Math 115, Handout 1: Problem-Solving Guidelines
Fall 2006, L. Peterson

This handout contains some suggestions and things to keep in mind when you are trying to solve a mathematical problem. Some of these things may not make much sense at first. But as the course progresses, I will refer to some of these points, and they should eventually become clearer.

1. Make sure that you understand what the problem is. Make sure that you understand all of the words in the problem. Do you know exactly what you are being asked to do? Do you know what assumptions you are allowed to make?

2. Draw pictures.

3. If the problem is difficult, try to first solve a simplified version of the problem.

4. Try to break complicated problems into two or more simpler problems.

5. Try to think of related problems that you have solved at some time in the past, and see if you can apply any ideas that you have already used.

6. If the problem statement is quite long, it may pay to extract key pieces of information from it. You may be able to do this by writing down a list of short key statements with lots of white space in between them.

7. Introduce meaningful notation. If the problem involves some unknown numbers, it may pay to use letters to stand for these numbers. It may then be helpful to set up an equation involving these unknown numbers.

8. Remember that some of the information given in the statement of the problem may not be relevant.

9. Is there any way to restate or reformulate the problem? Can you solve an equivalent problem which is simpler?

10. Some problems are very general. For example, some problems may ask you to say something about all students, or all possible poker hands, or all ways of arranging a set of building blocks. In such cases, it may be helpful to use more experimental techniques. For example, it may be helpful to begin by considering a small group of students rather than all students, or a specific poker hand rather than all hands, or a set of four or five building blocks rather than a much larger set. Then experiment with what you have and see if you learn anything that might help you to solve the problem as it was originally stated to you.

11. Could it be that the teacher made a mistake, and the problem is impossible to solve? If you think that this is the case, can you give a convincing argument to that effect? On the one hand, the teacher may indeed have made a mistake, and it may be impossible to solve the problem. On the other hand, even if the problem can in fact be solved, you may learn something by trying to to prove that it cannot be solved.

12. After you finally do solve the problem, look back on your work. Look for obvious errors. Does your solution seem plausible? For example, if the problem asks you to compute the distance from the Earth to the Moon, an answer of "315 feet" would clearly be wrong. Also ask yourself if you have learned anything by solving the problem. Have you learned anything that might be useful in solving future problems? Did you try any approaches to the problem that were unsuccessful? Did you make any mistakes that you would like to avoid in the future?

Much of the thinking that led to this handout was influenced by other sources. In particular, I have been influenced by the discussion of problem-solving techniques in the book Calculus, Fifth Edition, by James Stewart (Brooks/Cole, Belmont, California, 2003). See page 58 of this book. Stewart in turn acknowledges George Polya's book How To Solve It. A recent edition of Polya's book is available under the title How to solve it: A new aspect of mathematical method (Princeton University Press, Princeton, NJ, 2004). Polya's book is available in the Chester Fritz Library.