This schema is adapted from the book How to Solve
It, by George
Pòlya, published by Princeton University Press. Everyone who reads
this page should seriously consider buying the book. I consider this "fair
use" as an advertisement, rather than a copyright infringement. Please
don't sue me.
1. Understanding the Problem.
First you have to understand the
- 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.
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 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
- Go back to the 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?
Can 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
- Did you use all 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.
Carry out your
- 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
4. Looking back.
Examine the solution obtained.
- Can you check the result? Can you check the argument?
- Can you derive the result differently? Can you see it at a glance?
- Can you use the result, or the method, for some other problem?