udosuk wrote:The "Platinum trellis" (Pt), is the only grid (barring isomorphism) which contains NO fully-entwined pairs (out of 36 possible pairings)...

Yes.

And probably have the largest possible number of "2-digit unavoidables" (78)...

I'd be happier with "possibly" rather than "probably" but, yes, that's the gist.

And is conjectured to be one of the grids which could not give a 19-clue puzzle, but only 20-clue puzzles

I've finished my two branches of the 19 search, with no success. Assuming Moschopulus also turns up no 19s in his search then, yes, no 19s. Ocean's supplied a 20.

(how many such grids are there?)...

No idea.

Red Ed has also found the total of 127 grids which contain 36 (the maximum) fully-entwined pairs... Are they posted somewhere?

No, not posted yet. I keep meaning to generate some stats to go with them first. I'll get round to that soon, or just post the bare grids.

So these grids are supposed to have the smallest possible number of "2-digit unavoidables"?

The smallest number of minimal unavoidable sets comprising just 2 digits, yes. In a random grid, a pair of digits gives rise to 1, 2, 3 or 4 minimal unavoidable sets; in the 127 FE grids, each pair gives rise to just 1.

And are more probable to give 17-clue puzzles?

Mmm... that might be a "yes", but not in a useful way. You can generate all sorts of statistics that try to capture what is good about 17-capabable grids. The statistic, Unav, that counts the number of minimal 2-digit unavoidables is more-or-less as good as any other, in the sense that there's a

c. 80% chance that a 17-capable grid will "beat" a random grid by that measure. However, this only means that better Unav statistic => better chance of a 17,

not that better Unav statistic =>

good chance of a 17!!. In fact I would be surprised if any more 17-capable solution grids (beyond those derived from gfroyle's puzzles list) will be discovered.

The "most canonical grid" (123456789456789123789123456231564897564897231897231564312645978645978312978312645) is conjectured to NOT giving any 19-clue puzzle, but only ONE 20-clue puzzle (barring isomorphism)? Is it the ONLY one? Are there any grids which NO puzzles with 20 clues or fewer are found? Are there other grids which also give at most ONE 20-clue puzzle (up to isomorphism)?

Someone else had better comment on that. I'll just say that I am sure Ocean will find more 20s in the Pt grid if he goes looking, since if there was (essentially, up to 6-way automorphism) only one 20 then it would have been too much of an achievement to have found it so quickly ... even for Ocean's magic puzzle-discovery methods!