Problem Solving via Polya

8th Jan 2011

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Very cool. Some of those questions seem a little redundant, such as:

Have you seen it in another form?

An analogous problem?

Perhaps not the same, but reading the list made me wonder if it could be "simmered" a bit to distill the key points. In particular, I really liked the *Looking Back* section. Absolutely wonderful. It reminds me of my own post as well as many other LW posts: not attacking the strong points of a theory, but the weakest, being careful to avoid leaky generalizations, really knowing the purpose of your actions, internalizing vs. parroting, and not being so quick to assume you've thought of all the options.

I think the last section is a great set of questions to ask after coming to *any* decision and is certainly not isolated to mathematics! It, combined with the rest, seems like a nice recipe for both internalizing one's methods and data as well as trying to avoid duplicating efforts on related/similar issues. Thanks for sharing.

Very cool. Some of those questions seem a little redundant, such as:

Have you seen it in another form?

An analogous problem?

These aren't redundant in the context that Polya is talking about. In math, these are different. The first means the same problem but with different notation or some equivalent problem. The second means a problem that is similar in some way (say for example something over a finite field having an analog over the real numbers or rationals.)

Related to: Tips and Tricks for Answering Hard Questions

In How To Solve It Polya describes methods and heuristics intended to facilitate the solution of math problems. These are mostly conveyed in the form of self-questions that are aimed at inducing useful mental procedures, and subsequently developing awesome problem solving dispositions. Ultimately we should work from these dispositions directly. Polya advises us to use the questions only when progress is blocked; at other times our thoughts should flow naturally from our dispositions. I expect that his methods are useful outside of mathematics, and thought they might be of interest to people here.

Below is the summary given at the start of How To Solve It (with the exception of a few added notes). He breaks the problem solving process into four steps, with each step having a set of self-questions and heuristics. I've bolded parts that I thought were particularly useful. This is not meant to be an alternative to reading the book; I expect that reading his illustrative examples is somewhat important. But more important is working with these questions on problems in order to develop your own dispositions.

Understanding the problem

You have to

understandthe problem.Find a way to visualize the problem.Introduce suitable notation.Devising a plan

Find the connection between the data and the unknown. You may need to consider auxiliary problems. You should eventually obtain a

planof the solution.Here is a problem related to yours and solved before. Could you use it?Can you restate the problem?Can you imagine a more accessible related problem?Have you taken into account all essential notions involved in the problem?Carrying out the plan

Carry outyour plan.Try to prove formally what is seen intuitively and see intuitively what is proved formally.Progress is the mobilization and organization of our knowledge, the evolution of our conception of the problem, and increasing certainty of the solution plan.An increase in the completion of the connection between the data and the unknown is a sign of progress.Looking back

Examinethe solution obtained.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?