denis_berthier wrote:Fix some value of p (say 8 or 9 or more).
1) Consider all the known minimal puzzles with BxB ≥ p - in solution-minlex form
Take all their min-expands (only one isomorphic copy of each); we know they will all be in solution-minlex form and that they will all have their BxB ≥ p.
At this point, I think all the associated minimals with BxB ≥ p have already been found by previous searches.
2) Take all the 1-expands* of all those BxB ≥ p min-expands (again filtered so that there's only one isomorphic copy of each of them). Keep only those that still have BxB ≥ p.
Again, take the min-expands.
Again, we know they will all be in solution-minlex form and that they will all have their BxB ≥ p.
Question: have all the minimals (say with BxB ≥ p) been found?
I've done something close to what I mentioned in the above post. (There are many possible variants of such techniques.)
I fixed p = 10.
Note that all the search remains within the initial space of solution grids.
1) I started form the 1184 minimals B10B puzzles in the collection assembled by coloin here:
http://forum.enjoysudoku.com/the-bxb-classification-of-t-e-2-puzzles-t41922-105.html (This is not the full known collection, but not far from it; I didn't try to add the few newly found puzzles).
There are 581 non-isomorphic expansions by Singles (BRT-expands).
If you try to find all the associated minimals, each of them may be
a priori harder (due to the minimisation phase). (BRT-expansion alone can't make a puzzle simpler or harder.)
In our case, it appears that there are exactly 1184 minimal puzzles not in T&E(0) - the same ones as at the start.
Considering the search methods used for finding the original collection, this is not a big surprise.
2) After the above step, I computed all the 1-expands of the 581 non-isomorphic BRT-expands.
Note: I didn't prune the expansion phase with the condition that the expansions are in T&E(2) or that they have their BxB ≥ 10.
There are 29,049 such 1-expansions and only 28,853 up to isomorphisms (notice that there are not many isomorphisms at this point). The latter are also the non-isomorphic BRT+1-expansions of the original 1184 minimal B10B puzzles. Nothing can be said
a priori of the difficulties of these puzzles.
But it appears that there are no harder minimals, there are lots of much easier puzzles and there are exactly 1184 B10B puzzles - the same ones as at the start.
Considering the search methods used for finding the original collection, it was not granted that BRT+1-expansions wouldn't lead to some harder puzzle, but it isn't very surprising either because hard puzzles it's difficult to find harder puzzles.
The B10B sub-collection of minimals in the above-mentioned coloin's collection is closed under BRT+1 expansion followed by minimisation.* "easier" means wrt to T&E-depth and BxB classification within T&E(2).
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