**George Polya's tips for problem solving: **
(from *How to Solve It*)

### HOW TO SOLVE IT

##### 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 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

**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 arguments?

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?