The Art and Craft of Problem Solving

Chapter 2

Strategies for Investigating Problems

As we've seen, solving a problem is not unlike climbing a mountain. And for inex­

perienced climbers, the task may seem daunting. The mountain is so steep! There is

no trail! You can't even see the summit! If the mountain is worth climbing, it will

take effort, skill, and, perhaps, luck. Several abortive attempts (euphemistically called

"reconnaissance trips") may be needed before the summit is reached.

Likewise, a good math problem, one that is interesting and worth solving, will not

solve itself. You must expend effort to discover the combination of the right mathe­

matical tactics with the proper strategies. "Strategy" is often non-mathematical. Some

problem solving strategies will work on many kinds of problems, not just mathematical

ones.

For beginners especially, strategy is very important. When faced with a new and

seemingly difficult problem, often you don't know where to begin. Psychological

strategies can help you get in the right frame of mind. Other strategies help you start

the process of investigation. Once you have begun work, you may need an overall

strategic framework to continue and complete your solution.

We begin with psychological strategies that apply to almost all problems. These

are simple commonsense ideas. That doesn't mean they are easy to master. But once

you start thinking about them, you will notice a rapid improvement in your ability

to work at mathematical problems. Note that we are not promising improvement in

solving problems. That will come with time. But first you have to learn to really work.

After psychological strategies, we examine several strategies that help you begin

investigations. These too are very simple ideas, easy and often fun to apply. They

may not help you to solve many problems at first, but they will enable you to make

encouraging progress.

The solution to every problem involves two parts: the investigation, during which

you discover what is going on, and the argument, in which you convince others of

your discoveries. We discuss the most popular of the many methods of formal argu­

ment in this chapter. We conclude with a study of miscellaneous strategies that can be

used at different stages of a mathematical investigation.

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