Table of Contents
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