Serg wrote:Do you know an example of isomorphic transformation which transforms some valid X-sudoku solution grids to other valid X-sudoku solution grids, but breaks X-property for the rest valid X-sudoku solution grids?

Ok, let us call the set of 96 X-preserving transformations {X}, so that if G is SudokuX, then X[k](G) is SudokuX, for all k = 1 ... 96. Let's call such a set of 96 grids an "X-set".

Now, let {S} denote the set of standard Sudoku transformations, and let G be some SudokuX grid. You want a transformation S[j] (not in {X}) which for some SudokuX grids produces another SudokuX grid, but not always.

If H = S[j](G), and H is SudokuX, and S[j] is not in {X}, then clearly the equivalence class defined by {S} contains more than one "X-set".

For each X-set, there are 96 transformations that will convert members of one X-set to another, but will not always do so for SudokuX grids which are not in this equivalence class (ie are ED wrt G).

So these "X-set linking" transformations have the desired property. Also, they are the only cases possible, since no transformation in {S} can turn a SudokuX grid G into another SudokuX which is not isomorphic to G.

Here are some grids with more than one X-set. The number of SudokuX isotopes is indicated for each:

- Code: Select all
`123869457456237891789451236842793615567148923931526748278915364314672589695384172 # 288`

123869457456237891789451236542793618967148523831526749278915364314672985695384172 # 192

123869457456237891789451236542793618967148523831526749278914365315672984694385172 # 192

123869457456237891789451236845723619967148523231596748578912364312674985694385172 # 192

123869457456237891789451236245793618967148523831526749578912364312674985694385172 # 192

123698457456237891789451236264873915875149623931526784548712369312964578697385142 # 192

123698457456237891789451236962873514875149623341526789298714365514362978637985142 # 192

123698457456237891789451236964823715275149683831576924548712369312964578697385142 # 192

I haven't yet looked at the transformations involved. Nor do I yet know the maximum number of X-sets.