.

Following a suggestion from Leren , I have also been looking at standard SudokuX, ie Sudoku with distinct values along both diagonals. Mysteriously, while it is a popular variant, it seems, little is known about the numbers.

How many different SudokuX grids there? I think the answer is (up to relabelling) 153,255,603,504.

Given that number appears nowhere relevant on the Web, I am either wrong, or nobody has counted them before.

Ah, I hear you say, but how many essentially different SudokuX grids are there?

My guess is around a billion, but I do hope to come up with an accurate figure for the number of SX-different grids at some stage soon.

As we did for SudokuP, we define SX-equivalence in terms of the subset of Sudoku transformations that preserve the diagonal property.

There are, it seems, just 12 of these (x 8 for rotation/reflection, giving 96 in all). There are 6 ways to tweak a grid so that the diagonals in the corner boxes are permuted but not shifted.

We can also permute Band2/Stack 2 to give a 180-degree rotation of the center box.

It now becomes clear why nobody seems to know how many ED SudokuX grids there are. We can't use McGuire's method (Burnside's Lemma) because of the orbit connection problem. We can (and probably will) calculate the number of SX-different grids, but the exact number of ED grids will of course be less than that.

We were able to count ED SudokuP grids because the grids that needed checking are not too numerous. But we have here around 1,000 times the number of grids, so SudokuP methods that took just a few days would take years here (not to mention the 5TB of disk space we'd need to hold all the grids).

The only alternative appears to be checking the 5.47 billion ED Sudoku grids to deterimne which have SudokuX isotopes, but that's still a daunting task.

Minimum clue counts: it is known that 12-clue puzzles exist, so the first question here is, are there 11-clue SudokuX puzzles?

Finally, I determined the number of grids above by the '1's-template method (there are 1040 templates). Thanks largely to dobrichev's fabulous fsss2 solver, I counted them all in 4 threads x 6 hours.