Hi,

Mathimagics!

Mathimagics wrote:For the record, it was Serg, if I recall correctly, who explained to me in great detail why he thought he number of ED (aka S-different) SudokuP grids could not be established using Burnsides' Lemma counting method because there are S-transformations that produce SudokuP grids from non-SudokuP grids, and vice-versa.

I am very sorry. Yes, it was me who stated that Burnsides' Lemma was not applicable for SudokuP solution grids enumeration. But later I changed my mind. I published a post in the thread

SudokuP - Analysis, where I declared changing of my position:

Serg wrote:I have to make a surprise announcement.

I came to a conclusion, that it's not necessary to account for classical 3359232 VPT-transformations while counting essentially different SudokuP solution grids. So, 53,666,689 PF-different SudokuP solution grids is the number of essentially different SudokuP solution grids. It's not necessary to count connected orbits, etc. to refine this result.

You calculated "Number of PF-different SudokuP solution grids" using Burnside's Lemma, aren't you?

Because I treated number of PF-different SudokuP solution grids as the number of essentially different SudokuP solution grids and said that it's not necessary to count connected orbits it should be clear (as I thought) that I withdraw my objections to the application of Burnside's Lemma for ED SudokuP solution grids enumeration. You replied to this my post, and I thought you accepted change of my position

...

That time I used concepts of "content-independent" and "content-dependent" VPT. Now I have much more stronger arguments, why it's not necessary to account for classical 3359232 VPT-transformations while counting essentially different SudokuP (and X-sudoku) solution grids.

What is isomorphic transformation? It is transformation, which doesn't change essence of transformed object. Some external changes of transformed object can be accounted for by simple rules, such as relabelling, changing row/column numbers etc. But internal structure (essence) of the object must not be changed.

What is the essence of SudokuP (traditional Sudoku, X-sudoku, etc.) solution grid? I propose several variants of definitions. Choose definition you like.

1. The essence of SudokuP solution grid is a set of SudokuP (not necessary valid) puzzles, which has given solution grid as one of its solutions. One can consider distribution (histogram) of number of clues by number of puzzles as invariant, which must not be changed under isomorphic transformation.

2. The essence of SudokuP solution grid is a set of SudokuP minimal valid puzzles, having given solution grid as its solution. One can consider distribution (histogram) of number of clues by number of puzzles as invariant, which must not be changed under isomorphic transformation. Another possible invariant - list of MinClues for 3 bands, 3 stacks, 3 P-bands and 3 P-stacks (12 numbers). Numbers in MinClues list may be permuted, but according to some rules. These numbers may not disappear and may not appear from nowhere during transformations.

3. The essence of SudokuP solution grid is a set of grid's unavoidable sets. One can consider distribution (histogram) of UA sizes by number of UA sets as invariant, which must not be changed under isomorphic transformation.

When I objected to the application of Burnside's Lemma for ED SudokuP solution grids, I posted an example - MC grid (being valid SudokuP solution grid) can be transformed to another valid SudokuP solution grid by swapping rows r2/r3 only. I insisted that such transformation should be treated as isomorphic, because it transforms one SudokuP solution grid to another. But it turns out this transformation (swapping r2/r3) is not isomorphic!

- Code: Select all
` MinClues:`

MC grid Transformed grid

Band 1 6 6

Band 2 6 6

Band 3 6 6

Stack 1 6 4

Stack 2 6 4

Stack 3 6 4

P-band 1 6 6

P-band 2 6 3

P-band 3 6 3

P-stack 1 6 6

P-stack 2 6 6

P-stack 3 6 6

You can see that swapping rows r2/r3 is not isomorphic, because it changes internal structure (essence) of the MC grid.

Note. Typical UA set's permutation changes internal structure (essence) of the SudokuP (traditional Sudoku, X-sudoku, etc.) solution grid.

We should change terminology. When we consider solution grids we should say isomorphic

Structure Preserving Transformations, not Validity Preserving Transformations.

Serg