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I am looking for a way to put standard $9\times9$ Sudoku problems into a standard form by renaming the numbers ($9!$ possibilities), permuting the rows $6^4$ possibilities, and permuting the columns $6^4$ possibilities. This will be a tool to find out how difficult problems are different from easy problems. What features do difficult puzzles have different from easy ones?

By difficult, I mean difficult for MY tired ancient brain, not for any kind of algorithm. We are talking psychology, not computational complexity theory.

Arnaldo
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richard1941
  • 950
  • Note that you also have reflections and mirrorings, another $8$ possibilities. Then, of course, there are some boards for which different combinations of these changes lead to identical outcomes (other than the ones that obviously do, like swapping two rows, then rotating $90^\circ$ vs. first rotating, then swapping two columns), so you lose some possibilities again. It's not easy to count the number of essentially different valid boards. And then you have the question about what clues are filled in beforehand, which for each board can be done in a myriad of ways. – Arthur Dec 22 '16 at 18:47
  • It would be moreover an interesting problem, to see how many classes of filled sudokus exists with above changings as equivalence relation. – ctst Dec 22 '16 at 19:06
  • https://en.wikipedia.org/wiki/Mathematics_of_Sudoku gives an algorithm to do this, but this seems quite complicated. Considering the huge number of different sudoku equivalent classes, it appears to me that just storing them in the standard form ${x_i}_{i=1..81}$ and $x_i=0,1,2,3,4,5,6,7,8,9$ with $0$ for empty box is sufficient. – zwim Dec 22 '16 at 20:43
  • I was thinking of a partially filled in board that is offered as a puzzle. For ordering, you could put a zero in the empty cells. – richard1941 Dec 31 '16 at 00:24

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