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Only after several failed atI We thank Denise Hunter for telling us about this problem. 16 CHAPTER 2 STRATEG I E S FOR I NVESTI G AT I N G PROBLEMS tempts should you try to prove impossibility. If you cannot do so, then do not admit defeat. Go back to the problem later. Now let us try to solve the problem. It is helpful to try to loosen up, and not worry about rules or constraints. Wishful thinking is always fun, and often useful. For example, in this problem, the main difficulty is that the top boxes labeled A and C are in the "wrong" places.

Define g(m) = l vrnJ . Then what is g(n 2 + b) equal to? If b is "small enough," then g(n 2 + b) = n. We can easily make this more precise, either by experimenting or just thinking clearly about the algebra. For what values of m is g(m) = n ? The answer: n 2 :::; m < (n + 1 f = n2 + 2n + 1 . 2 . 2 STRATEG I ES FOR G ETT I N G STA RTED 33 In other words, g (n 2 + b) = n if and only if 0 :S b < 2n + 1 . " For example, consider the case where b = Then m = n 2 + 1 and g (m) = n and I. f(m) = m + g(m) = n 2 + 1 + n .

Nega­ tive, finite vs. infinite, etc. 26 CHAPTER 2 STRATEG I ES FOR I NVESTIGAT I N G 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. 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?

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