High clue tamagotchis

Everything about Sudoku that doesn't fit in one of the other sections

Re: High clue tamagotchis

Postby Serg » Mon Jan 13, 2014 8:58 pm

< post withdrown >
Last edited by Serg on Tue Jan 14, 2014 2:04 pm, edited 1 time in total.
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Postby dobrichev » Mon Jan 13, 2014 9:31 pm

Hi Serg,

These are the 3 new puzzles and their solutions.
If I am wrong (which isn't impossible), then please show me a secondary solution, a redundant clue, or which previously known puzzles they duplicate.

Code: Select all
........1..2.13.4.14..56237.1..75.82.8.3..1..2.5.81374.2..3...84...687238....74..       39
378249561562713849149856237613475982784392156295681374927534618451968723836127495

........1.12.34....3561.247..7.8..6..2.4....8.58.6.472.731..8.4.81.437262..8...1.       39
469728531712534689835619247147285963326497158958361472673152894581943726294876315

........1.12.34....356.1247..7..8.6..2.4....8.58..6472.731..8.4.81.437262..8...1.       39
469782531712534689835691247147258963326479158958316472673125894581943726294867315


GouinJP looked at the previously discussed article written by Havard in 2010 and not at the recently published puzzles. It is his right of course. Coloin's too. ;)
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Re: High clue tamagotchis

Postby ronk » Tue Jan 14, 2014 2:37 am

Serg wrote:... (maybe last 3 puzzles from 39puz340.txt file are new found puzzles, aren't they?).

In no particular order, they are puzzles #255, #279 and #280 in 39puz340.txt.
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Re: High clue tamagotchis

Postby Serg » Tue Jan 14, 2014 2:02 pm

Hi, Mladen!
Sorry, my statement that you wrote wrong puzzles in the post, containing minimal 39-clue puzzles number 338-340, was incorrect. Your post is correct. I was confused by puzzles order in the file 39puz340.txt (I supposed puzzles in the file are historically ordered).

Serg
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Re: High clue tamagotchis

Postby dobrichev » Tue Jan 14, 2014 3:19 pm

Here are the next 4
Code: Select all
.......12.....13.....23...4..5.6.2...7652.48128.1476.5.6.3....85.76..1.383.7145.6
.......12.....13.....23...4..5.6.2...7652.48128.1475.6.6.3....85.76..1.383.7146.5
........1.....123....23..4...5.6..2..6275.18.87.12.65..3657.4185.841637.7..3..5..
........1.....123....23..4...5.6..2..6257.18.87.12.65..3675.4185.841637.7..3..5..
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Re: High clue tamagotchis

Postby eleven » Tue Jan 14, 2014 6:43 pm

Congratulations Mladen,

on finding 7 new 39's.
I added them to the list in the first post (but i have to look for my old program to transform them to the other normalization too).

A very open question for me is, how many unknown puzzles are still out there. Compared to the 17 and 18 clue puzzles the space for the 39's is very big, and though there was already much effort to find them, it cannot be compared to the number of cpu's, which were burned for the 17 clue search.
So i can be quite sure, that not more than say 500 17 clues are missing (and i doubt, that anyone will run a program over years on a big cpu cluster for them), but here i can only say, that my interim estimate of 300 was wrong for sure :)
And i guess, that the first quantum computers will not be used to find them all ...
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Re: High clue tamagotchis

Postby Serg » Wed Jan 15, 2014 10:38 am

Hi, Mladen!
dobrichev wrote:Here are the next 4
Code: Select all
.......12.....13.....23...4..5.6.2...7652.48128.1476.5.6.3....85.76..1.383.7145.6
.......12.....13.....23...4..5.6.2...7652.48128.1475.6.6.3....85.76..1.383.7146.5
........1.....123....23..4...5.6..2..6275.18.87.12.65..3657.4185.841637.7..3..5..
........1.....123....23..4...5.6..2..6257.18.87.12.65..3675.4185.841637.7..3..5..

I confirm that these 4 puzzles are new minimal valid 39-clue puzzles. Congratulations!

Serg
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Re: High clue tamagotchis

Postby Serg » Sat Jan 18, 2014 4:50 pm

Hi, all!
I've analyzed published 344 minimal 39-clue puzzles for "twin" puzzles, and found that 120 puzzles only have no twins. I say 2 given puzzles are twins if the second puzzle can be produced from the first by some clues permutation preserving footprint of the puzzle. Simply speaking, such puzzles contain unhitted UA sets among their clues.

For example, puzzle #1 and puzzle #2 from the file 39puz340.txt forms "twin family".
Code: Select all
    Puzzle #1                 U4
. . . . . . . . .     . . . . . . . . .
. . . . . . . . 1     . . . . . . . . .
. . . 1 2 3 . . 4     . . . . . . . . .
. . 2 5 . 1 . 6 7     . . . . . . . . .
. 4 . . 7 . 5 . 3     . . . . . . . . .
5 7 . . 3 8 4 1 2     5 7 . . . . . . .
. 6 8 7 1 . 2 . 5     . . . . . . . . .
2 . 4 . . 5 . 7 8     . . . . . . . . .
7 5 . . 8 2 1 4 6     7 5 . . . . . . .

    Puzzle #2                 U4
. . . . . . . . .     . . . . . . . . .
. . . . . . . . 1     . . . . . . . . .
. . . 1 2 3 . . 4     . . . . . . . . .
. . 2 5 . 1 . 6 7     . . . . . . . . .
. 4 . . 7 . 5 . 3     . . . . . . . . .
7 5 . . 3 8 4 1 2     7 5 . . . . . . .
. 6 8 7 1 . 2 . 5     . . . . . . . . .
2 . 4 . . 5 . 7 8     . . . . . . . . .
5 7 . . 8 2 1 4 6     5 7 . . . . . . .

Both puzzles differs by 4 clues only, and these clues form unhitted U4 unavoidable set. So, puzzle #2 can be "mechanically" produced from puzzle #1 by UA set permutation and vice versa. Therefore it is not necessary to publish all puzzles belonging to the same "twin family". We can publish one representative puzzle for each "twin family" (it can be twin with the least minlex form, for example).

Here is "twin family" statistics for all 344 minimal 39-clue puzzles publised in this thread.
Code: Select all
Twins in a family    Number of families
                1    120
                2     51
                3     10
                4     15
                5      1
                6      3
                9      1
---------------------------------------
              Total: 201 families

You can see that 120 puzzle have no twin puzzles in 344-puzzles list (they form 1-puzzle "twin families"), 102 puzzles form 2-puzzle twin families (51 families), etc. Even one 9-puzzles twin family exist!

If we would exclude twins from 344-puzzles list we would get 201-puzzles list only.

Serg
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350

Postby dobrichev » Sat Jan 18, 2014 6:57 pm

Code: Select all
...........1.23.452465...31..78.2.1..68.....712.76..83.8437.1.261.2.8.747...4...8
...........1..2345.341562....6..785..4856.7.2.7.2.8....1762.58..8..7...165.8.14.7
.....1..2....3..4...52..13...6.1..7..58.6721.7.14826...6...47...84.763255.78.346.
.....1..2....2..3......314...4..56....5.6432767.23845..68.5721.5...1..7.71.84256.
.....1..2....2..3......314...4..56....5.6432767.23845..18.5726.5...1..7.76.84251.
.....1..2....3..4...52..13...6.1..7..58.7621.7.14826...6...47...84.673255.78.346.


These are the recent 39s and the list size now is 350 puzzles.
The updated mirror of the list is available here.
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Re: High clue tamagotchis

Postby dobrichev » Sat Jan 18, 2014 8:15 pm

eleven wrote:...A very open question for me is, how many unknown puzzles are still out there...

I have no answer. Comparing to 17s discovery history, there is much work to be done on 39s - algorithmic, organizational, and CPU-consuming.
The {-4+4} exercise resulted in a new island. The {-5+5} over this island resulted in one more island. With the existing software it is matter of days to apply {-5+5} to all known 39s on several dozens of cores.
Maybe we are in the middle of the way?
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Re: High clue tamagotchis

Postby dobrichev » Sat Jan 18, 2014 8:32 pm

Serg wrote:...Therefore it is not necessary to publish all puzzles belonging to the same "twin family"...

"Twin family" has very volatile definition. Actually the twins are much much more, but most of them are non-minimal. I.e. the process isn't symmetric.
For example, the following 27-clue puzzle is a minimized variant of one of the 39-clues' twins.
Code: Select all
...........1..34.6.4275...3....785.....3...486....2..1..8.......1.2.5.3..2..3.1.5

If I correctly interpreted blue's explanation of his searching methods, he exploited this fact in the opposite direction.
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Postby blue » Sat Jan 18, 2014 9:43 pm

dobrichev wrote:If I correctly interpreted blue's explanation of his searching methods, he exploited this fact in the opposite direction.

Yes. If a twin of a 39 reduced to one or more minimal 37's or 38's, or a twin of a 38 reduced to one or more minimal 37's, then I kept those around for further testing. On the flip side, I checked if adding (redundant) clues to 37's or 38's, produced 38's or 39's with minimal twins -- any variation that would be the symmetric counterpart of something that could happen doing the first kind of operation.
[ E.g. so that if a twin of a ("would be") 39 included a minimal 37, then the 37 could also be used to produce the 39 ].

Added: I may have gone as far as checking whether a "(minimal 37) + (2 redundant clues)" or "(minimal 38) + (1 redundant clue)" puzzle, gave a 39 with a twin that wasn't minimal itself, but did include a "new" minimal with size 37 or 38.

BTW: Congratulations, Mladen, on your new discoveries :!:
Nice work !
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Postby blue » Fri Feb 07, 2014 11:26 pm

For dobrichev:

Greetings !

    Are you still working on these ... the 39's ?
    If not, I have two 39's produced one of your latest six ... one new, and a "twin".

    Since I can only seem to produce 39's by "standing on the shoulders of giants" (as it were ...),
    I'ld rather see you announcing them, than me.
Best Regards,
Blue.
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Re: High clue tamagotchis

Postby dobrichev » Sat Feb 08, 2014 2:43 am

Hi blue,
I see no reason for you to delay adding these puzzles to the large list you already composed.

I have for 2 weeks running a 16-core machine which Intel kindly provided along with their latest development suite.

I closed the already published 350 39s up to {-5+5} and found nothing.
These are the clusters
Code: Select all
pass {-5}, src=350, children=197091603
150 clusters.
   .............................1..2.34.23.546177..36.8.2.32.7516817..3.2.55.82.6.73   =2
.................1...123..4..25.1.67.4..7.5.357..38412.6871.2.52.4..5.7875..82146
.................1...123..4..25.1.67.4..7.5.375..38412.6871.2.52.4..5.7857..82146
   ..........................1..1..2.3..24.31.565.3467.12.42.73568.7..4...33.5286.47   =8
........1.......2....123..4..25....6.5726..388.6437.52.4..7..6..7564.8136.8315.47
........1.......2....123..4..4.5..6..7564.81368.317.45.5726..38.6.7....282.435.76
........1......23.....12..4..45...6..7546.81268.127.45.5763..28.6..7...383.245.76
........1......23....123..4...5....6.5726.3.88.67345.2.4567.183.7..4.6..6.83154.7
........1.....2.3...413.2......5..6..5761..826.8274.15.4576.823.7.4....68.6325.74
........1.....2.3...413.2......5..6..5761..828.6274.15.4576.823.7.4....66.8325.74
........1....23.....21..34.....5..6..57.361826.8.7153..75.824162.671485.8....52..
........1....23.....21..34.....5..6..67.8153.8.5.36172.2681475..7...52..5.8.72416
   ..........................1..1..2.3..24.3156763..74.12.42.536783...4...55.6287.43   =2
........1.......2....123..4..25....6.67834.5285.26..37.76318.454.875.6135...4..7.
........1.......2....123..4..25....6.76834.5285.26..37.67318.454.875.6135...4..7.
   ..........................1..2..3.45.15.426374...651.2.27.84316.4..7.2.81.8.26.74   =1
........1.......2......3456..7.38.14.8.7.1.6.31.645.87.235.7.487...82..585.364.72
   .........................12.....3..4.3456.1785187.43.6.2..3.7...516.72.3.7325.641   =1
.......12.....34.....124.....5.....3.637415.887..35164.86.172.55....26..72..56831
   .........................12..1..3..4..3.4156745.678.31.28.35.463...2..5.54.816.23   =2
.....1..2....2..3......314...4..56....5.6432767.23845..18.5726.5...1..7.76.84251.
.....1..2....2..3......314...4..56....5.6432767.23845..68.5721.5...1..7.71.84256.
   .........................12..1..3..4.34..5.6726547..31.2..3..4..43.5278658674..23   =8
.......12.....134....23......5..3.6...7.5628.68.42753..5..4.6...74.6512886.71245.
.......12.....134....23......5..3.6...7.5628.68.42753.5...4.6..7.4.6512886.71245.
.......12.....134....23......5..3.6...7.6528.68.42753..5..4.6...74.5612886.71245.
.......12.....134....23......5..3.6...7.6528.68.42753.5...4.6..7.4.5612886.71245.
.....1..2....2..3....3..14...5..46...475.63286.827345..5.1...6..7.6.528.8.674251.
.....1..2....2..3....3..14...5..46...475.63288.627345..5.1...6..7.6.528.6.874251.
.....1..2....2..3....3..14...5..46...476.53286.827345..5.1...6..7.5.628.8.674251.
.....1..2....2..3....3..14...5..46...476.53288.627345..5.1...6..7.5.628.6.874251.
   .........................12..1..3..4.34.15.677.5.64.31.43.527862.7386.455...4..2.   =10
........1.....234...3.1.2....5..64...674.581.8.41.756..567.31283.826175.7..5...3.
........1....23.4...214......3.5..6..56.7138.7.8.3615..6571482.2.7.854168....25..
........1....23.4...214......3.5..6..65.7138.7.8.3615..5671482.2.7.854168....25..
........1....23.4...214......3.5..6..67.8135.8.5.3617..7681452.2.8.754165....27..
........1....23.4...214......3.5..6..75.3618.8.6.7135..27.85416.5...28..6.871452.
........1....23.4...214......5.3..6..67.8135.8.3.5617..2...57...7681452.5.8.72416
........1....23.4...214......5.3..6..67.8135.8.3.5617..5681472.2.8.754167....25..
........1....23.4...214......5.3..6..67.8135.8.3.5617..7681452.2.8.754165....27..
........1....23.4...214......5.3..6..73.5618.8.6.7135..27.85416.5...28..6.871452.
........1....23.4...214......5.3..6..73.5618.8.6.7135..57.824162....58..6.871452.
   .........................12..1..3..4.3415.678578.64.31.273.6.4..4352.867.5..4..2.   =4
........1.....123....23..4...5.6..2..6257.18.87.12.65..3675.4185.841637.7..3..5..
........1.....123....23..4...5.6..2..6275.18.87.12.65..3657.4185.841637.7..3..5..
.......12.....13.....23...4..5.6.2...7652.48128.1475.6.6.3....85.76..1.383.7146.5
.......12.....13.....23...4..5.6.2...7652.48128.1476.5.6.3....85.76..1.383.7145.6
   .........................12..1..34...24.5163..534.612..3.7...4..47.352865826.437.   =5
.......12.....34....4.1.3....5..26...627.58.17.81.62.5.4.5....6.863715.45.76.4183
.......12.....34....4.1.3....5..26...678.52.18.21.67.5.4.5....6.763815.45.86.4173
.......12.....34....4.1.3....5..26...721.68.58.67.52.1.576.41834..5....66.83715.4
.......12.....34....4.1.3....5..26...721.68.58.67.52.1.576.41834.83715.66..5....4
.....1..2....3..4...2...3.1..5..4..6.743.6.588.67.5.34.576.28132.8173.656..5...2.
   .........................12..1..34...24.5613..634.152..376.4285.4..3..7.682.7534.   =16
.......12.....34....4.1.3....5..26...627.58.18.71.62.5.765.41834.83715.65..6....4
.......12.....34....4.1.3....5..62...627.58.18.71.25.6.4.6....5.765.41835.83716.4
.......12.....34....4.1.3....5..62...627.58.18.71.25.6.765.41834..6....55.83716.4
.......12.....34....4.1.3....5..62...627.58.18.71.25.6.765.41834.83716.55..6....4
.......12.....34....4.1.3....5..62...627.58.18.71.26.5.765.41834.83715.65..6....4
.......12.....34....4.1.3....56....4.473815.66.85.4173.5...62...728.56.18.61.27.5
.......12.....34....4.1.3....56....4.473815.68.65.4173.5...62...728.56.16.81.27.5
.......12.....34....4.1.3....56....4.473815.68.65.4173.5...62...781.26.56.28.57.1
.......12.....34....4.1.3....56....4.473816.58.65.4173.5...62...781.25.66.28.57.1
.....1..2....3..4...2...3.1..5..4..6.647.5.388.73.6.54.765.28132.8173.655..6...2.
.....1..2....3..4...2...3.1..5..6..4.647.5.388.73.4.56.765.28132.8173.655..6...2.
.....1..2....3..4...2...3.1..5..6..4.647.5.388.73.4.65.765.28132.8173.565..6...2.
.....1..2....3..4...2...3.1..56...2..27183.568.65.2713.5...6..4.783.4.656.48.5.37
.....1..2....3..4...2...3.1..56...2..27183.656.85.2713.5...6..4.748.5.368.63.4.57
.....1..2....3..4...2...3.1..56...2..27183.658.65.2713.5...6..4.748.5.366.83.4.57
.....1..2....3..4...2...3.1..56...2..27183.658.65.2713.5...6..4.783.4.566.48.5.37
   .........................12..1..34...25.4.36..435.6127.3..6..7..763.5284582.7463.   =3
.......12.....13.....23...4..5..4.6..745.682368.372.45..76.5.38.6.7....185.143.76
.......12.....13.....23...4..5..4.6..746.582368.372.45..75.6.38.6.7....185.143.76
.......12.....134....23......5..6.2..6.7.5.838.7342.56.465.7831.7.4....55.8163.74
   .........................12..1..34...43.1562.72564.1...36.57284.7.3...6.58246.37.   =1
.......12.....13.....23.4....5..4.6..6782354.48..65237.764821..5...1.7..81..5762.
   .........................12..1.34.56.53..624..6.25.3.1..7.83.25.35462.78.8.57.4.3   =1
..............1.23.24..3.5..16..4.752.5.67..174..1.26..38.76512.62..8.3757..3.68.
   .........................12..3..4..1.14.56.73.763.1.45.3..4..2..426.3587687.25.34   =2
........1.....234...3.1.2....5..64...641.785.8.74.561..765.31283.827156.5..6...3.
........1....23.4...214......5.3..6..63.7518.7.8.6135..8671452.2.7.564185....26..
   .........................12..3..4..5.14.653.7.657.31.4.38.46271.4.3..8..6218.74.3   =2
.......12.....34....4.1.3....5..62...721.58.68.67.25.1.576.41834.83716.56..5....4
.....1..2....3..4...2...3.1..5..6..4.743.5.688.67.4.35.576.28132.8173.566..5...2.
   .........................12..3.14..5.456.7.31.6135..74.247.6.58.56.48.27.8.52..4.   =4
...........1.23456.45716.32....372.8...68..433..1.256...8.......1..78625.5.261.84
........1....23.4...45.13.2.16.572.454..126.77.2..61...63.7...845...87.38.7.354..
........1..2.34...34.1.5....2..41.5645..6721.6...5.7.4.7...8.6323..168.786..7312.
........1..2.34...34.1.5....6...7.832...137.673..8612.42..61.5865..4821.8...5.6.4
   ........................123....14..5.162.54.7.4567.2.1..47..8...63.487.2.875623.4   =3
...........1..2..3.4..5.126....71..8.1528.4.7.785.42.1..74..3...368257.4.54.376.2
...........1..2..3.4..5.126....71..8.1528.7.4.785.42.1..74..3...348256.7.56.374.2
...........1..2..3.4..5.126....71..8.1528.7.4.785.42.1..74..3...368254.7.54.376.2
   .......................1.23......4...1524.6376.453.21..5147.3864...1..7.7.638514.   =4
........1.12.34.5635..61742....86......47...8...5..46...1..5.7..27348.1553..17284
........1.12.34.5653..61742....86......47...8...5..46...1..5.7..27348.1535..17284
.......12.....134....23.5.......6.5..57348.268.65.2.73.75183.644.86.57316..4.....
.......12.....134....23.5.......6.5..57348.2686.5.2.73.75183.6448.6.57316..4.....
   .......................1.23.....4..5.46.7518285.612.74....4..5..64.578315..136.47   =3
.....1..2....3..4...5..2.31...163.78.7..281538...7..2...8..4..7.57316.8446..87315
.....1..2....3..4...5..2.31..6..4..7..7318.6448..67315.7..261535..183.766...7..2.
.....1..2....3..4...5..2.31..6.7..2..5.183.767...26153..7318.64.6...4..784..67315
   .......................1.23..1..45.2.25.16.474.7.5.13..52.4738.7.4.3821.8....5.74   =8
.............12.34.13....52..46.1..7.7842..6.1.6.78.45.3..87..6.8714..236.123..78
.............12.34.13....52..46.1..7.7842..6.1.6.87.45.3..78..6.8714..236.123..78
.............12.34.13....52..46.1..7.8742..6.1.6.78.45.3..87..6.7814..236.123..78
.............12.34.13....52..46.1..7.8742..6.1.6.87.45.3..78..6.7814..236.123..78
.............12.34.13....52..46.1.7..7842..6.1.6.78.45.3..87..6.8714..236.123..87
.............12.34.13....52..46.1.7..7842..6.1.6.87.45.3..78..6.8714..236.123..87
.............12.34.13....52..46.1.7..8742..6.1.6.78.45.3..87..6.7814..236.123..87
.............12.34.13....52..46.1.7..8742..6.1.6.87.45.3..78..6.7814..236.123..87
   .......................1.23..1.4...5.253.6.416.41.523..5263..74.7...4.5.4.65.7382   =2
.............12.34.15.3..2...3..6..7.4712.36.1.6.7324..7428165.5...6..7.6.1.574.2
...........1..2.34.23.51..6..7..84...8452.7.323.7.4.85.182.53.7.7..8...13.21.7.58
   .......................1.23..1.45.36.36712..4.5.6.3.7...517..62.62.54..71.72.6.85   =3
.............12.34.13....52..126..73.37148.2..6..37..8..68.1.47.7462..8.1.8.74.65
.............12.34.13....52..617..23.31248.6..7..36..8..78.1.46.6472..8.1.8.64.75
.............12.34.13....52..67.1.48.8462..7.1.7.84.65.3814..2..6..38..77.126..83
   .......................1.23..2..4..5.45.16.728765.2.4..54.68137.8..4..5.63.1.5.84   =2
.....1..2....3..4...52..13...6.1..7..58.6721.7.14826...6...47...84.763255.78.346.
.....1..2....3..4...52..13...6.1..7..58.7621.7.14826...6...47...84.673255.78.346.
   .......................1.23..2.45.61.4.6..35..56317.42..4.78.36.67.3418..8.1.6..4   =2
..............1.23.24.53.1...26.4.71.47135.62.6..7.34...53.6.87.765.813..8..17..6
..............1.23.24.53.1...26.4.71.47135.62.6..7.34...65.8.37.753.618..8..17..6
   .......................1.23..2.45.67.6...735..753.6142.5..1..862..6.857.68..542.1   =1
...........1..2.34..3.512....6.78..2.1.6.487..7812.3.6..75.64..1...476.36.421375.
   .......................1234....256.7..61.754..57.46312..47.285..75.1842..8..5...1   =1
.............12.34.13..45.2..16.34.7..6.7..2..7.421.65.3.2.67...67148.53.8..37.46
   .......................1234....52..1..56.73.2.7231.6.5.4.7..5...581364.7.67.458.3   =1
...........1..2.34.2..3.516....17..3.185.3.27.7.28..51..68...7..47325.68.82.76.45
   .......................1234....56..1..72..56..65718.23..84.715..5412..87.7..85.42   =1
...........1..2..3..3.45162.1...78.5374.58.218.5....3..67..43.813..86.744.8.7..16
   ......................12.34.....5..6.2567.14..6412357...826.35..3..57.6..563.14.7   =2
.............12.34.13.456.2..14....7.86.7142.7...28.61..526.7.8..8.5724..7..84.56
.............12.34.13.456.2..14.6..7.86.7142.7...28..1..526.7.8..8.5724..7..84.56
   ......................12.34....25.46..216.3.5.657341.2...68.5...5...1.68.86.57413   =4
...........1..2.34.2..156.7.372518.6.826....316..38....481.6..52...84...71.5234.8
...........1..2.34.2..516.7.372158.6.826....316..38....481.6..52...84...71.5234.8
...........1..2.34.3567...1.18.6.2...56.281433.2....68.87.4.3.21......875.3.87416
...........1..2.34.3567..1..18.6.2...56.281433.2....86.87.4.32.1......785.3.87461
   ......................12.34....352.6..34...75.6572..43.1..54.8..5817.462.7.28..51   =5
...........1..2.34.23.145....6.27.4..471.6...2184.5.76.325.1.87.752....318..73..5
...........1..2.34.23.145....6.27.4..471.6...2184.5.76.352.1.87.725....318..73..5
...........1..2.34.23.145....6.27.4..471.6.5.2184.5.76.325.1.87.752....318..73...
...........1..2.34.23.145....6.27.4..471.6.5.2184.5.76.352.1.87.725....318..73...
...........1.23.45.426...31.17..2.5.2.6.571844.5....72.7..6.4.81...7..636.4.38.17
   ......................12.34..1..5..6.63721.5875.86...3.48.57...1.724..8552.18..4.   =1
...........1..2.34.23.54.1..16....472.4.763.13.7.4.62..6...8..3132.67.857.8..516.
   ......................12.34..1..5.46.364275...5..613.2..3284..5..7.5..6..85.76423   =1
..............1.23.12.43.56..1.7...8..4..8532.28354.71..6.37.8..478..315.8.4.5.67
   ......................12.34..5..136..6...7.58.83.65417.27.568435.8...62.63..2...5   =1
...........1..2.34.23.54.1..16..78..275.861.33.8..5..6.6..4.38.132.684.78.4....61
   ......................12.34..5..346..26.51378.8..6...5.62.4..575.7.268438...756..   =4
...........1..2.34.34.51..2..5.673.8.768...2..83215.67..8.7.2...1.5.8..6.471268.3
...........1..2.34.34.51..2..5.673.8.768...2..83215.76..8.7.2...1.5.8..7.471268.3
...........1..2.34.34.51..2..5.678.3.768...2..83215.67..8.7.2...1.5.8..6.471263.8
...........1..2.34.34.51..2..5.678.3.768...2..83215.76..8.7.2...1.5.8..7.471263.8
   ......................12345..1.6..37.37.2156.65..7.2...73..46..16...7.5.542.86.73   =1
...........1..2.34.23451.6...7.2.84...87.5..3.32.48.57.1..8..75.7...438..865.7412
   ......................12345..3..6....64.7158387...5.6..26.4.7.83.8.2765.74..6.23.   =2
...........1..2345.34561.2.....57.68...2.87...7861.25...3.7..8..87125.3.1.68.357.
...........1..2345.34561.2.....57.68...2.87...8761.25...3.7..8..78125.3.1.68.357.
   ......................12345..6..7..3.73.24.5685..6..7..15..67.838.....616.7.81534   =3
.............12345.34.56..1.46...1.73.1..768.87..61534.63.7...84.8.257.37....84..
...........1.23.45.246.5.31.1..76.8..762.8..48.2.4..7..67.....81438.2..72.8.67.13
...........1.23.45.246.5.31.4..76..8.762.8.1.8.2.1...7.67....8.2.8.67.544158.2.7.
   ....................1..2.34....5.6...364.7.85.57.68.43.13.26458.2.1.536..6.8...12   =1
........1..1.2345.245.167.3..46.2....62.38.4.85..7.....172.....4.8.61.7.526.87.14
   ....................1..2345....67..3.2641.57.3.752.4.6..2.76854.6..5.73.7....46.1   =12
...........1..2345.23.451......67..3.3642.5.72.751.46...2.76854.6...47.17...5.63.
...........1..2345.23.451......67..3.364285.72.751.46...2.76.54.6...47.17...5.63.
...........1..2345.23.451......67..3.3652.4.72.741.56...2.76854.6...47.17...5.63.
...........1..2345.23.451......67..3.365284.72.741.56...2.76.54.6...47.17...5.63.
...........1..2345.23.451......67..3.3742.5.62.651.47...2.76854.7...46.16...5.73.
...........1..2345.23.451......67..3.374285.62.651.47...2.76.54.7...46.16...5.73.
...........1..2345.23.451......67..3.3752.4.62.641.57...2.76854.7...46.16...5.73.
...........1..2345.23.451......67..3.375284.62.641.57...2.76.54.7...46.16...5.73.
...........1.23.45452.1..36.278....41.8..7.5.54..3..78.7438.5.12......878.5.724.3
...........1.23.45452.1..36.278....41.8..7.5.54..3..78.7538.4.12......878.4.725.3
...........1.23.45452.1..36.278....41.8..7.5.54..3..87.7438.5.12......788.5.724.3
...........1.23.45452.1..36.278....41.8..7.5.54..3..87.7538.4.12......788.4.725.3
   ....................1.23.45..........6..32457542.673.8.16..857.425..68.18.7.1..64   =2
...........1..2.34.2345..16...12..47.1.7.5..8.72846..1..7.64.8...82.7.6526.58..73
...........1..2.34.2345..16...32..47.3.7.5..8.72846..3..7.64.8...82.7.6526.58..71
   ....................1.23.45........6..7.3651..65.72384.16.4.8.75....7.617.8.61453   =1
...........1..2.34.3214..56..7......24..8..75513.27.68.742...8.1.8.74..332581..47
   ....................1.23.45........6.627...8338.2.6.74.16372.585.718..6.82.6.5.1.   =1
...........1.23.45.2..641.3....7....71..38.5.8.3.45.17.56..782.1.8..2.7627..86531
   ....................1.23.45.....1..6.125.643..6423..51..7.5..6..263.7584.8.6.2.73   =1
...........1.23.45..2.4613.....6..7..173.2.682.6.87351..4..87...7823..141...74.83
   ....................1.23.45.....2..6.1..6745.3.615..27.38.7..6.1.7.86..46.4531.78   =1
...........1.23.45.24..5.31.....6.5..46.571.31..23..76.17..2.68.62.783144.8.6...7
   ....................1.23.45.....2..6.6..3745.8346.5.72.16.7..8.37..8..644.83.6.17   =2
...........1.23.45.24.6..31....3...7.7..8245.64275..83.17..8.6.28...6.744.627..18
...........1.23.45.42.5..36..4.....7.28..736.7.6132.84.1.2.8.73.8..75.1.2.731..58
   ....................1.23.45.....2.67.16.37.52.276..3....8.6..74.64.7853..753.4.86   =2
...........1.23456.45.172.3.....86....637..427.426.835..873.5.4.57.8.32.4....2..8
..........12.34.56.34.56127.....8.7..2.57.681.7.64.2.5.41.8.56..8.46..122....5..8
   ....................1.23.45.....2.67.2637.458.75.4632..17.68.34.6.7.....5.8..4.76   =2
........1....23.4..2415..36...5...78.57.81362.8673.15..4827..13.7...8...6.23...87
........1....23.4..2415..36...5...78.58.71362.7683.15..4728..13.8...7...6.23...87
   ....................1.23.45.....6..7.162.7.38.7831..6..27.64.5..65132.741.47....6   =3
..........12.34.56.3516.24.........7..761.4252..47.86...87..63..2.38..74.73.4658.
..........12.34.56.3516.42......7....7.61324.1..84.67...1.7658..5738..64.8.45.7..
..........12.34.56.3516.42......7....7.61354.1..84.67...1.7628..2738..64.8.42.7..
   ....................1.23.45.....6..7.267..358.753.2.64.17.6..83.6.8.751.5.8.3..76   =1
...........1.23.45.24.5..63..7.....84.2.38.768...72431.4.31..87.7..85.141.8..735.
   ....................1.23.45.....6..7.37258.646.8.7.52..1..672...7..1..568.6532.71   =1
...........1..2.34.3546..12..7....8..8.6.71.331685..27.4..8..71.78.4.3..1537.6.48
   ....................1.23.45.....6..7.37285.1661.37..82.73.62...16.5.8.7.5..73..64   =1
...........1.23.45.2..4613.....7.8..17..824532.8.35.71..4.68.1771....38.8.2..75.4
   ....................1.23.45.....6..7.37482.6161.37..84.73.64...16.2.8.7.2..73..56   =2
...........1.23.45.2..461.3.....78..17..823542.8.54.17..3.68.7171....4.88.2.7.53.
...........1.23.45.2..4613......78..17..823542.8.54.71..3.68.1771....48.8.2.7.5.3
   ....................1.23.45.....64.2.46.32.787.2.8..3..15..876..67.15.8448..6..51   =2
...........1..2.34.23.14.56...1.5.7..17.28.652.5.6...8.3427..8..72.81.4318..43..7
...........1..2.34.23.14.56...1.6.7..17.28.652.6.5...8.3427..8..72.81.4318..43..7
   ....................1.23.45..2..4..6.17.624.8.6.75.21..3..7..6..764.513.14..365.7   =1
...........1.23.45.45..132...4..6..2.172.8.64.6.41.73..5..6..7..7615.2.312.3.7.56
   ....................1.23.45..2..6..7.67.3215.13..5..6..76.815..2.8.6..7.31527.68.   =2
........1..2.34.5..65..2.34..724..6..8...3..72.6.78.43..83.1.7.6...87.127.142638.
........1..2.34.5..65..2.34..724..6..8...3..72.6.87.43..83.1.7.6...78.127.142638.
   ....................1.23.45..2.674.1.16....274.7..26...65.8..74124.765.87....516.   =2
...........1.23.45.24.1..63..52.1.78.18.57...2.78...5..83.72..61523.6.877..18..3.
...........1.23.45.24.6531..17...83.2...781.44.8.3..57.7...248.14..86.738....7.21
   ....................1.23.45..246..57.572316.46...571...1..724.6.7.....1.2.6.14.78   =1
........1....1234..12.34..5..5....6..76.254.828..46.5..612.8..45...618..82.453.16
   ....................1.23.45..26...57.46.721.371.38..6..68..7..41.42.8.7627..6..1.   =1
...........1.23.4524.5...36..72...681.4.58.7.82.73.5.4.123.5.87.8.17...37...82.5.
   ....................1.23.45..6.....7.1723..565..67.12..45.6.28..6834.5711...8..64   =1
........1..1.2345.245.61..3.576.23.83.8.7....426.38.75.7..8....5.2..78..8.42.6..7
   ....................1.23.45..6.....717.8.45.6583.7641.238.6715.6.5.8..7.7..3..6..   =1
...........1.23.45.421...63..78.....2.8.715.441.352.78.7.2.8.5.12.536.878....73..
   ....................1.23.45..6..245..2...7.61814.6527..67.8...413..7..8.4.8.3671.   =4
...........1.23.45..2.641.3..7.3.5.4.3..8..17615.473.8.78..6.5.12...8..65.6.728.1
...........1.23.45.24.153.6..7.5..63.5..8.4.7246.37.58.78..26..41...8..26...71.84
...........1.23.45.2415..63..671..84.87..2..614..6...7..823..5..5..81.7.61257..38
...........1.23.452451...63..7.8.....137.2.864.8.61.37.742....815283..748....7.5.
   ....................1.23.45..6..74.8.38.62.5757..186...83..456.6...8..7.7.5.368.4   =4
...........1.23.4524.5...13..6.72.584.86....772.38..61.6283...4.7...6.8.8.42.7.36
...........1.23.4524.5...13..6.72.584.86....772.83..61.6238...4.7...6.8.8.42.7.36
...........1.23.4524.5...13..6.72.5872.38..618.46....7.6283...4.7...6.8.4.82.7.36
...........1.23.4524.5...13..6.72.5872.83..618.46....7.6238...4.7...6.8.4.82.7.36
   ....................1.23.45.1...6.37.65.3721.3.7.8.5...54..8.7.186.7245.7.3.4.1..   =2
...........1.23.45.2..14.63.571...3.2.3..7..88163.2.74.752...8.1...75..66.24.1.57
...........1.23.45.2..14.63.572...8.1...75..66.24.1.57.751...3.2.3..7..88163.2.74
   ....................1.23.45.16..2.784.7.68...82.1..4.6.6.....8.134.865.778..516.4   =3
........1..2.13.4..512..3.6.....78...26.48.73.78325.64.1.7.24.8.8743....2...81.37
........1..2.13.4..512..3.6.....78...26.48.73.87325.64.1.7.24.8.7843....2...81.37
........1..2.13.4..512..3.6.....78...27.48.63.86325.74.1.7.24.8.7843....2...81.37
   ....................1.23456.....7.18.17.8264.4.8.6...7.5...87...726.4.858.4.7516.   =2
...........1..2345.341562....6..785..4856.7.2.7.2.8....1762.58..8..7...165.8.14.7
...........1.23.452465...31..78.2.1..68.....712.76..83.8437.1.261.2.8.747...4...8
   ....................1.23456....37.48...5..7...74.82563.173.5.84.4..78.1.6.8.1.37.   =7
...........1.23.452345...16..5..2.78.278.51.41.873..5..12....874732.8.618.....4..
...........1.23.452345...16..5..2.78.278.51.41.873..5..72....814132.8.678.....4..
...........1.23.452345...16..5..2.78.287.51.41.783..5..12....874832.7.617.....4..
...........1.23.452345...16.178...6.3.876.1.44.6.3..87.7...84..14237...88.3....71
...........1.23.452345...61..5..2.78.278.51.41.873..5..12....874732.8.168.....4..
...........1.23.452345...61..5..2.78.287.51.41.783..5..12....874832.7.167.....4..
...........1.23.452345...61..5..2.78.287.51.41.783..5..82....174132.7.867.....4..
   ....................1.23456....37145.7.......135.8476...7....2..18.6257.52..78631   =1
...........1.23.45.24.51.63...31..74.17.8.35.4...7...8..6..8..7.42.37.86.78.62431
   ....................1.23456..2..4..7.74.685.285..726....7.4.86.4....71.55.8.317.4   =2
..........12.34.56.3562.471...48...7.28.7614..7.....68.5.84.61..8...5...14.267.85
..........12.34.56.3562.471...84...7.28.7614..7.....68.5.48.61..8...5...14.267.85
   ....................1.23456..2..7.65..7.523.4.1563.72..4.3.5.7..58.71643.7..6....   =1
...........1..2.34.23456.71..5....8..18524.6726...8.4..3624...8.8..65..35.28.3.16
   ....................1.23456..4..7....78.32.4525..8.7...17..658.425.186.78...7..14   =4
.............12345134.56..2.....7..3.7..8.45.4.3.657.8.48..12.73.2.7..8.71..28534
.............12345134.56.2......7.3..7..8.5.44.3.6578..48..127.3.2.7...871..28453
.............12345134.65..2.....7..3.7..8.45.4.3.567.8.48..12.73.2.7..8.71..28534
.............12345134.65.2......7.3..7..8.5.44.3.5678..48..127.3.2.7...871..28453
   ....................1.23456..7..8.24.54.726..82..6.57..7....3..41..37.6553.6.174.   =1
...........1.23.45243..5.16.26..7.8.3.786...44...32.67.3425..711...7....7.231.45.
   ....................1.23456.4......7128.576.46.7.8.512.8..7236121..387..7.....2..   =1
...........1.23.45.24.516.3.12.....7.8713....6.3.72.8..483157.6.7...8..4136..7.58
   ...................12.34.56........7.21.47368.78316.2...7.8.63..8..5.7.223..71.85   =4
........1..1..234..35.14.26.....7....53248.67.78.5.234..742..8..8.7.5.1.52..81.73
........1..1..234..35.14.26.....7....53428.67.78.5.432..724..8..8.7.5.1.54..81.73
........1..1..234..53.14.26.....7....35248.67.78.5.234..742..8..8.7.5.1.52..81.73
........1..1..234..53.14.26.....7....35428.67.78.5.432..724..8..8.7.5.1.54..81.73
   ...................12.34.56........7.21.47538.57813.2..6.3.87.2.7..6.3..2.8.71.65   =2
........1..1..234..35.16.27.....8....53624.78.68.5.432.1.8.52.4.8.2...1.5.4.61.83
........1..1..234..35.61.27.....8....53624.78.68.5.432.1.8.52.4.8.2...1.5.4.16.83
   ...................12.34.56........7.253.7.68.73.6852..47.836.5.8..4..7.53.7.6.84   =1
........1..1.23.4..245.1.36..71...8..18.7245.24.8.5.17.7........82.57.644.62.8.75
   ...................12.34.56........7.2734..85.4578.32..71..3.6.2.641..734..67.21.   =2
...........1.23.45.24.561.3....7.....163.2.78.72.68.31.47..53..1..2.4.572.5.37.14
...........1.23.45.241.5.36.......7..16.785.4.47.51.68.725....31...374.24.3.12.57
   ...................12.34.56.......67..6..74.5.874.632..58143.72.712.85342..7.....   =1
........1..2..345..41.6532.....78.4..746.158.8.654.17...8...7...27.5683..6...7215
   ...................12.34.56.....1..7.473.25.1.51.472...6..7..2..78..36.5425.687.3   =1
...........1.23.45.24.5163......2.7..175.4.26.62.17..4.3..7.4...78..5.63146.38.57
   ...................12.34.56.....1..7.473.25.1.51.472...68..37.5.7..6..2.425.786.3   =1
........1..2..3.45.14.5263......1.7..275.4.16.61.27..4.38..5.67.7..3.4..246.78.53
   ...................12.34.56.....1.78.2786..31.81.4726..58.73..4.7.48....2.3..6.87   =1
...........1.23.45.24.156.3.....7.68..6.823.7.8713.4...35.71.84.78...5..14..587.6
   ...................12.34.56.....17.8.8735..2112..78.65.7..4....24.5.3..73.5.87.42   =2
...........1.23.45.245.6.13....78..4.782.415.4.26.5.78..7....3.15.3.2.872.3..75.1
...........1.23.45.245.6.13....78..4.782.415.4.26.5.87..7....3.15.3.2.782.3..75.1
   ...................12.34.56.....3..7..72.563..43.6758.....7.1...28.167.5.71452.68   =1
...........1.23.45.24.561.3..7285.34.3..1.7..482.375.1.46.7...8.785.2...2...68..7
   ...................12.34.56.....3..7.23.7854..752..6.3..7.8..6..81.674.553.4.27.8   =1
...........1.23.45.24.65.17..8....7.21..57.384.7.825...4...8..31.2.36.848.327.15.
   ...................12.34.56.....3..7.23.8754..572..6.3..8.7..6..71.684.553.4.27.8   =2
...........1.23.45.24.56.17..7....8.21..85.734.8.725...4..3...81.2.68.348.32.715.
...........1.23.45.24.56.17..8....7.21..75.834.7.825...4..3...81.2.68.348.32.715.
   ...................12.34.56.....3..7.34.7.58..78541.63.21..5.78.8..2..1.4.7.18.25   =4
........1..2.13.4..15.64.32....7...3.234.8.7..87.36425..8..5.1..51.87..427..41.58
........1..2.13.4..15.64.32....7...4.243.8.7..87.46325..8..5.1..51.87..327..31.58
........1..2.13.4..51.64.32....7...3.234.8.7..87.36425..8..5.1..15.87..427..41.58
........1..2.13.4..51.64.32....7...4.243.8.7..87.46325..8..5.1..15.87..327..31.58
   ...................12.34.56.....3..71.3.4758.7.815..43..7.15.68.81.764.56..4...7.   =3
.....1..2....2345..2456.1.3....7..2..726.53.44.6.327.5..7.....1.48.165.761..57.4.
.....1..2....3245..2456.1.3....7..2..726.53.44.6.237.5..7.....1.48.165.761..57.4.
.....1..2..3.2..4..5246..31..7....1..31.874.558..14.73.28.7..54.7...23..3.584..27
   ...................12.34.56.....5..7.25.47.8..7628..45..152..73.374.8...25..73.68   =2
..........12.34.56.352.614...7......1.47.3.8538.4.5.71.736.2.1.2.134..6764...7...
..........12.34.56.3526.4.1.27.43..535.6.27.44.6.7.....7.......2.8.165.756..271.8
   ...................12.34.56.....5..7.58.7362.72..86.35.86....7.24...73.837..48.62   =2
...........1.23.45.245.1.63.....7.5..8521..377.23.5.18.187.....2.6.38.7.3.71.2.86
...........1.23.45.245.1.63.178.....2.6.37.8.3.81.2.76.752...381....8.5.8.23.5.17
   ...................12.34.56.....5.7..25.87..4.7624..85..152..37.374.8...25..73.68   =1
..........12.34.56.345.61.2..7......1.67.3.2838.6.2.17.734.5..14.136..7556...7...
   ...................12.34.56.....5.7.1.7.4...54.5.78.212.3.6.1..6..42..37741.83.62   =1
........1..2.34.5..56.12.34.....3.7.2.3.78.655.7.2...34..26..176.1.4.5..725.81.46
   ...................12.34.56.....5.78..687.4...87.2.635.4851.367.71...5.46...4..81   =1
........1..2..3.45.43.5162..713...6.238.76.144.6.18....87...4..324.87.5.6...3..87
   ...................12.34.56.....7..8..7.8.56.58.36217..4..73.8..71.486358..6..74.   =1
........1..2.13.4..512.463...78......8..427..526.37.84.1842537..75..8.1.2...71..8
   ...................12.34.56.....7..8.274.856..5862.4.7.71.4..8.2...71.455..2.67.1   =1
...........1.23.45.24.516.3....7...8.7258.36..863.2.57.47..58..2...475.66..23..74
   ...................12.34.56.....7..8.482.3.752.7.8.36..81.42.3752..78..17.4.1..8.   =12
...........1.23.45.246.5.31..7....8..823.751441..58..7..87...56.465.2.78.7...64.3
...........1.23.45.2465..31..65...78.4873..56.7..6.4.3..7....8..8237.51441..85..7
...........1.23.45.2465..31..65...78.4876..53.7..3.4.6..7....8..8237.51441..85..7
...........1.23.45.2465..31..65...78.4876..53.7..384.6..7....8..8237.5144...85..7
...........1.23.45.2465..31..65.2.78.1873..56.7..6...3..7....8..8237.51414..85..7
...........1.23.45.2465..31..65.2.78.1876..53.7..3...6..7....8..8237.51414..85..7
...........1.23.45.2465..31..65.2.78.1876..53.7..38..6..7....8..8237.51.14..85..7
...........1.23.45.2465..31..67...58.4853..76.7..6.4.3..7....8..8237.51441..85..7
...........1.23.45.2465..31..67...58.4856..73.7..3.4.6..7....8..8237.51441..85..7
...........1.23.45.2465..31..67...58.4856..73.7..384.6..7....8..8237.5144...85..7
...........1.23.45.2465..31..7....8..8237.51441..85..7..85...76.4673..58.7..6.4.3
...........1.23.45.2465..31..7....8..8237.51441..85..7..87...56.4653..78.7..6.4.3
   ...................12.34.56.....7.68.4836..7..6728.3.4...85.7...751...83.81.73.25   =1
...........1.23.45.24.1563..47.5831.5...3748.8.3.....7.7....86.1.8.762.44.6.82.7.
   ...................12.34.56.....73...27.53.68.6328..75.46.725.1.7154.6..2...1...7   =4
...........1.23.45.24.516.3..7..6.3.2.6.375.44..2.57.6.42.183.7.783.24.11...7....
...........1.23.45.24.516.3..7..6.3.2.6.375.44..2.57.6.48.123.7.723.84.11...7....
...........1.23.45.246.5.31..7.3..6.1.65.2.732..7.615..123.7.84.782.431.4.......7
...........1.23.45.246.5.31..7.5...62..7.64.34.63.2.57.425.7.18.782.15.41......7.
   ...................12.34.56.....78.1.71.4856..863.1.47..87.3....2..8..7..374256.8   =2
...........1.23.45.23.451.6..6..74.1.17534.68.8..6..57..8.725...3.4.8..2.7235...4
...........1.23.4523..45.16..2.57.8.3..4.2.5757.3....41.3.7..6.72...6..886.234.71
   ...................12.34.56....173.2.218.3.677..4...8..48.7...51.75.8...25.341.78   =1
...........1.23.45.245.1.36...2.75.4.42.85.677...1..8..187....32.7.38...43.152.78
   ...................12.34.56...7..8.3.2834..17.73.8.64..571..4.8.6..58.71.8.47..6.   =1
...........1.23.4524..5631...5.37.8438..1.75.7.4.8.1....7....284.2.6857.8...724.1
   ...................12.34.56..1..37...27.85631.6..7..8..4...8..7.7832.5.42.6.47.18   =2
...........1.23.45.2456.13...73..8...1867.453.4..5...7.72.3658..8.2...7.4.678..21
...........1.23.45.2456.13...73..8...1867.453.4..5...7.72.3658..8.7...2.4.628..71
   ...................12.34.56..1..5..7.85.7164.76..48.1...8.2746..26..3.7857..8.2..   =1
...........1.23.45.2345..16..75...2..5..37.8131.28..57.7834..6.13.8...744...72..8
   ...................12.34.56..1..5.7..27..8..5.8.7..2...6857.43..734.6.822.4.83.67   =1
........1.12.3456.35..16427....48.52..56..78.82..7.6.4..14...7..8..6.1.55...8.246
   ...................12.34.56..1..7..8.27.856.186..2..7..76..3.8.153.4876.28..7.1..   =1
..........12.34.56.3516.4.2..1.47..8.2.68..4..8....6.7..8.75.6..5.81.7.417.426.85
   ...................12.34.56..1..7.35.7..1.6.8.856431.7..8.72.6..2746...3.4.3..7.2   =2
...........1.23.4524..51.63.2..37..81..5.2.373.71...5.41..7...67.2..6.8.8.6215.74
...........1.23.4524..51.63.2..37..81..5.2.373.71...5.41..7...67.6215.848.2..6.7.
   ...................12.34.56..1..7.45.65.41.7.74..2.6...2..735...83..2.6757.18.32.   =2
...........1.23.45.236.4.17..83.2.74.374....82...87.6..821.5..61.6.3845.3....6.81
...........1.23.45.2364..17..243..78.872....43...78.6..3815...61.6.8245.2...6..81
   ...................12.34.56..1..7.68.278.651..8..5.7....4.73.85.38615..7.7.4..63.   =1
..........12.34.56.3516.24...137.48..7..81.6..8342...7..76.382...874.6..3....8.74
   ...................12.34.56..1..75...4..6..7..87.4516...34.862..2..53.81.7862.3.5   =2
...........1.23.4523.4.5.61..2..7..63.65.2.877..3...5..2413..781.3.58..486.2.4.1.
..........12.34.56.345.621...1..75...48.6517..7..4..62..34.862..2..53..1.876..3.5
   ...................12.34.56..1..75.8.6..1..47.78425.61..7.43.8..2.5...73.83.7.4.5   =4
...........1.23.45.24.51.36.17.6...4.46132.872.8..7.6..752....3.8..15.7.1.2.7..58
...........1.23.45.24.51.36.17.6...4.48132.672.6..7.8..752....3.8..15.7.1.2.7..58
...........1.23.45264.51.37....15.8..12.8..568.52....3.26..8.7.1.8.7...44.7132.68
...........1.23.45264.51.37.12.8..566...15.8.8.52....3.2...8.7.1.8.7...44.7132.68
   ...................12.34.56..3..5..7.85.7.1.317..8..25.418.3.722.8.47..173..2..4.   =2
...........1.23.4526..47.13.1...8..454.2.1.8.8.2.64.5..56..2.381.8.3..6.32..865.1
...........1.23.4526..47.13.5...2.381.8.3..6.32..865.154.2.1.8.61...8..48.2.64.5.
   ...................12.34.56..3..7..8.87143.6515..8..7..28.61..7.7132.68.3...7..2.   =2
...........1.23.45.24.15.63..7.3..8..42.87536.8.5.2..7..8.71.54.1.3....847..583.1
...........1.23.45.24.65.13..7.32.81.1...75..28.5.1.37..8.7..5..42356.78.7...83.4
   ...................12.34.56..3..758..28.53.7457..486.2.8..7.4..2...8...773.46.82.   =1
...........1.23.4524..153.6..4.7.8..1.853..747...8..5..1...8.6747..615.88.6.5743.
   ...................12.34.56..3.174.5.7.8....35.1.43.78..7.8..2..284.1..715..72.84   =1
........1..2.34.5..63.152.4..1.7...8.285.31.7.7..8152..4...7.8..8735.4.223..4.7.5
   ...................12.34567........1.28..1453.51423.86..7.18.45.3524..78.8...7...   =1
..........12.34.56.3526.41...7....3..23.4768.186.2374..5..71.6..614.257..7..5....
   ...................12.34567........1.51623.78.83..165...5.42.86.2816..45.4...8.2.   =1
..........12.34.56.345.62.1..6..35.7..7.....4.43.7561..28.47165.7...2...4.1.587.2
   ...................12.34567.....8.....45167.8.81.2.45...3.8.64..68..32.524..658.3   =1
........1..1.23.4.452.16.73.....8....85234.6.2.6.7..38.18.427.6.27..1.846.4.8..1.
   ...................12.34567.....8....245167.8.81.2.45...3.8.64..68..5..314..638.5   =1
........1..1.23.4.452.16.37.....8....85274.632.6.3..78.1...2.84.28.417.66.4.8..1.
   ...................12.34567..1.456.8..87...1.46..18275..6..78...2..83756.8..5..2.   =3
........1..2.13.4.14..56237.1..75.82.8.3..1..2.5.81374.2..3...84...687238....74..
........1.12.34....356.1247..7..8.6..2.4....8.58..6472.731..8.4.81.437262..8...1.
........1.12.34....3561.247..7.8..6..2.4....8.58.6.472.731..8.4.81.437262..8...1.
   .................1..1.23.45........6.624.7.1881.2.647..36....5.15.3.2.672.86.513.   =2
........1..2.1345..31.452.6..3.576.2.2638.5.77.........18...7..2.7.381.436..7182.
........1.12.34.5..561.23....1...47..874.31.55.47.1.63.652.873..7.......2.83.75.6
   .................1..1.23.45.......6..47.365.26.5.72..3.58.1..761.638.2544...6.1.8   =1
........1.12.34.5.6.3.5124...6..7..4..7.45.26.2.3...7..6..1.78..7158.4622.8.7..13
   .................1..1.23.45......6...24.65.37.673.2.54..82.15.6.1..36..8.765.8.13   =1
........1.12.34.5..3651.2....37..86..68345.72.7..86..5.2147358..87.5..2.3....8...
   .................1..1.23.45.....6.7..47.38.626.327..84.74.6.2..1.63.2.573..7...16   =2
........1..1.2345..42.51..3..62..5.742..17.367...36.4..6..7.....84.627.52.7.85.64
........1..1.2345..42.51..3..62..7.542..17.367...36.4..6..7.....84.625.72.7.85.64
   .................1..1.23.45....36.7....2.4.6826.7.81.41.23.7.565.7.62.1.63.5....7   =3
........1..2.13.45.134.52....6.3..57.2.7.46.3.37.564.2.6.5.7....71.48...2.8.61.74
........1..2.13.45.134.52....6.3..57.2.7.46.3.37.564.2.6.5.7....78.41...2.1.68.74
........1..2.13.45.314.52....6.3..57.2.7.46.3.73.564.2.17.48....6.5.7...2.8.61.74
   .................1..1.23.45..2..64.3.357428.6.6.3.......7.6.1...162.45.752.1.76.4   =1
........1.23.14.56.17536.2.....6..8..864.3..53.2.856....13....8.38.415.22.4.581..
   .................1..1.23.45..2..64.3.357428.6.6.3.......7.6.1...162.75.452.4.16.7   =1
........1.23.14.56.17563.2......6.8..8643...53.2.586....13....8.38.451.22.4.815..
   .................1..1.23.45.1...4.67.3827..144.7...2.8.83.....61..76..8.7.6.38152   =3
........1..2.13.45.41.526...175.....2...67.184.8.215.7.2..8.7.6.76235.848.4..6...
........1..2.13.45.41.526...175.8...2...67.1.4.8.215.7.2..8.7.6.76235.848.4..6...
........1..2.13.45.41.526...175.8...2...67.184.8.215.7.2....7.6.76235.848.4..6...
   .................1..1.23.45.14.36.78.6387..148.........563...821.8.62.5.4..58.16.   =2
........1.12.34.5.5.3.16247....4..82..8..75..2.5.687.4.21.8347.3...71.2.8..4..1.3
........1.12.34.5.53..16247....4..82.8...75..25..687.4.21.8347.3...71.2.8..4..1.3
   .................1..1.23.45.1627..38.2...1.677.3..6.1..4..3...6.624.8.7.3.76.2.54   =4
..........12.34.56.3561.274..74.3.28..8.7.6...2.86..4..7.58.4.2.8...7.6.2.134..87
..........12.34.56.3561.274..74.3.28..8.7.6...2.86.74..7.58.4.2.8...7.6.2.134..8.
..........12.34.56.3561.472..74.3.28..8.7.6...2.86..4..7.58.2.4.8...7.6.2.134..87
..........12.34.56.3561.472..74.3.28..8.7.6...2.86.74..7.58.2.4.8...7.6.2.134..8.
   .................1..1.23456....17.4..17...5.24.823.1.7..6...7...75.68.2484..726.5   =1
........1.12.34.5..562.1.34..1..7.8..783.61.556.1.87.3.2..6...8.85...3..6.78.35.2
   .................1..1234.56....27.8..273.8.148.3.4...7.72.....81.8.73.6236.8...75   =1
..........12.34.56.3561.472..7.8...4..81.7.6..21.46.8..7.4...38.8..7.6.52.386.74.
   .................1..1234.56....27.8..284.3.754.78....3.84....6.21.6.8..77.6.42.18   =1
..........12.34.56.345.621...37...8...7.58..2.286.3.75.71.6582..8...75.12...8..67
   .................1..1234.56..2..7..8..7.42.6546.58...7..48...7..7842..1312..73.8.   =2
...........1.23456452.16..3..4..7.8...8.61.4551...86.7..5.8.7...87.325.424..7..38
...........1.23456452.16..3..4..7.8.51...86.77.8.61.45..5.8.7....7.325.424..7..38
   .................1..1234.56..2.43.75.5.6.7.3..7352.6.4.1......3.38.12.6.2.7.6..18   =1
........1..1.2345..421.5.63.....7....2835..74.742.85.6.1..72....8753..1.25.8.1.47
   .................1.12.34.56........7.283..56..37.56.28...8...1..71643.85.841.567.   =2
..........12.34.56.34.5612...1..7.6..76..8.12328.6...7..74.328..8..7....243.85.71
..........12.34.56.34.5612...14758...7..6851..8..1..6...8..7....2754368.34..8..75
   .................1.12.34.56........71.73.6.422.6.4713..214.8.755...7....7.8.53.14   =3
........1.12.34.56.3615.24...17......8741..2562..85.17..3.7.5.2.7.....6.26.54..73
........1.12.34.56.3651.24...17......8714..2562..85.17..3.7.5.2.7.....6.26.45..73
........1.12.34.56.3651.24...3.7.5.2.7.....6.26.45..73..71......8174..2562..85.17
   .................1.12.34.56......1...71.48365643.158...27.536.8.68..25..3...8...2   =1
..........12.34.56.345.62.1..13.75.2..7..1...3264...17.486.3.75.637.5.28.7....6..
   .................1.12.34.56......3....3472.652653...74..1.....7.3724..1.62871..43   =2
........1..2.3456..356.124......5.7..5736.12.2.3.17.5...4...7...71.4368.32..7641.
........1..2.3456..356.124......5.7..5736812.2.3.17.5...4...7...71.436..32..7641.
   .................1.12.34.56......3....3742.6526538..74..1.....7.3721..4.62.47..13   =3
........1..2.3456..3561.24.....5..7..573.612.2.3.71.5...4...7...71.634..328.4761.
........1..2.3456..3561.24.....5..7..573.612.2.3.71.5...4...7...71.6348.32..4761.
........1..2.3456..3561.24.....5..7..573.612.2.3.7185...4...7...71.634..32..4761.
   .................1.12.34.56.....1.7.12..576.867..2..15..6.43..723..75.6474.2....3   =9
........1..1.2345.264.15.37.26.51.48.482...151....8....8..3....4.2..6.836.3.825.4
........1..1.2345.264.51.37.26.15.48.482...151....8....8..3....4.2..6.836.3.825.4
........1..1.2345.462.15.37.26.51.48.482...151....8....8..3....2.3.865.46.4..2.83
........1..1.2345.462.15.37.26.51.48.482...151....8....8..3....2.4..6.836.3.825.4
........1..1.2345.462.51.37.26.15.48.482...151....8....8..3....2.3.865.46.4..2.83
........1..1.2345.462.51.37.26.15.48.482...151....8....8..3....2.4..6.836.3.825.4
........1..2.13.45.15.2436...4..7....7823..5452..4..37.51.82.73.873...1.2...71.8.
........1..2.13.45.15.2436...4..7....7823..5452..4..37.57.82.13.813...7.2...71.8.
........1..2.13.45.15.2436...4..7....8723..5452..4..37.51.82.73.783...1.2...71.8.
   .................1.12.34.56.....2.7..2735.6..6.1.7.52..48..3.67.7..4.8.51.6.8743.   =2
..........12.34.56.35.261.4..1.72.68.2...87.5.7846.21...3....8..8.64.53.15..836..
........1.12.34.5..63.1524...1..786..2..81.75.7845.1.2..754.6.3.3......72.6.735..
   .................1.12.34.56.....2.75.5734....2...7546..7821...412.4.3.873...87.1.   =2
..........12.34.56.35.16.27.....1.78.8736....1...876.2.4815..6.3...48..55.16.3.84
..........12.34.56.35.16.27.....3.78.8716....3...876.2.4835..6.1...48..55.36.1.84
   .................1.12.34.56.....3..7.7524..8332..7.5.4.47.8..6..6.4.2.782.8.67..5   =2
..........12.34.56.35.627.4..1.286.7.2..7..48.874.6.2...8..3....7364..811.4.8.36.
..........12.34.56.35.627.4..1.286.7.2..7..48.874.6.2...8..34...736...811.4.8.36.
   .................1.12.34.56.....6.78.6827.14..71..86.5..37.......7.8351418.4.2..7   =1
..........12.34.56.3562.41......5.78.5728.1.4.81..756...38.......8.7364117.4.2.8.
   .................1.12.34.56.....67...5738.6.4.63.47.85..1.7854...54.31.7.7..1...8   =2
........1..2.13.4514..2536..24.78.5.57.3.2.848...5.....8...7.1.2.7.8143.41.23...8
........1..2.13.4514..2536..24.78.5357.3.2.848...5.....8...7.1.2.7.814..41.23...8
   .................1.12.34.56.....7..8.27..816..8126..74..8.72.45.756..8..24..8..17   =2
..........12.34.56.35.1624......7.8..5814.76..71..84.5.274....8.8..7162.1.6.8.57.
..........12.34.56.35.1624......7.8..5834.76..73..84.5.274....8.8..7362.3.6.8.57.
   .................1.12.34.56.....7.8...734.6.5.456.87...21.73.68.78.6...24..8.25.7   =2
........1..1.2345..42.1563..1...784..78.3...64.6.81.73.2..58364.843....26....2.8.
........1..1.2345..42.1563..1...784..78.5...64.6.81.75.2..38564.845....26....2.8.
   .................1.12.34.56.....71.8.584.1.67.71.685....37.68...4..8..72.8724.6..   =1
..........12.34.56.35.627.4..732..48.2...86...834.61....8..7.61.5..8.4.7.71643.8.
   .................1.12.34.56..1..7.3..2..8.5.757..23.18.5...6.831...487.52.8.7.16.   =1
..........12.34.56.3526.41...1.47..8.2718..64.8.6.217..58..67...7.8...452...75.8.
   ................12..1.23.45..6.....1.1536.47.37.41..26.3768..54.625.4..75......6.   =1
........1.12.34.56.3561.24...6..3.7..2.56.4.35..47..62.51.8..24.7.1.....2.834..17

#clust   #puz
67   1
46   2
13   3
13   4
2   5
1   7
3   8
1   9
1   10
2   12
1   16
done


Some new 38s appeared and the count in my cache currently is 123 611. I searched around the new ones.

Now I am doing a systematic search at {-3+4} over all 38s and tomorrow the results from the first pass are expected to arrive.
{-3} on the 123 611 38s gave 998 209 586 multiple-solution 35s. Then {+1} gave 2 760 477 single-solution 36s and 1 730 599 663 multiple-solution. Next {+1} gave 1 472 240 single-solution 37s and 384 100 844 multiple-solution ones. The next {+1} to 38s is in progress.

While testing a piece of code, I took 1000 consecutive 37s and performed several {+1}. Curiously enough a new 39 appeared and this is the only one in my cache. It has no twins and has no neighbors at {-+5}. I don't intend to extrapolate. 8-)

P.S. From the 1 472 240 single-solution 37s 600 512 twins just appeared but I will involve them in the mill much later.
dobrichev
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Posts: 1316
Joined: 24 May 2010

Two more ...

Postby blue » Sat Feb 08, 2014 6:48 am

Hi dobrichev,

Interesting stuff -- especially the number of twins that you got from the 37's.

These are the two new ones.

Code: Select all
.....1..2....2..3......314...4..56....8.6432767.23845..85.1726.1...5..7.76.84251.
.....1..2....2..3......314...4..56....8.6432776.23845..85.1726.1...5..7.67.84251.

The first one is a "same shape" {-6,+6} change from this one in your last batch:

Code: Select all
.....1..2....2..3......314...4..56....5.6432767.23845..18.5726.5...1..7.76.84251.

The two solutions differ by a 9-cell UA, where 6 of the cells are clue cells.
That's how I got it -- changed the 6 clue cells to match the other version of the UA, and checked the result for being valid & minimal.
blue
 
Posts: 573
Joined: 11 March 2013

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