[hakank]

What I can see, of these 72 solutions, there are at most 9 distinct solutions when symmetries are removed (rotation, transpose, flipping). I might have missed some solution.

Though, of these 9 solutions there are more similarities, i.e. some of them are the same rows except that the positions are switched.

Some other thing. if the lines are sorted, there are only 4 distinct variants.

Also, the solution of the original Miracle Sudoku instance has some invariants, for example the rows are rotations of these numbers 1,5,9,4,8,3,7,2,6 and the columns are (when rows are sorted) rotations of the numbers 1..9.

I've added comments about this in my Picat model: http://hakank.org/picat/miracle_sudoku.pi

[ynik]

It's only a single pattern. Every solution has the 1,5,9,4,8,3,7,2,6 pattern (every step is +4 mod 9) in either the rows or the columns.

9 ways to transpose; 4 rotations, 2 for flipped or not. Multiply those, you get the full 72.

[hakank]

Which are the "9 ways to transpose"? In general there are 8 symmetries of a matrix, but you might also count reordering of rows/columns or blocks?

I agree that 1,5,9,4,8,3,7,2,6 pattern rotations is there somewhere, either in the columns or the rows of a solution, but that don't make all solutions the same.

[grogers]

Different people could reasonably disagree. The solutions with top rows:

159483726

594837261

Look pretty equivalent to me, which is how you get the 9 transpositions.

[hakank]

Yes, but the solutions are different if the order (permutation) of the rows with the pattern rotations are different. There is a huge number of ways the rows can be ordered (9!).