26 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS
- Begin to classify: is it a "to find" or "to prove" problem? Is the problem similar
to others you have seen? - Carefully identify the hypothesis and the conclusion.
- Try some quick preliminary brainstorming:
Think about convenient notation.
Does a particular method of argument (see Section 2.3) seem plausible?
Can you guess a possible solution? Trust your intuition!
Are there key words or concepts that seem important? For example, might
prime numbers or perfect squares or infinite sequences play an important
role?
When you finish this (and don't rush!), go back and do it again. It pays to reread
a problem several times. As you rethink classification, hypothesis and conclusion, ask
yourself if you can restate what you have already formulated. For example, it may
seem that the hypothesis is really trivial, and you just repeat it verbatim from the state
ment of the problem. But if you try to restate it, you may discover new information.
Sometimes just reformulating hypothesis and conclusion with new notation helps (for
example, Example 1.2. 1 on page 4). Also, notice how restating helps one to solve the
Census-Taker problem (Example 1.1. 3 on page 2). More subtly, recall Example 2.1. 7
on page 20, which involved the sequence I, 11 ,21, 121 1, 1112 21, .... Normally, one
reads a problem silently. But for many people, reciting the sequence out loud is just
enough of a restatement to inspire the correct solution (as long as a number such as
"1211" is read "one-two-one-one," not "one thousand, two hundred and eleven").
When looking at the conclusion of the problem, especially of a "to find" problem,
sometimes it helps to "fantasize" an answer. Just make something up, and then reread
the problem. Your fantasy answer is most likely false, and rereading the problem with
this answer in mind may help you to see why the answer is wrong, which may point
out some of the more important constraints of the problem.
Don't spend too much time on orientation. You are done once you have a clear
idea of what the problem asks and what the given is. Promising guesses about answers
or methodology are bonuses, and nothing you should expect. Usually they require
more intensive investigation.I'm Oriented. Now What?
At this point, you understand what the problem is asking and you may have some
ideas about what to do next. More often than not, this involves one or more of the four
basic "startup" strategies that we have seen, penultimate step, get your hands dirty,
wishful thinking and make it easier. Let's discuss these in more detail.Get Your Hands Dirty: This is easy and fun to do. Stay loose and experi
ment. Plug in lots of numbers. Keep playing around until you see a pattern.
Then play around some more, and try to figure out why the pattern you see is
happening. It is a well-kept secret that much high-level mathematical research
is the result of low-tech "plug and chug" methods. The great Carl Gauss, widely