Something has been missing in this discussion.

Earlier,

Mathimagics wrote:

The elements of the SudokuP symmetry group should be a subset of the elements of the Sudoku symmetry group, or all is lost!

In the end, that doesn't turn out to be the case.

The group that has been under discussion, is the intersection the usual Sudoku group, and the (full) SudokuP group.

Like the Sudoku group, the SudokuP group is strictly larger than the intersection.

(There are transformations that map SudokuP grids to SudokuP grids, but usually destroy the integrity of Sudoku grids).

The full group, has 8 * 6^4 elements.

One of the missing transformations ... F, I've been calling it ... transforms each band, like this:

- Code: Select all
`+-------+-------+-------+ +-------+-------+-------+`

| a b c | d e f | g h i | | a b c | j k l | s t u |

| j k l | m n o | p q r | -> | d e f | m n o | v w x |

| s t u | v w x | y z ? | | g h i | p q r | y z ? |

+-------+-------+-------+ +-------+-------+-------+

It's like a "transpose" for the positions of the (box) mini-rows.

It maps rows to boxes, boxes to rows, columns to box positions, and box positions to columns.

GAP string for F:

(4,10)(5,11)(6,12)(7,19)(8,20)(9,21)(16,22)(17,23)(18,24)(31,37)(32,38)(33,39)(34,46)(35,47)(36,48)(43,49)(44,50)(45,51)(58,64)(59,65)(60,66)(61,73)(62,74)(63,75)(70,76)(71,77)(72,78)

Adding "F" to the list of generators for the intersection, gives a set of generators for the full SudokuP group.

The set {D,F}, where D is the usual "transpose" operation, generates a group of size 8, that accounts for the factor of 8, in "(8 * 6^4)".

The most interesting elements, are E := F * D * F, and G := D * F * D.

G is like F, but it acts on stacks instead of bands.

E is like D, in that they're conjugates.

E maps rows to rows, and columns to columns ... 123456789 -> 147258369.

In addition, it maps boxes to box positions, and vica-versa.

It's like a "b/p" transpose, as opposed to an "r/c" transpose.

The remaining elements can be written as: D * E, D * F, and D * G.

(D * E), like D, E, F and G, has order 2 -- applying it twice, restores the original grid.

The other two have order 4. (One is the inverse of the other).

Have fun with that !

Cheers,

Blue.