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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.
"[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
"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."--Scientific Monthly
"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
"Every mathematics student should experience and live this book"--Mathematics Magazine
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 | ||
Pt. 1 | 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 |
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 |
12 | Example | 13 |
13 | Looking back | 14 |
14 | Example | 16 |
15 | Various approaches | 19 |
16 | The teacher's method of questioning | 20 |
17 | Good questions and bad questions | 22 |
18 | A problem of construction | 23 |
19 | A problem to prove | 25 |
20 | A rate problem | 29 |
Pt. II | How to solve it | |
A dialogue | 33 | |
Pt. 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 problems? | 75 | |
Decomposing and recombining | 75 | |
Definition | 85 | |
Descartes | 92 | |
Determination, hope, success | 93 | |
Diagnosis | 94 | |
Did you use all the data? | 95 | |
Do you know a related problem? | 98 | |
Draw a figure | 99 | |
Examine your guess | 99 | |
Figures | 103 | |
Generalization | 108 | |
Have you seen it before? | 110 | |
Here is a problem related to yours and solved before | 110 | |
Heuristic | 112 | |
Heuristic reasoning | 113 | |
If you cannot solve the proposed problem | 114 | |
Induction and mathematical induction | 114 | |
Inventor's paradox | 121 | |
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 | 160 | |
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 | |
Specialization | 190 | |
Subconscious work | 197 | |
Symmetry | 199 | |
Terms, old and new | 200 | |
Test by dimension | 202 | |
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 | |
Pt. IV | Problems, hints, solutions | |
Problems | 234 | |
Hints | 238 | |
Solutions | 242 |
Overview
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...