| 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 cross-references) | 73 |
| Corollary | 73 |
| Could you derive something useful from the data? | 73 |
| Could you restate the problem? (contains only cross-references) | 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 cross-references) | 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 cross-references) | 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? | 213 |
| Wisdom of proverbs | 221 |
| Working backwards | 223 |
| PART IV. PROBLEMS, HINTS, SOLUTIONS | |
| Problems | 234 |
| Hints | 238 |
| Solutions | 242 |