How to Guard an Art Gallery and Other Discrete Mathematical Adventures [NOOK Book]

Overview

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 tough-to-teach subject to life using ...

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How to Guard an Art Gallery and Other Discrete Mathematical Adventures

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Overview

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 tough-to-teach 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.

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Editorial Reviews

Choice

A valuable reference for instructors teaching these topics.

Mathematical Reviews

Accessible and engaging, with many examples, pithy section titles, exercises, historical notes, and a bibliography for further reading.

— Matthias Beck

The Physics Teacher

Seven great chapters that make discrete mathematics much more relevant to the real world.

— John L. Hubisz

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Product Details

  • ISBN-13: 9780801897047
  • Publisher: Johns Hopkins University Press
  • Publication date: 8/7/2009
  • Sold by: Barnes & Noble
  • Format: eBook
  • Pages: 272
  • Sales rank: 1,044,989
  • File size: 22 MB
  • Note: This product may take a few minutes to download.

Meet the Author

T. S. Michael is an associate professor of mathematics at the United States Naval Academy.

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Table of Contents

Preface ix

1 How to Count Pizza Pieces 1

1.1 The Pizza-Cutter's Problem 1

1.2 A Recurring Theme 4

1.3 Make a Difference 7

1.4 How Many Toppings? 9

1.5 Proof without Words 12

1.6 Count 'em and Sweep 14

1.7 Euler's Formula for Plane Graphs 16

1.8 You Can Look It Up 20

1.9 Pizza Envy 21

1.10 Notes and References 22

1.11 Problems 24

2 Count on Pick's Formula 33

2.1 The Orchard and the Dollar 33

2.2 The Area of the Orchard 34

2.3 Twenty-nine Ways to Change a Dollar 37

2.4 Lattice Polygons and Pick's Formula 42

2.5 Making Change 46

2.6 Pick's Formula: First Proof 48

2.7 Pick's Formula: Second Proof 53

2.8 Batting Averages and Lattice Points 56

2.9 Three Dimensions and N-largements 58

2.10 Notes and References 65

2.11 Problems 66

3 How to Guard an Art Gallery 73

3.1 The Sunflower Art Gallery 73

3.2 Art Gallery Problems 75

3.3 The Art Gallery Theorem 81

3.4 Colorful Consequences 83

3.5 Triangular and Chromatic Assumptions 86

3.6 Modern Art Galleries 88

3.7 Art Gallery Sketches 89

3.8 Right-Angled Art Galleries 93

3.9 Guarding the Guards 96

3.10 Three Dimensions and the Octoplex 102

3.11 Notes and References 106

3.12 Problems 107

4 Pixels, Lines, and Leap Years 113

4.1 Pixels and Lines 113

4.2 Lines and Distances 116

4.3 Arithmetic Arrays 118

4.4 Bresenham's Algorithm 123

4.5 A Touch of Gray: Antialiasing 124

4.6 Leap Years and Line Drawing 125

4.7 Diophantine Approximations 128

4.8 Notes and References 134

4.9 Problems 135

5 Measure Water with a Vengeance 139

5.1 Simon Says: Measure Water 139

5.2 A Recipe for Bruce Willis 142

5.3 Skew Billiard Tables 144

5.4 Big Problems147

5.5 How to Measure Water: An Algorithm 148

5.6 Arithmetic Arrays: Climb the Staircase 151

5.7 Other Problems to Pour Over 155

5.8 Number Theory and Fermat's Congruence 160

5.9 Notes and References 164

5.10 Problems 165

6 From Stamps to Sylver Coins 169

6.1 Sylvester's Stamps 169

6.2 Addition Tables and Symmetry 173

6.3 Arithmetic Arrays and Sylvester's Formula 176

6.4 Beyond Sylvester: The Stamp Theorem 180

6.5 Chinese Remainders 186

6.6 The Tabular Sieve 188

6.7 McNuggets and Coin Exchanges 190

6.8 Sylver Coinage 194

6.9 Notes and References 196

6.10 Problems 198

7 Primes and Squares: Quadratic Residues 207

7.1 Primes and Squares 207

7.2 Quadratic Residues Are Squares 208

7.3 Errors: Detection and Correction 212

7.4 Multiplication Tables, Legendre, and Euler 216

7.5 Some Square Roots 221

7.6 Marcia and Greg Flip a Coin 224

7.7 Round Up at the Gauss Corral 226

7.8 It's the Law: Quadratic Reciprocity 232

7.9 Notes and References 239

7.10 Problems 240

References 245

Index 251

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