TITLE: How many knight's tours are there?
QUESTION [25 upvotes]: The knight's tour is a sequence of 64 squares on a chess board, where each square is visted once, and each subsequent square can be reached from the previous by a knight's move.  Tours can be cyclic, if the last square is a knight's move away from the first, and acyclic otherwise.
There are several symmetries among knight's tours.  Both acyclic and cyclic tours have eight reflectional symmetries, and cyclic tours additionally have symmetries arising from starting at any square in the cycle, and from running the sequence backwards.
Is it known how many knight's tours there are, up to all the symmetries?

REPLY [14 votes]: This problem seems to be have been solved recently by Alex Chernov, see sequence A165134 in OEIS, who claims that there are 19,591,828,170,979,904 open tours (where rotations and reflections are counted separately).