This article is in two parts, the first of which is a do-it-yourself operation, in which I'll show you 35 examples of patterns that seem to appear when we look at several small values of n, in various problems whose answers depend on n. The question will be, in each case: do you think that the pattern persists for all n, or do you believe that it is a figment of the smallness of the values of n that are worked out in the examples? Caution: examples of both kinds appear; they are not all figments! In the second part I'll give you the answers, insofar as I know them, together with references. Try keeping a scorecard: for each example, enter your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all. This first part contains no information; rather it contains a good deal of disinformation. The first part contains one theorem:
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Richard K. Guy (1988) studied this question.
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