Jamison Rios

2022-06-30

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?

Gornil2

Expert

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.
Finding the exact number of open tours (not cyclic/reentrant) was open, but was estimated to be about ${10}^{15}$ or $2×{10}^{16}$.

Ximena Skinner

Expert