How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library)

Cover......Page 1
Title......Page 4
Copyright......Page 5
Contents......Page 12
From the Preface to the First Printing......Page 6
From the Preface to the Seventh Printing......Page 9
Preface to the Second Edition......Page 10
"How to Solve It" list......Page 17
Foreword......Page 20
Introduction......Page 26
2. Questions, recommendations, mental operations......Page 32
3· Generality......Page 33
5· Teacher and student. Imitation and practice......Page 34
6. Four phases......Page 36
7· Understanding the problem......Page 37
8. Example......Page 38
9. Devising a plan......Page 39
10. Example......Page 41
11. Carrying out the plan......Page 43
12. Example......Page 44
13· Looking back......Page 45
14· Example......Page 47
15· Various approaches......Page 50
16. The teacher's method of questioning......Page 51
17· Good questions and bad questions......Page 53
18. A problem of construction......Page 54
19. A problem to prove......Page 56
20. A rate problem......Page 60
A dialogue......Page 64
Analogy......Page 68
Auxiliary elements......Page 77
Auxiliary problem......Page 81
Bolzano......Page 88
Bright idea......Page 89
Can you check the result?......Page 90
Can you derive the result differently?......Page 92
Can you use the result?......Page 95
Carrying out......Page 99
Condition......Page 103
Could you derive something useful from the data?......Page 104
Decomposing and recombining......Page 106
Definition......Page 116
Descartes......Page 123
Determination, hope, success......Page 124
Diagnosis......Page 125
Did you use all the data?......Page 126
Do you know a related problem?......Page 129
Examine your guess......Page 130
Figures......Page 134
Generalization......Page 139
Here is a problem related to yours and solved before......Page 141
Heuristic......Page 143
Heuristic reasoning......Page 144
Induction and mathematical induction......Page 145
Inventor's paradox......Page 152
Is it possible to satisfy the condition?......Page 153
Look at the unknown......Page 154
Modern heuristic......Page 160
Notation......Page 165
Pappus......Page 172
Pedantry and mastery......Page 179
Practical problems......Page 180
Problems to find, problems to prove......Page 185
Progress and achievement......Page 188
Puzzles......Page 191
Reductio ad absurdum and indirect proof......Page 193
Routine problem......Page 202
Rules of style......Page 203
Separate the various parts of the condition......Page 204
Setting up equations......Page 205
Signs of progress......Page 209
Specialization......Page 221
Subconscious work......Page 228
Symmetry......Page 230
Terms, old and new......Page 231
Test by dimension......Page 233
The future mathematician......Page 236
The intelligent problem-solver......Page 237
The intelligent reader......Page 238
The traditional mathematics professor......Page 239
Variation of the problem......Page 240
What is the unknown?......Page 245
Why proofs?......Page 246
Wisdom of proverbs......Page 252
Working backwards......Page 256
PART IV. PROBLEMS, HINTS, SOLUTIONS......Page 264
Problems......Page 265
Hints......Page 269
Solutions......Page 273