How to Solve It
(Princeton University Press
is a wonderful little book by
mathematician George Pólya
. It is
a hard problem
and is cited by almost anyone studying the science/art
of problem solving
. His program is useful for any
HOW TO SOLVE IT, G. Pólya - Summary
Understanding the Problem
Devising a Plan
- Find the connection between the data and the unknown. You may be obliged to
consider auxiliary problems if an immediate connection cannot be found. You
should obtain eventually a plan of the solution.
- Have you seen it before? Or have you seen the same problem in a slightly
- Do you know a related problem? Do you know a theorem that could be useful?
- Look at the unknown! And try to think of a familiar problem having the same or a
- Here is a problem related to yours and solved before. Could you use it? Could
you use its result? Could you use its method? Should you introduce some
auxiliary element in order to make its use possible?
- Could you restate the problem? Could you restate it still differently?
Go back to definitions.
- If you cannot solve the proposed problem try to solve first some related
problem. Could you imagine a more accessible related problem? A more general
problem? A more special problem? An analogous problem? Could you solve a part of
the problem? Keep only a part of the condition, drop the other part; how far is
the unknown then determined, how can it vary? Could you derive something useful
from the data? Could you think of other data appropriate to determine the
unknown? Could you change the unknown or the data, or both if necessary, so that
the new unknown and the new data are nearer to each other?
- Did you use all the data? Did you use the whole condition? Have you taken into
account all essential notions involved in the problem?
Carrying Out the Plan
- Carrying out your plan of the solution, check each step. Can you see clearly
that the step is correct? Can you prove that it is correct?