by **ravel** » Tue Jan 16, 2007 9:12 pm

Probably this is not the right thread for saying this, because it is about symmetries we can find for the givens of sudokus, not about those in solution grids. But i want to take the opportunity to ask, what the right academic words are for what i mean (looking to wikipedia was more confusing than clarifying).

There are well known transformations for sudokus (including solution grids), that keep all essential properties, some say, their mathematical equivalence, some call such puzzles equivalent, some isomorph.

They also keep all symmetry properties (when there are some) in a mathematical sense, though you might not spot them at the first glance.

E.g. if you have a diagonally symmetrical grid, you can transform it non trivially (where non trivial means, that the transformed puzzle does not have each cell with the same number at the same place again), by mirroring and exchanging the numbers in mirrored cells, to the same grid, where 9 cells (on the diagonal) keep on the same place. The same is true for all equivalent grids. For all of them there exists a transformation (maybe a combination of basic transformations), that maps them to the same grid with exactly 9 cells at the same place and 36 pairs with exchanged numbers.

This is one sample for what i called "the puzzle maps to itself".

I suppose this is an automorphism.

Beside of the diagonally symmetry we know the 180 degree rotational symmetry (where only 1 cell keeps its place), the threefolded symmetry over 3 chunks (samples by Mauricio) or triples of boxes in the way arranged, that we recently saw by gurth (no cell keps on its place), further 90 degree symmetry (again with 1 cell unchanged). Are there more kinds of symmetry ?

All equivalent grids to those keep this property, just the cells are distributed in another way.

Now iff a grid has such symmetry properties, some of the equivalent grids are equal, for diagonal symmetry there are always pairs of them, for threefold puzzles triples and so on. Therefore the size of the equivalence class is only half or third etc. of the maximum size. This is, what my quick calculation for the number of grids with symmetry properties is based on.

So i hope someone with better background will let me know, if it is correct, what i said here, and if there are better terms for that.