From the Preface to the First Printing 
v 
From the Preface to the Seventh Printing 
viii 
Preface to the Second Edition 
ix 
"How To Solve It" list 
xvi 
Introduction 
xix 

PART I. IN THE CLASSROOM 
Purpose 
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 

Main divisions, main questions 
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 

More examples 
18. A problem of construction 
23 
19. A problem to prove 
25 
20. A rate problem 
29 

PART II. HOW TO SOLVE IT 
A dialogue 
33 

PART 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 (contains only crossreferences) 
73 
Corollary 
73 
Could you derive something useful from the data? 
73 
Could you restate the problem? (contains only crossreferences) 
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 (contains only crossreferences) 
99 
Examine your guess 
99 
Figures 
108 
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 
l23 
Look at the unknown 
120 
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 (contains only crossreferences) 
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 problemsolver 
206 
The intelligent reader 
207 
The traditional mathematics professor 
208 
Variation of the problem 
209 
What is the unknown? 
214 
Why proofs? 
213 
Wisdom of proverbs 
221 
Working backwards 
223 

PART IV. PROBLEMS, HINTS, SOLUTIONS 
Problems 
234 
Hints 
238 
Solutions 
242 