[SOLVED] How To Solve It G. Polya

How To Solve It G. Polya

Understanding 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?

Devising A Plan

Second.

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 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 known 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

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?

Looking Back

Fourth.

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?