The Art and Craft of Problem Solving

Chapter 3

Tactics for Solving Problems

Now we tum to the tactical level of problem solving. Recall that tactics are broadly

applicable mathematical methods that often simplify problems. Strategy alone rarely

solves problems; we need the more focused power of tactics (and often highly spe­

cialized tools as well) to finish the job. Of the many different tactics, this chapter will

explore some of the most important ones that can be used in many different mathemat­

ical settings.

Most of the strategic ideas in Chapter 2 were plain common sense. In contrast,

the tactical ideas in this chapter, while easy to use, are less "natural," as few people

would think of them. Let's return to our mountaineering analogy for a moment. An

important climbing tactic is the rather non-obvious idea (meant to be taken literally):

Stick your butt out!

The typical novice climber sensibly hugs the rock face that he is attempting to climb,

for it is not intuitive to push away from the rock. Yet once he grits his teeth and pushes

out his rear end, a miracle happens: the component of gravity that is perpendicular to

the rock rises, which increases the frictional force on his feet and immediately pro­

duces a more secure stance.

Before After

Likewise, you may find that some ofthe tactical ideas below are peCUliar. But once

you master them, you will notice a dramatic improvement in your problem solving.

Many fundamental problem-solving tactics involve the search fo r order. Often

problems are hard because they seem "chaotic" or disorderly; they appear to be miss­

ing parts (facts, variables, patterns) or the parts do not seem connected. Finding (and

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