Snow, Chess and Knight Tours

 We had our first snow of the year last night, but by this afternoon it had all thawed, before I could take a photo to prove it. So I had no trouble getting out to Crewe Chess Club this evening, to play third board in an F-team match against Alsager, which I managed to win against a young opponent. 

I'm thinking of entering the Blackpool Chess event next month, but I'm not sure I fancy traipsing about the seafront in the icy wind, and the Hotel costs are getting higher, and there are still threats of rail strikes, which put me off entering the Hastings tournament earlier this month. 

Professor Knuth sent me an email that included a knight's tour sent to him for his birthday on January 10. This is a 10x10 tour of the type showing a queen on cell 85 (his age) and guarding all the cells with prime numbers, and various other features. I will put it in the Figured Tours page shortly. 

I've been spending an inordinate amount of time checking up on the 6x7 symmetric knight tours. There was an error in the 266 diagrams of the Sulian (axial symmetric) tours, and it took me quite a while to find the duplicate among them, since the total should be 265 according to Knuth. I did this by classifying them by geometrical properties like number of straight line moves. It turned out that tour 166 in the list was the same as 165. 

Similarly when I drew out diagrams for the Eulerian tours I found 262, which was one short. I had to try check most of the enumeration again before I found the missing one. Now I have to arrange them to appear on the corrected website pages, but I'm somewhat undecided on the best arrangement method. Probably I will group together those that have the same angles in the centre cells.

Here is a puzzle to solve based on the search. Complete the tour, given the four pairs of moves in the top corner-adjacent cells. I published this also in Mathstodon but it attracted no interest there. There are twelve such choices for the near-corner angles that have a unique solution.

Some cases have no solution. Others have up to 30 solutions, so this method of enumerating the tours does not provide a good way of classifying them.