Preface for Instructors and Other Teachers 1 About This Book... xvii

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1 Preface for Instructors and Other Teachers xvii 1 About This Book.... xvii 2 How tousethis Book xx 2.1 A Start on Discovery-Based Learning..... xxi 2.2 Details of Conducting Group Work xxiii 3 Chapter and Bonus-Section Dependencies xxvi Preface for Students and Other Learners xxix 1 About This Book (and about Learning Mathematics)... xxix 2 How tousethis Book xxx 2.1 How to Use This Book in a Class xxxii 2.2 How to Use This Book for Self-Study..... xxxii 3 Tips forreading Mathematics xxxiii 4 Problem-Solving Prompts xxxiv 5 Tips forwriting Mathematics... xxxv Acknowledgments xxxix I Theme: TheBasics 1 1 CountingandProofs Introduction and Summary Try This! Let s Count The Sum and Product Principles Preliminaries on Proofs and Disproofs Pigeons and Correspondences Where to Go from Here Problems That Use Counting or Proofs Instructor Notes vii

2 viii Contents 2 SetsandLogic Introduction and Summary Sets Making New Sets from Scratch Finding Sets inside Other Sets Proof Technique: Double-Inclusion Making New Sets from Old Looking at Sets Logic Combining Statements Restriction of Variables via Quantifiers Negation Interactions Try This! Problems on Sets and Logic Proof Techniques: Not! Try This! A Tricky Conundrum Where to Go from Here Bonus: Truth Tellers Problems about Sets and Logic Instructor Notes GraphsandFunctions Introduction and Summary Function Introdunction Try This! Play with Functions and Graphs Play with Functions Play with Graphs A Dot Game Functions and Counting Graphs: Definitions and Examples Isomorphisms Graphs: Operations and Uses Sets and Graphs Have Some Things in Common How Are Graphs Useful? Try This! More Graph Problems Ramseyness Where to Go from Here

3 ix 3.11 Bonus: Party Tricks Bonus 2: Counting with the Characteristic Function Problems about Graphs and Functions Instructor Notes Induction Introduction and Summary Induction Summation Notation Induction Types and Styles Try This! Induction More Examples The Best Inducktion Proof Ever Try This! More Problems about Induction Are They or Aren t They? Resolving Grey Ducks Where to Go from Here Bonus: Small Crooks Bonus 2: An Induction Song Problems That Use Induction Instructor Notes Potential Practice Proof Problems Algorithms withciphers Introduction and Summary Algorithms Conditionals and Loops Efficiency Algorithms and Existence Proofs Modular Arithmetic (and Equivalence Relations) Cryptography: Some Ciphers Shift Ciphers Atbash Ciphers The Vigenère Cipher Decryption and the Real World Try This! Encryptoequivalent Modulalgorithmic Problems Where to Go from Here

4 x Contents 5.7 Bonus: Algorithms for Searching Graphs Bonus 2: Pigeons and Divisibility Problems about Algorithms, Modular Arithmetic, and Ciphers Instructor Notes II Theme: Combinatorics BinomialCoefficientsandPascal striangle Introduction and Summary You Have a Choice Try This! Investigate a Triangle Pascal s Triangle Overcounting Carefully and Reordering at Will Try This! Play with Powers and Permutations Binomial Basics Combinatorial Proof Try This! Pancakes and Proofs Where to Go from Here Bonus: Sorting Bubbles in Order of Size Bonus 2: Mastermind One Strategy for Playing Mini-Project Problems Binomially Combinatorial in Nature Instructor Notes Balls andboxesandpie:countingtechniques Introduction and Summary Combinatorial Problem Types Try This! Let s Have Some PIE Combinatorial Problem Solutions and Strategies Strategy: Slots Strategy: Stars and Bars Solutions to Problem Types Denouement: Bijective Counting, Again Let s Explain Our PIE!

5 xi 7.6 Try This! What Are the Balls and What Are the Boxes? And Do You Want Some PIE? Where to Go from Here Bonus: Linear and Integer Programming Problems about Balls, Boxes, and PIEs Instructor Notes Recurrences Introduction and Summary Fibonacci Numbers and Identities Recurrences and Integer Sequences and Induction Try This! Sequences and Fibonacci Identities Naive Techniques for Finding Closed Forms and Recurrences Arithmetic Sequences and Finite Differences Try This! Recurrence Exercises Geometric Sequences and the Characteristic Equation Try This! Find Closed Forms for These Recurrence Relations! Where to Go from Here Bonus: Recurring Stories Recurring Problems Instructor Notes CuttingUp Food: CountingandGeometry Introduction and Summary Try This! Slice Pizza (and a Yam) Pizza Numbers Try This! Spaghetti, Yams, and More Yam, Spaghetti, and Pizza Numbers Let s Go for It! Hyperbeet Numbers Where to Go from Here Bonus: Geometric Gems Problems That Combine Combinatorial Topics Instructor Notes

6 xii Contents III Theme: Graph Theory Trees Introduction and Summary Basic Facts about Trees Try This! Spanning Trees Spanning Tree Algorithms Greedy Algorithms Binary Trees Try This! Binary Trees and Matchings Matchings Backtracking Where to Go from Here Bonus: The Branch-and-Bound Technique in Integer Programming Tree Problems Instructor Notes Euler s Formula and Applications Introduction and Summary Try This! Planarity Explorations Planarity A Lovely Story Or, Are Emus Full?: A Theorem and a Proof Applications of Euler s Formula Try This! Applications of Euler s Formula Where to Go from Here Bonus: Topological Graph Theory Problems about Planar Graphs Instructor Notes Graph Traversals Introduction and Summary Try This! Euler Traversals Euler Paths and Circuits

7 xiii 12.4 Dijkstra s Algorithm, with sides of Hamilton Circuits and the Traveling Salesperson Problem Try This! Do This! Try This! Where to Go from Here Bonus: Digraphs, Euler Traversals, and RNA Chains Bonus 2: Network Flows Bonus 3: Two Hamiltonian Theorems Problems with Traversing Instructor Notes Graph Coloring Introduction and Summary Try This! Coloring Vertices and Edges Vertex Coloring Edge Coloring More on Vertex Coloring More on Edge Coloring Introduction to Coloring Coloring Bounds Applications of Vertex Coloring Try This! Let s Think about Coloring Coloring and Things (Graphs and Concepts) That Have Come Before Let s Color the Edges of Complete Graphs Let s Color Bipartite Graphs Add a Condition, Get a Different Bound Greedy Matchings Where to Go from Here Bonus: The Four-Color Theorem Colorful Problems Instructor Notes IV Other Material Probability and Expectation Introduction and Summary What Is Probability, Exactly?

8 xiv Contents 14.3 High Expectations You Are Probably Expected to Try This! Conditional Probability and Independence The Helpfulness of PIE in the Real World of Probability Independence versus Exclusivity Try This!..., Probably, Under Certain Conditions Higher Expectations That s Wild! (A Hint at the Probabilistic Method) Where to Go from Here Bonus: Ramsey Numbers and the Probabilistic Method Expect Problems, Probably Instructor Notes Fun with Cardinality Introduction and Summary Read This! Parasitology, the Play Scene 1: The Storage Coordinator Scene 2: The Taxonomist Scene 3: The Café Scene 4: Cataloguing How Big Is Infinite? Try This: Investigating the Play Questions about Sample Storage More Questions about Sample Storage Questions about CaféConversations Indiscrete Questions How High Can We Count? The Continuum Hypothesis Where to Go from Here Bonus: The Schröder Bernstein Theorem Infinitely Large Problems Instructor Notes A Additional Problems 459 B Solutionsto CheckYourself Problems 487

9 xv C TheGreekAlphabetandSomeUsesfor SomeLetters 517 D List of Symbols 519 Glossary 523 Bibliography 537