Problem Solving Heuristics

We confront problems all the time in our lives, and success in life is largely determined by how we confront and resolve these problems. Thus, it’s important to know how to approach problem solving.That’s a big issue that I don’t feel capable of tackling, so let’s consider a more specialized question: How does someone approach solving a mathematical problem? From answers to this question, we can hopefully extrapolate about the real world.

The most important skill in mathematical problem solving isn’t knowledge of theorems and facts; instead, it’s more important to have the right mindset for approaching a problem. Here’s some general advice:

Before beginning to approach a mathematical problem, the best problem solvers are psychologically prepared. They are confident in their own ability to solve problems, and they are not worried about failure. In the meantime, they are willing to try new and creative attacks. They are persistent and never give up, and they are also courageous — willing to try any technique that might seem to succeed.

  • Be confident. Believe in yourself.
  • Be creative.
  • Be persistent.
  • Be courageous: “Fearless courage is the foundation to all success.”
  • If you worry then you will fail… so don’t worry!
  • Be brave and try it.

When confronted with a problem, first try to understand it. There are a number of typical strategies for doing this:

  • What are we trying to find?
  • What information are we given?
  • What conditions do we have on the problem?
  • Get your hands dirty: plug in numbers, draw a diagram, do some simple calculations.

Then, try to find a solution.

  • Try some special cases: plug in simple numbers, extremal values, guess and check.
  • Estimate. Order of magnitude estimate, draw a careful diagram.
  • Choose convenient notation.
  • Pursue symmetry, in geometry, in algebraic expressions, and in notation.
  • Find a pattern, make a hypothesis. Can you prove it?
  • Work backwards, go back to the definitions, wishful thinking.
  • Have you used all of the data and conditions? Can you do a more general problem?
  • Change the data and the conditions. How much can they vary?
  • Have you seen the problem before? Do you know of a related problem?
  • Look at the problem in a different way: convert between different topics in math.
  • Divide and conquer: Separate into easier subproblems and attack each separately.
  • “If you can’t solve a problem, then there is an easier problem you can solve: find it.”

Once you’ve solved the problem, check your answer.

  • Check your work. Are you sure that every step is correct? Is every step valid?
  • Estimate. Is the answer reasonable?
  • Do you see another way to solve the problem? Does it give the same result?

These problem solving heuristics do not consider any particular formula, theorem, or method; they apply to every problem. By being independent of specific techniques, they transcend such techniques and become universal; some of these heuristics (“Be brave and try it”, for example) could also apply to the real world.

For more information on problem solving heuristics, see George Pölya’s classic book How to Solve It. I leave you with a quotation from his book:

A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.
— George Pölya, How to Solve It

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One response to “Problem Solving Heuristics

  1. Elliott December 20, 2010 at 6:01 pm

    Excellent post! Some more things you can do after you’ve solved the problem:
    – See what happens when you relax some of the hypotheses
    – Check if the dimensions make sense (ok ok this one is for physics, not math problems)
    – Consider limiting cases and see if it matches your physical intuition (same comment)

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