The New Sudoku Players' Forum

[dobrichev]

Hi blue,

I confirm your 2 puzzles are new valid minimal 39s. Congratulations!

Your approach is very interesting to me, especially if you are not joking and the method is really productive.
I did similar (or the same) stuff in my early attempts in finding low-clue puzzles.
In my understanding, this is a mixture between a fixed pattern and very similar solution grid. At the time I played with this, I had some understanding that "similar grids" can be found by permuting the givens within an UA, but didn't realize that when entire UA is covered by givens a valid puzzle is guaranteed, i.e. the twins concept.
I, maybe wrongly, found this method unproductive, particularly due to the difficulties in finding UA with the then software.
The theoreticians could name these "almost twins" and depending on details to extend to "born from almost married" and/or "almost pregnant girl", shortly ATAMAP, ATNM, ATNP, etc.

[dobrichev]

Below are 61 new 39s, making the entire list 413 puzzles long.

Code: Select all

.............12.34..15...26...4...78.7.8.13.28.32.7.41..7..8.6.3.6145.875.87.6413
.............12.34..15...26...4...78.732.8.418..7.13.2..8..7.6..36145.87.578.6413
...........1..2.34.2345..16..75...8..85.2764161284..57.38.7.1...7.....681.6.8..73
...........1..2.34.2345..16..75...8..85.2764363284..57.18.7.3...7.....683.6.8..71
...........1..2.34.3.451.26...2.7.65..6.45.7..7561.4.2..713.2.8.83.24..71..7.8.43
...........1..2.34.3245..16..7..4...2135.8.4754.27..81..8..5.6..258.6.7336.72...8
...........1..2.34.3245..16..7..4...24.57..815132.8.47..8..5.6..258.6.7336.72...8
...........1..2.34.3245..16..75...8..85.2764332684..57.18.7.3...7.....686.3.8..71
...........1..2.34.3245..61..7..4...2135.8.4754.27..18..8..5..6.258.6.7336.72..8.
...........1..2.34.3245..61..7..4...24.57..185132.8.47..8..5..6.258.6.7336.72..8.
...........1.23.45.234..6.1..7..4.68..8.721.4.1.8..7...725.8416.8624.5.71.5..7.8.
...........1.23.45.24.51.36....3...7.721.5..31...6745..16.78..4.47512.682.8..6.7.
...........1.23.45.24.613.7.48.167..2.7..8..661.2.74.8.8..3.5..1...85..44.51.28.3
...........1.23.45.24.6173..43..8..6.68.32.5.2.5...38..8...751.1.7.8..6.452.1687.
...........1.23.45.241.5.36..75......15832.67236.7..58.4..17.8..723.8..41..2...73
...........1.23.45.245.6.13....67..8.762.8.518.235..67.47.3..8.1.8.7253.2..8...7.
...........1.23.45.245.6.13....67..8.762.8.548.235..67.17.3..8.2..8...7.4.8.7253.
...........1.23.45.2456.31....7..8...4863..57.76.85134.1...8.7..8735.4.14...7.58.
...........1.23.45.42.5..36..6....7..27.85.644.8.7651..1.5...87.7483..51.8..174.3
...........1.23.45.42.5..36..6....7..27.85.644.8.7651..1483..57.7.5...81.8..174.3
...........1.23.45236.457.1....86...6.235.4.88.32.4.76.24..8..71.7.328.43.8...1..
...........1.23.45236.457.1....86...6.235.4.88.32.4.76.27..8..41.4.328.73.8...1..
...........1.2345645..16.27....62.84.84.5.6.26...8..7..48.357.11.7..8..553...1.48
...........1.23456452.61..3....7..84.7..8.6..8..1.67.5..3..7.48.47.385615.8.1.3.7
..........12.34.56.35.2641...1.67.8..27.8.16..8...2..7..8.43.71.736..8..1.4.786.3
..........12.34.56.35.2641...1.78.6..27.6.18..8...2..7..8.43.71.736..8..1.4.876.3
..........12.34.56.35.621.4..1.27.68.2..8.7...87..6..1.53..86.7.786.531.1...73.8.
..........12.34.56.35.621.4..3..7.8..58.4...7.7..834.5.21.7..48.8741.52.5...287..
..........12.34.56.351.62.4.....7..8..7.18462..846..7..7364.8...8.7.1..52.1.83.47
..........12.34.56.351.62.4.....7..8..7.18642..864..7..7346.8...8.7.1..52.1.83.67
..........12.34.56.3516.24...1.7.4....8.26.17.2741.86..7..83..4.8.6...7.2.374.68.
..........12.34567.345.6182.......4..48.652.11...48.75..1.8375..7365..18.8......3
..........12.34567.345.6182......84..48.652.11...48.75..1.8375..7365..1..8......3
..........12.34567.345.6281.......3..83.651.22...83.75..8.....4.2..4875..4765..28
..........12.34567.345.6281......83..83.651.22...83.75..8.....4.2..4875..4765..2.
..........12.34567.345.6812......4...48.65.211...487.6..1.8367..7365.1.8.8......3
..........12.34567.345.6812......48..48.65.211...487.6..1.8367..7365.1...8......3
..........12.34567.345.6821......3...83.65.122...837.6..8.....4.2..4867..4765.2.8
..........12.34567.345.6821......38..83.65.122...837.6..8.....4.2..4867..4765.2..
........1.....2.3...3145.26...456.78..8..7.1..478.1653.7.2.83653.6.74.828.2.....7
........1..1.23.45.52.4163........7..73264.58825.17.6..37.8....21..3..8.5.8.7231.
........1..1.23456.24.517.3..6......4..37..6553..86.74.4..3..1..63.185471.5..7..2
........1..2.13.4.13..452.6..3..7....27.3146.61.4.27.3.5..7..2..76..85.428..546.7
........1..2.13.4.13..542.6..3.7.....27.3146.61.42.7.3.5...7.2..76.8.5.428..456.7
........1..2.13.45.614.2.3......74...28534..77..26...3..5.26.74..67.53..2.73415.6
........1..2.1345614.256..7.....7....27.3158441.528.73....65..8...3...6..6..72345
........1..2.1345614.256..7.....7...2.7.3158441.528.73....65..8...3...6.6...72345
........1..2.1345614.526..7.....7....27.3158441.258.73....65..8...3...6..6..72345
........1..2.1345614.526..7.....7...2.7.3158441.258.73....65..8...3...6.6...72345
........1..2.34.5.36.51..24..7.8....62.37..1883.....72.53..8.472.6.43.8548.75.2..
........1..2.34.5.36.51..24..7.8....62.37..1883.....72.73..8.452.6.43.8748.75.2..
........1.12.34.5.53..162.4..3.7.8.5.856.3.2772..8.4...5...7...27.36.5.83.1..8.72
.......12.....134....23......3..5.6..6742853.58..6327..5..1.6...7684215.8...56427
.......12.....134....23......3..5.6..7642853.58..6327..5..1.6...6784215.8...56427
.......12.....134....23......5..6.3..7342865.68..5327..5784216..6..1.5..8...65427
.....1..2....2..3....3..14...15...6..6784251.58.1.627..5...36...7628435.8..6.5427
.....1..2....2..3....3..14...15...6..6784251.85.1.627..7625438..8...36..5..6.8427
.....1..2....2..3....3..14...15...6..7684251.58.1.627..5...36...6728435.8..6.5427
.....1..2....2..3....3..14...15...6..7684251.85.1.627..6725438..8...36..5..6.8427
.....1..2....2..3....3..14...5..36...7.6.54288.627435..571.628.1.854276.6..7...1.
.....1..2....2..3....3..14...5..36...7.6.54288.627435..571.628.1.874256.6..5...1.


Latest 2 39s from blue were not rediscovered on this pass.
I did {-4+4} to the new 39s and checked them for twins.
46 appeared from 38 {-3+4} search, the rest came easily after {-3+3}. None was found at {-4+4}.
Now the seed for the second pass is in preparation.

[blue]

Below are 61 new 39s, making the entire list 413 puzzles long.

Fantastic results ! Congratulations again

Your approach is very interesting to me, especially if you are not joking and the method is really productive.
I did similar (or the same) stuff in my early attempts in finding low-clue puzzles.

It wasn't a joke, but it isn't as productive as I once thought it was.
[ It didn't produce anythng new (on its own) from your last batch of 61 39's. ]

It's something that I had in my code before you first wrote about the "twins" possiblity.
In my code, I only look for size <= 9 UA sets, and 2-row/col or 2-digit UA sets, to work with.
A lot of what it catches, is just {-1,+1}, {-2,+2} or simple "twins" transformations.
Beyond that, some of the catches (for 2-xxx UA's) are equivalent to a "twins" transformation,
followed by a Sudoku isomorphism.

On the other hand, besides just making the UA change and checking for a valid, minimal puzzle,
I also check to see if, for example, a 39, with a UA change, produces a valid but non-minimal puzzle,
that can (still) be reduced to a minimal 37 or 38 -- one that I can feed back into the testing.
That probably extends the utility of the concept, somewhat ... but it's hard to tell.

To try to gauage its efficacy, I've run the 413 39's and the 121038 38's at your "google" site, through
the code, and collected the puzzles into "classes" doing this kind of thing:
1) start with each puzzle in a class by itself
2) test each puzzle with for kind of transformations that I descibed (collecting only "same sized" results).
3) merge classes when a puzzle in one class produces a puzzle in different class, and

1. the number of cells that changed values is > 2 ... otherwise it would be equivalent to a {-1,+1} or {-2,+2} change
   (which is as far as I've ever gone with {-n,+n}).
2. (optionally) the change is not equivalent to a "twins" transformation followed by an (optional) Sudoku transformation.

The results for 39's were:

Code: Select all

Ignoring changes that are equivalent to a twins tranformation plus an (optional) Sudoku isomorrphism:

413 inputs
402 classes
 
 count | size
-------+------
   391 | 1
    11 | 2

Allowing those changes:

413 inputs
242 classes

 count | size
-------+------
   147 | 1
    59 | 2
     9 | 3
    21 | 4
     1 | 5
     3 | 6
     2 | 7

For 38's:

Code: Select all

Ignoring changes that are equivalent to a twins tranformation plus an (optional) Sudoku isomorrphism:

121038 inputs
117106 classes

 count | size
-------+------
113585 | 1
  3263 | 2
   130 | 3
   113 | 4
     7 | 5
     7 | 6
     1 | 8

Allowing those changes:

121038 inputs
65176 classes

 count | size
-------+------
 36027 |  1
 18797 |  2
  3581 |  3
  4034 |  4
   523 |  5
  1140 |  6
   125 |  7
   434 |  8
    99 |  9
    99 | 10
    13 | 11
   139 | 12
     9 | 13
    24 | 14
    18 | 15
    46 | 16
     5 | 17
    12 | 18
     1 | 19
    14 | 20
     3 | 21
     5 | 22
     1 | 23
     9 | 24
     4 | 25
     3 | 26
     1 | 27
     1 | 28
     3 | 30
     1 | 32
     2 | 34
     2 | 36
     1 | 78



On a side note, I've started a process using your last batch of recent 61 39's, as "seed".
It produced 4 new 39's "early" ... via 38's ... but nothing since then. You'll probably produce those too.
Now it's on a run of 37's and 38's that seems like it won't stop before my internal cache of 250000 puzzles fills up -- 1139 38's and 29873 37's so far. Added: that stopped shortly after my writing, with 1195 38's in all, and no more 39's
By way of contrast, your previous 14 39's, lead to only 666 puzzles in the 37-39 range, including the two 39's.

[dobrichev]

Hi,

I updated the mirror of the high-clue puzzles collections here, including the 38-givens.

The results for 39's were:...

So, for the 39s distribution by such classification gives nothing and for 38s probably nothing, right?
I mean inspecting distribution by, say, (MD5(puzzle) mod 413) would result in similar result.

The {-5+5} distribution until closure for the latest 63 39 is

Code: Select all

#clust   #puz
16   1
13   2
1   4
1   8
1   9

with details below

Hidden Text: Show

Code: Select all

pass {-5}, src=63, children=35901310
32 clusters.
   .........................12..1..3..4..5.4167846.785.31.2..3..6.5.3.26.4764.517.23   =2
.....1..2....2..3......314...1.5..6..6784251.58..1627..7623845.4....57..8...74326
.....1..2....2..3......314...1.5..6..7684251.58..1627..6723845.4....57..8...74326
   .........................12..1..3..4.45267.3137..41.65.3..8..4..54726.837...34526   =9
.......12.....134....23......3..5.6..6742853.58..6327..5..1.6...7684215.8...56427
.......12.....134....23......3..5.6..7642853.58..6327..5..1.6...6784215.8...56427
.......12.....134....23......5..6.3..7342865.68..5327..5784216..6..1.5..8...65427
.....1..2....2..3....3..14...15...6..6784251.58.1.627..5...36...7628435.8..6.5427
.....1..2....2..3....3..14...15...6..6784251.85.1.627..7625438..8...36..5..6.8427
.....1..2....2..3....3..14...15...6..7684251.58.1.627..5...36...6728435.8..6.5427
.....1..2....2..3....3..14...15...6..7684251.85.1.627..6725438..8...36..5..6.8427
.....1..2....2..3....3..14...5..36...7.6.54288.627435..571.628.1.854276.6..7...1.
.....1..2....2..3....3..14...5..36...7.6.54288.627435..571.628.1.874256.6..5...1.
   .......................1.23.....4.56..5.768.268..52.47.5..6..3.16..3547883.417.65   =2
.............12.34..15...26...4...78.7.8.13.28.32.7.41..7..8.6.3.6145.875.87.6413
.............12.34..15...26...4...78.732.8.418..7.13.2..8..7.6..36145.87.578.6413
   .......................1.23..2..4..5..5.12.676817.5.421.45.36783...4..5.5.81.6.34   =4
........1..2.1345614.256..7.....7....27.3158441.528.73....65..8...3...6..6..72345
........1..2.1345614.256..7.....7...2.7.3158441.528.73....65..8...3...6.6...72345
........1..2.1345614.526..7.....7....27.3158441.258.73....65..8...3...6..6..72345
........1..2.1345614.526..7.....7...2.7.3158441.258.73....65..8...3...6.6...72345
   .......................1234....56123..6.4.5...351274.6.5.7126.86.8..5...7.1.6835.   =1
........1.....2.3...3145.26...456.78..8..7.1..478.1653.7.2.83653.6.74.828.2.....7
   ......................12.34.....5..6.152.6347.6314.5...2..58.63.5.6.14.8.8642.75.   =1
...........1..2.34.3.451.26...2.7.65..6.45.7..7561.4.2..713.2.8.83.24..71..7.8.43
   ......................12345....67..3..6...57.37.25.4.651...673.6.3.7125474..2.6.1   =1
........1..2.13.45.614.2.3......74...28534..77..26...3..5.26.74..67.53..2.73415.6
   ....................1.23.45.....4..6.17.36.24.462.71.8.68.7.25.1...6..877.5.82.61   =2
...........1.23.45.42.5..36..6....7..27.85.644.8.7651..1.5...87.7483..51.8..174.3
...........1.23.45.42.5..36..6....7..27.85.644.8.7651..1483..57.7.5...81.8..174.3
   ....................1.23.45.....6..7..2.5748..78412.56....6..7..36.745.8.875316.4   =2
...........1..2.34.2345..16..75...8..85.2764161284..57.38.7.1...7.....681.6.8..73
...........1..2.34.2345..16..75...8..85.2764363284..57.18.7.3...7.....683.6.8..71
   ....................1.23.45.....6..7..6.5248..82417.56....7..6..37.645.8.685317.4   =1
...........1..2.34.3245..16..75...8..85.2764332684..57.18.7.3...7.....686.3.8..71
   ....................1.23.45.....64...1734..56.4651273...8..75...62.3587..7.28..63   =2
...........1..2.34.3245..61..7..4...2135.8.4754.27..18..8..5..6.258.6.7336.72..8.
...........1..2.34.3245..61..7..4...24.57..185132.8.47..8..5..6.258.6.7336.72..8.
   ....................1.23.45.....64...1734..56.6451273...8..75...26.3587..7.28..63   =2
...........1..2.34.3245..16..7..4...2135.8.4754.27..81..8..5.6..258.6.7336.72...8
...........1..2.34.3245..16..7..4...24.57..815132.8.47..8..5.6..258.6.7336.72...8
   ....................1.23.45..6..4..7.1..7.56.73..6..14.68.37451.7.48.6..1.3.56.78   =1
...........1.23.45.234..6.1..7..4.68..8.721.4.1.8..7...725.8416.8624.5.71.5..7.8.
   ...................12.34.56.....1..7.476...2.12547..68.74.18.3.23.746.8558.3.....   =1
...........1.23.45.24.6173..43..8..6.68.32.5.2.5...38..8...751.1.7.8..6.452.1687.
   ...................12.34.56.....3..7.27158.43.387..512..6.81.75.71..54...8.47..6.   =1
...........1.23.45.241.5.36..75......15832.67236.7..58.4..17.8..723.8..41..2...73
   ...................12.34.56.....5..7.2734...53...7614..63.87..1.71423.6828..6..7.   =1
...........1.23.45.24.51.36....3...7.721.5..31
Total time 170.300 seconds.
...6745..16.78..4.47512.682.8..6.7.
   ...................12.34.56.....7..8.2786..45.8524367..3..8.7...783.2.642...76.8.   =1
...........1.23.45.2456.31....7..8...4863..57.76.85134.1...8.7..8735.4.14...7.58.
   ...................12.34.56.....75...573.2.611.3.5672..34..817..71.43.822...7..4.   =2
........1..2.13.4.13..452.6..3..7....27.3146.61.4.27.3.5..7..2..76..85.428..546.7
........1..2.13.4.13..542.6..3.7.....27.3146.61.42.7.3.5...7.2..76.8.5.428..456.7
   ...................12.34.56.....758..58.61.7272..586.4.83.7.4..17.64.82.2...8...7   =2
...........1.23.45.245.6.13....67..8.762.8.518.235..67.47.3..8.1.8.7253.2..8...7.
...........1.23.45.245.6.13....67..8.762.8.548.235..67.17.3..8.2..8...7.4.8.7253.
   ...................12.34.56.....78...28....67.74.68.25.317..68..876.15.22...83.71   =2
........1..2.34.5.36.51..24..7.8....62.37..1883.....72.53..8.472.6.43.8548.75.2..
........1..2.34.5.36.51..24..7.8....62.37..1883.....72.73..8.452.6.43.8748.75.2..
   ...................12.34.56..1..5..7.2..7..68.7826.5.1..36.8.7..4..57.83.8734.6.5   =2
..........12.34.56.351.62.4.....7..8..7.18462..846..7..7364.8...8.7.1..52.1.83.47
..........12.34.56.351.62.4.....7..8..7.18642..864..7..7346.8...8.7.1..52.1.83.67
   ...................12.34.56..1.43.67.3...75.1.765.134...8.2..7..274.56.8.6..784..   =1
...........1.23.45.24.613.7.48.167..2.7..8..661.2.74.8.8..3.5..1...85..44.51.28.3
   ...................12.34567........8..8.23.54135.48.26.87....1.32..7..855.1.82.73   =1
........1..1.23.45.52.4163........7..73264.58825.17.6..37.8....21..3..8.5.8.7231.
   ...................12.34567.....3.58..8.1.6.4.56.48.31.87......52..81.766.147..85   =2
...........1.23.45236.457.1....86...6.235.4.88.32.4.76.24..8..71.7.328.43.8...1..
...........1.23.45236.457.1....86...6.235.4.88.32.4.76.27..8..41.4.328.73.8...1..
   .................1....23456..2........7.4256.56..37.42.75.8..146...1.87.8.1.74625   =1
...........1.23456452.61..3....7..84.7..8.6..8..1.67.5..3..7.48.47.385615.8.1.3.7
   .................1..1.23.45.....6..7.162.7.5..275...16..83...64.64.52.3813.6.4.72   =8
..........12.34567.345.6182.......4..48.652.11...48.75..1.8375..7365..18.8......3
..........12.34567.345.6182......84..48.652.11...48.75..1.8375..7365..1..8......3
..........12.34567.345.6281.......3..83.651.22...83.75..8.....4.2..4875..4765..28
..........12.34567.345.6281......83..83.651.22...83.75..8.....4.2..4875..4765..2.
..........12.34567.345.6812......4...48.65.211...487.6..1.8367..7365.1.8.8......3
..........12.34567.345.6812......48..48.65.211...487.6..1.8367..7365.1...8......3
..........12.34567.345.6821......3...83.65.122...837.6..8.....4.2..4867..4765.2.8
..........12.34567.345.6821......38..83.65.122...837.6..8.....4.2..4867..4765.2..
   .................1..1234.56....27.63.3...8..77.236...4.18.4..7.2.781..4554..72.1.   =1
...........1.2345645..16.27....62.84.84.5.6.26...8..7..48.357.11.7..8..553...1.48
   .................1..1234.56..7.28..5.3247..688..3...7..73.8...41..7...8.2.8543.17   =1
..........12.34.56.3516.24...1.7.4....8.26.17.2741.86..7..83..4.8.6...7.2.374.68.
   .................1.12.34.56........76.8.175.27...82614..6.2..7.17...5..82.4.73165   =1
........1..1.23456.24.517.3..6......4..37..6553..86.74.4..3..1..63.185471.5..7..2
   .................1.12.34.56.....3..7.83.2746.72.54...8..1.72.8..384...7.27.3.86.4   =1
........1.12.34.5.53..162.4..3.7.8.5.856.3.2772..8.4...5...7...27.36.5.83.1..8.72
   .................1.12.34.56..1..3.78.7..8...43.8.47.1..37..8.6515.3.6.878...751..   =2
..........12.34.56.35.2641...1.78.6..27.6.18..8...2..7..8.43.71.736..8..1.4.876.3
..........12.34.56.35.621.4..1.27.68.2..8.7...87..6..1.53..86.7.786.531.1...73.8.
   .................1.12.34.56..1.57..8.27.8.1.5.8...2.7...8.43.17.735..8..1.4.7853.   =2
..........12.34.56.35.2641...1.67.8..27.8.16..8...2..7..8.43.71.736..8..1.4.786.3
..........12.34.56.35.621.4..3..7.8..58.4...7.7..834.5.21.7..48.8741.52.5...287..

#clust   #puz
16   1
13   2
1   4
1   8
1   9

I think these clusters are disjoint from those for the first 350 39s but this isn't proven. I can't calculate clusters for the whole set with my tool since the intermediate results exceed the available 64GB RAM.

On the other hand, besides just making the UA change and checking for a valid, minimal puzzle,
I also check to see if, for example, a 39, with a UA change, produces a valid but non-minimal puzzle,
that can (still) be reduced to a minimal 37 or 38 -- one that I can feed back into the testing.

I have no serious arguments, but such seed is probably cancerogenous, leading to the detached non-unique space.
The first pass produced 127560 multiple-solutions minimal 39s, 238 such 40s, and no 41s. I don't intend to use them as seed.
Contrary, the single shots in the opposite direction you are doing could be useful.

I started second pass from about 1.6e9 minimal multiple-solution 35s. It will take weeks to complete.

[dobrichev]

Distribution of 413 39s by pattern

Code: Select all

#patterns  #39 puzzles
      1     9
      1     8
      1     7
      3     6
      1     5
     21     4
     11     3
     58     2
    133     1

Distribution of 131511 38s by pattern

Code: Select all

#patterns  #38 puzzles
      1      78
      1      37
      4      36
      3      34
      2      33
      3      32
      3      30
      5      28
      3      26
      5      25
      7      24
      2      23
      7      22
      6      21
     11      20
     12      18
      8      17
     59      16
     21      15
     35      14
     10      13
    168      12
     13      11
    131      10
    132       9
    517       8
    150       7
   1267       6
    647       5
   4470       4
   4047       3
  20154       2
  36601       1

The one with 78 puzzles has crazy row & column (but not box) distribution

Code: Select all

.........
..1.11.11
.11..1.11
........1
.1..11.11
.11.11111
..1.11.11
.1..11.1.
.1111..11

Does 78 puzzles per 38-clue class become familiar?

[blue]

On the other hand, besides just making the UA change and checking for a valid, minimal puzzle,
I also check to see if, for example, a 39, with a UA change, produces a valid but non-minimal puzzle,
that can (still) be reduced to a minimal 37 or 38 -- one that I can feed back into the testing.

I have no serious arguments, but such seed is probably cancerogenous, leading to the detached non-unique space.
The first pass produced 127560 multiple-solutions minimal 39s, 238 such 40s, and no 41s. I don't intend to use them as seed.
Contrary, the single shots in the opposite direction you are doing could be useful.

I wasn't talking about multi-solution puzzles there, and so I'm a little confused by your comments.

BTW: the "opposite direction" shots have proved useful, working from your ~10K new 38's.
I have 4+19=23 additional 39's now, that I haven't posted -- all based on your good work.

I started second pass from about 1.6e9 minimal multiple-solution 35s. It will take weeks to complete.

I'm surprized that it's so many. Did you fold 36-1's, 37-2's back in, in addition to 38-3's ?
I would enjoy reading some time, some details about how you're dealing with such a large volume of puzzles.

Does 78 puzzles per 38-clue class become familiar?

Yes ! It splits into two "twin-connected" classes with sizes 50 and 28.
There's at least one bridge that's of the kind of transformation that I was talking about.
The first one I found (with code), used a 9-cell UA, where 8 are clue cells.
Bridges with smaller UA's may exist as well.

[dobrichev]

...
I wasn't talking about multi-solution puzzles there, and so I'm a little confused by your comments.

I simply was wrong, ignore it.

BTW: the "opposite direction" shots have proved useful, working from your ~10K new 38's.
I have 4+19=23 additional 39's now, that I haven't posted -- all based on your good work.

Very good news! Congratulations!
I have no time to experiment with code and stake on old software and methods. I am not surprised at all that you with your innovative and smart methods found new 39s. The good surprise is their growth rate of 6% per day.

I started second pass from about 1.6e9 minimal multiple-solution 35s. It will take weeks to complete.

I'm surprized that it's so many. Did you fold 36-1's, 37-2's back in, in addition to 38-3's ?

Yes I did and maybe will suffer from that.
My initial goal was to do {-5+5} on 39s till closure, and {-4+5} on 38s till closure.
Then, as always happens, I found that it is sin to ignore 37s ... and even 36s.
It is well known that search around 39s will be most productive, followed by 38s, and then 37s.
I can't predict when 38s will close at {-4+4}, especially when, like in weather prediction, the problem continuously receives external data/noise from inconsistent methods like twins checking.

I would enjoy reading some time, some details about how you're dealing with such a large volume of puzzles.

Will do it soon.

Does 78 puzzles per 38-clue class become familiar?

Yes ! It splits into two "twin-connected" classes with sizes 50 and 28.
There's at least one bridge that's of the kind of transformation that I was talking about.
The first one I found (with code), used a 9-cell UA, where 8 are clue cells.
Bridges with smaller UA's may exist as well.

I rather asked whether your cluster consists of puzzles with the same pattern or this is coincidence.
I am interested in your results on classifications and "bridges" which I have no s/w to do myself. If it is easy to explain of course.
For example I did {-5+5} on the 39s partitioning them by clusters; and processed a cluster at once this saving some work on the intermediate repetitive rediscoveries.

[blue]

Does 78 puzzles per 38-clue class become familiar?

Yes ! It splits into two "twin-connected" classes with sizes 50 and 28.
There's at least one bridge that's of the kind of transformation that I was talking about.
The first one I found (with code), used a 9-cell UA, where 8 are clue cells.
Bridges with smaller UA's may exist as well.

I rather asked whether your cluster consists of puzzles with the same pattern or this is coincidence.

It does. I should have mentioned that first.

I am interested in your results on classifications and "bridges" which I have no s/w to do myself. If it is easy to explain of course.
For example I did {-5+5} on the 39s partitioning them by clusters; and processed a cluster at once this saving some work on the intermediate repetitive rediscoveries.

I'm not sure if or how any of this would apply to the {-5,+5} thing (not that it couldn't), or if I'll be reviewing a bunch of stuff that you already know, but ...

Have a look at these two Wikipedia pages

Connected component (graph_theory)
Disjoint-set data structure

What I was calling a "twins-connected class", would be a closed component of an edge graph, where verticies are canonicalized (minimal) puzzles, and there's an edge between two verticies when one puzzle can be transformed to the other by applying a "twins" operation (that produces a minimal puzzle) and canonicalizing the result. Two puzzles would be in the same class, (same "connected component") iff they are connected by a path of edges like that.

Also, have a look back near the middle of this post. The "merging of classes" that I was talking about there, corresoponds to the Union operation that's discussed in the "disjoint-set data structure" (wiki) page.

About "bridges": If a new type of edge is introduced into the graph (e.g. corresponding to a new way of producing one puzzle from another), then each class/"connected component" for the new graph, ends up being a (disjoint) union of classes from the original graph. What I meant by a "bridge", would correspond to one of the new-style edges that connects a pair of puzzles that were in different classes/components in the original graph (but now, due to that edge, are now in the same class, in the modified graph). Unfortonately, to identify one, you need to know which classes the puzzles were in, in the original graph.

[ There's a different kind of graph that can be produced in that situation. It has classes/components from the original graph as verticies, and an edge between verticies exists, whenever one of the "bridging" edges from above, connects a pair of puzzles, one from each vertex's (defining) class -- a "multi-lane bridge", possibly. A union of the classes corresponding to the verticies in a connected component of that kind of graph, is a connected component of the modified original graph, and vica-versa. ]

[dobrichev]

I'm not sure if or how any of this would apply to the {-5,+5} thing (not that it couldn't), or if I'll be reviewing a bunch of stuff that you already know, but ...

For {-5,+5} partitioning by criterion "two puzzles are connected if they have a common morph with 5 givens removed" takes about 15 minutes calculations and 53 GB of RAM for the 350 39-clue puzzles using this years old code

Hidden Text: Show

Code: Select all

#include <stdio.h>
#include <map>
#include <set>
#include "ch81.h"

using namespace std;

struct ch81RefList;

struct clSizes : public map<int, int> {};

struct ch81RefList : public set<ch81RefList*> {
   const ch81 * cluster;
   ch81RefList() {
      cluster = NULL;
   }
};
struct clusters : public map<ch81, puzzleSet> {};

struct ch81WithReferences : public map<ch81, ch81RefList> {};

void setCluster (const ch81 * theCluster, ch81RefList &references) {
   for(ch81RefList::iterator p1 = references.begin(); p1 != references.end(); p1++) {
      if((*p1)->cluster == NULL) {
         (*p1)->cluster = theCluster;
         setCluster(theCluster, **p1);
      }
   }
}

void clusterize () {
   puzzleSet m0; //given list, using old code to read, then copy to list.
   ch81WithReferences list, m1; //given & minusN list
   clusters cl; //clusters' details
   clSizes cs; //cluster sizes
   
   //load the whole list
   m0.loadFromFile(stdin); //load puzzles
   for(puzzleSet::const_iterator p = m0.begin(); p != m0.end(); p++) {
      list[*p]; //add keys with empty lists with references to {-N} children
   }
   m0.clear();

   //first pass: do {-1} and store references
   for(ch81WithReferences::iterator p = list.begin(); p != list.end(); p++) {
      ch81 puz = p->first; //structure copy
      int t;
      for(int i = 0; i < 81; i++) {
         //apply {-1}
         if(0 != (t = puz.chars[i])) {
            puz.chars[i] = 0; //clear the given
            //apply {-2}
            for(int j = i + 1; j < 81; j++) {
               int tt = puz.chars[j];
               if(tt) {
                  puz.chars[j] = 0;
                  //apply {-3}
                  for(int k = j + 1; k < 81; k++) {
                     int ttt = puz.chars[k];
                     if(ttt) {
                        puz.chars[k] = 0;
                        //apply {-4}
                        for(int l = k + 1; l < 81; l++) {
                           int tttt = puz.chars[l];
                           if(tttt) {
                              puz.chars[l] = 0;
                              //apply {-5}
                              for(int m = l + 1; m < 81; m++) {
                                 int ttttt = puz.chars[m];
                                 if(ttttt) {
                                    puz.chars[m] = 0;
                                    /////
                                    ch81 puzCanon;
                                    subcanon(puz.chars, puzCanon.chars);
                                    ch81RefList &puzInChildren = m1[puzCanon]; //add canonicalized puzzle in children
                                    puzInChildren.insert(&(p->second)); //add reference to source
                                    p->second.insert(&puzInChildren); //in source, add reference to child
                                    /////
                                    puz.chars[m] = ttttt; // restore the {-5} given
                                 }
                              }
                              puz.chars[l] = tttt; // restore the {-4} given
                           }
                        }
                        puz.chars[k] = ttt; // restore the {-3} given
                     }
                  }
                  puz.chars[j] = tt; // restore the {-2} given
               }
            }
            puz.chars[i] = t; //restore the {-1} given
         }
      }
   }
   printf("pass {-5}, src=%d, children=%d\n", (int)list.size(), (int)m1.size());
   //second pass: obtain clusters
   for(ch81WithReferences::iterator p = m1.begin(); p != m1.end(); p++) {
      ch81RefList &parents = p->second;
      const ch81 *theCluster = parents.cluster;
      if(theCluster == NULL) {
         theCluster = &(p->first);
         setCluster(theCluster, parents);
      }
   }
   //third pass: order clusters
   for(ch81WithReferences::const_iterator p = list.begin(); p != list.end(); p++) {
      puzzleSet &ps = cl[*(p->second.cluster)];
      ps.insert(p->first);
   }
   printf("%d clusters.\n", (int)cl.size());
   //fourth pass: output
   for(clusters::const_iterator p = cl.begin(); p != cl.end(); p++) {
      char buf[81];
      p->first.toString(buf);
      printf("\t%81.81s\t=%d\n", buf, (int)p->second.size());
      cs[(int)p->second.size()]++;
      for(puzzleSet::const_iterator pp = p->second.begin(); pp != p->second.end(); pp++) {
         pp->toString(buf);
         printf("%81.81s\n", buf);
      }
   }
   //fifth pass: output sizes
   printf("\n#clust\t#puz\n");
   for(clSizes::const_iterator p = cs.begin(); p != cs.end(); p++) {
      printf("%d\t%d\n", p->second, p->first);
   }
   cl.clear();
   cs.clear();
   printf("done\n");
}


Seems we are aligned in the understanding of partitioning.

One thing still confuse me. You use phrases "xxx transformation followed by a Sudoku isomorphism" and "xxx transformation followed by an (optional) Sudoku transformation". Do both phrases simply mean that you always map the result to the class representative in order to match classes?

How you keep the results? Are you use puzzles with tags, or database, or separate lists for each class with metadata for bridges?

Do you think that identifying a significant "bridge" could help later, say by digging deeply around it?

[dobrichev]

I would enjoy reading some time, some details about how you're dealing with such a large volume of puzzles.

The magic is the "merge" algorithm and the GNU tools sort and comm that use it.
I am using sorted files of canonical class representative of puzzles.
For {-n+m} operation the input is a puzzle list which during the reading is canonicalized and duplicates are removed, and the output is a file with mixed-size irreducible canonicalized puzzles of size (-n+1) up to (-n+m) tagged with the number of givens. I use the number of givens for faster later filtering by "grep" command.
I am setting +m so that eventual 40 or 41-clue puzzles to be found. At the final "+" stages the amount of the irreducible multiple-solution puzzles is small so that large +m don't affect the performance.

For the general 38-clue and 39-clue extension I use {+1} at a time and the following data model.

File family "35.doneup" ... "41.doneup" - ordered list of multiple-solution irreducible puzzles that have been checked "up" by adding one more clue in all possible ways. One puzzle per line followed by <tab> and one unused tag with the number of givens.

File family "36" ... "40" - ordered list of unique minimal puzzles of this size. Same format. I see that the file "40" is empty.

File family "35.doingup", etc. - a temporary file with newly discovered multilple-solution irreducible puzzles that should be checked "up". Prior to processing the large file is split into parts "35.doingup.part00", etc. in order to fit the maximal batch size. For 35 to 36 transfer, for 64 GB of RAM and 32 threads, I use batch size of 150 millions of puzzles which takes about 50 GB RAM, 9.5 hours processing and 15 minutes disk i/o.
The splitting command is "split -d -l150000000 35.doingup 35.doingup.part".
The tool produce ordered list of single-solution irreducible to the stdout and multiple-solution list to the stderr. the outpus is about twice as the input (12.75 GB input, 23 GB + 0.3 GB output).
Then the results are merged and here the ordered list takes place. The "sort" command has "merge" option and in this mode no sotrting is performed but several already sorted files are merged into one, optionally with duplicates removed. The command is "sort -m -u file1 file2 > outfile". Some Windows implementations of this command work well only when 2 input files are given and use temporary copies when more than 2 files are given. Linux implementation performs well doing all the job in one pass.
Then the results are split into "36.new" and "36.doingup" by removal of already processed/known puzzles from previous passes. Again the ordered lists take place. The command "comm" takes 2 ordered files and depending on parameters outputs puzzles from either first, second, or both files. It does it in one pass. In our case, "comm -13 knowns justfound > new" subtracts knowns from justfound producing new.
This the seed for the next step "up" is prepared.
The newly found unique puzzles are involved to "twin" check and {-n} transformations for the next pass.
For each size, the already used seed is subtracted from the newly obtained seed. At early stages (35s, 36s, and even 37s) a very small portion of the seed is duplicated so keeping strict track what has been already processed is probably not necessary. Repeating some amount of the processing could take less time than comparing the huge files, even on single pass.

This is the script for the 35 to 36 transfer that is currently running.

Hidden Text: Show

Code: Select all

/dobrichev/tools/plus1/plus1 < 35.doingup.part00 > 35.doneup.part00.ss 2> 35.doneup.part00.ms
/dobrichev/tools/plus1/plus1 < 35.doingup.part01 > 35.doneup.part01.ss 2> 35.doneup.part01.ms
/dobrichev/tools/plus1/plus1 < 35.doingup.part02 > 35.doneup.part02.ss 2> 35.doneup.part02.ms
/dobrichev/tools/plus1/plus1 < 35.doingup.part03 > 35.doneup.part03.ss 2> 35.doneup.part03.ms
/dobrichev/tools/plus1/plus1 < 35.doingup.part04 > 35.doneup.part04.ss 2> 35.doneup.part04.ms
/dobrichev/tools/plus1/plus1 < 35.doingup.part05 > 35.doneup.part05.ss 2> 35.doneup.part05.ms
/dobrichev/tools/plus1/plus1 < 35.doingup.part06 > 35.doneup.part06.ss 2> 35.doneup.part06.ms
/dobrichev/tools/plus1/plus1 < 35.doingup.part07 > 35.doneup.part07.ss 2> 35.doneup.part07.ms
/dobrichev/tools/plus1/plus1 < 35.doingup.part08 > 35.doneup.part08.ss 2> 35.doneup.part08.ms
/dobrichev/tools/plus1/plus1 < 35.doingup.part09 > 35.doneup.part09.ss 2> 35.doneup.part09.ms
/dobrichev/tools/plus1/plus1 < 35.doingup.part10 > 35.doneup.part10.ss 2> 35.doneup.part10.ms
sort -m -u 35.doneup.part00.ss 35.doneup.part01.ss 35.doneup.part02.ss 35.doneup.part03.ss 35.doneup.part04.ss 35.doneup.part05.ss 35.doneup.part06.ss 35.doneup.part07.ss 35.doneup.part08.ss 35.doneup.part09.ss 35.doneup.part10.ss | comm -13 36 - > 36.new
sort -m -u 35.doneup.part00.ms 35.doneup.part01.ms 35.doneup.part02.ms 35.doneup.part03.ms 35.doneup.part04.ms 35.doneup.part05.ms 35.doneup.part06.ms 35.doneup.part07.ms 35.doneup.part08.ms 35.doneup.part09.ms 35.doneup.part10.ms | comm -13 36.doneup - > 36.doingup


Note on the last 2 lines that I am saving one huge file write/read using pipe. The "-" parameter is not well documented for the "comm" command but it works just like in "sort" command.
The next commands I will execute (after seeing everything goes OK) are to merge files 35.doneup with 35.doingup into 35.doneup.

Code: Select all

sort -m -u 35.doneup 35.doingup > 35.tmp
rm 35.doneup
rm 35.doingup
mv 35.tmp 35.doneup

For stripping the tags (when I have to merge files with tags with files w/o tags) the command is "cut -f1 infile > outfile".
For adding tag with puzzle size and for sorting a possibly unsorted file, I relaxed "minus n" tool to accept 0 as parameter. In this mode it reads, canonicalizes, removes duplicates, and writes ordered puzzles with size tag.

It is paradoxical but the solver I used for {+} operations works only with pencilmarks, zeroing them when the cell is solved, and never knows the solution. The "twin" testing is done on another machine.

[dobrichev]

Thanks to the server from the Intel's Lab, the next load of 54 39-clue puzzles is ready.

Code: Select all

.............12.34.12.356.7..6..8....841.67.317.3248.6.2..4...8.482.137.7.1.834..
.............12.34.13..4562..7.834.5..87.5.2.53.241.78.7..382...8...7.4.35.42..87
.............12.34.13..4562..7.834.5..87.5.2.53.241.87.7..382...8...7.4.35.42..78
.............12.34.13.4562.....7..8...7.382.6.8625174...81.437..3..27..8.71.834.2
.............12.34.13.4562.....7..8...7.382.6.8625174...81.437..31.874.2.7..23..8
.............12.34.13.4562.....7..8...7.832.6.8625174...81.437..3..27..8.71.384.2
.............12.34.13.4562.....7..8..6825174..7..382.6..7.23..8.31.874.2.8.1.437.
.............12345134.5...2..6....7.37..6185441..876.325..7643.6.3.....774..2.5.6
...........1..2..3.23.45.61.57..4.1831..78.5.8...5.3.7.7...3.861.5.8.7.26.8.27135
...........1..2..3.23.45.61.57..4.1831..78.5.8...5.3.7.7...3.86135.8.7.26.8.271.5
...........1..2..3.23.45.61.57..4.3813..78.5.8...5.1.7.7...1.863.5.8.7.26.8.27315
...........1..2..3.23.45.61.57..4.3813..78.5.8...5.1.7.7...1.86315.8.7.26.8.273.5
...........1..2.34.2536.71...462.87..87..3..62.6.871.3.1824.36..4.....8.6.2.384.1
...........1..2.34.34.51.26.13.78..2.8251..737.52.3..8.481.5.67.5..87.4.1.7....8.
...........1.23.45.23.4516..37.8265..8536..7.2....78...5823..17.72....8.31..785..
...........1.23.45.24.153.6.1736...8.831526..2.6.87....782.....14..387..6.2.71.8.
...........1.23.45.24.6.31...7.8.1...38.7...414.35.78..83..7.5141253.87.7.5..84..
...........1.23.45.24.6.31...7.8.1...38.7...414.35.87..83..7.5141253.78.7.5..84..
...........1.23.45.2451..36..243..67.7...8..44.627.38...5.8..7...73.2.582.875.6.3
...........1.23.45.2451..36..243..67.7...8..44.627.38...53.2.78..7.8..5.2.875.6.3
...........1.23.4523..45.16..2..7.8..7.3.2.543..48...7.86234.711.3.78.6.72.1....8
...........1.23.4523..45.16..3.7..8..2.5.8..77..23..54.12.87.6.37.1....88.6352.71
...........1.23.4523.4.5.16..3.76.8..2853..676..2.8.5..1.35...8.82.61..43..8.2.71
...........1.23.45263.4517..253...8.1.8.57...637.84.51.76..281.312.78...8.......7
...........1.23.45263.4517..352...8.1.8.57...627.84.51.76..281.312.78...8.......7
...........1.23456.24561..7..2.157.8.8.67.12.1.7.82645..6.385.4..8.......4..568..
..........12.34.56.34.56712.....852...8.7...12.1..3.87.83.6..751.73.5.6862..8.1..
..........12.34.56.34.56712.....852...8.7...12.13.5.87.83.6..751.75.3..862..8.1..
..........12.34.56.34.56712.....852...8.7...12.15.3.87.83.6..751.73.5..862..8.1..
..........12.34.56.34.57218........1.46..852.12.54.6.7..1.758.2.7..8.16.28..6..75
..........12.34.56.35.72148........5.63.8.41.5.14.36.7..7..856..5..478.11.8..6.74
..........12.34.56.3516.4.2....7...8.28.13.67.73.8624..576..8.4.8....62.2...48.75
........1.....2.3....13..24..5.6..1..17248.56.6.51.472.7.8....353.62..8768.37.245
........1.....234..2314..5.....6..7..327.416..6728143...84.56...5682..142...1658.
........1.....234..2314..5.....6..7..3728146..627.413...84.56...5682..142...1658.
........1.....234..23154..6.....7....3752641..524187.3.7..6.....962451.72.57.16..
........1....2345..24.15.36.47.82.633.2..6.488...3.....75..8.1.2.8.51..441.2.7.85
........1....2345..24.15.36.57..8.1.28..51..44.12.7.85.72.86.4334...2.688...3....
........1..1.23.4.24..513........6...52.367.46.4.72.35.7..1..6.1.8.674534.638.1.7
........1..1.2345.4521.6.3......71..1.723...52.56.174...8....7.52.7.83.47.43.25.8
........1..1.2345.4526.1.3......71..1.723...52.51.674...8....7.52.7.83.47.43.25.8
........1..1.2345.462.51.73...23..48.8.......2.4.86.35.48.12.671.7.6..8.62...8.14
........1..1.23456.42.51.3...4.1657.1...7....7.63.51.4.6...2...2.8.673454.75..6.2
........1..1.23456.42.51.3...4.17..51...6..7.7.63.51.4.6...2...2.8.765434.75..6.2
........1..1.23456.42.513....4.17..51...6.7..7.63.5.14.6...2...2.8.765434.75...62
........1..1.23456.42.513....6.1574.1...7....7.43.6.15.6...2...2.8.675344.75...62
........1..2.13.45.134.56....7..85...2853.7.4.3..47.86.7.35..68.86.74...3.18..4.7
........1.12.34.5..36.17.24..83...1..51.482..32..51.68.6347..82.8..6....2.5..3.46
........1.12.34.5..36.17.42..34.5.27.7..8....26..73.84.28.514...5134..783....8.1.
........1.12.34.5..5361.427..71...8..81.4...252.87.1.4.3...7....783..2452..48.7.3
........1.12.34.5..5361.427..81......71.4.28.52.87.1.4.3...7....873..5422..48.7.3
........1.12.34.5..651.23.4..6.47.8..2.3.8..7.7826..43.578.3.1..8.4..7..2.1.7..38
........1.23.14.5..165732....8.......714358.22...81.75.67.485...8.3.6..73.2.57..8
........1.23.1456.1.456723...162.8..3.2.81..68....51....8.5...4.37.486.54....638.


[dobrichev]

23 more. All 490 are closed up to {-5,+5}.

Code: Select all

..............1.23124.3..56.17.8...52.8..4..743517..68.71.43.823....8.7.8.271..34
..............1.23124.3..56.17.8...52.8..4..743571..68.71.43.823....8.7.8.217..34
...........1..2.34.3245.1....67..8...2..65.47.7382461..17.46..8.652.8.7128..7..6.
...........1.23.45.426...31..5.....7.6753.48242..67.53.14.76..8.76.5..1425..1..7.
...........1.23.4524..516.3..4.378.676...8.5.8.2.65.74.2..8...71.8.7.4..47..125.8
...........1.23.45243.5..16....7..8.1..3.2.573.784..6243.28..717.2..4..881..37..4
...........1.23.45243.5..16.2..7..8.1..3...573.784..6243.28..717.2..4..881..37..4
........1..1.23.4..25.61.3........7..52.76.83.87342..5.18.347...7.2.8.142.4.17.58
........1..1.23.4..52.61.3........7..25.76.83.87342..5.18.347...7.2.8.142.4.17.58
........1..1.23.45.251.43.........6..572.6..3.623485.7.1..72.36.7643.1..2..6.17.4
........1..1.23.45.521.43.........6..263475.8.852.6..3.1..82.36.6843.1..2..6.18.4
........1..1.2345.24.16.37..26...78.4..67....5.7.326.4.5231.84.1.428..3.8...5.1..
........1..1.2345.24.16.37..26...78.4..67....5.7.326.4.5421.83.1.238..4.8...5.1..
........1..1.2345.24.16.73..26...87.4..67....5.7.326.4.5421.38.1.238..4.8...5.1..
........1..1.2345.245.6137...4....8..872.65.452...8..7.186.2...452.1786.7...8.1..
........1..2.13.4..51.26.3........7..25.67.83.87342..5.18.347...7.2.8.142.4.71.58
........1..2.13.4.13..542.6..6..5.7.32..7.8..7...38.62..8342.17.1..674.8.7..8162.
........1..2.1345..41.26.3........7..24.67.83.87352..4.18.357...7.2.8.152.5.71..8
........1..2.3456.35.16.42......3.7.23..8715687..162.3.4..75.125.....7..72..416..
........1..2.3456.35.16.42......3.7.27..861.383..17256.4..75.125.....7..72..416..
........1.12.34.5..36.5124...1..758..2..18.74.78.4.1.2..74..8...8...3.252.3.854.7
.....1..2.12...34..35.42.16..7.2.....287.5.3435.4.8.27.6.1......81.564735....4.61
.....1..2.12...34..35.42.61..7.2.....287.5.3435.4.8.27.6.1......81.564735....4.16

[eleven]

Great work, Mladen,
I never thought, that mankind would find so many real 39's.

[dobrichev]

Thank you!

[dobrichev]

Code: Select all

...........1..2..3.23.45.16....748.1.78..634.1.4..8.27.17.2.5.8.82.5713.3.5.8..72
...........1..2..3.23.45.16....748.1.78..634.1.4..8.72.17.2.5.8.82.5713.3.5.8..27
...........1..2..3.23.45.16.17.2.5.8.82.5713.3.5.8..27.3..7.8.1.78..634.1.4..8.72
...........1..2..3.23.45.16.17.2.5.8.82.5713.3.5.8..72.3..7.8.1.78..634.1.4..8.27
...........1..2..3.23.45.16.17.2853..82.5.1.73.5.7..28.3..8.7.1.78..634.1.4..7.82
...........1..2..3.23.45.16.17.2853..82.5.1.73.5.7..82.3..8.7.1.78..634.1.4..7.28
...........1..2.34.32154.67..52.6..8..8.15.7.21.4.8.56.83.2..4.1..8.3..552..41.83
...........1..2.34.32154.67..52.8..6..8.15.7.21.4.6.58.83.2..4.1..8.3..552..41.83
...........1..2345.32.45.67.......1..14.2857332..51.86..3.8..54.4651..38.8...46.1
...........1..2345.32.4567........1..14.2856332..5178...3.8.45..4751.83..8...41.7
...........1.23.45.2.14..63..2.17.86.8.4.2..717.38...4..8....7.254.71.3871.83..52
...........1.23.45.2.14..63..2.17.86.8.4.2..717.38...4..82...7..54.71.3871.83..52
...........1.23.45.245.6.31.......7..172.5.644.26.715...3..8.17.783....61467.2.83
...........1.23.45.245.6.31.......7..172.5.644.26.715...3.68.17.783.....1467.2.83
...........1.23.45.245.6.31.......7..172.5.644.26.715...3.68.17.783....614.7.2.83
...........1.23.45.245.6.31.......7..176.2.544.27.516...3..8.17.783....61462.7.83
...........1.23.45.245.6.31.......7..176.2.544.27.516...3.68.17.783.....1462.7.83
...........1.23.45.245.6.31.......7..176.2.544.27.516...3.68.17.783....614.2.7.83
...........1.23.45.246.5.31.......7..172.6.544.25.716...3..8.17.783....61467.2.83
...........1.23.45.246.5.31.......7..172.6.544.25.716...3.68.17.783.....1467.2.83
...........1.23.45.246.5.31.......7..172.6.544.25.716...3.68.17.783....614.7.2.83
...........1.23.45.246.5.31.......7..175.2.644.27.615...3..8.17.783....61462.7.83
...........1.23.45.246.5.31.......7..175.2.644.27.615...3.68.17.783.....1462.7.83
...........1.23.45.246.5.31.......7..175.2.644.27.615...3.68.17.783....614.2.7.83
...........1.23.4524..561.3....37.81.8.5..7..1.7.68534.14.823.77....5.188...7.4.6
...........1.23456452.16.73..8.3.5.4.241...385...8.71..8.....4.2.5..8..7347.61.85
...........1.23456452.16.73..8.3.5.4.241...385...8.71..8.....4.2.7..8..5345.61.87
...........1.23456452.61.73..8.3.5.4.241...385...8.71..8.....4.2.5..8..7347.16.85
...........1.23456452.61.73..8.3.5.4.241...385...8.71..8.....4.2.7..8..5345.16.87
..........12.34.56.3516.24...174.8...27.1846..8...6.7...867.3.4.73..168.1...83.2.
..........12.34.56.3526.174....8..4..284.76.554..26.81...8...6..5..73..8.836..517
........1..1.2345..42.15..3.....67...6..72..47.4.5..6..1658.3.73.7.6184.4.8.37.16
........1..1.2345..42.51..3....6.7...6..72..47.4..5.6..165.83.73.7.1684.4.8.37.16
........1..1234.56.625.173...835..6..236.85.76....2.83.1.8......8..2..152.91.5.78
........1..2.1345..41.25..3.....67...6..72..47.4.5..6..1658.3.73.7.6184.4.8.37.16