Mathematics of Sudoku I

In an earlier article [1] of Bertram Felgenhauer and the second author, the number of different Sudoku grids was computed to be N0 = 6670903752021072936960 ≈ 6.671 × 10 . However, distinct solutions were treated as different, even when one could be transformed into another by some symmetry. However, you may feel that two grids should not be counted differently if, say, the second is just the first rotated by 90 degrees. Similarly, one can reflect a grid to get another valid grid – should these be counted differently? And one can relabel all the entries (exchanging 1s and 2s, for example) to give another grid. In this note, we want to explain how to refine the counting method of [1] to find how many essentially different grids there are, if we allow various possible symmetries.