The art and craft of problem solving

Content: Cover --
Title Page --
Copyright --
Preface to the Third Edition --
Contents --
1 What This Book Is About and How to Read It --
1.1 "Exercises" vs. "Problems" --
1.2 The Three Levels of Problem Solving --
1.3 A Problem Sampler --
1.4 How to Read This Book --
2 Strategies for Investigating Problems --
2.1 Psychological Strategies --
Mental Toughness: Learn from PoÌ#x81
lya's Mouse --
Creativity --
2.2 Strategies for Getting Started --
The First Step: Orientation --
I'm Oriented. Now What? --
2.3 Methods of Argument --
Common Abbreviations and Stylistic Conventions --
Deduction and Symbolic Logic --
Argument by Contradiction --
Mathematical Induction --
2.4 Other Important Strategies --
Draw a Picture! --
Pictures Don't Help? Recast the Problem in Other Ways! --
Change Your Point of View --
3 Tactics for Solving Problems --
3.1 Symmetry --
Geometric Symmetry --
Algebraic Symmetry --
3.2 The Extreme Principle --
3.3 The Pigeonhole Principle --
Basic Pigeonhole --
Intermediate Pigeonhole --
Advanced Pigeonhole --
3.4 Invariants --
Parity --
Modular Arithmetic and Coloring --
Monovariants --
4 Three Important Crossover Tactics --
4.1 Graph Theory --
Connectivity and Cycles --
Eulerian and Hamiltonian Walks --
The Two Men of Tibet --
4.2 Complex Numbers --
Basic Operations --
Roots of Unity --
Some Applications --
4.3 Generating Functions --
Introductory Examples --
Recurrence Relations --
Partitions --
4.4 Interlude: A few Mathematical Games --
5 Algebra --
5.1 Sets, Numbers, and Functions --
Sets --
Functions --
5.2 Algebraic Manipulation Revisited --
The Factor Tactic --
Manipulating Squares --
Substitutions and Simplifications --
5.3 Sums and Products --
Notation --
Arithmetic Series --
Geometric Series and the Telescope Tool --
Infinite Series --
5.4 Polynomials --
Polynomial Operations --
The Zeros of a Polynomial --
5.5 Inequalities. Fundamental Ideas --
The AM-GM Inequality --
Massage, Cauchy-Schwarz, and Chebyshev --
6 Combinatorics --
6.1 Introduction to Counting --
Permutations and Combinations --
Combinatorial Arguments --
Pascal's Triangle and the Binomial Theorem --
Strategies and Tactics of Counting --
6.2 Partitions and Bijections --
Counting Subsets --
Information Management --
Balls in Urns and Other Classic Encodings --
6.3 The Principle of Inclusion-Exclusion --
Count the Complement --
PIE with Sets --
PIE with Indicator Functions --
6.4 Recurrence --
Tiling and the Fibonacci Recurrence --
The Catalan Recurrence --
7 Number Theory --
7.1 Primes and Divisibility --
The Fundamental Theorem of Arithmetic --
GCD, LCM, and the Division Algorithm --
7.2 Congruence --
What's So Good About Primes? --
Fermat's Little Theorem --
7.3 Number Theoretic Functions --
Divisor Sums --
Phi and Mu --
7.4 Diophantine Equations --
General Strategy and Tactics --
7.5 Miscellaneous Instructive Examples --
Can a Polynomial Always Output Primes? --
A Combinatorial Proof of Fermat's Little Theorem --
Sums of Two Squares --
8 Geometry for Americans --
8.1 Three "Easy" Problems --
8.2 Survival Geometry I --
Points, Lines, Angles, and Triangles --
Parallel Lines --
Circles and Angles --
Circles and Triangles --
8.3 Survival Geometry II --
Area --
Similar Triangles --
Solutions to the Three "Easy" Problems --
8.4 The Power of Elementary Geometry --
Concyclic Points --
Area, Cevians, and Concurrent Lines --
Similar Triangles and Collinear Points --
Phantom Points and Concurrent Lines --
8.5 Transformations --
Symmetry Revisited --
Rigid Motions and Vectors --
Homothety --
Inversion --
9 Calculus --
9.1 The Fundamental Theorem of Calculus --
9.2 Convergence and Continuity --
Convergence --
Continuity --
Uniform Continuity --
9.3 Differentiation and Integration --
Approximation and Curve Sketching. The Mean Value Theorem --
A Useful Tool --
Integration --
Symmetry and Transformations --
9.4 Power Series and Eulerian Mathematics --
Don't Worry! --
Taylor Series with Remainder --
Eulerian Mathematics --
Beauty, Simplicity, and Symmetry: The Quest for a Moving Curtain --
References --
Index --
EULA.