{"product_id":"the-art-and-craft-of-problem-solving-isbn-9781119239901","title":"The Art and Craft of Problem Solving","description":"Appealing to everyone from college-level majors to independent learners, \u003cb\u003e\u003ci\u003eThe Art and Craft of Problem Solving, 3rd Edition\u003c\/i\u003e\u003c\/b\u003e introduces a problem-solving approach to mathematics, as opposed to the traditional exercises approach. The goal of The Art and Craft of Problem Solving is to develop strong problem solving skills, which it achieves by encouraging students to do math rather than just study it. Paul Zeitz draws upon his experience as a coach for the international mathematics Olympiad to give students an enhanced sense of mathematics and the ability to investigate and solve problems. \u003cp\u003e\u003cb\u003e1 What This Book Is About and How to Read It 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 “Exercises” vs. “Problems” 1\u003c\/p\u003e \u003cp\u003e1.2 The Three Levels of Problem Solving 3\u003c\/p\u003e \u003cp\u003e1.3 A Problem Sampler 6\u003c\/p\u003e \u003cp\u003e1.4 How to Read This Book 9\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Strategies for Investigating Problems 12\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Psychological Strategies 12\u003c\/p\u003e \u003cp\u003eMental Toughness: Learn from Pólya’s Mouse 13\u003c\/p\u003e \u003cp\u003eCreativity 15\u003c\/p\u003e \u003cp\u003e2.2 Strategies for Getting Started 23\u003c\/p\u003e \u003cp\u003eThe First Step: Orientation 23\u003c\/p\u003e \u003cp\u003eI’m Oriented. Now What? 24\u003c\/p\u003e \u003cp\u003e2.3 Methods of Argument 37\u003c\/p\u003e \u003cp\u003eCommon Abbreviations and Stylistic Conventions 37\u003c\/p\u003e \u003cp\u003eDeduction and Symbolic Logic 38\u003c\/p\u003e \u003cp\u003eArgument by Contradiction 39\u003c\/p\u003e \u003cp\u003eMathematical Induction 42\u003c\/p\u003e \u003cp\u003e2.4 Other Important Strategies 49\u003c\/p\u003e \u003cp\u003eDraw a Picture! 49\u003c\/p\u003e \u003cp\u003ePictures Don’t Help? Recast the Problem in Other Ways! 51\u003c\/p\u003e \u003cp\u003eChange Your Point of View 55\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Tactics for Solving Problems 58\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Symmetry 59\u003c\/p\u003e \u003cp\u003eGeometric Symmetry 60\u003c\/p\u003e \u003cp\u003eAlgebraic Symmetry 64\u003c\/p\u003e \u003cp\u003e3.2 The Extreme Principle 70\u003c\/p\u003e \u003cp\u003e3.3 The Pigeonhole Principle 80\u003c\/p\u003e \u003cp\u003eBasic Pigeonhole 80\u003c\/p\u003e \u003cp\u003eIntermediate Pigeonhole 82\u003c\/p\u003e \u003cp\u003eAdvanced Pigeonhole 83\u003c\/p\u003e \u003cp\u003e3.4 Invariants 88\u003c\/p\u003e \u003cp\u003eParity 90\u003c\/p\u003e \u003cp\u003eModular Arithmetic and Coloring 95\u003c\/p\u003e \u003cp\u003eMonovariants 97\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Three Important Crossover Tactics 105\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Graph Theory 105\u003c\/p\u003e \u003cp\u003eConnectivity and Cycles 107\u003c\/p\u003e \u003cp\u003eEulerian and Hamiltonian Walks 108\u003c\/p\u003e \u003cp\u003eThe Two Men of Tibet 111\u003c\/p\u003e \u003cp\u003e4.2 Complex Numbers 116\u003c\/p\u003e \u003cp\u003eBasic Operations 116\u003c\/p\u003e \u003cp\u003eRoots of Unity 122\u003c\/p\u003e \u003cp\u003eSome Applications 123\u003c\/p\u003e \u003cp\u003e4.3 Generating Functions 128\u003c\/p\u003e \u003cp\u003eIntroductory Examples 129\u003c\/p\u003e \u003cp\u003eRecurrence Relations 130\u003c\/p\u003e \u003cp\u003ePartitions 132\u003c\/p\u003e \u003cp\u003e4.4 Interlude: A few Mathematical Games 138\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Algebra 143\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Sets, Numbers, and Functions 143\u003c\/p\u003e \u003cp\u003eSets 143\u003c\/p\u003e \u003cp\u003eFunctions 145\u003c\/p\u003e \u003cp\u003e5.2 Algebraic Manipulation Revisited 147\u003c\/p\u003e \u003cp\u003eThe Factor Tactic 148\u003c\/p\u003e \u003cp\u003eManipulating Squares 149\u003c\/p\u003e \u003cp\u003eSubstitutions and Simplifications 150\u003c\/p\u003e \u003cp\u003e5.3 Sums and Products 157\u003c\/p\u003e \u003cp\u003eNotation 157\u003c\/p\u003e \u003cp\u003eArithmetic Series 158\u003c\/p\u003e \u003cp\u003eGeometric Series and the Telescope Tool 158\u003c\/p\u003e \u003cp\u003eInfinite Series 161\u003c\/p\u003e \u003cp\u003e5.4 Polynomials 164\u003c\/p\u003e \u003cp\u003ePolynomial Operations 165\u003c\/p\u003e \u003cp\u003eThe Zeros of a Polynomial 165\u003c\/p\u003e \u003cp\u003e5.5 Inequalities 174\u003c\/p\u003e \u003cp\u003eFundamental Ideas 174\u003c\/p\u003e \u003cp\u003eThe AM-GM Inequality 177\u003c\/p\u003e \u003cp\u003eMassage, Cauchy-Schwarz, and Chebyshev 181\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Combinatorics 189\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Introduction to Counting 189\u003c\/p\u003e \u003cp\u003ePermutations and Combinations 189\u003c\/p\u003e \u003cp\u003eCombinatorial Arguments 192\u003c\/p\u003e \u003cp\u003ePascal’s Triangle and the Binomial Theorem 193\u003c\/p\u003e \u003cp\u003eStrategies and Tactics of Counting 195\u003c\/p\u003e \u003cp\u003e6.2 Partitions and Bijections 197\u003c\/p\u003e \u003cp\u003eCounting Subsets 197\u003c\/p\u003e \u003cp\u003eInformation Management 200\u003c\/p\u003e \u003cp\u003eBalls in Urns and Other Classic Encodings 203\u003c\/p\u003e \u003cp\u003e6.3 The Principle of Inclusion-Exclusion 207\u003c\/p\u003e \u003cp\u003eCount the Complement 207\u003c\/p\u003e \u003cp\u003ePIE with Sets 208\u003c\/p\u003e \u003cp\u003ePIE with Indicator Functions 212\u003c\/p\u003e \u003cp\u003e6.4 Recurrence 215\u003c\/p\u003e \u003cp\u003eTiling and the Fibonacci Recurrence 215\u003c\/p\u003e \u003cp\u003eThe Catalan Recurrence 217\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Number Theory 224\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Primes and Divisibility 224\u003c\/p\u003e \u003cp\u003eThe Fundamental Theorem of Arithmetic 224\u003c\/p\u003e \u003cp\u003eGCD, LCM, and the Division Algorithm 226\u003c\/p\u003e \u003cp\u003e7.2 Congruence 232\u003c\/p\u003e \u003cp\u003eWhat’s So Good About Primes? 233\u003c\/p\u003e \u003cp\u003eFermat’s Little Theorem 234\u003c\/p\u003e \u003cp\u003e7.3 Number Theoretic Functions 236\u003c\/p\u003e \u003cp\u003eDivisor Sums 237\u003c\/p\u003e \u003cp\u003ePhi and Mu 238\u003c\/p\u003e \u003cp\u003e7.4 Diophantine Equations 242\u003c\/p\u003e \u003cp\u003eGeneral Strategy and Tactics 242\u003c\/p\u003e \u003cp\u003e7.5 Miscellaneous Instructive Examples 249\u003c\/p\u003e \u003cp\u003eCan a Polynomial Always Output Primes? 249\u003c\/p\u003e \u003cp\u003eIf You Can Count It, It’s an Integer 250\u003c\/p\u003e \u003cp\u003eA Combinatorial Proof of Fermat’s Little Theorem 250\u003c\/p\u003e \u003cp\u003eSums of Two Squares 251\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Geometry for Americans 258\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Three “Easy” Problems 258\u003c\/p\u003e \u003cp\u003e8.2 Survival Geometry I 259\u003c\/p\u003e \u003cp\u003ePoints, Lines, Angles, and Triangles 260\u003c\/p\u003e \u003cp\u003eParallel Lines 262\u003c\/p\u003e \u003cp\u003eCircles and Angles 265\u003c\/p\u003e \u003cp\u003eCircles and Triangles 267\u003c\/p\u003e \u003cp\u003e8.3 Survival Geometry II 271\u003c\/p\u003e \u003cp\u003eArea 271\u003c\/p\u003e \u003cp\u003eSimilar Triangles 275\u003c\/p\u003e \u003cp\u003eSolutions to the Three “Easy” Problems 277\u003c\/p\u003e \u003cp\u003e8.4 The Power of Elementary Geometry 283\u003c\/p\u003e \u003cp\u003eConcyclic Points 284\u003c\/p\u003e \u003cp\u003eArea, Cevians, and Concurrent Lines 287\u003c\/p\u003e \u003cp\u003eSimilar Triangles and Collinear Points 290\u003c\/p\u003e \u003cp\u003ePhantom Points and Concurrent Lines 293\u003c\/p\u003e \u003cp\u003e8.5 Transformations 297\u003c\/p\u003e \u003cp\u003eSymmetry Revisited 297\u003c\/p\u003e \u003cp\u003eRigid Motions and Vectors 299\u003c\/p\u003e \u003cp\u003eHomothety 306\u003c\/p\u003e \u003cp\u003eInversion 308\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Calculus 316\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 The Fundamental Theorem of Calculus 316\u003c\/p\u003e \u003cp\u003e9.2 Convergence and Continuity 318\u003c\/p\u003e \u003cp\u003eConvergence 319\u003c\/p\u003e \u003cp\u003eContinuity 324\u003c\/p\u003e \u003cp\u003eUniform Continuity 325\u003c\/p\u003e \u003cp\u003e9.3 Differentiation and Integration 329\u003c\/p\u003e \u003cp\u003eApproximation and Curve Sketching 329\u003c\/p\u003e \u003cp\u003eThe Mean Value Theorem 332\u003c\/p\u003e \u003cp\u003eA Useful Tool 335\u003c\/p\u003e \u003cp\u003eIntegration 336\u003c\/p\u003e \u003cp\u003eSymmetry and Transformations 338\u003c\/p\u003e \u003cp\u003e9.4 Power Series and Eulerian Mathematics 342\u003c\/p\u003e \u003cp\u003eDon’t Worry! 342\u003c\/p\u003e \u003cp\u003eTaylor Series with Remainder 344\u003c\/p\u003e \u003cp\u003eEulerian Mathematics 347\u003c\/p\u003e \u003cp\u003eBeauty, Simplicity, and Symmetry: The Quest for a Moving Curtain 350\u003c\/p\u003e \u003cp\u003eReferences 355\u003c\/p\u003e \u003cp\u003eIndex 357\u003c\/p\u003e \u003cp\u003ePaul Zeitz studied history at Harvard and received a Ph.D. in mathematics from the University of California, Berkeley. He currently is an associate professor at the University of San Francisco. He won the USA Mathematical Olympiad (USAMO) and was a member of the first American team to participate in the International Mathematical Olympiad (IMO) in 1974. Since 1985, he has composed and edited problems for several national math contests, including the USAMO and helped train several American IMO teams, most notably the 1994 \"Dream Team\" which, for the first time in history, achieved a perfect score. In 2003, he received the Deborah Tepper Haimo award, a national teaching award for college and university math, given by the Math Association of America.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47990159638757,"sku":"NP9781119239901","price":110.0,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781119239901.jpg?v=1761786731","url":"https:\/\/k12savings.com\/products\/the-art-and-craft-of-problem-solving-isbn-9781119239901","provider":"K12savings","version":"1.0","type":"link"}