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How to Solve It; A New Aspect of Mathematical Method (Princeton Science Library) ~ Usually ships in 24 hours

George Polya, Gyorgy Polya / Paperback / Published 1988


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George Polya(1887-1985) is a famous mathematician, who get importance results in probability, analysis, number theory, geometry, combinatorics and mathematical physics. His book "How to Solve It" was probably the most significant contribution to heuristic since Descartes' Discourse on Method. In it he structured the problem-solving process into four stages:

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  Understanding the Problem

  1. Devising a Plan
  2. Carrying out the Plan
  3. Looking Back

For each stage George Polya supplied a series of questions that help to solve the problem. Some of them are:

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  What is the unknown?

  1. what are the data?
  2. Could you restate the problem?
  3. Is there a related problem that has been solved before?

More full list of questions can be found in http://www.math.utah.edu/~alfeld/math/polya.html. The examples in the book are drawn mostly from elementary math, but the method is quite general and applies to nearly every problem one might encounter. I read it when I was 15 years old and this was the only mathematical book that I really liked. It contains really amazing insights into problem solving skills. BTW  it was George Polya, who said "There are many questions which fools can ask that wise men cannot answer." (See H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988).

IMHO the book is much more valuable to programmers than all modern "pattern" stuff (although the latter is useful too).  The only review of the book that I found was a review in DDJ.  BTW Microsoft used to give this book to all of its new programmers. Probably not any more ;-).

See also: Synthesis of Research on Problem Solving.

27-190 Discrete Structures in Computer Science Techniques of Proof and Problem Solving

Some famous quotations that belongs to George Polya: 

Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.

D. J. Albers and G. L. Alexanderson, Mathematical People, Boston: Birkhäuser, 1985.

Mathematics consists of proving the most obvious thing in the least obvious way.

In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.

The American Mathematical Monthly, v. 100, no. 3.

The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face the blackboard and to turn his back to the class. He writes a, he says b, he means c; but it should be d. Some of his sayings are handed down from generation to generation:

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  "In order to solve this differential equation you look at it till a solution occurs to you."

  1. "This principle is so perfectly general that no particular application of it is possible."
  2. "Geometry is the science of correct reasoning on incorrect figures."
  3. "My method to overcome a difficulty is to go round it."
  4. "What is the difference between method and device? A method is a device which you used twice."

How to Solve It. Princeton: Princeton University Press. 1945.

Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. ... A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.

How to Solve It. Princeton: Princeton University Press. 1945

Last modified: February 28, 2008