How to Solve it by Polya - Summary

How to Solve it - by Polya

A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive facilities, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.

First principle: Understand the problem - Most obvious and most neglected Can you state the problem in your own words? Do you understand all the words used in stating the problem? What are you trying to find or do? What are the unknowns? What information do you obtain from the problem? Is there enough information to enable find the solution? Is any further information needed? Can you think of a picture or diagram that might help you understand the problem? Do you need to ask the question to get the answer?

Devising a Plan Make a guess and check. Make an orderly list. Look for a pattern. Eliminate possibilities Use direct reasoning - Can you find a problem related to yours that has already been solved and use that to solve your problem? Use an anology - Can you find a problem analogous to your problem and solve that? Solve a simpler problem - Can you find a subproblem or side problem whose solution will help you solve your problem? Divide and conquer. Consider special cases. Make a table. Make a diagram. Use a model - Can you find a problem more general than your problem? Can you add some new element to your problem to get closer to a solution? Work backward. Make the system fail Quit Thinking and Look

Carrying out the Plan Implement the strategy. Change one thing at a time. Check each step of the plan as you proceed. Keep an accurate record of your work. (An audit trail). Check the plug. Check for elementary stuff.

Looking Back Examine the results and verify. Interpret the solution. Does your answer make sense? Is it reasonable? Determine whether there is another method of finding the solution. Get a fresh view. If you haven’t fixed, it ain’t fixed.

Quite often, when an idea that could be helpful presents itself, we do not appreciate it, for it is so inconspicuous. The expert has, perhaps, no more ideas than the inexperienced, but appreciates more what he has and uses it better. - George Polya, How to Solve it: A New Aspect of Mathematical Method (1957)

Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice. . . . if you wish to learn swimming you have to go in the water, and if you wish to become a problem solver you have to solve problems.- Mathematical Discovery