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. Contents
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 Related Topics
Algebra, Discrete Mathematics, Geometry