Further Travels with My Ant

A recurring theme of this book has been computer-generated mysteries. Examples are sequences defined by simple rational recursions whose terms turn out to be integers with interesting but unexplained divisibility properties or geometric configurations that exist although there are no proofs of existence. In most of the examples, the reported mysteries have remained unsolved and in some cases may in fact be, in a suitable sense, unsolvable. It is therefore gratifying to be able to present an elegant solution of a previously described mystery. An especially pleasing feature of this solution is that the breakthrough became possible by drawing the right picture. Once the picture is drawn, it becomes clear what must be proved, after which further study of the picture gives the clue for constructing the proof. It turns out that at one point one needs to use the Jordan curve theorem for a special class of closed curves.