A cyclic knight's tour is a sequence of knight's moves that visit every square of an n x n chessboard, returning to the first square. These are some images from Ian Parberry's research on cyclic knight's tours in the 1990s. See these papers for more information. Some of the tours are generated by a random walk, one is generated by a neural network, some are generated by a divide-and-conquer algorithm that gives the tours a tiled look. Some are symmetric under rotations.

The following small knight's tour was used as part of the base of the recursion in a divide-and-conquer algorithm described in Ian Parberry, "An Efficient Algorithm for the Knight's Tour Problem", Discrete Applied Mathematics, Vol. 73, pp. 251-260, 1997. [pdf].

 

The following small knight's tour was used as part of the base of the recursion in a divide-and-conquer algorithm described in Ian Parberry, "An Efficient Algorithm for the Knight's Tour Problem", Discrete Applied Mathematics, Vol. 73, pp. 251-260, 1997. [pdf].

 

The following small knight's tour was used as part of the base of the recursion in a divide-and-conquer algorithm described in Ian Parberry, "An Efficient Algorithm for the Knight's Tour Problem", Discrete Applied Mathematics, Vol. 73, pp. 251-260, 1997. [pdf]. You may have noticed by now that all of the closed knight's tours you have seen so far (6x6, 8x8, and 10x10) have even sides. That's because there can be no knight's tours on an odd-sided board (to see this, consider the fact that on a real chessboard, the squares visited in a knight's tour must alternate between black and white).

 

Closed knight's that are invariant under a 90° rotation exist on nxn boards for all even n at least 6 that is not divisible by 4 (Dejter, 1983). This tour is from Ian Parberry, "An Efficient Algorithm for the Knight's Tour Problem", Discrete Applied Mathematics, Vol. 73, pp. 251-260, 1997. [pdf].

 

Closed knight's that are invariant under a 180° rotation exist on nxn boards for all even n at least 10. This tour is from Ian Parberry, "An Efficient Algorithm for the Knight's Tour Problem", Discrete Applied Mathematics, Vol. 73, pp. 251-260, 1997. [pdf], as are the following 14x14, 16x16, 18x18, and 20x20 knight's tours.

 

 

 

 

 

The following was the largest closed knight's tour I could generate at the time using the Hopfield network of Takefuji and Lee. More details appear in Ian Parberry, "Scalability of a Neural Network for the Knight's Tour Problem", Neurocomputing, Vol. 12, pp. 19-34, 1996. [pdf]

 

The following was generated by a divide-and-conquer algorithm described in "An Efficient Algorithm for the Knight's Tour Problem", Discrete Applied Mathematics, Vol. 73, pp. 251-260, 1997. [pdf].

 

The following was generated by a divide-and-conquer algorithm described in "An Efficient Algorithm for the Knight's Tour Problem", Discrete Applied Mathematics, Vol. 73, pp. 251-260, 1997. [pdf].

The following was generated by a random walk algorithm due to Euler in 1795 modifed with a heuristic due to Warnsdorf in 1823. It appears in Ian Parberry, "Scalability of a Neural Network for the Knight's Tour Problem", Neurocomputing, Vol. 12, pp. 19-34, 1996. [pdf]