**1 What This Book Is About and How to Read It 1** 1.1 “Exercises” vs. “Problems” 1

1.2 The Three Levels of Problem Solving 3

1.3 A Problem Sampler 6

1.4 How to Read This Book 9

**2 Strategies for Investigating Problems 12**

2.1 Psychological Strategies 12

Mental Toughness: Learn from Pólya’s Mouse 13

Creativity 15

2.2 Strategies for Getting Started 23

The First Step: Orientation 23

I’m Oriented. Now What? 24

2.3 Methods of Argument 37

Common Abbreviations and Stylistic Conventions 37

Deduction and Symbolic Logic 38

Argument by Contradiction 39

Mathematical Induction 42

2.4 Other Important Strategies 49

Draw a Picture! 49

Pictures Don’t Help? Recast the Problem in Other Ways! 51

Change Your Point of View 55

**3 Tactics for Solving Problems 58**

3.1 Symmetry 59

Geometric Symmetry 60

Algebraic Symmetry 64

3.2 The Extreme Principle 70

3.3 The Pigeonhole Principle 80

Basic Pigeonhole 80

Intermediate Pigeonhole 82

Advanced Pigeonhole 83

3.4 Invariants 88

Parity 90

Modular Arithmetic and Coloring 95

Monovariants 97

**4 Three Important Crossover Tactics 105**

4.1 Graph Theory 105

Connectivity and Cycles 107

Eulerian and Hamiltonian Walks 108

The Two Men of Tibet 111

4.2 Complex Numbers 116

Basic Operations 116

Roots of Unity 122

Some Applications 123

4.3 Generating Functions 128

Introductory Examples 129

Recurrence Relations 130

Partitions 132

4.4 Interlude: A few Mathematical Games 138

**5 Algebra 143**

5.1 Sets, Numbers, and Functions 143

Sets 143

Functions 145

5.2 Algebraic Manipulation Revisited 147

The Factor Tactic 148

Manipulating Squares 149

Substitutions and Simplifications 150

5.3 Sums and Products 157

Notation 157

Arithmetic Series 158

Geometric Series and the Telescope Tool 158

Infinite Series 161

5.4 Polynomials 164

Polynomial Operations 165

The Zeros of a Polynomial 165

5.5 Inequalities 174

Fundamental Ideas 174

The AM-GM Inequality 177

Massage, Cauchy-Schwarz, and Chebyshev 181

**6 Combinatorics 189**

6.1 Introduction to Counting 189

Permutations and Combinations 189

Combinatorial Arguments 192

Pascal’s Triangle and the Binomial Theorem 193

Strategies and Tactics of Counting 195

6.2 Partitions and Bijections 197

Counting Subsets 197

Information Management 200

Balls in Urns and Other Classic Encodings 203

6.3 The Principle of Inclusion-Exclusion 207

Count the Complement 207

PIE with Sets 208

PIE with Indicator Functions 212

6.4 Recurrence 215

Tiling and the Fibonacci Recurrence 215

The Catalan Recurrence 217

**7 Number Theory 224**

7.1 Primes and Divisibility 224

The Fundamental Theorem of Arithmetic 224

GCD, LCM, and the Division Algorithm 226

7.2 Congruence 232

What’s So Good About Primes? 233

Fermat’s Little Theorem 234

7.3 Number Theoretic Functions 236

Divisor Sums 237

Phi and Mu 238

7.4 Diophantine Equations 242

General Strategy and Tactics 242

7.5 Miscellaneous Instructive Examples 249

Can a Polynomial Always Output Primes? 249

If You Can Count It, It’s an Integer 250

A Combinatorial Proof of Fermat’s Little Theorem 250

Sums of Two Squares 251

**8 Geometry for Americans 258**

8.1 Three “Easy” Problems 258

8.2 Survival Geometry I 259

Points, Lines, Angles, and Triangles 260

Parallel Lines 262

Circles and Angles 265

Circles and Triangles 267

8.3 Survival Geometry II 271

Area 271

Similar Triangles 275

Solutions to the Three “Easy” Problems 277

8.4 The Power of Elementary Geometry 283

Concyclic Points 284

Area, Cevians, and Concurrent Lines 287

Similar Triangles and Collinear Points 290

Phantom Points and Concurrent Lines 293

8.5 Transformations 297

Symmetry Revisited 297

Rigid Motions and Vectors 299

Homothety 306

Inversion 308

**9 Calculus 316**

9.1 The Fundamental Theorem of Calculus 316

9.2 Convergence and Continuity 318

Convergence 319

Continuity 324

Uniform Continuity 325

9.3 Differentiation and Integration 329

Approximation and Curve Sketching 329

The Mean Value Theorem 332

A Useful Tool 335

Integration 336

Symmetry and Transformations 338

9.4 Power Series and Eulerian Mathematics 342

Don’t Worry! 342

Taylor Series with Remainder 344

Eulerian Mathematics 347

Beauty, Simplicity, and Symmetry: The Quest for a Moving Curtain 350

References 355

Index 357