The amount of knight's scenic tours exist?
The knight's scenic tour is a series of 64 squares on a chess board, where each square is visted as soon as, and also each succeeding square can be gotten to from the previous by a knight's action. Scenic tours can be cyclic, if the last square is a knight's action far from the first, and also acyclic or else.
There are numerous proportions amongst knight's scenic tours. Both acyclic and also cyclic scenic tours have 8 reflectional proportions, and also cyclic scenic tours in addition have proportions emerging from beginning at any kind of square in the cycle, and also from running the series in reverse.
Is it well-known the amount of knight's scenic tours there are, approximately all the proportions?
I was lately stunned to uncover that it's in fact not recognized (modify : see listed below). The variety of shut knight's scenic tours (cyclic) was calculated in the 1990s, making use of binary decision diagrams. There are 26,534,728,821,064 shut routed knight's scenic tours, and also the variety of undirected ones is half that or 13,267,364,410,532. If you count equivalence courses under turning and also representation, there are a little greater than 1/ 8th of that : 1,658,420,855,433.
(Loebbing and also Wegener (1996) created a paper "The Number of Knight's Tours Equals 33,439,123,484,294 — Counting with Binary Decision Diagrams" ; the number in the title in the blunder, as they mentioned in a comment to their paper. Brendon McKay individually computed the proper number with an additional method, and also the initial writers appear to have later found the very same solution.)
Locating the specific variety of open scenic tours (not cyclic/reentrant) was open, but was estimated to be around 10 15 or 2 × 10 16 .
Edit : Please see and also upvote the solution by customer rantonse listed below, mentioning that Alex Chernov shows up to have actually computed the complete variety of knight's scenic tours (consisting of non - cyclic ones).