# Pearls of Discrete Mathematics

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## Book Description

__Methods Used to Solve Discrete Math Problems__

Interesting examples highlight the interdisciplinary nature of this area

**Pearls of Discrete Mathematics** presents methods for solving counting problems and other types of problems that involve discrete structures. Through intriguing examples, problems, theorems, and proofs, the book illustrates the relationship of these structures to algebra, geometry, number theory, and combinatorics.

Each chapter begins with a mathematical teaser to engage readers and includes a particularly surprising, stunning, elegant, or unusual result. The author covers the upward extension of Pascal’s triangle, a recurrence relation for powers of Fibonacci numbers, ways to make change for a million dollars, integer triangles, the period of Alcuin’s sequence, and Rook and Queen paths and the equivalent Nim and Wythoff’s Nim games. He also examines the probability of a perfect bridge hand, random tournaments, a Fibonacci-like sequence of composite numbers, Shannon’s theorems of information theory, higher-dimensional tic-tac-toe, animal achievement and avoidance games, and an algorithm for solving Sudoku puzzles and polycube packing problems. Exercises ranging from easy to challenging are found in each chapter while hints and solutions are provided in an appendix.

With over twenty-five years of teaching experience, the author takes an organic approach that explores concrete problems, introduces theory, and adds generalizations as needed. He delivers an absorbing treatment of the basic principles of discrete mathematics.

## Table of Contents

**Counting: Basic**

Subsets of a Set

Pascal’s Triangle

Binomial Coefficient Identities

**Counting: Intermediate**

Finding a Polynomial

The Upward-Extended Pascal’s Triangle

Recurrence Relations and Fibonacci Numbers

**Counting: Advanced**

Generating Functions and Making Change

Integer Triangles

Rook Paths and Queen Paths

**Discrete Probability**

Probability Spaces and Distributions

Markov Chains

Random Tournaments

**Number Theory**

Divisibility of Factorials and Binomial Coefficients

Covering Systems

Partitions of an Integer

**Information Theory**

What Is Surprise?

A Coin-Tossing Game

Shannon’s Theorems

**Games**

A Little Graph Theory Background

The Ramsey Game

Tic-Tac-Toe and Animal Games

**Algorithms**

Counters

Listing Permutations and Combinations

Sudoku Solving and Polycube Packing

**Appendix A: Hints and Solutions to Exercises**

**Appendix B: Notation **

**Bibliography **

**Index**

## Author(s)

### Biography

**Martin Erickson** is a professor of mathematics at Truman State University.

## Reviews

…this book will remind you why you think mathematics is fun. … the real strengths of this text would be self-study for the motivated student or a source of interesting examples for a more traditional course in various areas of discrete mathematics. … The motivated student will learn much new mathematics if they work through this book carefully. For a student curious about exactly which topics make up the field of discrete mathematics, a more cursory trip through this book will do a good job of answering the question. For the library with a collection in recreational mathematics, this book will serve as a nice bridge to the ‘more serious’ associated areas of mathematics. And finally, for the professional mathematician, using this book as bedtime reading just might remind you of why you found math fun in the first place.

—Robert A. Beezer, University of Puget Sound,SIAM Review, Vol. 52, Issue 3, 2010The book will be beneficial to many undergraduates in mathematics. In fact, many sections are within reach of some advanced high school students who would like to explore interesting topics in mathematics. … Summing Up: Recommended.

—CHOICE, April 2010, Vol. 47, No. 08The book under review is a friendly volume in discrete mathematics, which covers various classical topics, such as counting, probability and number theory, and more recent topics, such as information theory, game theory and algorithms. As far as I know, this book is one of only a few at this level that covers such recent topics. Because it covers algorithms, … the book may be interesting for students interested in computer programming. … This is essentially a textbook for undergraduates, and perhaps it can be useful for high school students as well. … because of the variety of topics, teachers will have many choices of what to cover for students in applied fields. …

—Mehdi Hassani,MAA Reviews, December 2009