Analogy

Kapitel in diesem Buch

1. Frontmatter i
2. From the Preface to the First Printing v
3. From the Preface to the Seventh Printing viii
4. Preface to the Second Edition ix
5. Contents xi
6. "How to Solve It" list xvi
7. Foreword xix
8. Introduction xxv
9. Part I. In the Classroom
10. Purpose 1
11. Main divisions, main questions 5
12. More examples 23
13. Part II. How to Solve it
14. A dialogue 33
15. Part III. Short Dictionary of Heuristic
16. Analogy 37
17. Auxiliary elements 46
18. Auxiliary problem 50
19. Bolzano 57
20. Bright idea 58
21. Can you check the result? 59
22. Can you derive the result differently? 61
23. Can you use the result? 64
24. Carrying out 68
25. Condition 72
26. Contradictory 73
27. Corollary 73
28. Could you derive something useful from the data? 73
29. Could you restate the problem? 75
30. Decomposing and recombining 75
31. Definition 85
32. Descartes 92
33. Determination, hope, success 93
34. Diagnosis 94
35. Did you use all the data? 95
36. Do you know a related problem? 98
37. Draw a figure 99
38. Examine your guess 99
39. Figures 103
40. Generalization 108
41. Have you seen it before? 110
42. Here is a problem related to yours and solved before 110
43. Heuristic 112
44. Heuristic reasoning 113
45. If you cannot solve the proposed problem 114
46. Induction and mathematical induction 114
47. Inventor's paradox 121
48. Is it possible to satisfy the condition? 122
49. Leibnitz 123
50. Lemma 123
51. Look at the unknown 123
52. Modern heuristic 129
53. Notation 134
54. Pappus 141
55. Pedantry and mastery 148
56. Practical problems 149
57. Problems to find, problems to prove 154
58. Progress and achievement 157
59. Puzzles 160
60. Reductio ad absurdum and indirect proof 162
61. Redundant 171
62. Routine problem 171
63. Rules of discovery 172
64. Rules of style 172
65. Rules of teaching 173
66. Separate the various parts of the condition 173
67. Setting up equations 174
68. Signs of progress 178
69. Specialization 190
70. Subconscious work 197
71. Symmetry 199
72. Terms, old and new 200
73. Test by dimension 202
74. The future mathematician 205
75. The intelligent problem-solver 206
76. The intelligent reader 207
77. The traditional mathematics professor 208
78. Variation of the problem 209
79. What is the unknown? 214
80. Why proofs? 215
81. Wisdom of proverbs 221
82. Working backwards 225
83. Part IV. Problems, Hints, Solutions
84. Problems 234
85. Hints 238
86. Solutions 242