Hi, Denis!

denis_berthier wrote:Serg wrote:Such patterns were discussed in the thread

Fully symmetrical invalid patterns in 2017. Two lists of "basic" patterns were published there - 25 invalid patterns hot having valid puzzles and 49 valid patterns having valid puzzles.

This will be of great help for the present thread and I'll use this before publishing more about the other Families.

To be more precise, skipping the beginning of the thread, the main results are here:

http://forum.enjoysudoku.com/fully-symmetrical-invalid-patterns-t33569-53.html and in the next 4 or 5 posts. Can you confirm that those are the final lists of valid and invalid fully-symmetrical patterns?

My post dated by "Sun Mar 12, 2017 8:32 pm" (the topmost post at the page

http://forum.enjoysudoku.com/fully-symmetrical-invalid-patterns-t33569-53.html) contains attached files "fullsym_patterns_invalid.zip" (full list of 871 essentially different invalid fully symmetrical patterns) and "fullsym_patterns_valid.zip (full list of 5145 essentially different valid fully symmetrical patterns, in total - 6016 ed fully symmetrical patterns). Those lists are final.

denis_berthier wrote:Serg wrote:If any fully symmetrical pattern is subset of some "basic" invalid pattern, it has no valid puzzles. If any fully symmetrical pattern is superset of "basic" valid patterns, it has at least one valid puzzle.

When you say "it has valid puzzles", do you mean "not necessarily minimal ones", "not necessarily symmetrically-minimal ones" or merely (as I understand it) valid ones (i.e.single-solution)?

Can you confirm that the above referenced patterns are resp. maximal/minimal-symmetric?

When I am saying "(some) pattern has valid puzzles" I assume "not necessarily minimal valid puzzles". Valid puzzles, as usual, must have unique solution. Published lists of 25 symmetrically maximal fully symmetrical invalid patterns are symmetrically maximal in the sense that addition of any clues

preserving full symmetry makes those patterns valid. Any fully symmetrical invalid pattern must be subset of some pattern(s) from this list. But all 25 patterns are independent, i.e. the aren't subsets of each other. Similarly the list of 49 symmetrically minimal fully symmetrical valid patterns becomes invalid by removal of any clues preserving their full symmetry.

Some of symmetrically maximal invalid patterns are maximal in general sense, i.e. they becomes valid by adding any one clue (not necessarily preserving full symmetry). For example, well-known

- Code: Select all
` Magic Pattern`

x x x x x x x x x

x x x . . . x x x

x x x . . . x x x

x . . . . . . . x

x . . . x . . . x

x . . . . . . . x

x x x . . . x x x

x x x . . . x x x

x x x x x x x x x

is maximal in general sense.

Saying about minimality of valid puzzles produced by symmetrically minimal valid patterns. When those patterns are minimal in general sense, i.e. they become invalid by removing any one clue, they have

minimal valid puzzles only. More detailed it's discussed in the thread

Minimal patterns.

Serg