· Summary
taken from G. Polya, “How to Solve It”, 2nd ed., Princeton
University Press, 1957, ISBN 0-691-08097-6.
1.
UNDERSTAND THE PROBLEM
· First. You
have to understand the problem.
· What is the
unknown? What are the data? What is the condition?
·
Is it possible to satisfy the condition?
Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
· Draw a
figure. Introduce suitable notation.
· Separate the
various parts of the condition. Can you
write them down?
2. DEVISING
A PLAN
· Second. Find the connection between the data and the
unknown. You may be obliged to consider
the auxiliary problems if an immediate connection cannot be found. You should eventually obtain a plan
of the solution.
·
Have you seen it before? Or have you
seen the same problem in a slightly different form?
· 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
similar unknown.
· 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 then the unknown?
Could you change the unknown or data, or both if necessary, so that the
new unknown and the new data are nearer to each other?
·
Did you use all of the data? Did you
use the whole condition? Have you taken
into account all essential notions involved in the problem?
3. CARRYING OUT THE PLAN
· Third. Carry out your 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?
4. LOOKING
BACK
· Fourth. Examine the solution obtained.
· Can you check
the result? Can you check the
argument?
· Can you derive
the solution differently? Can you see
it at a glance?
· Can you use the
result, or the method, for some other problem?
Obtained from http://www.math.utah.edu/~alfeld/math/polya.html