How to solve it.A new aspect of math--G.Polya.pdf

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Cover

Title

Contents

From the Preface to the First Printing

From the Preface to the Seventh Printing

Preface to the Second Edition

"How to Solve It" list

Foreword

Introduction

PART I. IN THE CLASSROOM

Purpose

1. Helping the student

2. Questions, recommendations, mental operations

3· Generality

4· Common sense

5· Teacher and student. Imitation and practice

Main divisions, main questions

6. Four phases

7· Understanding the problem

8. Example

9. Devising a plan

10. Example

11. Carrying out the plan

12. Example

13· Looking back

14· Example

15· Various approaches

16. The teacher's method of questioning

17· Good questions and bad questions

More examples

18. A problem of construction

19. A problem to prove

20. A rate problem

PART II. HOW TO SOLVE IT

A dialogue

PART III. SHORT DICTIONARY OF HEURISTIC

Analogy

Auxiliary elements

Auxiliary problem

Bolzano

Bright idea

Can you check the result?

Can you derive the result differently?

Can you use the result?

Carrying out

Condition

Contradictory†

Corollary

Could you derive something useful from the data?

Could you restate the problem?†

Decomposing and recombining

Definition

Descartes

Determination, hope, success

Diagnosis

Did you use all the data?

Do you know a related problem?

Draw a figure†

Examine your guess

Figures

Generalization

Have you seen it before?

Here is a problem related to yours and solved before

Heuristic

Heuristic reasoning

If you cannot solve the proposed problem

Induction and mathematical induction

Inventor's paradox

Is it possible to satisfy the condition?

Leibnitz

Lemma

Look at the unknown

Modern heuristic

Notation

Pappus

Pedantry and mastery

Practical problems

Problems to find, problems to prove

Progress and achievement

Puzzles

Reductio ad absurdum and indirect proof

Redundant†

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

Terms, old and new

Test by dimension

The future mathematician

The intelligent problem-solver

The intelligent reader

The traditional mathematics professor

Variation of the problem

What is the unknown?

Why proofs?

Wisdom of proverbs

Working backwards

PART IV. PROBLEMS, HINTS, SOLUTIONS

Problems

Hints

Solutions

How to Solve It

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How to Solve It A New Aspect of Mathematical Method G. POLYA With a new foreword by john H. Conway Princeton University Press Princeton and Oxford

Copyright© 1945 by Princeton University Press Copyright© renewed 1973 by Princeton University Press Second Edition Copyright© 1957 by G. Polya Second Edition Copyright © renewed 1985 by Princeton University Press All Rights Reserved First Princeton Paperback printing, 1971 Second printing, 1973 First Princeton Science Library Edition, 1988 Expanded Princeton Science Library Edition, with a new foreword by John H . Conway, 2004 Library of Congress Control Number 2004100613 ISBN-13: 978-0-691-11966-3 (pbk.) ISBN-10: 0-691-11966-X (pbk.) British Library Cataloging-in-Publication Data is available Printed on acid-free paper. oo psi. princeton.edu Printed in the United States of America 3 5 7 9 10 8 6 4

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 curios ity 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 experi ences 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 opportu nity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his oppor tunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowl edge, and helps them to solve their problems with stimu lating 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 singular opportunity. This opportu nity is lost, of course, if he regards mathematics as a subject in which he has to earn so and so much credit and which he should forget after the final examination 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 v

vi From the Preface to the First Printing 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 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 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 prob lems, 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 reflec tion. The space devoted by popular newspapers and magazines to crossword puzzles and other riddles seems to show that people spend some time in solving unprac-

From the Preface to the First Printing vii tical problems. Behind the desire to solve this or that problem that confers 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 adays 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 mathe matics in the making appears as an experimental, in ductive science. Both aspects are as old as the science of mathematics itself. But the second aspect is new in one respect; mathematics "in statu nascendi," in the process of being invented, has never before been presented in quite this manner to the student, or to the teacher him self, 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 belong ing 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 exten sive book by the author which is on the way to com pletion. Stanford University, August I, I944

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