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.
The tools he employs systematically are two:
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!
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
PART I. IN THE CLASSROOM
1. Helping the student
2. Questions, recommendations, mental operations
4. Common sense
5. Teacher and student. Imitation and practice
Main divisions, main questions
6. Four phases
7. Understanding the problem
9. Devising a plan
11. Carrying out the plan
13. Looking back
15. Various approaches
16. The teacher’s method of questioning
17. Good questions and bad questions
18. A problem of construction
19. A problem to prove
20. A rate problem
PART II. HOW TO SOLVE IT
PART III. SHORT DICTIONARY OF HEURISTIC
Can you check the result?
Can you derive the result differently?
Can you use the result?
Could you derive something useful from the data?
Could you restate the problem?†
Decomposing and recombining
Determination, hope, success
Did you use all the data?
Do you know a related problem?
Draw a figure†
Examine your guess
Have you seen it before?
Here is a problem related to yours and solved before
If you cannot solve the proposed problem
Induction and mathematical induction
Is it possible to satisfy the condition?
Look at the unknown
Pedantry and mastery
Problems to find, problems to prove
Progress and achievement
Reductio ad absurdum and indirect proof
Rules of discovery
Rules of style
Rules of teaching
Separate the various parts of the condition
Setting up equations
Signs of progress
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?
Wisdom of proverbs
PART IV. PROBLEMS, HINTS, SOLUTIONS