Hi

Serg,

JPF wrote:Serg wrote:So, I can state that every fully symmetric pattern is "subset" or "superset".

I checked and I can confirm your results:

Every fully symmetric pattern is a subset of the 25 "invalid" patterns* or a superset of one of the 49 "valid" patterns.Well done! A clever way to structure the set of 6016 ed-fully symmetric patterns.

I had done that confirmation, too.

Serg wrote:blue wrote:To avoid (possible) confusion in the future, it might be good to explain things in a little more depth.

I am ready to discuss this project in details (if there will be someone's interest).

blue wrote:Explain how these two (valid) puzzles, fit into your outline ...

- Code: Select all
`4 7 . | . . . | . 5 3`

2 . 5 | . . . | 8 . 7

. 3 . | . . . | . 6 .

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

. . . | 5 7 2 | . . .

. . . | 4 . 3 | . . .

. . . | 1 8 9 | . . .

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

. 8 . | . . . | . 7 .

3 . 2 | . . . | 9 . 8

1 6 . | . . . | . 3 4

Your second example (cited above) is more complicated than the first one. But it is superset of F77 pattern. Swap columns c4/c5 of your example to detect that resulting pattern is superset of F77.

F77 too ... Interesting

It wasn't as complicated as I thought.

I saw it as a "superset" (in quotes) of F55.

Swapping rows 2&3, rows 7&8, columns 1&2 and columns 8&9, shows the connection with F55.

The point that I was trying to bring out, about those two patterns, was that none of thier "actual" subsets, both 1) has a valid puzzle, and 2) is fully symmetric. Similarly, neither of them can be mapped to another form that 1) is fully symmetric, and 2) is an "actual" superset of one of the 49 "valid" shapes. On the other hand, each has at least one subset that 1)

does have valid puzzles, 2) is

not fully summetric 3)

is, as it turns out,

isomorphic to a fully symmetric pattern. Similarly (again), each one

can be mapped to a form that 1) is

not fully symmetric, but 2)

is an "actual" superset of one of the 49 "valid" shapes.

The "(possible) confusion in the future", that I was worried about, involved the possiblity that someone might not understand that those kinds of details were in play.

Like

JPF, I applaud your use of the 6106 "ED" shapes, and the corresponding interpretations for the "subset"/"superset" relations.

For fixing the "core" sets of 25 and 49 puzzle shapes, it's the best approach.

Also, your mention of a "magic-60" list, wouldn't be affected by the concerns that I mentioned.

---

As food for thought ... there is another way to look at things:

- There is a universe of "fully symmetric shapes", and corresponding puzzles (valid or not).
- There is a set of transformations ... a subset of the usual set of VPT's ... that includes only the transformations that map fully symmetric shapes (in general), to fully symmetric shapes. In particular, each one, maps every fully symmetric puzzle, to a fully symmetric puzzle, and (so) every fully symmetric shape, to a fully symmetric shape.
- For those transformations, a fully symmetric superset/subset, of a fully symmetric shape ... is always mapped to a fully symmetric subset/superset of the image of the initial shape.
- There is a corresponding notion of "ED"-vs-"EE", and/or "non-isomorphic"-vs-"isomorphic", for puzzles and/or shapes. Two puzzles/shapes are "isomorphic"/"EE", if and only if they are related by one of the transformations in (2). [ Note "EE", is short for "essentially equivalent". ]
- In that view:
- The number of "ED" shapes, is 6528 (rather than 6106).
- The number of "ED maximal invalid shapes", is 29 (rather than 25).

Each "invalid" fully symmetric shape is an (actual) fully symmetric subset, of one of the shapes that is "EE" (in the restricted sense) to a shape from that list. - The number of "ED minimal valid shapes", is 57 (rather than 49).

Each "valid" fully symmetric shape is an (actual) fully symmetric superset, of one of the shapes that is "EE" (in the restricted sense) a shape from that list.

The 4 extra "maximal invalid" shapes, are due to pairs like these two, that are "EE" in the usual sense, but not "EE" in the restricted sense.

- Code: Select all
` F7 F7a`

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 . x . . x . . x .

Note: The 2nd one has fully a fully symmetric superset, that (in the restricted sense) is not "EE/isomorphic" to any fully symmetric superset of the first shape. (Add r19c19 to the 2nd shape). That's meant to show that there's at least

some reason to think that the shapes couldn't possibly be "essentially equivalent".

For the 8 extra "minimal valid" shapes, 4 of them are the "2nd half" of pairs like the one above.

The other 4 include the two that I mentioned earlier, and the shapes for these puzzles:

- Code: Select all
`. 7 3 | . . . | 5 9 .`

8 . 5 | . . . | 7 . 4

2 4 . | . . . | . 8 3

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

. . . | 7 . 4 | . . .

. . . | . 9 . | . . .

. . . | 5 . 2 | . . .

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

9 2 . | . . . | . 4 6

1 . 6 | . . . | 9 . 7

. 3 4 | . . . | 1 2 .

. 9 2 | . . . | 8 1 .

1 . 3 | . . . | 4 . 7

6 7 . | . . . | . 3 2

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

. . . | 1 7 4 | . . .

. . . | 6 . 8 | . . .

. . . | 2 5 3 | . . .

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

7 6 . | . . . | . 8 3

3 . 1 | . . . | 2 . 9

. 2 9 | . . . | 6 7 .

Cheers,

Blue.