tarek wrote: .... At the moment we have the following possible continuity planes explored:

1. Cylinder: Vertical or Horizontal

2. Toroid: Vertical + Horizontal Cylinders

3. Moebius: Vertical or Horizontal

4. Projective Plane: Vertical and Horizontal Moebius

5. Klein bottle: Cylinder in one direction + Moebius in the other perpendicular direction.

....

Here's another possibility. Call it the Moebius Corkscrew (or Wormscrew).

When you get to the top of each column of 9, glue it to the bottom of the

next column. Wrap this idea around, so that the top of column 9 also glues to the bottom of column 1.

This gives you a strip 81 cells high by 1 cell wide, with the top and bottom edges glued together, if you look at it vertically, or a 9x9 grid if you look at it horizontally. At least I think that's what it would be.

No repeats allowed in the rows. In each column, any two occurrences of any digit must be at least 7 (or some similar number) positions away from each other, i.e. there must be at least 6 other digits in between any two occurrences of the same digit.

I guess you could also have a Projective Plane Corkscrew, which I can't visualize, where each row would be glued to the next in the same manner as the columns. I guess this would be 81 by 1 if viewed horizontally, or 1 by 81 if viewed vertically. But my mind boggles.

Then you could forget the chess altogether and get back to actual Sudoku (sort of).

Bill Smythe