumeration problems (e.g., simple counting techniques with sums, products, and inclusion-exclusion; permutations; combinations; generating functions) and the proofs of a number of results. At the same time, the chapter includes three probability distributions from statistical physics, for example, but devotes relatively little attention to recurrence relations. Only a few well-known problems and results from graph theory, such as the Konigsberg bridge problem and Cayley's Theorem, are included. …