### 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
Introduction

## PART 1. IN THE CLASSROOM

Purpose
1. Helping the student
2. Questions, recommendations, mental operations
3. Generality
4. Common sense
5. Teacher and student. Imitation and practice
Main divisions, main questions
1. Four phases
2. Understanding the problem
3. Example
4. Devising a plan
5. Example
6. Carrying out the plan
7. Example
8. Looking back
9. Example
10. Various approaches
11. The teacher's method of questioning
12. Good questions and bad questions
More examples
1. A problem of construction
2. A problem to prove
3. A rate problem

A dialogue <

## PART Ill. SHORT DICTIONARY OF HEURISTIC

Analogy
Auxiliary elements
Auxiliary problem
Bolzano
Bright idea
Can you check the result?
Can you derive the result differently?
Can you use the result?
Carrying out
Condition
Corollary
Could you derive something useful from the dat
Could you restate the problem?t
Decomposing and recombining
Definition
Descartes
Determination, hope, success
Diagnosis
Did you use all the data?
Do you know a related problem?
Draw a figuret
Figures
Generalization
Have you seen it before?
Here is a problem related to yours and solved before
Heuristic
Heuristic reasoning
If you cannot solve the proposed problem
Induction and mathematical induction
Is it possible to satisfy the condition?
Leibnitz
Lemma
Look at the unknown
Modern heuristic
Notation
Pappus
Pedantry and mastery
Practical problems
Problems to find, problems to prove
Progress and achievement
Puzzles
Reductio ad absurdum and indirect proof
Redundantt
Routine problem
Rules of discovery
Rules of style
Rules of teaching
Separate the various parts of the condition
Setting up equations
Signs of progress
Specialization
Subconscious work
Symmetry
Terms, old and new
Test by dimension
The future mathematician
The intelligent problem-solver
Variation of the problem
What is the unknown?
Why proofs?
Wisdom of proverbs
Working backwards

## PART IV. PROBLEMS, HINTS, SOLUTIONS

Problems
Hints
Solutions 