A source on this, Lagarias, J., American Mathematical Monthly 92 (1985), 3-23, has:
"The conjecture is ultimately periodic for all n and such that there is only one final cycle 1 -> 4 -> 2 -> 1."
No one knows: this hasn't been resolved.
The city of Königsberg now Kaliningrad, Russia was on both sides of the Pregel River. It iincluded two large islands, connected to each other, or to the two mainland portions of the city, by seven bridges. The problem to devise a walk through the city that would cross each of those bridges once and only once was resolved by Euler, who proved that the problem has no solution.
|©2006 Allen Klinger|