### 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*

- Helping the student
- Questions, recommendations, mental operations
- Generality
- Common sense
- Teacher and student. Imitation and practice

*Main divisions, main questions*

- Four phases
- Understanding the problem
- Example
- Devising a plan
- Example
- Carrying out the plan
- Example
- Looking back
- Example
- Various approaches
- The teacher's method of questioning
- Good questions and bad questions

*More examples*

- A problem of construction
- A problem to prove
- A rate problem

## PART II. HOW TO SOLVE IT

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

Contradictory

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

Examine your guess

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

Inventor's paradox

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

The intelligent reader

The traditional mathematics professor

Variation of the problem

What is the unknown?

Why proofs?

Wisdom of proverbs

Working backwards

## PART IV. PROBLEMS, HINTS, SOLUTIONS

Problems

Hints

Solutions

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Copyright © 1996-2017 Alexander Bogomolny