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2021 November 10 |
Problem solving

This book is an **essential tool** **for** anyone who intends to approach systematically **problem solving** related to the most diverse situations.

The author is **George Polya**, a Hungarian mathematician who taught first at ETH Zürich in Switzerland and then at Stanford University in the U.S.; **the book** was first published in 1945 and represents **the first** important **effort to teach a method for solving problems.**

How does Professor Polya approach the task?

He starts from his experiences in teaching mathematics to **identify first the mental processes** and **then the strategies** that can help us to give a solution to a complex situation.

who are not familiar with mathematics.

The tools he employs systematically are two:

**questions**, which are**essential to remove preconceptions and expand**the space for**investigation**. As you go through the book, you will often encounter questions such as:*What is the unknown?**What information is available?**Do you know of a related problem for which the solution is known?*

**heuristics**,**strategies that help solve problems and discover new insights**, valuable in the**innovation of any kind**. Finding new solutions by analogy, through the introduction of auxiliary variables, or by reducing the main problem into problems of lower complexity and easier solution, are examples of heuristic strategies beautifully explained in the book.

As you can see below in the table of contents, you will find **plenty of practical examples** in the book, with **exercises and solutions. **

I found on YouTube a video from 1966 in which Professor Polya himself explains his strategies in problem-solving: here it is!

I recommend reading the book to a large audience:

- teachers;
- those who deal with problem solving and innovation for professional reasons;
- people who simply want to improve their thinking effectiveness.

However, others will certainly not mind.

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

**PART I. 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*

6. Four phases

7. Understanding the problem

8. Example

9. Devising a plan

10. Example

11. Carrying out the plan

12. Example

13. Looking back

14. Example

15. Various approaches

16. The teacher’s method of questioning

17. Good questions and bad questions

*More examples*

18. A problem of construction

19. A problem to prove

20. A rate problem

**PART II. HOW TO SOLVE IT**

A dialogue

**PART III. 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 data?

Could you restate the problem?†

Decomposing and recombining

Definition

Descartes

Determination, hope, success

Diagnosis

Did you use all the data?

Do you know a related problem?

Draw a figure†

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

Redundant†

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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