How to Solve It. A New Aspect of Mathematical Method Princeton

139 Pages·2004·6.44 MB·English

by  G. Polya

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