How to Solve It

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.

Strategies that can be adopted even by people
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.

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

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