How to Solve It: A New Aspect of Mathematical Method

G. Polya

Published by Princeton University Press 1945

How to Solve It

A New Aspect of Mathematical Method

G. Polya
Preface by: John H. Conway

Volume 34 in the series Princeton Science Library

https://doi.org/10.1515/9781400828678

Overview
Contents

About this book

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.

Author / Editor information

George Polya (1887–1985) was one of the most influential mathematicians of the twentieth century. His basic research contributions span complex analysis, mathematical physics, probability theory, geometry, and combinatorics. He was a teacher par excellence who maintained a strong interest in pedagogical matters throughout his long career. Even after his retirement from Stanford University in 1953, he continued to lead an active mathematical life. He taught his final course, on combinatorics, at the age of ninety. John H. Conway is professor emeritus of mathematics at Princeton University. He was awarded the London Mathematical Society's Polya Prize in 1987. Like Polya, he is interested in many branches of mathematics, and in particular, has invented a successor to Polya's notation for crystallographic groups.

Reviews

"In an age that all solutions should be provided with the least possible effort, this book brings a very important message: mathematics and problem solving in general needs a lot of practice and experience obtained by challenging creative thinking, and certainly not by copying predefined recipes provided by others. Let's hope this classic will remain a source of inspiration for several generations to come."---A. Bultheel, European Mathematical Society

"Any young person seeking a career in the sciences would do well to ponder this important contribution to the teacher's art."---A. C. Schaeffer, American Journal of Psychology

"I recommend it highly to any person who is seriously interested in finding out methods of solving problems, and who does not object to being entertained while he does it."

"Every mathematics student should experience and live this book"

"[This] elementary textbook on heuristic reasoning, shows anew how keen its author is on questions of method and the formulation of methodological principles. Exposition and illustrative material are of a disarmingly elementary character, but very carefully thought out and selected."---Herman Weyl, Mathematical Review

"Every prospective teacher should read it. In particular, graduate students will find it invaluable. The traditional mathematics professor who reads a paper before one of the Mathematical Societies might also learn something from the book: 'He writes a, he says b, he means c; but it should be d.'"---E. T. Bell, Mathematical Monthly

Topics

Frontmatter Publicly Available Download PDF	i
From the Preface to the First Printing Publicly Available Download PDF	v
From the Preface to the Seventh Printing Publicly Available Download PDF	viii
Preface to the Second Edition Publicly Available Download PDF	ix
Contents Publicly Available Download PDF	xi
"How to Solve It" list Publicly Available Download PDF	xvi
Foreword John H. Conway Publicly Available Download PDF	xix
Introduction Publicly Available Download PDF	xxv
Part I. In the Classroom
Purpose Requires Authentication Unlicensed Licensed Download PDF	1
Main divisions, main questions Requires Authentication Unlicensed Licensed Download PDF	5
More examples Requires Authentication Unlicensed Licensed Download PDF	23
Part II. How to Solve it
A dialogue Requires Authentication Unlicensed Licensed Download PDF	33
Part III. Short Dictionary of Heuristic
Analogy Requires Authentication Unlicensed Licensed Download PDF	37
Auxiliary elements Requires Authentication Unlicensed Licensed Download PDF	46
Auxiliary problem Requires Authentication Unlicensed Licensed Download PDF	50
Bolzano Requires Authentication Unlicensed Licensed Download PDF	57
Bright idea Requires Authentication Unlicensed Licensed Download PDF	58
Can you check the result? Requires Authentication Unlicensed Licensed Download PDF	59
Can you derive the result differently? Requires Authentication Unlicensed Licensed Download PDF	61
Can you use the result? Requires Authentication Unlicensed Licensed Download PDF	64
Carrying out Requires Authentication Unlicensed Licensed Download PDF	68
Condition Requires Authentication Unlicensed Licensed Download PDF	72
Contradictory Requires Authentication Unlicensed Licensed Download PDF	73
Corollary Requires Authentication Unlicensed Licensed Download PDF	73
Could you derive something useful from the data? Requires Authentication Unlicensed Licensed Download PDF	73
Could you restate the problem? Requires Authentication Unlicensed Licensed Download PDF	75
Decomposing and recombining Requires Authentication Unlicensed Licensed Download PDF	75
Definition Requires Authentication Unlicensed Licensed Download PDF	85
Descartes Requires Authentication Unlicensed Licensed Download PDF	92
Determination, hope, success Requires Authentication Unlicensed Licensed Download PDF	93
Diagnosis Requires Authentication Unlicensed Licensed Download PDF	94
Did you use all the data? Requires Authentication Unlicensed Licensed Download PDF	95
Do you know a related problem? Requires Authentication Unlicensed Licensed Download PDF	98
Draw a figure Requires Authentication Unlicensed Licensed Download PDF	99
Examine your guess Requires Authentication Unlicensed Licensed Download PDF	99
Figures Requires Authentication Unlicensed Licensed Download PDF	103
Generalization Requires Authentication Unlicensed Licensed Download PDF	108
Have you seen it before? Requires Authentication Unlicensed Licensed Download PDF	110
Here is a problem related to yours and solved before Requires Authentication Unlicensed Licensed Download PDF	110
Heuristic Requires Authentication Unlicensed Licensed Download PDF	112
Heuristic reasoning Requires Authentication Unlicensed Licensed Download PDF	113
If you cannot solve the proposed problem Requires Authentication Unlicensed Licensed Download PDF	114
Induction and mathematical induction Requires Authentication Unlicensed Licensed Download PDF	114
Inventor's paradox Requires Authentication Unlicensed Licensed Download PDF	121
Is it possible to satisfy the condition? Requires Authentication Unlicensed Licensed Download PDF	122
Leibnitz Requires Authentication Unlicensed Licensed Download PDF	123
Lemma Requires Authentication Unlicensed Licensed Download PDF	123
Look at the unknown Requires Authentication Unlicensed Licensed Download PDF	123
Modern heuristic Requires Authentication Unlicensed Licensed Download PDF	129
Notation Requires Authentication Unlicensed Licensed Download PDF	134
Pappus Requires Authentication Unlicensed Licensed Download PDF	141
Pedantry and mastery Requires Authentication Unlicensed Licensed Download PDF	148
Practical problems Requires Authentication Unlicensed Licensed Download PDF	149
Problems to find, problems to prove Requires Authentication Unlicensed Licensed Download PDF	154
Progress and achievement Requires Authentication Unlicensed Licensed Download PDF	157
Puzzles Requires Authentication Unlicensed Licensed Download PDF	160
Reductio ad absurdum and indirect proof Requires Authentication Unlicensed Licensed Download PDF	162
Redundant Requires Authentication Unlicensed Licensed Download PDF	171
Routine problem Requires Authentication Unlicensed Licensed Download PDF	171
Rules of discovery Requires Authentication Unlicensed Licensed Download PDF	172
Rules of style Requires Authentication Unlicensed Licensed Download PDF	172
Rules of teaching Requires Authentication Unlicensed Licensed Download PDF	173
Separate the various parts of the condition Requires Authentication Unlicensed Licensed Download PDF	173
Setting up equations Requires Authentication Unlicensed Licensed Download PDF	174
Signs of progress Requires Authentication Unlicensed Licensed Download PDF	178
Specialization Requires Authentication Unlicensed Licensed Download PDF	190
Subconscious work Requires Authentication Unlicensed Licensed Download PDF	197
Symmetry Requires Authentication Unlicensed Licensed Download PDF	199
Terms, old and new Requires Authentication Unlicensed Licensed Download PDF	200
Test by dimension Requires Authentication Unlicensed Licensed Download PDF	202
The future mathematician Requires Authentication Unlicensed Licensed Download PDF	205
The intelligent problem-solver Requires Authentication Unlicensed Licensed Download PDF	206
The intelligent reader Requires Authentication Unlicensed Licensed Download PDF	207
The traditional mathematics professor Requires Authentication Unlicensed Licensed Download PDF	208
Variation of the problem Requires Authentication Unlicensed Licensed Download PDF	209
What is the unknown? Requires Authentication Unlicensed Licensed Download PDF	214
Why proofs? Requires Authentication Unlicensed Licensed Download PDF	215
Wisdom of proverbs Requires Authentication Unlicensed Licensed Download PDF	221
Working backwards Requires Authentication Unlicensed Licensed Download PDF	225
Part IV. Problems, Hints, Solutions
Problems Requires Authentication Unlicensed Licensed Download PDF	234
Hints Requires Authentication Unlicensed Licensed Download PDF	238
Solutions Requires Authentication Unlicensed Licensed Download PDF	242