Hi,
blue!
blue wrote:A topic for another day: The '560' number for quarter turn symmetry, is also the number of general minlex solution grids (i.e. of 5.4e9), that have at least one transformation to a form that shows quarter turn symmetry. The numbers that eleven has given, and possibly Serg too, depending on how he used 'gridchecker' , seem to be that kind of number. That isn't quite what I've been calculating, which is more like the number of "symmetrically minlex" solution grids, that actually do show the symmetry. The difference would come down to what kinds of transformations are allowed, in producing the "minlex" grids from initial grids showing the symmetry -- any transformation at all, or only those that preserve the corresponding shape symmetry. For arbitrary symmetry types, it isn't clear to me, that the two kinds of numbers should match. Maybe I'm missing something ?
Let's consider all possible solution grids (6 x 10^21). Some of them have "quarter turn symmetry" property, so they form subset of all possible solution grids. Let's consider now all possible VPTs (3359232). Some of them preserve quarter turn symmetry, i.e.
any quarter turn symmetric grid transformed by these VPTs will remain quarter turn symmetric grid (but maybe another quarter turn symmetric grid). For example, half turn rotation forever preserves quarter turn symmetry. Other VPTs may preserve symmetry for
some quarter turn symmetric grid, but destroy symmetry for the rest symmetric grid, and can transform some non-symmetric grids to symmetric grid. My point is that it's no matter what kind of VPTs are considered. If two different quarter turn symmetric grids have the same minlex form, such grids can be transformed to each other by some VPT, so
they are isomorphic. Hence it's no sense to consider both grids because they are
similar. So, we can consider, say, the one having less "minlex quarter turn symmetric form" and discarding other (isomorphic) grid.
Sorry for saying trivial things. Why we usually avoid considering isomorphic solution grids? Because they are equivalent. So, if we know properties of the one of them, we know properties of others. For example, let a grid contains two 17-clue puzzles exactly. Then all its isomorphs will contain two 17-clue puzzles each, and each puzzle of isomorphs can be produced from considered puzzle by the transformation producing given isomorph.
I use the following method for symmetric grids processing.
1. Produce (by specially written program) list L1 of possible quarter turn symmetric grids in "minlex quarter turn symmetric form" (having 1,2,...9 in central box B5 and one digit "5" in the rest of the boxes). At this stage isomorphs are not filtered out.
2. Produce (by gridchecker) list L2 of "true" minlex forms for given symmetric grids list L1.
3. Start with empty "unique true minlex forms" list L3 and empty "ED quarter turn symmetric grids" list L4.
4. For each grid in L1 list check - is true minlex form of the grid (corresponding position in L2 list) in "unique true minlex forms" list L3? If "no" - put the grid in "minlex quarter turn symmetric form" to output list L4 and put true minlex form of the grid to L3 list. If "yes" - do nothing.
5. At the end we'll have output "ED quarter turn symmetric grids" list L4 in "minlex quarter turn symmetric form".
Serg