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112 Combinatorial Problems from the AwesomeMath Summer Program

Vlad Matei and Elizabeth Reiland
XYZ Press
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Mehdi Hassani
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Combinatorics is the mathematical study of countable discrete structures and the art of solving problems having to do with counting. It is an essential part of mathematical competitions at every level. Nevertheless, intelligent ideas to solve such problems are rarely discussed in textbooks, being hidden instead at the heart of their solutions.

The book under review aims to introduce students and instructors some introductory to intermediate topics in combinatorics, based on solved problems selected from the AwesomeMath Summer Program. It consists of several chapters discussing the mathematics needed to solve Olympiad combinatorial problems. The fundamental topics covered by this book are basic ideas on counting, permutations and combinations, stars and bars and multinomials, the principle of inclusion-exclusion, Pascal’s triangle, and the binomial theorem.

There are chapters on standard topics such as induction, recurrence relations, graph theory, invariants, combinatorial geometry, generating functions, and probabilistic methods. One chapter discusses the method of obtaining combinatorial identities by counting something in more than one way. Another discusses non-trivial ways of applying the pigeonhole principle.

There are several solved examples selected from various competitions. Also, the book contains four concluding chapters giving introductory and advanced problems and their solutions.

The only prerequisites to study the book are a background in arithmetic and some basic algebra. I believe that this will be a useful source for a wide range of introductory to intermediate audiences. Instructors preparing students for mathematical competitions will find the book very suitable for that purpose. Finally, this book can be used as a problem source parallel to a standard course in undergraduate Discrete Mathematics.

Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His fields of interest are Elementary, Analytic and Probabilistic Number Theory.