The Art and Craft of Problem Solving, Second Edition by Zeitz

By Zeitz

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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 sp ent thinking about a problem is always time worth spent. Even if you seem to make no progress at all. 1 (Russia, 1 995 ) The sequence aO , a1 , a2 , satisfies . . am + n + am - n = "2 (a2m + a2n ) for all nonnegative integers m and n with m 2 n . If a1 1 , determine a1995 . 1 (1) = Partial Solution : Equation ( 1 ) i s hard to understand without experimentation.

The answer is simple: Toughen up, loosen up , and practice. Toughen up by gradually increasing the amount and difficulty of your problem solving work. Loosen up by deliberately breaking rules and consciously opening yourself to new ideas (including shamelessly appropriating them ! ) . Don 't be afraid to play around, and try not to let failure inhibit you. Like P6lya's mouse, several failed attempts are perfectly fine, as long as you keep trying other approaches. And unlike P6lya's mouse, you won 't die if you don 't solve the problem.

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. 2. 1 on page 4). Also, notice how restating helps one to solve the Census-Taker problem (Example 1 . 3 on page 2). More subtly, recall Example 2. 7 on page 20, which involved the sequence I , 1 1 , 2 1 , 1 2 1 1 , 1 1 1 22 1 , .

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