Polya goes into great depths of revealing the intrinsic notions of
heuristics and analogy and the role they played in solving problems of
mathematical and non mathematical nature. In this journey Polya
wanted to remind us that asking the right questions is even more
important than the solution itself. According to Polya there are 4
steps we should follow in order to solve our problems:
1. Understanding the problem
To understand the problem
What is the unknown?
ask questions.
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.
Have you seen it before?
Consider auxiliary problems if an immediate connection
Or have you seen the same problem in a slightly different form?
cannot be found.
Do you know a related problem?
Eventually obtain a plan of the solution.
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 part of the problem?
Keep only a part of the condition, drop the other part; how far is 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 unkown and the new data are nearere 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?
3. Carrying out the plan
Carry out the plan
Carrying out the 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
Examine the
obtained solution
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?
