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[ ripped by themusicgod1 ] How To Solve It --------------- A New Aspect of Mathematical Method G. POLYA Stanford University Illegally transcribed by : Prophet themusicgod1 Second Edition Princeton University Press Princeton, New Jersey Copyright 1945 by Princeton University Press Copyright (C) renewed 1973 by Princeton University Press Second Edition Copyright (C) 1957 by G. Polya All Rights Reserved L.C. Card: 79-160544 ISBN 0-691-02356-5 (paperback edn.) ISBN 0-691-08097-6 (hardcover edn.) First Princeton Paperback Printing, 1971 Second Printing, 1973 This book is sold subject to the condition that it shall not, by way of trade, be lent, resold, hired out, or otherwise disposed of without the publisher's consent, in any form of bind- binding or cover other than that in which it is published. Printed in the United States of America by Princeton University Press, Princeton, New Jersey If you find any spelling mistakes in this text, please correct them, and increment this number by a random amount: 000000000000004 If you do find any errors in this document, please feel free to email me at themusicgod1@schnits.org. This text is going to be written in simple HTML, instead of text, because it is required to keep with the continuity that this book appears to intend. It may be possible to read this book in TXT format, but it would lose much of its quality. Apologies to those without HTML interpreters. From the Preface to the First Printing A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such expert experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime. Thus, a teacher of mathematics has a great opportunity. If he fills his alloted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge edge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking. Also a student whose college curriculum includes some mathematics has a single opportunity. This opportunity is lost, of course, if he regards mathematics as a subject in which he has to earn so and so much credit as quickly as possible. The opportunity may be lost even if the student has some natural talent for mathematics because he, as everybody else, must discover his talents and tastes; he cannot know that he likes raspberry pie if he has never tasted raspberry pie. He may manage to find out, however, that a mathematics problem may be as much fun as a crossword puzzle, or that vigorous mental work may be an exercise as desirable as a fast game of tennis. Having tasted the pleasure in mathematics he will not forget it easily and then there is a good chance that mathematics will become something for him : a hobby, or a tool of his profession, or his profession, or a great ambition. The author remembers the first time when he was a student himself, a somewhat ambitious student, eager to under -stand a little mathematics and physics. He listened to lectures, read books, tried to take in the solutions and facts presented, but there was a question that disturbed him again and again: "Yes, the solution seems to work. it appears to be correct; but how is it possible to invent such a solution? Yes, this experiment system seems to work, this appears to be a fact; but how can people discover such facts? And how could I invent or discover such things by myself?" Today the author is teaching mathematics in a university; he thinks or hopes that some of his more eager students ask similar questions and he tries to satisfy their curiosity. Trying to understand not only the solution of this or that problem but also the motives and procedures of the solution, and trying to explain these motives and procedures to others, he was finally led to write the present book. He hopes that it will be useful to teachers who wish to develop their students' ability to solve problems, and to students who are keen on developing their own abilities. Although the present book pays special attention to the requirements of students and teachers of mathematics, it should interest anybody concerned with the ways and means of invention and discovery. Such interest may be more widespread than one would assume without reflection. The space devoted by popular newspapers and magazines to crossword puzzles and other riddles seems to show that people spend some time in solving unpractical problems. Behind the desire to solve this or that problem that conveys no material advantage, there may be a deeper curiosity, a desire to understand the ways and means, the motives and procedures, of solution. The following pages are written somewhat concisely, but as simply as possible, and are based on a long and serious study of methods of solution. This sort of study, called heuristic by some writers, is not in fashion now; but has a long past, and, perhaps, some future. Studying the methods of solving problems, we perceive another face of mathematics. Yes, mathematics has two faces; it is the rigorous science of Euclid but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science ; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself. But the second aspect is new in one respect; mathematics "in statu mascendi", in the process of being invented, has never before been presented in quite this manner to the student, or to the teacher himself, or to the general public. The subject of heuristic has manifold connections; mathematicians, logicians, psychologists, educationalists, even philosophers may claim various parts of it as belonging to their special domains. The author, well aware of the possibility of criticism from opposite quarters and keenly conscious of his limitations, has one claim to make; he has some experience in solving problems and in teaching mathematics on various levels. The subject is more fully dealt with in a more extensive book by the author which is on the way to completion. From the Preface to the Seventh Printing I am glad to say that I have now succeeded in fulfilling at least in part, a promise given in the preface to the first printing: The two volumes Induction and Analogy in Mathematics and Patterns of Plausible Inference which constitute my recent work Mathematics and Plausible Reasoning continue the line of thinking begun in How to Solve It. Zurich, August 30, 1954 Preface to the Second Edition The present second edition adds, besides a few minor improvements, a new fourth part, "Problems, Hints, Solutions." As this edition was being prepared for print, a study appeared (Educational Testing Service, Princeton, N.J.; cf. Time, June 18, 1956) which seems to have formulated a few pertinent observations. They are not new to the people in the know, but it was high time to formulate them for the general public: "... mathematics has the dubious honour of being the least popular subject in the curriculum ... Future teachers pass through the elementary schools learning to detest mathematics... They return to the elementary school to teach a new generation to detest it." I hope that the present edition, designed for wider diffusion, will convince some of its readers that mathematics, besides being a necessary avenue to engineering jobs and scientific knowledge, may be fun and may also open up a vista of mental activity on the highest level. Zurich, June 30, 1956 Contents From the Preface to the First Printing v From the Preface to the Seventh Printing viii Preface to the Second Edition ix "How To Solve It" list xvi Introduction xix PART I. IN THE CLASSROOM 1. Helping the student 1 2. Questions, recommendations, mental operations 1 3. Generality 2 4. Common sense 3 5. Teacher and student, Imitation and practice 3 Main divisions, main questions 6. Four phases 5 7. Understanding the problem 6 8. Example 7 9. Devising a plan 8 10. Example 10 11. Carrying out the plan 12. Example 13. Looking back 14. Example 16 15. Various approaches 19 16. The teacher's method of questioning 20 17. Good questions and bad questions 22 More Examples 18. A problem of construction 23 19. A problem to prove 25 20. A rate problem 29 PART II. HOW TO SOLVE IT A dialogue 33 PART III. SHORT DICTIONARY OF HEURISTIC Analogy 37 Auxiliary elements 46 Auxiliary problem 50 Bolzano 57 Bright idea 58 Can you check the result? 59 Can you derive the result differently? 61 Can you use the result? 64 Carrying out 68 Condition 72 Contradictory 73 Corollary 73 Could you derive something useful from the data? 73 Could you restate the problem? 75 Decomposing and recombining 75 Definition 85 Descartes 92 Determination,Hope,Success 93 Diagnosis 93 Did you use all the data? 95 Do you know a related problem? 98 Draw a figure 99 Examine your guess 99 Figures 103 Generalisation 108 Have you seen it before? 108 Here is a problem related to yours and solved before 110 Heuristic reasoning 112 If you cannot solve the proposed problem 114 Induction and mathematical induction 114 Inventor's paradox 114 Is it possible to satisfy the condition? 122 Leibnitz 123 Lemma 123 Look at the unknown 123 Modern heuristic 129 Notation 134 Pappus 141 Pedantry and mastery 148 Practical Problems 149 Problems to find, problems to prove 154 Progress and achievement 157 Puzzles 157 Reductio ad Absurdum and indirect proof 162 Redundant 171 Routine problem 171 Rules of discovery 172 Rules of style 172 Rules of teaching 173 Separate the various parts of the condition 173 Setting up equations 174 Signs of progress 178 Specialisation 190 subconscious work 197 Symmetry 199 Terms, old and new 200 Test by dimension 200 The future mathematician 205 The intelligent problem-solver 206 the intelligent reader 207 The traditional mathematics professor 208 Variation of the problem 209 What is the unknown 214 Why proofs? 215 Wisdom of proverbs 221 Working Backwards 225 PART IV. PROBLEMS, HINTS, SOLUTIONS Problems 234 Hints 238 Solutions 242 HOW TO SOLVE IT UNDERSTANDING THE PROBLEM First. You have to understand the problem What is the unknown? What are the data? What is the condition? Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? contradictory? Draw a figure. Introduce suitable notation. Separate the various parts of the condition. Can you write them down? DEVISING A PLAN Second Find the connection between the data