Magic Wizard Tour Problem Continued
The original double latin square by E. T. Parker in the 1960 paper, differs from the version shown as the frontispiece in the Coxeter/Ball Mathematical Recreations, in being more geometrically regular:
This has only 15 'Witch' moves that pass over the centres of cells: 06-07 (N), 09-10 (R), 25-26 (N), 32-33 (N), 36-37 (B), 40-41 (N), 48-49 (C), 54-55 (B), 57-58 (C), 62-63 (Z), 69-70 (R), 86-87 (A), 89-90 (N), 96-97 (N), 98-99 (B). Where the letters indicate tthe directions of the moves: A = antelope, B = bishop, C = camel, N = knight, Z = zebra.
Working from this by permuting the ranks and files several times I have arrived at the following square:
This has only 4 Witch moves: 44-45 (N), 51-52 (Z), 94-95 (B), 99-00 (N), shown by the straight lines. One of these is the closure move 99-00. So regarded as an open tour it uses only three Witch moves. Can this be further improved?
What is in effect the same double latin square can be presented in other forms by permutation of the left or right digits (since a latin square is just a pattern independent of the actual symbols used). But whether a better permutation can be chosen to make a Wizard tour more likely is not clear to me.
It seems that every version will have one vertical and one horizontal move lke 09-10 and 69-70 in these examples, or 49-50 and 79-80 in the previous examples.