G. Pólya, How to Solve It
Summary taken from G. Pólya, "How to Solve It", 2nd ed.,
Princeton University Press, 1957, ISBN 0-691-08097-6.
- UNDERSTANDING THE PROBLEM
- First. You have to understand the
- What is the unknown? What are the data? What is the
condition? What needs to be found?
- 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
- 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 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
- 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
- Can you check the result? Can you check the
- Can you derive the solution differently? Can you see
it at a glance?
- Can you use the result, or the method, for some other