sciguy47 wrote:These two are connected in a very special way.

They acually represent the same tesseract in 4D, but have been "sliced" and laid out in different ways.

Aaahh, I get it. Your grids can be addressed by coordinates (R,C,r,c) where R,C specify the 3x3 box and r,c specify the cell within that box. Write out the first grid as 81 5-tuples (R,C,r,c,d) where d is the digit at position (R,C,r,c). For example, the leading diagonal is:

(1,1,1,1,9)

(1,1,2,2,4)

(1,1,3,3,7)

(2,2,1,1,5)

(2,2,2,2,6)

(2,2,3,3,2)

(3,3,1,1,3)

(3,3,2,2,1)

(3,3,3,3,8)

The second grid just takes the (R,C,r,c,d) list from the first and reads it back as (r,c,C,R,d).

You can get away with this because you've insisted on the condition, not normally found in sudoku, that no digit appears in the same position in more than one box. There are 201105135151764480 grids that satisfy that criterion, or about 0.003% of the full set of 6.67x10^21 unrestricted grids.

Since you've gone further and put 1-9 down the two major diagonals as well, it might interest you to know (or it might not!) that there appear to be only 48534220224000 grids with that property in addition to the property mentioned previously. Any of these will preserve the diagonal feature under the particular transformation (R,C,r,c,d) -> (r,c,C,R,d) that you used.