by [anonymous]
4 min read8th Jan 20114 comments


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 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?
  • Find a way to visualize the problem.
  • Introduce suitable notation.
    • Good notation should be unambiguous, pregnant, easy to remember, and easy to recognize.
    • Good notation should avoid harmful second meanings and take advantage of useful second meanings.
    • The order and connection of signs should suggest the order and connection of things.
  • Separate the various parts of the condition. Can you write them down?

Devising a plan

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

  • Have you seen the problem before?
  • Have you seen it in another 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?
    • Can you introduce an auxiliary element to make its use possible?
    • Can you use it to make plausible conjectures?
  • Can you restate the problem?
  • Go back to definitions.
  • Can you imagine a more accessible related problem?
  • A more general problem?
  • A more special problem?
  • An analogous problem?
  • Try varying the problem to facilitate new associations with past knowledge. Varying the problem can also help one maintain interest.
  • Can you solve part of the problem?
  • Keep only 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 and/or the data so that they are closer 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?
  • Can you generalize from a consideration of special cases?
  • Can you refute a conjecture by considering special cases?
  • How can you attain a result of this kind?
  • What causes produce such a result?
  • What do people normally do to obtain such a result?
  • Persevere through unsuccess, appreciate small advances, wait for the essential idea, and then concentrate fully when it appears.

Carrying out the plan

Carry out your plan.

  • We may use heuristic arguments when devising formal arguments as we use scaffolding to support a bridge during construction.
  • When devising a plan of the solution don't be afraid of using heuristic arguments; anything is right that leads to the right idea.
  • Can you see clearly that each step is correct?
  • Can you prove it is correct?
  • 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.
  • The absence of signs helps save us effort while their presence can cause us to correctly concentrate our effort.
  • It takes experience to learn to interpret signs correctly.

Looking back

Examine the solution obtained.

  • Can you check the result?
  • Consider special cases of the result to see it they make sense.
  • Can you check the argument?
  • Try to examine the weakest point of the argument first.
  • Introduce variation in your review of the problem to avoid stumbling in the same places.
  • Can you derive the result differently?
  • Try interpreting parts of the result differently. This may lead to a larger re-interpretation that inspires a different derivation.
  • Can you see it at a glance?
  • Can you use the result, or the method, for some other problem?
  • Create new related problems through generalization, specialization, analogy, and decomposing and recombining that may be solved similarly.

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4 comments, sorted by Click to highlight new comments since: Today at 8:11 AM

Thanks for the distillation. Polya's been on my reading list for a long time, but is still not nearing the top, and I appreciate the useful preview.

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.)

I appreciate the explanation, especially when considering a math context (which is the intended context anyway, but I was thinking generally with my comment).