Polya's four steps to solving a problem


George Polya (1887-1985), a Hungarian mathematician, wrote "How to solve it." for high school students in 1957.

Here is his four step method.

Understand the problem
 Read the problem over carefully and ask yourself:  Do I know the meaning of all the words?  What is being asked for?  What is given in the problem?  Is the given information sufficient (for the solution to be unique)?  Is there some inconsistent or superfluous information which is given? By way of checking your understanding, try restating the problem in a different way.
Design a plan for solving the problem

In essence, decide how you are going to work on the problem.  This involves making some choices about what strategies to use.  

Some possible strategies are:

  • Draw a picture or diagram --  making a picture which relates the information given to what is asked for can often lead to a solution.
  • Make a list -- this is a strategy which is especially useful  in problems where you need to count the members of a set.
  • Solve smaller versions of the problem and look for a pattern --  almost any problem can be made simpler in some way. By working out simpler versions, you can often see patterns which help solve the original problem.
  • Decompose the problem -- Many problems can be broken into a  series of smaller problems. This strategy can turn a problem which on first glance seems intractable into something more doable.
  • Use variables and write an equation -- the method of algebra. Very useful in a lot of problems.
Carry out the plan: Spend a reasonable amount of time trying to solve the problem using your plan.  If you are not successful, go back to step 2.  If you run out of strategies,  go back to step 1.  If you still don't have any luck, talk the problem over with a classmate.
Look back: Did your solution work? If not, start again. After you have a proposed solution, check your solution out.  Is it reasonable?  Is it unique? Can you see an easier way  to solve the problem?  Can you generalize the problem?  Return to the original problem.

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