(1+BRT) expansion paths within T&E(n) and beyond

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Re: (1+BRT) expansion paths within T&E(n) and beyond

Postby denis_berthier » Fri Jul 25, 2025 3:30 pm

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Let's make the challenge harder, by replacing strict-expands by (1+BRT)-expands.

Find some minimal puzzle P0.
n=1: Compute all the minimals of BRT-expand(P0). Among them, find some minimal puzzle P1 such that BRT-expand(P0) is a (1+BRT)-expand of BRT-expand(P1).
n=2: Compute all the minimals of BRT-expand(P1). Among them, find some minimal puzzle P2 such that BRT-expand(P1) is a (1+BRT)-expand of BRT-expand(P2).
...
The question is the same, how far can one go?

Said otherwise:
Code: Select all
P0 minimal      P'0=BRT-expand(P0)
P1 minimal      P'1 = BRT-expand(P1)       P''1 = some 1-expand of P'1      BRT-expand(P''1) = P'0
P2 minimal      P'2 = BRT-expand(P2)       P''2 = some 1-expand of P'2      BRT-expand(P''2) = P'1
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Re: (1+BRT) expansion paths within T&E(n) and beyond

Postby P.O. » Sat Jul 26, 2025 1:33 pm

if i understood correctly
Hidden Text: Show
Code: Select all
......2.5...8..9......3....9.1....6....72....6....5......1...4.52........3.......   17c minimal
489671235213854976765239418951483762348726159672915384897162543526347891134598627   81c brt-expansion p0

.......35..38..9.676.2.....9.1..3..2.4...61.......5..4..71..54..2...7............   24c minimal
...6712352138549767652394..9.1483..2.48.261..6.2.15..4..716254..2...7........8.2.   48c brt-expansion p1

....712.5..38...76.6....4..9....3..2.4....1..6.2..5.....71.254..2............8...   25c minimal
...671235213854976765..94..9.1483..2.48..61..6.2.15..4..716254..2...7........8.2.   45c brt-expansion p2

......2.5.13....7676....4..9..4....2.48..61....2.15......16254......7........8.2.   27c minimal
...671235213854976765..94..9.1483..2.48..61..6.2.15..4...16254..2...7........8.2.   44c brt-expansion p3

......23..1.85..7..65...4..9..483....48...1..6.2.1.......16254......7..........2.   26c minimal
...671235213854.7.765..94..9.1483..2.48..61..6.2.15..4...16254..2...7........8.2.   42c brt-expansion p4

.....1.352..8.4.7..65...4..9.14.3..2.48..61..............16254......7..........2.   26c minimal
...671235213854.7.765..94..9.1483..2.48..61......15..4...16254......7........8.2.   39c brt-expansion p5

......23521.8.4.7.7.5...4..9..483....48..61..............16254......7..........2.   26c minimal
...671235213854.7.765..94..9.1483....48..61......15..4...16254......7........8.2.   38c brt-expansion p6

...6....52.3.5..7.7.5..94..9..4.3....48..61......1.......16.54......7........8.2.   25c minimal
...6712352.3854.7.7.5..94..9..483....48..61......15..4...16254......7........8.2.   35c brt-expansion p7

...6...3.2.38...7.7.5..9...9.........48..61......15..4...16254......7........8.2.   25c minimal
...67.2352.385..7.7.5..9...9..483....48..61......15..4...16254......7........8.2.   32c brt-expansion p8

...6....52.3....7.7.5..9...9...83....48...1.......5..4...16.54......7........8.2.   23c minimal
...67.2352.3....7.7.5..9...9..483....48..61......15..4...16254......7........8.2.   30c brt-expansion p9

...6....52.3....7.7.5..9...9...83....48..61..........4...16.54......7........8.2.   23c minimal
...67.2352.3....7.7.5..9...9..483....48..61..........4...16254......7........8.2.   28c brt-expansion p10

this reads: 1brt-expansion of p1 with value 5 in cell 29 = p0 / (...)
Code: Select all
p1 + (29 5) = p0 / p2 + (22 2) = p1 / p3 + (57 7) = p2 / p4 + (16 9) = p3 / p5 + (48 2) = p4
p6 + (36 2) = p5 / p7 + (11 1) = p6 / p8 +  (6 1) = p7 / p9 + (13 8) = p8 / p10 + (50 1) = p9
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Re: (1+BRT) expansion paths within T&E(n) and beyond

Postby denis_berthier » Sat Jul 26, 2025 2:15 pm

P.O. wrote:if i understood correctly
Code: Select all
......2.5...8..9......3....9.1....6....72....6....5......1...4.52........3.......   17c minimal
489671235213854976765239418951483762348726159672915384897162543526347891134598627   81c brt-expansion p0

.......35..38..9.676.2.....9.1..3..2.4...61.......5..4..71..54..2...7............   24c minimal
...6712352138549767652394..9.1483..2.48.261..6.2.15..4..716254..2...7........8.2.   48c brt-expansion p1


this reads: 1brt-expansion of p1 with value 5 in cell 29 = p0 / (...)
Code: Select all
p1 + (29 5) = p0


There must be an error here. r2c9 is already equal to 6 in P1.
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Re: (1+BRT) expansion paths within T&E(n) and beyond

Postby P.O. » Sat Jul 26, 2025 2:22 pm

(29 5) means n5r4c2
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Re: (1+BRT) expansion paths within T&E(n) and beyond

Postby denis_berthier » Sat Jul 26, 2025 2:45 pm

P.O. wrote:(29 5) means n5r4c2

OK, sorry for the misunderstanding.
and congrats for such a long chain.

Have you tried what your algorithm gives when you restrict it to T&E(3) or T&E(2)?
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Re: (1+BRT) expansion paths within T&E(n) and beyond

Postby P.O. » Sat Jul 26, 2025 3:17 pm

i haven't tried either te2 or te3 puzzles
naively, the idea is to start with a minimal puzzle whose brt-expansion has a large number of redundant values, and then find minimal puzzles whose brt-expansion reduces the number of redundant values as slowly as possible
17c puzzles that are solved by singles are good candidates for this, an exhaustive search of these puzzles, if possible, should find even longer chains
i'll see if i can find some good starting points for te2 and te3 puzzles
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Re: (1+BRT) expansion paths within T&E(n) and beyond

Postby P.O. » Mon Jul 28, 2025 4:04 pm

from Blue's 780 te2 on your github pages
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..............1..2.34....5.......3....1..5....6.....4.....3..7...2.4...1.8.26....   18c minimal
21..53..4.5.4.1.32.34.2615.52.6.431...13.5.26.63.1254....138275372549..1185267493   55c brt-expansion p0

....53...........2.34..6.........3....1..5....6..1..4....1..2.5..2549..1.8..6....   22c minimal
21..53..4.5.4.1.32.34.2615.52.6.431...13.5.26.63.1.54....138275372549..1185.6.493   52c brt-expansion p1

.....3......4.......4.2.15....6.4.....1....26..3.1.54....13..7.3.25....1.8..6.49.   26c minimal
21..53..4.5.4.1.32.34.2615.52.6.431...13.5.26..3.1.54....138275372549..1185.6.493   51c brt-expansion p2

21..............3...4.261..5..6........3...26....1.54.....3..7..725.9.....5...4.3   24c minimal
21..53..4.5.4.1.32.34.2615.52.6.43....13.5.26..3.1.54....138275372549...185.6.493   49c brt-expansion p3

21..53..4........2.34...1..5..6.43.......5.26....1..........27.3.2549.....5.6..9.   27c minimal
21..53..4.5.4.1.32.34..615.5..6.43....13.5.26..3.1.54....138275372549...185.6.493   47c brt-expansion p4

2...5.....5.....32..4...1.....6.4.....1....26..3.1.54......8.7.37.5......85.6.49.   26c minimal
21..53..4.5.4.1.32.34..615.5..6.43....13.5.26..3.1.54....138.7537.549...185.6.493   45c brt-expansion p5

21..53..........32..4...1..5..6........3...26....1.54........7..7.549...18..6..93   26c minimal
21..53..4.5.4.1.32..4..615.5..6.43....13.5.26....1.54....138.75.7.549...185.6.493   42c brt-expansion p6

21..53..4.......32..4...1..5..6.43.....3...26....1.5.....138.7..7.......1.5.6.49.   28c minimal
21..53..4.5.4.1.32..4..615.5..6.43....13.5.26....1.54....138.75.7.549...1.5.6.49.   40c brt-expansion p7

Code: Select all
p1 + (51 2) = p0 / p2 + (47 6) = p1 / p3 + (35 1) = p2 / p4 + (23 2) = p3
p5 + (61 2) = p4 / p6 + (48 3) = p5 / p7 + (74 8) = p6
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Re: (1+BRT) expansion paths within T&E(n) and beyond

Postby denis_berthier » Tue Jul 29, 2025 5:09 am

.
good. Again, I didn't "expect so long chains in T&E(2) (because of the constraint with minimal puzzles).

See http://forum.enjoysudoku.com/post354007.html#p354007 for a topological interpretation of the challenge.
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Re: (1+BRT) expansion paths within T&E(n) and beyond

Postby P.O. » Tue Jul 29, 2025 4:56 pm

from this collection
with te3 puzzles the second constraint, 1brt-expansion = previous brt-expansion sometimes fails, which i have not observed for te1 and te2 puzzles
Hidden Text: Show
Code: Select all
98.76.5..7.5.......648...9.59....87...8....4.......6...7.6.5......32.4......749..   27c minimal
98.76.5.47.5.......6485.79.59...687.6.8..7.4..47...6..47.6.5......32.4.7...1749..   38c brt-expansion

98..6.5.47.5.......6.8...9.59....87...8..7....4....6..4..6.5......32...7....749..   27c minimal
98.76.5.47.5.......6485.79.59...687.6.8..7....47...6..47.6.5......32...7...1749..   36c brt-expansion (44 4)

98.7..5.47.5.......6.8...9.59...68....8.......47...6..4..6.5......32...7....749..   27c minimal
98.7..5.47.5.......6485.79.59...687.6.8..7....47...6..47.6.5......32...7...1749..   35c brt-expansion (5 6)

98....5.4..5.......6.8..79.59...68....8.......47...6..4..6.5......32...7....749..   26c minimal
98....5.47.5.......6485.79.59...68..6.8.......47...6..47.6.5......32...7...1749..   32c brt-expansion (4 7)

98....5.4..5.......6.8..79.59...68....8.......47...6..47...5......32...7...1.49..   26c minimal
98....5.47.5.......6485.79.59...68..6.8.......47...6..47...5......32...7...1749..   31c brt-expansion (58 6)
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Re: (1+BRT) expansion paths within T&E(n) and beyond

Postby denis_berthier » Tue Jul 29, 2025 5:49 pm

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Difficulty of satisfying the additional constraint may be due to the rarity of minimal T&E(3) puzzles.

As for the shorter sequence, I'm not surprised either: the expansion paths are shorter also in T&E(3).
.
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Re: (1+BRT) expansion paths within T&E(n) and beyond

Postby denis_berthier » Sat Aug 09, 2025 7:42 am

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As explained in previous posts, I have a systematic procedure for generating extreme B T&E(1) puzzles from minimal T&E(3) ones, or from their min-expands. As previously, I consider "extreme B" to mean "B ≥ 12" (justified by the unbiased B distribution).
Reminder: the procedure is very simple (but time consuming in the "compute minimals" part): take the 1-expands of the T&E(3) min-expands, filter those in T&E(1) and with B ≥ 12, compute the minimals of the latter and filter those in T&E(1) (which are guaranteed to have B ≥ 12). Notice that, for each puzzle in T&E(3), the whole process remains within the same solution grid (in particular, there's no {-p +q} search.
As mentioned in previous posts, the mean throughput is 1.28 B12+ puzzle for each T&E(3) min-expand. As far as I know, this is the first time some kind of extreme puzzle can be obtained from another kind in a systematic way. What's also noticeable is, no similar procedure works to obtain high BxB puzzles in T&E(2).

I've just published on my Google drive a zipped collection of 504,481 such puzzles with their B ratings (corresponding to only a part of mith's T&E(3) collection):
https://drive.google.com/file/d/1wSkSokvR8maE9yQJI3_X1NSb5ank35IU/view?usp=share_link

The distribution of B ratings in % (in the [12 24] interval is as follows:
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12      13      14      15     16     17     18     19     20      21      22       23        24
45.02   25.81   14.09   8.88   3.70   1.54   0.62   0.23   0.091   0.018   0.0040   0.00079   0.00040


For the most extreme values, this means
Code: Select all
3126 B18 puzzles
1141 B19 puzzles
461  B20 puzzles
92   B21 puzzles
25   B22 puzzles
4    B23 puzzles
2    B24 puzzles

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