One might conjecture that there is an interesting fact concerning each of the positive integers. Here is a ‘proof by induction’ that such is the case. Certainly, 1, which is a factor of each positive integer, qualifies, as do 2, the smallest prime; 3, the smallest odd prime; 4, Bieberbach’s number; etc. Suppose the set S of positive integers concerning each of which there is no interesting fact is not vacuous, and let k be the smallest member of S. But this is a most interesting fact concerning k! Hence S has no smallest member and therefore is vacuous. Is the proof valid?

— Edwin F. Beckenbach, “Interesting Integers,” American Mathematical Monthly, April 1945

The coastline of Nova Scotia was once frequented by pirates, and people occasionally dig for buried pirate treasure. On a local radio program a few years ago I heard an interview with someone who had done a study of attempts to find pirate treasure. He claimed that in most of the cases in which treasure was actually found, it was in a place where treasure-hunters had dug before, rather than in a brand new, previously undug, location. Past diggers simply hadn’t dug deep enough. The previous digger had, in fact, often stopped just short of the treasure. If the previous digger had dug a little deeper than he did, he would have found it.

The interviewer asked him what advice he would give to treasure hunters on the basis of this study; and, producing an interesting application of induction, he lamely suggested that diggers should dig a little deeper than they in fact do. Can you see why this advice is impossible to follow?

— Robert M. Martin, There Are Two Errors in the the Title of This Book, 2002