How To Solve It A New Aspect of Mathematical Mgthoa! G . POLYA Stanford U,ni versity SECOND EDITION Princeton University Press Princeton, New Jersey Cupyright ig gg by Princeton University Press Copyright @ renewed 1973 by Princeton University Press From the Preface to the First Printing Second Edition Copyright @19I 57 by G. Polya All Rights Reserved A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your L.C. Card: 79-I 60544 problem may be modest; but if it challenges your curia- ISBN 0-69I -02356-1, (paperback edn.) jty and brings into play your inventive faculties, and if ISBN 0-69I -08097-6 (hardcover edn.) you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experi- ences at a susceptible age may create a taste for mental First Princeton Paperback Printing, 197 I work and leave their imprint on mind and character for a lifetime. - Second Printing, 197 3 Thus, a teacher of mathematics has a great cpportu- nity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intel1ectual development, and misuses his appor- tunity. But if he challenges the curiosity of his students This book is sold subject to the condition that by setting them problems proportionate to their knowl- it shall not, by way of trade, be lent, resold, edge, and helps them to solve their problems with stimu- hired out, or otherwise disposed of witllout lating questions, he may give them a taste for, and some the publisher's consent, in any form of bind means of, independent thinking. ing or cover other than that in which it is Also a student whose college curriculum indudes some published, mathematics has a singular opportunity. This opportu- nity is lost, of course, if he regards mathematics as a subject in which he has to earn so md so much aedit and which he shouId forget after the final examination as quickly as possible. The opportunity may be lost even Printed in the United States of America if the student has some natural talent fox mathematics by princeton University press, princeton, New Jersev bemuse 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 V vi From the Preface to the Firsi Printing From the Preface to the First Printing vii work may be an exercise as desirable as a fast game of tennis. Having tasted the pleasure in mathematics he will tical problems. Behind the desire to solve this or that poblem that confers no material advantage, there may not forget it easily and then there is a good chance that be a deeper curiosity, a desire to understand the ways and mathematics will become something for him: a hobby, or means, the motives and procedures, of solution. a tool of his profession, ar his profession, or a great The following pages are written somewhat concisely, ambition. The author remembers the time when he was a student but as simply as possible, and are based on a long and serious study of methods of solution. This sort of study, himself, a somewhat ambitious student, eager to under- called heuristic by some writers, is not in fashion now- stand a little mathematics and physics. He listened to adays but has a long past and, perhaps, some future. lectures, read books, tried to take in the solutions and Studying the methods of solving problems, we perceive facts presented, but there was a question that disturbed another face of mathematics. Yes, mathema tics has two him again and again: "Yes, the solution seems to work, faces; it is the rigorous science of Euclid but it is also it appears to be correct; but how is it possible to invent something else. Mathematics presented in the Euclidean such a solution? Yes, this experiment seems to work, this appears to be a fact; but how can people discover such way appears as a systematic, deductive science; but mathe- matics in the making appears as an experimental, in- facts? And how could I invent or discover such things by ductive science. Both aspects are as old as the science of myself?"T oday the author is teaching ma thematics in a mahematics itself. But the second aspect is new in one university; he thinks or hopes that some of his more eager respect; mathematics "in statu nascendi," in the process students ask similar questions and he tries to satisfy their of being invented, has never before been presented in curiosity. Trying to understand not only the solution of this or that problem but also the motives and procedures quite this manner to the student, or to the teacher him- of the solution, and trying to explain these motives and self, or to the general public. The subject of heuristic has manifold connections; procedures to others, he was finally led to write the mathematicians, logicians, psychologists, educationalists, present book. He hopes that it will be useful to teachers even philosophers may claim various parts of it as belong- who wish to develop their students' ability to soIve prob- ing to their special domains. The author, well aware of lems, and to students who are keen on developing their the possibility of criticism from opposite quarters and own abilities. keenly conscious of his limitations, has one claim to Although the present book pays special attention to the make: he has some experience in solving problems and requirements of students and teachers of mathematics, it in teaching rnathernatirs on various levels. should interest anybody concerned with the ways and The subject is more fully deaIt with in a more exten- means of invention and discovery. Such interest may be sive book by the author which is on the way to corn- more widespread than one would assume without reflec- pletion. tion. The space devoted by popular newspapers and magazines to crossword puzzles and other riddles seems Stanford Uniuersity, August 1, 1944 to show that people spend same time in solving unprac- viii From the Preface to the Smenth Printing Preface to the Second Edition ix From the Preface to the Seventh Printing Preface to the Second Edition I am glad to say that I have now succeeded in fulfilling, The present second edition adds, besides a few minor at least in part, a promise given in the preface to the improvements, a new fourth part, "Problems, Hints, first printing: The two volumes Induction and Analogy Solutions." in Mathematics and Patterns of Plausible Inference which As this edition was being prepared for print, a study constitute my recent work Mathema tics and Plausible appeared {Educational Testing Service, Prince ton, N.J.; Reasoning continue the li~ieo f thinking begun in How cf. Time, June 18, 1956) which seems to have formu- to Solue It. lated a few pertinent observations-they are not new to the people in the know, but it was high time to fornu- Zurich, Augesst 30, 1954 . . late them for the general public-: " . mathematics has the dubious honor of being the least popular subject in . . the curriculum . Future teachers pass through the . elementary schooIs learning to detest mathematics . . They return to the elementary school to teach a new peration to detest it." I hope that the present edition, designed for wider diffusion, will convince some of its readers that mathe- matics, 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 Cont ents From the Preface to the First Printing v From the Preface to the Seventh Printing VAl*.l l Preface to the Second Edition ix "How To Solve It" list xvi Introduction xix BART I. IN THE CLASSROOM Purpose 1. Helping the student I 2. Ques tioras, recommendations, mental operations I g. Generality P 4. Common sense 4 5. Teacher and student. Imitation and practice 3 Main divisions, main questions 6. Four phases 7. Understanding the problem 8. Example g. Devising a plan lo. Example 11. Carrying out the plan xi -L xii Contents Contents 12. Example 1 3 Condition 3 . Looking back 14 ~ontradictory? 14. Example 16 Corollary 15. Various approaches '9 Could you derive something useful from the data? 16. The teacher's method of questioning 20 Could you restate the problern?-f 17. Good questions and bad questions 21 Decomposing and recombining Mom examples Descartes Determination, hope, success 18. A probIem of construction Diagnosis lg. A problem to prove Did you use all the data? 40. A rate problem Do you know a related problem? Draw a figuret PART 11. HOW TO SOLVE IT Examine your guess A dialogue . Figures Generalization Have you seen it before? PART 111. SHORT DICTIONARY :Here is a problem related to yours OF HEURISTIC and solved before , Analqg Auxiliary eIernen~ peuristic reasoning 1r3 Auxiliary problem #f you cannot solve the proposed problem 114 Bolzano -& ;$*duction and mathematical induction 114 Bright idea L-*tsl& ven tor's paradox 1s 1 Can you check the result? it possible to satisfy the condition? lee Can you derive the result differently? ibnitz 123 Can you use the result? mma 123 Carrying out t Contains only cross-references. xiv Contents Contents Look at the unknown variation of the problem Modern heuristic What is the unknown? Notation why proofs? Papp us Wisdom of proverbs Pedantry and mastery Working backwards Practical problems Problems to find, problems to prove PART IV. PROBLEMS, HINTS, SOLUTIONS Progress and achievement Puzzles Problems Reductio ad absurdurn and indirect proof Hints Redundant? Solutions Routine problem Rules of discovery Rules of style Rules of teaching Separate the various parts of the condition Setting up equations Signs of progress Specialization Subconscious work Symmetry Terns, old and new Test by dimension The future mathematician The intelligent problem-soher The intelligent reader The traditional mathematics professor t Contains only cross-references. HOW T O SOLVE I T UNDERSTANDING THE PROBLEM What is the unknown? What are the data? What is the condition? First. Is it possible to satisfy the condition? Is the condition sufficier~t to You have to understand determine the unknown? Or is it insufficient? Or redundant? Or the problem, contradictary? Draw a figure. Introduce suitable notation. Separate the various parts of the condition. Can you write them down? DEVISING A PLAN Have you seen it before? Or have you seen the same problem in a Second. slightly different form? Find the connection between Do you know a related problem? Do you know a theorem that could the data and the unknown. be useful? You may be obliged Look at the unknown! And try to think of a familiar problem having to consider auxiliary problems the same or a similar unknown. if an immediate connection cannot be found. Here is a problem related to yours and solved before. Could you use it? You should obtain eventually Could you use its result? Could you use its method? Should you intro- a blan of the solution. duce some auxiliary element in order to make its use powible? Could you restate the problem? Could you restate it still differently? Go back to definitions. B. ++P -d "?~<*'>,*r*i d whti th'e p p dpr oblem uy to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you soIve a part of the problem? Keep only a part of the condi- tion, drop the other part; how far is the unknown then determined, hour can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or the data, or both if necessary, so that the ncw unknown and the new data are nearer to each other? Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem? CARRYING OUT THE PLAN Third. Carrying out your plan of the solution, check each str:fi. Can you see Carry out your plan. clearly that the step is correct? Can you prove that it is correct? LOOKTNG RACK Can you cheek the result? Can you chcck the argument? Fourth. --- Can you derite the result diffcrcntly? Can you see it at a gIance? Examine the solution obtained. Can you use the result, or thc method, for some other prohlum? Introduction The folIowing considerations are grouped around the preceding list of questions and suggestions entitled "How Solve It." Any question or suggestion quoted from it will be printed in italics, and h e whole list wiII be referred to simply as "the list" or as "our list." The following pages will discuss the purpose of the list, illustrate its practical use by examples, and explain the underlying notions and mental operations. By way of ~reliminarye xplanation, this much may be said: If, using them properly, you address these questions and suggestions to yourself, they may help you to solve your problem. If, using them properly, you address the same questions and suggestions to one of your students, you may help him to solve his problem. The book is divided into four parts. The title of the first part is "In the Classroom." It contains twenty sections. Each section will be quoted by ih number in heavy type as, for instance, "section 7." ktions 1 to 5 discuss the "Purpose" of our list in gem era1 terms. Sections 6 to 17 explain what are the "Main bivisions, Main Questions" of the list, and discuss a first practical example. Sections 18, 19, 20 add "More Ex- amples." The title of the very short second part is "How to Solve It." It is written in dialogue; a somewhat idealized eacher answers short questions of a somewhat idealized +dent. he third and most extensive part is a "Short Diction- f Heuristic"; we shall refer to it as the "Dictionary." xix
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