Hi,

Mathimagics!

I have too many questions, as usual.

Mathimagics wrote:I propose a simple Canonical Form algorithm for SudokuP solution grids.

It has the advantage of simplicity of implementation, and gives a box1-normal form.

What do you mean, saying "box1-normal form"?

Mathimagics wrote:If X is the grid in question, we first convert it to gsf minlex form (eg: using Michael Deverin's function).

For what purpose do you convert given SudokuP solution grid to minlex form?

Mathimagics wrote:We then apply band2/3 and stack 2/3 permutations in a fixed order, selecting the first case that has SudokuP property. If we use the same order of test and permute operations, we will produce the same result.

Is this coset representative transformations set (mentioned earlier 1296 transformations)?

Why don't you account for F/G-transformations?

I have a counter proposal.

I pondered about necessary modification of your method to count

naturally different SudokuP (valid) solution grids. Here is verbal description of the algorithm.

1. Initially we have the list of all essentially different regular sudoku solution grids (5472730538 solution grids).

2. Using 1296 coset representative transformations, we check images of all these representative transformations (transformations are repeatedly applied to given regular sudoku solution grid).

3. If an image of representative transformation is valid SudokuP solution grid, we produce "brother" SudokuP solution grids, applying set of 4 F/G-transformations (10368/2592). So, at the end of this procedure we have 4 "brother" SudokuP solution grids, including starting image of applying

representative transformation (result of applying transformation "Do nothing", participating F/G-transformations group).

4. New found 3 "brother" SudokuP solution grids are reduced to minlex form. (Minlex form of starting image of applying representative transformation is already known).

5. Different minlex forms (among 4 "brother" SudokuP solution grids) are added to set of "brother" minlex forms.

6. Count (different) minlex forms in the set of "brother" minlex forms after cycle done by 1296 coset representative transformations. Now we have

N - number of interconnected sudoku solution grid's orbits for given sudoku solution grid (one of 5472730538 solution grids).

7. We must increase resulting number of naturally different SudokuP solution grids by

fraction 1/N.

8. After processing of all essentially different sudoku solution grids (5472730538 grids) we'll get number of naturally different SudokuP solution grids.

Sum of fractions looks like very strange, but it should work.

Serg

[Edited. I corrected an error - F/G-transformations group contains 8 elements, not 4 elements, as was written earlier.]

[Edited 2. I was wrong again - we need no overall F/G-transformations group (containing 8 elements), it is sufficient to consider 4 elements only (representatives for partitioning PVP Group to cosets by 2592-elements subgroup).]