The Art and Craft of Problem Solving

2.1 PSYCHOLOGICAL STRATEGIES 15

It isn't hard to acquire a modest amount of mental toughness. As a beginner, you

most likely lack some confidence and powers of concentration, but you can increase

both simultaneously. You may think that building up confidence is a difficult and

subtle thing, but we are not talking here about self-esteem or sexuality or anything

very deep in your psyche. Math problems are easier to deal with. You are already

pretty confident about your math ability or you would not be reading this. You build

upon your preexisting confidence by working at first on "easy" problems, where "easy"

means that you can solve it after expending a modest effort. As long as you work on

problems rather than exercises, your brain gets a workout, and your subconscious gets

used to success. Your confidence automatically rises.

As your confidence grows, so too will your frustration threshold, if you gradually

increase the intellectual "load." Start with easy problems, to warm up, but then work on

harder and harder problems that continually challenge and stretch you to the limit. As

long as the problems are interesting enough, you won't mind working for longer and

longer stretches on them. At first, you may bum out after 15 minutes of hard thinking.

Eventually, you will be able to work for hours single-mindedly on a problem, and keep
other problems simmering on your mental backbumer for days or weeks.
That's all there is to it. There is one catch: developing mental toughness takes
time, and maintaining it is a lifetime task. But what could be more fun than thinking
about challenging problems as often as possible?
Here is a simple and amusing problem, actually used in a software job interview,
that illustrates the importance of confidence in approaching the unknown.^1
Example 2.1.1 Consider the following diagram. Can you connect each small box on
the top with its same-letter mate on the bottom with paths that do not cross one another,
nor leave the boundaries of the large box?

B

Solution: How to proceed? Either it is possible or it is not. The software company's

personnel people were pretty crafty here; they wanted to see how quickly someone

would give up. For certainly, it doesn't look possible. On the other hand, confidence

dictates that

Just because a problem seems impossible does not mean that it is im­

possible. Never admit defeat after a cursory glance. Begin optimistically;

assume that the problem can be solved. Only after several failed at-

I We thank Denise Hunter for telling us about this problem.