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
Subsets of a Set
Binomial Coefficient Identities
Finding a Polynomial
The Upward-Extended Pascal’s Triangle
Recurrence Relations and Fibonacci Numbers
Generating Functions and Making Change
Rook Paths and Queen Paths
Probability Spaces and Distributions
Divisibility of Factorials and Binomial Coefficients
Partitions of an Integer
What Is Surprise?
A Coin-Tossing Game
A Little Graph Theory Background
The Ramsey Game
Tic-Tac-Toe and Animal Games
Listing Permutations and Combinations
Sudoku Solving and Polycube Packing
Appendix A: Hints and Solutions to Exercises
Appendix B: Notation
Martin Erickson is a professor of mathematics at Truman State University.
…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, 2010
The 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. 08
The 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