The New Sudoku Players' Forum

[P.O.]

a value that is placed collects all the values of its BRT-expansion minus those already placed so a value that has in its BRT-expansion all the other values regardless of its position in the path will collect all the remaining values
it is unlikely that from the 36 values i have i will get a longer expansion path than the one you found because in this selection there are 10 such values so there are 26 values left to organize into expansion steps before ending with any of the 10
what is important in this exercise is to have a good starting selection

here the BRT-expansion of your values:

Hidden Text: Show

Code: Select all

1: ((13 7))
1: ((17 2))
1: ((20 9))
1: ((23 3))
1: ((26 6))
1: ((36 7))
1: ((42 4))
1: ((51 8))
1: ((52 4))
1: ((56 4))
1: ((66 9))
2: ((10 4) (20 9))
2: ((15 9) (20 9))
2: ((24 2) (20 9))
2: ((79 9) (66 9))
2: ((80 4) (56 4))
3: ((4 4) (10 4) (20 9))
3: ((14 8) (13 7) (23 3))
3: ((27 4) (26 6) (23 3))
4: ((1 1) (10 4) (20 9) (2 2))
4: ((2 2) (1 1) (10 4) (20 9))
4: ((48 7) (36 7) (74 7) (56 4))
4: ((74 7) (56 4) (48 7) (36 7))
6: ((41 7) (48 7) (36 7) (74 7) (13 7) (56 4))
9: ((38 1) (1 1) (10 4) (48 7) (74 7) (20 9) (2 2) (56 4) (36 7))


[denis_berthier]

it is unlikely that from the 36 values i have i will get a longer expansion path than the one you found

it's not only unlikely but impossible. My algorithm finds all the longest expansion paths.[/quote]
But I was talking of your example with 2x4 backdoors
.

[P.O.]

My algorithm finds all the longest expansion paths

so it is not this algorithm that built this expansion

[denis_berthier]

My algorithm finds all the longest expansion paths

so it is not this algorithm that built this expansion

The longest expansion path for the minimal puzzle has 18 expansion steps.It's clearly not the one you mention. It's the result of the full expansion process.

If you're alluding to another of my algorithms that didn't find the shortest 1-step path out of T&E(3), it's a completely separate process, a small process coming after the full expansion one, just as a fallout of it. There was a bug in it, now corrected. Fortunately, I hadn't done lots of calculations with it yet. I was and I still am more interested in the longest paths than in the shortest ones.
.

[denis_berthier]

.
I now have explored 20,000 solution grids and their 1,240,589 minimal puzzles and corresponding 192,391 min-expands in the T&E(3) collection mentioned in the first post of this thread.
By searching the minimals of the 1-expands of the min-expands, I've found 236,000+ puzzles in T&E(1) with B ≥ 12.

The new max value for B is 24, for a puzzle proposed for solution here: http://forum.enjoysudoku.com/b24-t45793.html
.
.

[denis_berthier]

.

The new max value for B is 24, for a puzzle proposed for solution here: http://forum.enjoysudoku.com/b24-t45793.html

This is still very far from Mauricio's puzzle with B = 30 mentioned before. Do you have any relative statistics?

Short answer: no.

Longer answer:
The unbiased distribution of the B or W ratings has been given in [HCCS2, table 2.2] (note: it was already given in [CRT] and [PBCS]), based on the controlled-bias collection (https://github.com/denis-berthier/Controlled-bias_Sudoku_generator_and_collection). This table shows a 0.007% estimate for all the puzzles with B ≥ 10. Needless to say this eliminates any practical possibility of doing unbiased stats for B > 10.

As for the B ≥ 12 puzzles I'm finding, they are the result of a precise process, which guarantees a very strong bias at every stage (start with puzzles in T&E(3)...).
What I'll soon be able to get is their distribution.
(I'll develop function to do stats not only on each sample of solution grids, but also on several samples at the same time. This is somehow required by my decision top make all calculations solution grid by solution grid.)
As for now, I can only say: I found 236,000+ puzzles with B ≥ 12 and only 1 with B = 24.

Note also that, at the start of my process, I consider only 1-expands in T&E(1) of min-expands in T&E(3) and I keep only those with B ≥ 12. If instead of this I used the BRT-expands of these 1-expands and/or if I kept for instance all those with B ≥ 10, I'd probably get many more B12+ puzzles - but that'd make the computations much longer.
.

[denis_berthier]

.
In the 5th post of this thread and in the second last one above, I explained how T&E(3) minimal puzzles, after (BRT+1)-expansion and then minimisation, frequently give rise to very high B ratings in T&E(1) (with a max value obtained for B = 24).
I also applied the same procedure to puzzles in T&E(2) with BxB ≥ 7 (http://forum.enjoysudoku.com/1-brt-expansion-paths-within-t-e-n-and-beyond-t45647-32.html) and I got similar results.

I didn't try farther, because I thought this success was due to the persistence of a tridagon or to some degenerated remains of one (all the T&E(3) and B7B+ minimal puzzles have a tridagon).
However, in order to make a substantial conjecture about it, it was necessary to try to start from other puzzles. For that purpose, I started from the ph2010 T&E(2) database and I tried the B6B puzzles (111 puzzles for 107 different solution grids). Note that none of these puzzles has a tridagon. And here my big surprise: I find a puzzle in B27. See here: http://forum.enjoysudoku.com/b27-t45797.html.
Initial non-conjecture disproved.

My next explanation for the many B12+ puzzles is merely, when you dig in the (BRT+1)-vicinity of T&E(3) or T&E(2) puzzles, there are very high B rating T&E(1) puzzles, because the latter are more frequent than the former.

Note: I've launched similar calculations for the 2204 minimal puzzles of ph2010 in B5B. Be patient, because such calculations are very time consuming - but as B rating calculations are ongoing, I can already see values ≥ 18.
.

[P.O.]

for this puzzle i did not find an expansion larger than 18 steps, i found an expansion of 18 steps with the same 25 values ​​but a slightly different distribution

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..3...789.56......7.81..5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3...   29c   min-expand
..3...789.56......7981..5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3...   30c   p1
..3...789.56......7981.25...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3...   31c   p2
..3...789456......7981.25...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3...   32c   p3
..3...789456......7981.25...345..8..8.5...6.2.6.....5.3...9..7.589.47...6....3...   33c   p4
..34..789456......7981.25...345..8..8.5...6.2.6.....5.3...9..7.589.47...6....3...   34c   p5
..34..789456......7981.25...345..8..8.5...6.2.6.....5.3...9..7.589.47...6....39..   35c   p6
..34..789456......7981.25...345..8..8.5..46.2.6.....5.3...9..7.589.47...6....39..   36c   p7
..34..789456..9...7981.25...345..8..8.5..46.2.6.....5.3...9..7.589.47...6....39..   37c   p8
..34..789456..9...7981.25...345..8..8.5..46.2.6...8.5.3...9..7.589.47...6....39..   38c   p9
..34..789456..9...7981.25...345..8..8.5..46.2.6...845.3...9..7.589.47...6....39..   39c   p10
..34..789456..9...7981.25...345..8..8.5..46.2.6...845.34..9..7.589.47...6....39..   40c   p11
..34..789456..9.2.7981.25...345..8..8.5..46.2.6...845.34..9..7.589.47...6....39..   41c   p12
..34..789456..9.2.7981325...345..8..8.5..46.2.6...845.34..9..7.589.47...6....39..   42c   p13
..34..789456..9.2.7981325...345..8.78.5..46.2.6...845.34..9..7.589.47...6....39..   43c   p14
.234..789456..9.2.7981325...345..8.78.5..46.2.6...845.34..9..7.589.47...6....39..   44c   p15
1234..789456..9.2.7981325...345..8.78.5..46.2.6...845.34..9..7.589.47...6....39..   45c   brt
1234..789456.89.2.7981325...345..8.78.5..46.2.6...845.34..9..7.589.47...6....39..   46c   p16
1234..789456789.2.7981325...345..8.78.5..46.2.6...845.34..9..7.589.47...6....39..   47c   brt
1234..789456789.2.79813256..345..8.78.5..46.2.6...845.34..9..7.589.47...6....39..   48c   p17
1234..789456789.2.798132564.345..8.78.5..46.2.6...845.34..9..7.589.47...6....394.   50c   brt
1234..789456789.2.798132564.345..8.78.5..46.2.6...845.34..9..7.589.47...67...394.   51c   p18
1234..789456789.2.798132564.345..8.7815.746.2.67..845.34..9..7.589.47...67...394.   54c   brt


i tried with the puzzle B27, and i found an expansion of 19 steps so far
the distribution of its values: (2 1 1 1 1 1 1 1 1 1 1 2 2 1 2 1 2 3 9)

Hidden Text: Show

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......789..6...1.....2..6....19..5..8....3....4..7....3....8.....21....6.7..4....   21c   min-expand
......789..6...1.....2..6....19..5..8....3....4..7....3....8.....21....667..4....   22c   p1
......789..6...1.....2..6....19..5..8....3....4..7....31...8.....21....667..4....   23c   brt
......789..6...1.....2..6....19..5..86...3....4..7....31...8.....21....667..4....   24c   p2
......789..6...1.....2..6...319..5..86...3....4..7....31...8.....21....667..4....   25c   p3
......789..6...1.....2..6...319..5..86...3....4..7....31...8.....21....667.34....   26c   p4
......789..6...1.....2..6...319..5..86...3.9..4..7....31...8.....21....667.34....   27c   p5
.2....789..6...1.....2..6...319..5..86...3.9..4..7....31...8.....21....667.34....   28c   p6
.2....789..6...12....2..6...319..5..86...3.9..4..7....31...8.....21....667.34....   29c   p7
12....789..6...12....2..6...319..5..86...3.9..4..7....31...8.....21....667.34....   30c   p8
12....789..6...12....2..6...319..5..86...3.9..4..7....31...8.....21....667.34..1.   31c   p9
12....789..6...12....2..6...319..5..86...3.9..4..7....31...89....21....667.34..1.   32c   p10
12....789..6...12....2..6...319..5..86...3.9..4..7..6.31...89....21....667.34..1.   33c   p11
12....789..6...12....2..6...319..5.886...3.9..4..7..6.31...89....21....667.34..1.   34c   p12
12....789..6...12....2..6...319..5.886...3.9..4.87..6.31...89....21....667.34..1.   35c   brt
12....789..67..12....2..6...319..5.886...3.9..4.87..6.31...89....21....667.34..1.   36c   p13
12....789..67..12....2..6...319..5.886...3.9..4.87..6.31...89....21.7..667.34..1.   37c   brt
12....789..67..12....2..6...319..5.886...3.9..4.87..6.31.6.89....21.7..667.34..1.   38c   p14
12....789..67..12....2..6...3196.5.886...3.9..4.87..6.31.6.89....21.7..667.34..1.   39c   p15
12...6789..67..12....2..6...3196.5.886...3.9..4.87..6.31.6.89....21.7..667.34..1.   40c   brt
12...6789..67..12....2..6...3196.5.886...3.9..4.87..6.31.6289....21.7..667.34..1.   41c   p16
12...6789..67..12....2..6...3196.5.886...3.9..4.87..6.31.6289....21.7.3667.34..1.   42c   p17
12...6789..67..12....2..6...3196.5.886...3.9..4.87.36.31.6289....21.7.3667.34..1.   43c   brt
12...6789..67..12....2.16...3196.5.886...3.9..4.87.36.31.6289....21.7.3667.34..1.   44c   p18
12...6789..67..12....2.16...3196.5.886..13.9..4.87.36131.6289....21.7.3667.34..1.   46c   brt
12...6789..67..12....2.16...3196.5.886..13.9..4.87.36131.6289.7..21.7.3667.34..1.   47c   p19
12.4.6789..67..12.7..2.16..231964578867513.9..4.87236131.6289.7..21.7.3667.34..1.   55c   brt


[denis_berthier]

for this puzzle i did not find an expansion larger than 18 steps, i found an expansion of 18 steps with the same 25 values ​​but a slightly different distribution

As some steps can be permuted, no wonder there can be more than one expansion path.

i tried with the puzzle B27, and i found an expansion of 19 steps so far

Good. I haven't computed the expansions of the high B T&E(1) puzzles obtained at the end of my search process, but I'm not surprised that they allow long expansion paths. Generally speaking, puzzles in T&E(1) have much longer paths (in the mean) than those in T&E(3 or 2).
.

[P.O.]

for puzzle B24 i found an expansion path of 20 steps
its distribution: (1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 9 3)

Hidden Text: Show

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12....7.94.........8..2..4.21.3........6.....9...42.17.9...847.7......9884...71.2   29c   min-expand
12....7.94.........8..2..4.21.3........6.....9...42317.9...847.7......9884...71.2   30c   p1
12....7.94.........8..2..4.21.3........6.....9..842317.9...847.7......9884...71.2   31c   p2
12....7.94.........8.72..4.21.3........6.....9..842317.9...847.7......9884...71.2   32c   p3
12....7.94.........8972..4.21.3........6.....9..842317.9...847.7......9884...71.2   33c   p4
12....7.94.........8972..4121.3........6.....9..842317.9...847.7......9884...71.2   34c   p5
12....7.94.........89723.4121.3........6.....9..842317.9...847.7......9884...71.2   35c   p6
12....7.94.........89723.4121.3...6....6.....9..842317.9...847.7......9884...71.2   36c   p7
12....7.94.........89723.4121.3...6....6.....9..842317.9...847.7......9884.9.71.2   37c   p8
12....7.94..1......89723.4121.3...6....6.....9..842317.9...847.7......9884.9.71.2   38c   p9
12....7.94..1......89723.4121.3...6.3..6.....9..842317.9...847.7......9884.9.71.2   39c   p10
12....7.94..1......89723.4121.3...6.3..6.....9..842317.9...847.73.....9884.9.71.2   40c   p11
123...7.94..1......89723.4121.3...6.3..6.....9..842317.9...847.73.....9884.9.71.2   41c   p12
123...7.94..1......89723.4121.3...6.3..6.1...9..842317.9...847.73.....9884.9.71.2   42c   p13
123...7.94..1......89723.4121.37..6.3..6.1...9..842317.9...847.73.....9884.9.71.2   43c   p14
123...7.94..1......89723.4121.37..6.37.6.1...9..842317.9...847.73.....9884.9.71.2   44c   p15
123...7.94.71......89723.4121.37..6.37.6.1...9..842317.9...847.73.....9884.9.71.2   45c   brt
123...7.94.71..2...89723.4121.37..6.37.6.1...9..842317.9...847.73.....9884.9.71.2   46c   p16
123...7.94.71..2...89723.4121.37..6.37.6.1.2.9..842317.9...847.73.....9884.9.71.2   47c   brt
123...7.94.71..2...89723.4121.37..6.37.6.1.2.9..842317.9...847.73...4.9884.9.71.2   48c   p17
1234..7.94.71..2...89723.4121.37..6.37.6.1.2.9..842317.9...847.73...4.9884.9.71.2   49c   brt
1234.67.94.71..2...89723.4121.37..6.37.6.1.2.9..842317.9...847.73...4.9884.9.71.2   50c   p18
1234.67.94.71.92...89723.4121.37..6.37.6.1.2.9..842317.9...847.73...4.9884.9.71.2   51c   p19
1234.67.94.71.92...89723.412183759643746918259..842317.9...847.73...4.9884.9.71.2   59c   brt
1234.67.94.71892...89723.412183759643746918259..842317.9...847.73...4.9884.9.71.2   60c   p20
1234567894.71892...89723.412183759643746918259..842317.9...847.73...4.9884.9.71.2   62c   brt


[denis_berthier]

.
One of the themes of this thread is, non-minimal puzzles can be as interesting as minimal ones, in particular some special kinds of non-necessarily-minimals, those that stand on either side of the T&E(n) to T&E(p) borders, n≠p, n,p = 0, 1, 2 3. In my terminology, such puzzles are either T&E(n)-expands or 1-expands of T&E(n)-expands.

In [HCCS2, chapter 6, "Across and along the T&E(n) borders"], I've already granted much attention to such puzzles.
Here's now an example of interesting puzzles reached by applying 1-expansion to T&E(2)-expands.

As of now, the largest known finite B rating was 30 and only two T&E(1) puzzles were known with this rating, namely:

Code: Select all

.....1..2....3..4...15..6....71..8...2..9...71....4.5...86......4...7.9.3...5....; Mauricio; 9.6; B30; W∞
.....1..2....3..4...56..7....6...5...1......37..8...9...9..5.8..2..4....3..7..9..; 1to9only; 9.6; B30; W∞

or, in gsf''s solution minlex form for better comparison with the forthcoming ones:
1...5...9..71..2...8...2.4......3..8...7..6...9..4..1.5......2..4......3..62..8..; Mauricio; 9.6; B30; W∞
.....67..4...8...3..82...1.2.....5....1.4..9..3......6..6.1....8..9...2..7...5...; 1to9only; 9.6; B30; W∞

In previous posts, I've given a few minimal puzzles with B ratings close to 30. Here are now two (non-minimal) puzzles with B rating 30. They are 1-expands of min-expands of minimal puzzles in B5B:

Code: Select all

1......8..5....1.3..9..24....4.7.9..8.......1.3.....5...2..7.......246.....59....; 9.6; B30;  W∞
...4..7......8...6..9..3.5.....7.4.8.8.6.......5.9..1.5......2...2.3...19....1...; 9.7; B30;  W∞

originating in the two T&E(2) B5B minimals:
1......8..5......3..9..24....4.7.9..8.......1.3.....5...2..7.......246.....59....
...4..7......8...6..9..3.5.....7.4.8.8.6.......5....1.5......2...2.3...19....1...

Note that the two B30 puzzles have no minimals in T&E(1).
.

[P.O.]

for Mauricio B30 i found an expansion path of 22 steps:
(1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 2 3 1 2 2 2)

Hidden Text: Show

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.....1..2....3..4...15..6....71..8...2..9...71....4.5...86......4...7.9.3...5....   23c   min-expand
.....1..2....36.4...15..6....71..8...2..9...71....4.5...86......4...7.9.3...5....   24c   p1
.....1..2....36.4...15..6....716.8...2..9...71....4.5...86......4...7.9.3...5....   25c   p2
.....1..2....36.4...15..6....716.82..2..9...71....4.5...86......4...7.9.3...5....   26c   p3
.....1..2....36.4...15..6....716.82..2..9...71....495...86......4...7.9.3...5....   27c   p4
.....1..2....36.4...15..6....716.82..2..9...718...495...86......4...7.9.3...5....   28c   p5
.....1..2....36.4...15..6....716.82..2.89...718...495...86......4...7.9.3...5....   29c   p6
.....1..28...36.4...15..6....716.82..2.89...718...495...86......4...7.9.3...5....   30c   p7
.....1..28...36.4...15..6.9..716.82..2.89...718...495...86......4...7.9.3...5....   31c   p8
.6...1..28...36.4...15..6.9..716.82..2.89...718...495...86......4...7.9.3...5....   32c   p9
.6...1..28.2.36.4...15..6.9..716.82..2.89...718...495...86......4...7.9.3...5....   33c   p10
.6...1..28.2.36.4...15..6.9..716.82..2.89...718...495...86......4.3.7.9.3...5....   34c   p11
.6...1..28.2.36.4..315..6.9..716.82..2.89...718...495...86......4.3.7.9.3...5....   35c   p12
.6...1..28.2.36.4..315..6.9..716.82..2.89...718...495...86.9....4.3.7.9.3...5....   36c   p13
.6...1..28.2.36.4..315..6.9..716.82..2.89...718...495...86.9....4.3.7.9.3.9.5....   37c   p14
.6...1..28.2.36.4..315..6.9..716.82..2.89...718...495...86.9....4.3.7.9.3.9.58...   38c   p15
.6...1..28.2.36.4..315.26.9..716.82..2.89...718...495...86.9....4.3.7.983.9.58...   40c   brt
.6...1..28.2.36.4..315.26.9..716.82..2.89...718...495...86.9....4.317.983.9.58...   41c   p16
.6..81..28.2.36.4..315.26.9..716.82..2.89...718...495...86.9....4.317.983.9.58...   42c   p17
.6..81..28.2.36.4..315.2689..716.82..2.89...718...495...86.9....4.317.983.9.58...   43c   brt
.6..81..28.2.36.4..315.2689..716.82..2.89...718...495...86.9....4.317.983.9.58.6.   44c   p18
.6..81..28.2.36.4..315.2689..716.82..2.89...7183..4956..86.9....4.317.983.9.58.6.   46c   brt
.6..81..28.2.36.4..315.2689..716.82..2.89..17183..4956..86.9....4.317.983.9.58.6.   47c   p19
.6..81..28.2.36.4..315.2689..716.82..2.89..17183..4956..86.9....46317.983.9.58.6.   48c   p20
.6..81..28.2.36.4..315.2689..716.82.62.89..17183..4956..86.9....46317.983.9.58.6.   49c   brt
.6..81..28.2.3614..315.2689..716.82.62.89..17183..4956..86.9....46317.983.9.58.6.   50c   p21
.6..81..28.2.36145.315.2689..716.82.62.89..17183..4956..86.9....46317.983.9.58.6.   51c   brt
.6..81..28.2.36145.315.2689..716.82.62.89..17183..4956..86.9....463175983.9.58.6.   52c   p22
.6..81..28.2.36145.315.2689..716.82.62.89..17183..4956..86.9...2463175983.9.58.6.   53c   brt


[P.O.]

i'm testing eleven’s collection with '1-expansion to T&E(2)-expands', i frequently find expansion paths >= 20 steps, with a maximum of 24 steps so far

Code: Select all

1....6....571.......9...51..4....3.......8..29..7...6......24..5..6...9.....3..28    #e127 + (80 2) 23 steps
1...5...9..67..2.......2....3..1...4.....8.2.8..9...6...8...47.54......39...4....   #e1590 + (46 8) 23 steps
1...56.......8.13..8.2.1..62...6...3..59...........4..3...17..8.......7.8...2...1   #e9937 + (5 5)  24 steps


1....6.......8.13..8.231..62...6...3..59...........4..3...17..8.......7.8...2...1   #e9937 + (23 3) 24 steps
(1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 6 8 3)


Hidden Text: Show

Code: Select all

1....6.......8.13..8.231..62...6...3..59...........4..3...17..8.......7.8...2...1   24c   min-expand
1....6.......8.13..8.231..62...6...3..594..........4..3...17..8.......7.8...2...1   25c   p1
1....6.......8.13..8.231..62...6...3..594..........4..3...17..8.......7.8...2.3.1   26c   p2
1....6.......8.13..8.231..62...6...3..594........7.4..3...17..8.......7.8...2.3.1   27c   p3
1.3..6.......8.13..8.231..62...6...3..594........7.4..3...17..8.......7.8...2.3.1   28c   p4
1.3..6.......8.13..8.231..62...6...3..594........7.4.53...17..8.......7.8...2.3.1   29c   p5
1.3..6.....6.8.13..8.231..62...6...3..594........7.4.53...17..8.......7.8...2.3.1   30c   p6
1.3..6.....6.8.13..8.231..62...6...3..594........724.53...17..8.......7.8...2.3.1   31c   p7
1.3..6.....6.8.13..8.231..62...6...3..594........724.53...176.8.......7.8...2.3.1   32c   p8
1.3..6.....6.8.13..8.231..62...6...3..594......8.724.53...176.8.......7.8...2.3.1   33c   p9
1.3..6.....6.8.13..8.231..62...6...3..594......8.724.53.2.176.8.......7.8...2.3.1   34c   p10
1.3..6.....6.8.13..8.231..62...6...3..594....9.8.724.53.2.176.8.......7.8...2.3.1   35c   p11
1.3..6.....6.8.13..89231..62...6...3..594....9.8.724.53.2.176.8.......7.8...2.3.1   36c   p12
1.3..6.....6.8.13..89231..62...6...3..594....9.8.724653.2.176.8.......7.8...2.3.1   37c   p13
1.3..6.....6.8.13..89231..62...6...36.594....9.8.724653.2.176.8.......7.8...2.3.1   38c   p14
1.3..6.....6.8.13..89231..62...65..36.594....9.8.724653.2.176.8.......7.8...2.3.1   39c   p15
1.3..6.....6.8.13..89231..62...65..36.594....9.8.724653.2.176.8.......7.8.7.2.3.1   40c   p16
1.3..6.....6.8.13..89231..62...65..36.594....9.8.724653.2.176.8..1....7.8.7.2.3.1   41c   p17
1.3..6.....6.8.13..89231..62.4.65..36.594....9.8.724653.2.176.8..1....7.8.7.2.3.1   42c   brt
1.3..6.....6.8.13..89231..6274.65..36.594....9.8.724653.2.176.8..1....7.8.7.2.3.1   43c   p18
1.3..6.....6.8.13..89231..6274.65..36.594...79.8.724653.2.176.8..1....7.8.7.2.3.1   44c   p19
1.3..6.....6.8.13..89231..6274.659.36.594...79.8.724653.2.176.8..1....7.8.7.2.3.1   45c   p20
1.3..6.....6.8.13..89231..6274.659.36.594...79.8.724653.2.176.8.61....7.8.7.2.3.1   46c   p21
1.3..6.....6.8.13..89231..6274.659.36.594...79.8.724653.2.176.8.61....7.8.762.3.1   47c   brt
1.3..6.....6.8.13..89231..6274.659.36.594...79.8.724653.2.176.8.61....7.8.76243.1   48c   p22
1.3.56..9..6.8913..89231..6274.659.36.594...79.8.724653.25176.8.61.9..7.8.76243.1   53c   brt
1.3.56..9..6.8913..89231..62748659.36.594...79.8.724653.25176.8.61.9..7.8.76243.1   54c   p23
1.3.56..9..6.8913..89231..6274865913615943..79381724653.25176.8.61398.7.8.76243.1   61c   brt
1.3.56..9..6.8913..89231..6274865913615943.279381724653.25176.8.61398.7.8.76243.1   62c   p24
1.3.56.89..6.8913..89231..62748659136159438279381724653.25176.8.61398.7.8.76243.1   64c   brt


[denis_berthier]

i'm testing eleven’s collection with '1-expansion to T&E(2)-expands', i frequently find expansion paths >= 20 steps, with a maximum of 24 steps so far

1-expands of T&E(2)-expands are in T&E(1) or T&E(0). Those in T&E(1) are likely to behave like puzzles in T&E(1) and to have long expansion paths.

At the bottom of T&E(1), there are three kinds of puzzles:
- minimals
- 1-expands of T&E(2)-expands
- 1-expands of T&E(3)-expands

It would be interesting to see if they have the same distribution of lengths for their longest expansion paths. I may check this later, but for now my Mac is saturated with other calculations. So, if you want to do it, I suggest you use the cbg-000 collection (purged of its T&E(0) puzzles) for the minimals.
.

[P.O.]

So, if you want to do it, I suggest you use the cbg-000 collection (purged of its T&E(0) puzzles) for the minimals

unfortunately i can't do that, i'm too slow and even if i find long expansion paths there is no guarantee that they are the longest because i sample a lot
i tried a few and the best result i got was 18 steps:
(1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 3 5)

Hidden Text: Show

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1......89.57......8..2...5...5..39.1.1...4....6.51.......3..62.64..9...37........   25c   minimal   #16845
1......89.57......8..2...5..75..39.1.1...4....6.51....5..3..62.64..9...37......9.   28c   min-expand
12.....89.57......8..2...5..75..39.1.1...4....6.51....5..3..62.64..9...37......9.   29c   p1
12.....89.57......8..2...5..75..39.1.1...4....6.51....5..3..62.64..9..137......9.   30c   p2
12.....89.57......8..2...5..75..39.1.1...4....6.51....5..34.62.64..9..137......9.   31c   p3
12.....89.57......8..2...5..75..39.1.1...4....6.51....5..34.62.642.9..137......9.   32c   p4
12.....89.57......8..2...5..75..39.1.1...4....6.51....5..34.627642.9..137......9.   33c   p5
12.....89.57......8..2...5..75..39.1.18..4....6.51....5..34.627642.9..137......9.   34c   p6
12.....89.57......8..2...5..75..39.1.18..4....6.51....5..34.627642.9..137.....49.   35c   p7
12.....89.57......8..2...5..75..39.1.18..4....6.51....5..34.62764279..137.....49.   36c   p8
12.4...89.57......8..2...5..75..39.1.18..4....6.51....5..34.62764279..137.....49.   37c   p9
12.4..789.57......8..2...5..75..39.1.18..4....6.51....5..34.62764279..137.....49.   38c   p10
12.4..789.57......8..2..15..75..39.1.18..4....6.51....5..34.62764279..137.....49.   39c   p11
12.4..789.57......8..2..15..75..39.1.189.4....6.51....5..34.62764279..137.....49.   40c   p12
12.4..789.57......8..2..15..75..39.13189.4....6.51....5..34.62764279..137.....49.   41c   p13
12.4..789.571.....8..2..15..75..39.13189.4....6.51....5..34.62764279..137.....49.   42c   p14
12.4..789.571.....8..2..15..75..39.13189.4....6.51....5..34.62764279..137..6..49.   43c   p15
12.4..789.571.....8..2..15..758.39.13189.4....6.51....5..34.62764279..137..6..49.   44c   brt
12.4..789.5718....8..2..15..758.39.13189.4....6.51....5..34.62764279..137..6..49.   45c   p16
12.4..789.5718....8..2..15.2758.39.13189.4....6.51....5..34.62764279..137..6..49.   46c   p17
12.4..789.5718....8..2..15.2758639413189.4....6.51....5..34.62764279..137..6..49.   48c   brt
12.4..789.5718....8..2..15.2758639413189.4....6451....5..34.62764279..137..6..49.   49c   p18
12.4..789457189...8..2..1542758639413189.4...96451....5..34.62764279..137..6..49.   53c   brt     


I just found a 20-step expansion path
(2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 1 2 17)

Hidden Text: Show

Code: Select all

....5....4.67....379...2...21...56....9....5..6....9....4.2...66.2.143.....5.3...   26c   minimal   #1303     
....5....4.67....379...2...21...56...49....5..6....9....4.27..66.2.143.....563...   29c   min-expand
....5....4.67....379...2...21..956...49....5..6....9....4.27..66.2.143.....563...   30c   p1
....5....4.678...379...2...21..956...49....5..6....9....4.27..66.2.143.....563...   31c   brt
....5.7..4.678...379...2...21..956...49....5..6....9....4.27..66.2.143.....563...   32c   p2
....5.7..4.678...379...2...217.956...49....5..6....9....4.27..66.2.143.....563...   33c   p3
....5.7..4.678...379...2...217.956...49.7..5..6....9....4.27..66.2.143.....563...   34c   p4
....5.7..4.678...379...2...217.956...49.7..5..6....9...34.27..66.2.143.....563...   35c   p5
....5.7..4.678...379...2...217.956...49.7..5..6....9...34.27..66.2.143..9..563...   36c   p6
....5.7..4.678...379...2...217.956...49.7..5..6....9.7.34.27..66.2.143..9..563...   37c   p7
....5.7..4.678...379...2...217.956..349.7..5..6....9.7.34.27..66.2.143..9..563...   38c   p8
....5.7..4.6789..379...2...217.956..349.7..5..6....9.7.34.27..66.2.143..9..563...   39c   p9
....5.7..4.6789..379...2...217.956..349.7..5..6.2..9.7.34.27..66.2.143..9..563...   40c   p10
....5.7..4.6789..379...2...217.956..349.7.25..6.2..9.7.34.27..66.2.143..9..563...   41c   p11
....5.7..4.6789..379...2.6.217.956..349.7.25..6.2..9.7.34.27..66.2.143..9..563...   42c   p12
....5.7..4.6789..379...2.6.217.956..349.7.25..6.2..9.7.34.27..66.2.143..9..563..2   43c   p13
....5.7..4.6789..379...2.6.217.956..349.7.25..6.2..9.7.34.27..66.2.143..9..5634.2   44c   p14
....5.7.94.6789..379...2.6.217.956..349.7.25..6.2..9.7.34.27..66.2.143..9..5634.2   45c   p15
....5.7.94.6789..379...2.6.217.956..34967.25..6.2..9.7.34.27..66.2.143..9..5634.2   46c   p16
....567.94.6789..379...2.6.217.956..34967.25..6.2..9.7.34.27..66.2.143..9..5634.2   47c   brt
....567.94.6789..379...2.6.217.956..34967825..6.2..9.7.34.27..66.2.143..9..5634.2   48c   p17
....567.94.6789..379...2.6.217.956..349678251.6.2.19.7.34.27..66.2.143..9..5634.2   50c   brt
....567.94.6789..379.1.2.6.217.956..349678251.6.2.19.7.34.27..66.2.143..9..5634.2   51c   p18
1...567.94.6789..379.1.2.6.217.956..349678251.6.2.19.7.34.27..66.2.143..9..5634.2   52c   p19
1...567.94.6789..379.1.2.6.217.956..349678251.6.2.19.7.34.27..66.2.143..9.15634.2   53c   brt
1...567.94.6789..379.1.2.6.217.956..349678251.6.2.19.7.34.27..66.2.143..9.1563472   54c   p20
12..567.945678912379.1.256.217.956..3496782518652.19.7534927816672814395981563472   70c   brt