Proofs that Really Count. The Art of Combinatorial Proof.
The goal of the Dolciani Mathematical Expositions series is to foster the ideal of excellence in mathematical exposition by publishing books selected for their lucid expository style and stimulating mathematical content. The books are intended to be sufficiently elementary for the undergraduate and even the mathematically inclined high-school students. The book under review fully fits these intentions. In many situations and from different points of view it demonstrates the power of the combinatorial proof by counting in two different ways. Most of material is really suitable for advanced high-school classes and the book itself could do a great job of attracting bright students to mathematics. More mature mathematicians will also find themselves amused by unexpected twists - "mathemagical" tricks - in surprisingly simple proofs of not so innocent looking formulas. The topics covered in the text include the basic combinatorial number theory such as (generalized) Fibonacci, Lucas, Stirling and harmonic numbers, the inclusion-exclusion principle, binomial coefficients, linear recurrences and continued fractions.