The Art and Craft of Problem Solving

2.2 STRATEGIES FOR GETTING STARTED 27

regarded as one of the greatest mathematicians in history (see page 67), was a

big fan of this method. In one investigation, he painstakingly computed the

number of integer solutions to.x2 + y^2 ::; 90,000.^3

Penultimate Step: Once you know what the desired conclusion is, ask your­

self, "What will yield the conclusion in a single step?" Sometimes a penultimate

step is "obvious," once you start looking for one. And the more experienced

you are, the more obvious the steps are. For example, suppose that A and Bare

weird, ugly expressions that seem to have no connection, yet you must show

that A = B. One penultimate step would be to argue separately that A 2 B

AND B 2 A. Perhaps you want to show instead that A I-B. A penultimate step

would be to show that A is always even, while B is always odd. Always spend

some time thinking very explicitly about possible penultimate steps. Of course,

sometimes, the search for a penultimate step fails, and sometimes it helps one

instead to plan a proof strategy (see Section 2.3 below).

Wishful Thinking and Make it Easier: These strategies combine psychology

and mathematics to help break initial impasses in your work. Ask yourself,

"What is it about the problem that makes it hard?" Then, make the difficulty

disappear! You may not be able to do this legally, but who cares? Temporarily

avoiding the hard part of a problem will allow you to make progress and may

shed light on the difficulties. For example, if the problem involves big, ugly

numbers, make them small and pretty. If a problem involves complicated alge­

braic fractions or radicals, try looking at a similar problem without such terms.

At best, pretending that the difficulty isn't there will lead to a bold solution,

as in Example 2.1.1 on page 15. At worst, you will be forced to focus on the

key difficulty of your problem, and possibly formulate an intermediate ques­

tion, whose answer will help you with the problem at hand. And eliminating

the hard part of a problem, even temporarily, will allow you to have some fun

and raise your confidence. If you cannot solve the problem as written, at least

you can make progress with its easier cousin!

Here are a few examples that illustrate the use of these strategies. We will not con­

centrate on solving problems here, just making some initial progress. It is important

to keep in mind that any progress is OK. Never be in a hurry to solve a problem! The
process of investigation is just as important. You may not always believe this, but try:

Time spent thinking about a problem is always time worth spent. Even

if you seem to make no progress at all.

Example 2.2.1 (Russia, 1995 ) The sequence aO , a1 ,a 2 , ... satisfies

1

am+n +am-n = "2 (a 2 m +a 2 n)

for all nonnegative integers m and n with m 2 n. If a1 = 1, determine a19 (^95).
(1)
Partial Solution: Equation (1) is hard to understand without experimentation.

Let's try to build up some values of an. First, we keep things simple and try m = n = 0,

(^3) The answer is 282 , 697 , in case you are interested. See [ 21 ], p. 33.