Sudoku: 1 Solution, 0.6T Variations?

The other day, after working on a few Sudokus and really getting the hang of it :-), I thought to apply a little combinatoric; i.e., how many variations of the same Sudoku are possible?

Well, you can easily swap complete rows or columns — within each of the respective 3x blocks to leave their integrity intact — and you can swap the complete column or row with the 3x blocks.

E.g., you can swap row 2 with row 1 or row 3, or column 7 with column 8 or column 9; or you can swap the rows 4 – 6 with rows 1 – 3 or rows 7 – 9, or columns 7 – 9 with columns 1 – 3 or columns 4 – 6.

So… swapping the rows within the 3x boundaries, and there are three of those; then the three row blocks themselves; ditto for the columns and column blocks:

(3!)³ • (3!) • (3!)³ • (3!)

or

1’679’616 combinations

Here my basic Sudoku I developed „on the fly“:

jm's Sudoku

But then, the show won’t stop here!  🙂

Consider you can also swap the numbers; e.g., exchange 1 with 7 and/or 3 with 9, and what not, resulting to

9!

or

additional 363’880 combinations

Leaves me with the one question: is my Sudoko shown — under consideration of the various combinations listed — the only one?

©  August 2009 Jürgen Menge, San José

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