Problem Solving Strategies

There are no hard and fast rules that will ensure success in solving problems. However, it is possible to outline some general steps in the problem solving process and to give some principles that may be useful in the solution of certain problems. These steps are just common sense made explicit. They have been adapted from George Polya’s book How to Solve It.

A.    Understand the Problem

The first step is to read the problem and make sure that you understand it clearly. Ask yourself the following questions:

·                  What is the unknown?

·                  What are the given quantities?

·                  What are the given conditions?

            For many problems it is useful to draw a diagram and identify the given and required quantities  

            on the diagram. Usually it is necessary to introduce suitable notation. In choosing symbols for  

            the unknown quantities we often use letters such as a, b, c, m, n, x, and y, but in some cases it             helps to use initials as suggestive symbols; for instance, V for volume or t for time.

B.    Think of a Plan

            Find a connection between the given information and the unknown that will enable you to            
            calculate the unknown. It often helps to ask explicitly, “How can I relate the given to the      
            unknown?” If you don’t see a connection immediately, the following ideas may be helpful in        
            devising a plan:

·                  Try to Recognize Something Familiar – Relate the given situation to previous    knowledge. Look at the unknown and try to recall a more familiar problem that has a             similar unknown.

·                  Try to Recognize Patterns – Some problems are solved by recognizing that some kind of pattern is occurring. The pattern could be geometric, or numerical, or algebraic. If you          can see regularity or repetition in a problem, you might be able to guess what the            continuing pattern is and then prove it.

·                  Use Analogy - Try to think of an analogous problem, that is, a similar problem, a related problem, but one that is easier than the original problem. If you can solve a similar,   simpler problem, then it might give you the clues you need to solve the original, more        difficult problem. For instance, if a problem involves very large numbers, you could first try a similar problem with smaller numbers. Or if the problem involves three-dimensional     geometry, you could look for a similar problem in two-dimensional geometry. Or if the           problem you start with is a general one, you could first try a special case.

·                  Introduce Something Extra – It may sometimes be necessary to introduce something  new, an auxiliary aid, to help make the connection between the given and the unknown. For instance, in a problem where a diagram is useful the auxiliary aid could be a new line drawn in the diagram. In a more algebraic problem it could be a new unknown related to the original unknown.

·                  Take Cases – We may sometimes have to split a problem into several cases and give a different argument for each of the cases. For instance, we often have to use this strategy in dealing with absolute value.

·                  Work Backward – Sometimes it is useful to imagine that your problem is solved and work backward, step by step, until you arrive at the given data. Then you may be able to           reverse your steps and thereby construct a solution to the original problem. This procedure is commonly used in solving equations. For instance, in solving the equation 3x – 5 = 7, we suppose that x is a number that satisfies 3x – 5 = 7 and work backward. We add 5 to each side of the equation and then divide each side by 3 to get x = 4. Since each of these steps can be reversed, we have solved the problem.

·                  Establish Subgoals – In a complex problem it is often useful to set subgoals (in which  the desired situation is only partially fulfilled). If we can first reach these subgoals, then we may be able to build on them to reach our final goal.

·                  Indirect Reasoning – Sometimes it is appropriate to attack a problem indirectly. In using proof by contradiction to prove that P implies Q we assume that P is true and Q is        false and try to see why this can’t happen. Somehow we have to use this information and arrive at a contradiction to what we absolutely know is true.

·                  Mathematical Induction – In proving statements that involve a positive integer n, it is  frequently helpful to use the following principle:

§  Principle of Mathematical Induction – Let S_n be a statement about the positive integer n. Suppose that

1.    S₁ is true.

2.    S_k+1 is true whenever S_k is true.

                                    Then S_n is true for all positive integers n.

C.   Carry Out the Plan

            In Step B a plan was devised. In carrying out that plan we have to check each stage of the plan and
            write the details that prove that each stage is correct.

D.   Look Back

            Having completed our solutions, it is wise to look back over it, partly to see if we have made        
            errors in the solution and partly to see if we can think of an easier way to solve the problem.             Another reason for looking back is that it will familiarize us with the method of solution and this             may be useful for solving a future problem. 


           As Descartes said, 

 “Every problem that I solved became a rule which served afterwards to solve other problems.”

Source: Calculus: Concepts and Contexts 2nd Edition by James Stewart