How many knight's tours are there?

Solution 1:

I was recently surprised to discover that it's actually not known (edit: see below). The number of closed knight's tours (cyclic) was computed in the 1990s, using binary decision diagrams. There are 26,534,728,821,064 closed directed knight's tours, and the number of undirected ones is half that or 13,267,364,410,532. If you count equivalence classes under rotation and reflection, there are slightly more than 1/8th of that: 1,658,420,855,433.

(Loebbing and Wegener (1996) wrote 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 mistake, as they pointed out in a comment to their paper. Brendon McKay independently computed the correct number with another method, and the original authors seem to have later found the same answer.)

Finding the exact number of open tours (not cyclic/reentrant) was open, but was estimated to be about 10¹⁵ or 2×10¹⁶.

Edit: Please see and upvote the answer by user rantonse below, pointing out that Alex Chernov appears to have calculated the total number of knight's tours (including non-cyclic ones).

Solution 2:

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).