How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library)

Description

A perennial bestseller by eminent mathematician G. Polya, How to Solve It will show anyone in any field how to think straight.In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out--from building a bridge to winning a game of anagrams. Generations of readers have relished Polya's deft--indeed, brilliant--instructions on stripping away irrelevancies and going straight to the heart of the problem.In this best-selling classic, George Plya revealed how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out--from building a bridge to winning a game of anagrams. Generations of readers have relished Plya's deft instructions on stripping away irrelevancies and going straight to the heart of a problem. How to Solve It popularized heuristics, the art and science of discovery and invention. It has been in print continuously since 1945 and has been translated into twenty-three different languages. Plya was one of the most influential mathematicians of the twentieth century. He made important contributions to a great variety of mathematical research: from complex analysis to mathematical physics, number theory, probability, geometry, astronomy, and combinatorics. He was also an extraordinary teacher--he taught until he was ninety--and maintained a strong interest in pedagogical matters throughout his long career. In addition to How to Solve It, he published a two-volume work on the topic of problem solving, Mathematics of Plausible Reasoning, also with Princeton. Plya is one of the most frequently quoted mathematicians, and the following statements from How to Solve It make clear why: "My method to overcome a difficulty is to go around it." "Geometry is the science of correct reasoning on incorrect figures." "In order to solve this differential equation you look at it till a solution occurs to you." Cover page......Page 1 Title page......Page 4 From the Preface to the First Printing......Page 6 From the Preface to the Seventh Printing......Page 9 Preface to the Second Edition......Page 10 Contents......Page 12 "How to Solve It" list......Page 17 Foreword by John H. Conway......Page 20 Introduction......Page 26 2. Questions, recommendations, mental operations......Page 32 3. Generality......Page 33 5. Teacher and student. Imitation and practice......Page 34 6. Four phases......Page 36 7. Understanding the problem......Page 37 8. Example......Page 38 g. Devising a plan......Page 39 10. Example......Page 41 11. Carrying out the plan......Page 43 12. Example......Page 44 13. Looking back......Page 45 14. Example......Page 47 15. Various approaches......Page 50 16. The teacher's method of questioning......Page 51 17. Good questions and bad questions......Page 53 18. A problem of construction......Page 54 19. A problem to prove......Page 56 20. A rate problem......Page 60 PART II. HOW TO SOLVE IT A dialogue......Page 64 Analogy......Page 68 Auxiliary elements......Page 77 Auxiliary problem......Page 81 Bolzano......Page 88 Bright idea......Page 89 Can you check the result?......Page 90 Can you derive the result differently?......Page 92 Can you use the result?......Page 95 Carrying out......Page 99 Condition......Page 103 Could you derive something useful from the data?......Page 104 Decomposing and recombining......Page 106 Definition......Page 116 Descartes......Page 123 Determination, hope, success......Page 124 Diagnosis......Page 125 Did you use all the data?......Page 126 Do you know a related problem?......Page 129 Examine your guess......Page 130 Figures......Page 134 Generalization......Page 139 Here is a problem related to yours and solved before......Page 141 Heuristic......Page 143 Heuristic reasoning......Page 144 Induction and mathematical induction......Page 145 Inventor's paradox......Page 152 Is it possible to satisfy the condition?......Page 153 Look at the unknown......Page 154 Modern heuristic......Page 160 Notation......Page 165 Pappus......Page 172 Pedantry and mastery......Page 179 Practical problems......Page 180 Problems to find, problems to prove......Page 185 Progress and achievement......Page 188 Puzzles......Page 191 Reductio ad absurdum and indirect proof......Page 193 Routine problem......Page 202 Rules of style......Page 203 Separate the various parts of the condition......Page 204 Setting up equations......Page 205 Signs of progress......Page 209 Specialization......Page 221 Subconscious work......Page 228 Symmetry......Page 230 Terms, old and new......Page 231 Test by dimension......Page 233 The future mathematician......Page 236 The intelligent problem-solver......Page 237 The intelligent reader......Page 238 The traditional mathematics professor......Page 239 Variation of the problem......Page 240 What is the unknown?......Page 245 Why proofs?......Page 246 Wisdom of proverbs......Page 252 Working backwards......Page 256 PART IV. PROBLEMS, HINTS, SOLUTIONS......Page 264 Problems......Page 265 Hints......Page 269 Solutions......Page 273