What is the maximum number of pizza slices one can get by making four straight cuts through a circular pizza? How does a computer determine the best set of pixels to represent a straight line on a computer screen? How many people at a minimum does it take to guard an art gallery?Discrete mathematics has the answer to these -- and many other -- questions of picking, choosing, and shuffling. T. S. Michael's gem of a book brings this vital but toughtoteach subject to life using examples from real life and popular culture. Each chapter uses one problem -- such as slicing a pizza -- to detail key concepts about counting numbers and arranging finite sets. Michael takes a different perspective in tackling each of eight problems and explains them in differing degrees of generality, showing in the process how the same mathematical concepts appear in varied guises and contexts. In doing so, he imparts a broader understanding of the ideas underlying discrete mathematics and helps readers appreciate and understand mathematical thinking and discovery.This book explains the basic concepts of discrete mathematics and demonstrates how to apply them in largely nontechnical language. The explanations and formulas can be grasped with a basic understanding of linear equations.
Preface1. How to Count Pizza Pieces1.1. The Pizza-Cutter's Problem1.2. A Recurring Theme1.3. Make a Difference1.4. How Many Toppings?1.5. Proof without Words1.6. Count 'em and Sweep1.7. Euler's Formula for Plane Graphs1.8. You Can Look It Up1.9. Pizza Envy1.10. Notes and References1.11. Problems2. Count on Pick's Formula2.1. The Orchard and the Dollar2.2. The Area of the Orchard2.3. Twenty-nine Ways to Change a Dollar2.4. Lattice Polygons and Pick's Formula2.5. Making Change2.6. Pick's Formula: First Proof2.7. Pick's Formula: Second Proof2.8. Batting Averages and Lattice Points2.9. Three Dimensions and N-largements2.10. Notes and References2.11. Problems3. How to Guard an Art Gallery3. The Sunflower ArtGallery3.1. The Sunflower Art Gallery3.2. Art Gallery Problems3.3. The Art Gallery Theorem3.4. Colorful Consequences3.5. Triangular and Chromatic Assumptions3.6. Modern Art Galleries3.7. Art Gallery Sketches3.8. Right-Angled Art Galleries3.9. Guarding the Guards3.10. Three Dimensions and the Octoplex3.11. Notes and References3.12. Problems4. Pixels, Lines, and Leap Years4.1. Pixels and Lines4.2. Lines and Distances4.3. Arithmetic Arrays4.4. Bresenham's Algorithm4.5. A Touch of Gray: Antialiasing4.6. Leap Years and Line Drawing4.7. Diophantine Approximations4.8. Notes and References4.9. Problems5. Measure Water with a Vengeance5.1. Simon Says: Measure Water5.2. A Recipe for Bruce Willis5.3. Skew Billiard Tables5.4. Big Problem5.5. How to Measure Water: An Algorithm5.6. Arithmetic Arrays: Climb the Staircase5.7. Other Problems to Pour Over5.8. Number Theory and Fermat's Congruence5.9. Notes and References5.10. Problems6. From Stamps to Sylver Coins6.1. Sylvester's Stamps6.2. Addition Tables and Symmetry6.3. Arithmetic Arrays and Sylvester's Formula6.4. Beyond Sylvester: The Stamp Theorem6.5. Chinese Remainders6.6. The Tabular Sieve6.7. McNuggets and Coin Exchanges6.8. Sylver Coinage6.9. Notes and References6.10. References7. Primes and Squares: Quadratic Residues7.1. Primes and Squares7.2. Quadratic Residues Are Squares7.3. Errors: Detection amd Correction7.4. Multiplication Tables, Legendre, and Euler7.5. Some Square Roots7.6. Marcia and Greg Flipa Coin7.7. Round Up at the Gauss Corral7.8. It's the Law: Quadratic Reciprocity7.9. Notes and References7.10. ProblemsReferencesIndex
""Accessible and engaging, with many examples, pithy section titles, exercises, historical notes, and a bibliography for further reading.""
