The New Sudoku Players' Forum

[denis_berthier]

I cannot recall any specific case of puzzles with the same solution but different B7B+ ratings. To draw a definitive conclusion on this, larger numbers of new grids would have to be investigated.

I can't see any a priori reason why this would be impossible. What happens very often is the BRT-expand of a B7B+ minimal puzzle having B6B- minimals.
But who knows if you consider only B7B+ puzzles; then, very strange things happen, such as the resolution paths having a gap of eliminations between B7B and their BxB.

Just seeing your second post.

Some second thoughts about different BxB ratings for a given solution. There are now a large number of minimals of non minimal B7B+ puzzles available. Has someone tested if among those minimals there are some with different BxB ratings? I imagine that this could be the case for 'higher' expansions.

If you don't restrict the minimals to those in B7B+, you'll easily find counter-examples, even puzzles in T&E(1).
I've done a quick statistical study of the minimals of BRT and BRT+1 expands of B7B+ minimals. See details in [HCCS2], sections 6.2.3 and 6.2.4. For the 3 known B14B, we can already say that all the minimals of their BRTR+1 expands are in B14B or in B6B- or T&E(1). I'll see if I can extract relevant information for other individual puzzles.

Or is this theoretically impossible?

Probably like many questions in Sudoku, if true of all the known examples: impossible to prove.

I can imagine that some 'higher' expansion has a lower BxB rating. In that case you would have different BxB rating for the same solution. Or must the search be limited to minimal puzzles or min_expands?

Between the characteristics of minimals of BRT and BRT+1 expansions, I didn't see much difference.

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[denis_berthier]

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After reviewing my calculations for coloin's BXB-Mastermins collection of 3362 B7B+ minimal puzzles (Sept. 2024), it appears I had computed the following - which doesn't appear in [HCCS2] (July 2024).
I studied the B10B and B9B sub-collections, their BRT and BRT+1 expansions and the minimals of the latter two.

1) B10B: 1,184 puzzles
- 581 non-isomorphic BRT-expands
- giving rise to 1,184 minimal puzzles (which can only be the same as at the start): the B10B sub-collection is closed under BRT-expansion and minimisation

-28,853 non-isomorphic BRT+1-expands
- giving rise to 105,521 minimals
- only 63,693 of which are at T&E-depth 2
- with the following BxB distribution:

Code: Select all

(X-distribution-p-to-q "Coloin-2024-09-03-B7B/B10B/U-BRT+1-TE2-BxB.txt" 63693 0 10)
0: 0
1: 319
2: 33833
3: 25543
4: 2374
5: 254
6: 154
7: 0
8: 0
9: 32
10: 1184

No new B10B appears, but 32 B9B have appeared.

As BRT+1 expansion doesn't produce new solution grids, the conjecture is disproved!


2) B9B: 1,163 puzzles
- 419 non-isomorphic BRT-expands
- giving rise to 1,163 minimal puzzles (which can only be the same as at the start): the B9B sub-collection is closed under BRT-expansion and minimisation
I haven't studied the BRT+1 expansions

3) B8B: 244 puzzles
- 86 non-isomorphic BRT-expands
- giving rise to 244 minimal puzzles (which can only be the same as at the start): the B8B sub-collection is closed under BRT-expansion and minimisation
.

[hendrik_monard]

deleted post

[hendrik_monard]

3 new solutions, bringing the total to 804.

Code: Select all

1.3.56....5718...668.3.7..5.71.68...36.5.1...5.8.3....71...........1.9.7.......2.   10
12.456......1.9.......3....2.5....6..14.6.82.68.......54.6..1.....5...48...2..65.   10
12..56....57.8...66.8..7..................39.....218.7.618.2...5.2.7.1...8..1..2.   7


[coloin]

I also expected more new solution grids. That there are so few can perhaps be explained by the fact.......

ahhh I see now that there are some puzzles in your recent list which were already known. I guess its difficult to purge the expanded forms, but easier to purge the minimals.
Expanding the puzzles to maximal BXB in my hands isnt completely reliable..perhaps rarely missing a few clues at the top end.
This happened here, the duplicate puzzles are all in the BxB 10 with 10 twins ... 18 twins found with the extra clue.

Code: Select all

...4.67..4..18.2.6....27....96.......48261...5.1...6.....81.4.2..46.2..1..2.7486. #10#12#   720 FNBP C33/M2.74.88
...4.67..4..18.2.6....27....96.......48261...5.1...6.....81.4.2..46.2.71..2.7486. #10#18#   734 FNBP C34/M2.91.72

I guess what we need is a complete file of minimal puzzles and a complete file of max expands...[ as denis suggests !] so will combine them

[denis_berthier]

Expanding the puzzles to maximal BXB in my hands isnt completely reliable..

Here is a reliable procedure (k fixed):
Let S be a set of minimal puzzles with BxB = k
1) expand all of the puzzles in S by Singles : you can be sure they all have BxB = k
2) compute all the 1-expands of all the previous puzzles, they can only have BxB ≤ k; keep only those with BxB = k
3) if the above remaining set is not empty, goto 1

I guess what we need is a complete file of minimal puzzles and a complete file of max expands...[ as denis suggests !]

Indeed, we need 3 sets: minimals, min-expands, max-expands
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[denis_berthier]

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I've now checked the full coloin's Mastermind and Masterminds4 collections of minimal B7B+ puzzles according to the following process:
- split each collection into sub collections of constant BxB
- for each sub-collection:
--- compute the BRT-epxands (Singles-expands)
--- eliminate redundancies (using gsf's software)
--- generate all the minimal puzzles (with no redundancies, using gsf's software)
--- question : do we get puzzles with a different BxB?
--- answer: NO ! Indeed, we get exactly the same number of puzzles as at the start (which can therefore only be isomorphic forms of the original ones).
Conclusion : for these collections, digging for minimals in the BRT-vicinity of a minimal puzzle doesn't produce puzzles with a different BxB classification.
(As mentioned in a previous post, replacing BRT-vicinity by a larger BRT+1-expansion wouldn't preserve this result.)
The result is undoubtedly related to the presence of a tridagon, but why it is true is still eluding me.
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[hendrik_monard]

Here is a reliable procedure (k fixed):
Let S be a set of minimal puzzles with BxB = k
1) expand all of the puzzles in S by Singles : you can be sure they all have BxB = k
2) compute all the 1-expands of all the previous puzzles, they can only have BxB ≤ k; keep only those with BxB = k
3) if the above remaining set is not empty, goto 1

I guess what we need is a complete file of minimal puzzles and a complete file of max expands...[ as denis suggests !]

Indeed, we need 3 sets: minimals, min-expands, max-expands
.

In the mean time, a quick procedure for checking if a new solution has been found is available:
1) transform the puzzle to the solution minlex isomorph (if it is not already done)
2) solve the puzzle
3) check if the solution is not in the B7B+ solutions database, currently available on my google drive at the link provided on December 19th.
That's how I do it.
I tried to add the 5 recently found new solutions to this database, but it seems to be impossible. So it looks like I'll have to change the link after each extension of the database, unless someone knows if and how an existing document on the google drive can be modified.
.