ronk wrote:I don't think they're as different as you make it sound.
In order to have "clue value symmetry", you first need "clue positional symmetry". And the symmetry doesn't need to be perfect, i.e., the "symmetric distance" may be greater than zero.
In my previous post I also wrote:(Although all automorphic puzzles should be in theory morphable into a form with a certain clue-pattern/shape symmetry, we generally aren't putting much interest in that aspect.)
I thought I made that point already, perhaps you missed that part?
While I said "two different topics", I didn't imply they're "two totally irrelevant topics".
However, the automorphic puzzles/solutions we're studying, has the property that you can scramble them randomly that no visual "clue positional symmetry" appears anymore, but the automorphism still exists.
Also currently in this thread a heavy focus is on the solution grids (or so it seems to me). But I think the studying of "clue positional symmetry" isn't too relevant to the solution grids (please correct me if I'm wrong).
Also, some of the "clue positional symmetries", such as horizontal/vertical reflections, has no automorphism to match.
Bottomline: if you replace all the clues of an automorphic puzzle by asterisks and perform the same row/column transformations that morph it back, you will find a certain "automorphism symmetry" on the asterisks, such as this one:
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.*..**...
..**.*...
*..**....
....*..**
...*..**.
.....**.*
...*...*.
.....**..
....*...*
However, I don't think a lot of players consider the clue pattern above "symmetrical", and probably not in the sense of gfroyle's paper.
That said, it would be very nice if you or someone else can somehow connect these 2 concepts together and develop something helpful for cracking very diabolic/evil puzzles with heavily scrambled hidden automorphisms.