Hi,

Mathimagics!

Mathimagics wrote:Hey, I figured it out! It's rather magical, but yes, as Serg said, there are just 1296 isomorphisms we need to check in order to determine if a non-SudokuP grid X has a SudokuP-isomorph.

I should explain - what I meant when I was saying about checking 1296 transformations instead of checking 3359232 transformations from VPT Group.

DefinitionSudokuP Validity Preserving Group or PVP Group is group of transformations preserving validity of any valid SudokuP puzzle or solution grid. This group is generated by following set of transformations and is subgroup of VPT Group.

1. Transposing.

2. Permutations of 3 bands.

3. Permutations of 3 stacks.

4. The same permutations of rows in every band.

5. The same permutations of columns in every stack.

Totally we have 2 x 6^4 = 2592 isomorphic transformations.

Suppose one wants to check - what images (i.e. results of grid's transformation) of given valid SudokuP solution grid are valid under the action of VPT Group (3359232 transformations)? It's no sense to check results of PVP Group, because images of those 2592 transformations will be definitely valid SudokuP solution grids. It's not so obvious, but can be easily proven, that applying PVP Group to

invalid SudokuP solution grid will give invalid SudokuP solution grids only (2592 grids at most).

Now let's consider VPT Group partition by left cosets (Group Theory terminology) of PVP Group. VPT Group is divided by 1296 cosets, each coset containing 2592 transformations. Each coset has its own representative

t (transformation from VPT Group), that produces all coset's transformations by applying

t to each PVP Group transformation (coset contains elements:

t p1,

t p2, ...

t p2592). So, we have 1296 representatives (including "Do nothing" transformation) for VPT Group partition by left cosets of PVP Group. If we want to check what images of given valid SudokuP solution grid are valid under the action of VPT Group, we need to check images of applying 1296 representatives only to given valid SudokuP solution grid.

Those 1296 representatives can form group if PVP Group is normal (hello, GAP!).

Serg