SudokuX (Diagonal Sudoku)

For fans of Killer Sudoku, Samurai Sudoku and other variants

Re: SudokuX (Diagonal Sudoku)

Postby Leren » Tue Dec 10, 2019 3:38 am

Hi Tarek,

Perhaps you can send me your email address via the forum system and I can send you the list. The text file is about 34 MB but an Excel file is smaller at 13 MB, so you can nominate your preferred format.

BTW The "official" count is currently 422,969 but is rising slowly (about 700/week) as I have been forced into a [+-3] search for new puzzles. I'll never actually finish this (or even make a serious dent in it), but I'm hoping that the supply of new puzzles will just dry up and I can use a statistical argument to say that I have "probably" found "most" puzzles.

Leren
Leren
 
Posts: 5117
Joined: 03 June 2012

Re: SudokuX (Diagonal Sudoku)

Postby tarek » Thu Dec 12, 2019 7:54 pm

With that variants enabled Sukaku explainer I've attempted to rate large collections of known variants and the 12 clue sudoku X is a good example of that. Thanks to Leren for supplying the 400,000+ puzzles that are still being updated.

I managed to iron out a few issues with the help of the puzzle variety but there were very tough puzzles in the collection that unfortunately will make me curtail the full task and possible re-rate with the nested chains disabled.

Here are some monster puzzles rated. The 1st part shows the options used with sukaku explainer. The puzzle will be followed by the ER/EP/ED the the short description of these techniques with that ER/EP/ED rating
Code: Select all
v1.14.2 2019-12-10 S t0 T X1 B2
The following techniques where set and unset:
01, Set   Hidden Single
02, Set   Direct Pointing
03, Set   Direct Hidden Pair
04, Set   Naked Single
05, Set   Direct Hidden Triplet
06, Unset Pointing & Claiming
07, Set   Generalized Intersections
08, Unset Naked Pair
09, Set   Generalized Naked Pair
10, Set   X-Wing
11, Set   Hidden Pair
12, Unset Naked Triplet
13, Set   Generalized Naked Triplet
14, Set   Swordfish
15, Set   Hidden Triplet
16, Set   Scraper, Kite, Turbot
17, Set   XY-Wing
18, Set   XYZ-Wing
19, Set   WXYZ-Wing
20, Set   Unique Rectangle / Loop
21, Unset Naked Quad
22, Set   Generalized Naked Quad
23, Set   Jellyfish
24, Set   Hidden Quad
25, Unset 3 Strong-linked Fishes
26, Set   Generalized Naked Quintuple
27, Set   VWXYZ-Wing
28, Set   Bivalue Universal Grave
29, Unset 4 Strong-Linked Fishes
30, Set   Aligned Pair Exclusion
31, Unset 5 Strong-Linked Fishes
32, Unset 6 Strong-Linked Fishes
33, Set   Forcing Chains & Cycles
34, Set   Aligned Triplet Exclusion
35, Set   Nishio Forcing Chains
36, Set   Multiple Forcing Chains
37, Set   Dynamic Forcing Chains
38, Set   Dynamic Forcing Chains (+)
39, Set   Nested Forcing Chains
 techs111110101110111111110111011101001111111 f%g;%r/%p/%d;%S/%T/%U;%e o-
.......................1..2.......3.1..2.4.........56.7.......8.3.....5..........;12.0/1.2/1.2;DCFC+DFC/HS/HS;12m45s
.......................1..21................345.6..............2.3.........7..86.;12.0/1.2/1.2;DCFC+DFC/HS/HS;7m47s
......................1................2.3.45..1..6...34......2....78............;12.5/1.5/1.5;DCFC+DFC/HS/HS;46m41s
......................1...2....2.....3.....4..5..6.......4.7...8.......1.....3...;12.5/1.2/1.2;DCFC+DFC/HS/HS;40m16s
......................1...2....3.....4.....5.....6.......5.7.8.2.......1.....4...;12.0/1.2/1.2;DCFC+DFC/HS/HS;6m30s
......................1...2....3.....4.....5..6..7.......8.5...2.......1...4.....;12.5/1.2/1.2;DCFC+DFC/HS/HS;23m54s
......................1..23....4.....5.....6.....7.......6.8...3.......7.....5...;12.2/1.2/1.2;DCFC+DFC/HS/HS;10m23s



The solver has been attempting to solve for hours the following puzzle but hasn't given up yet which is at odds with the longest time required by by one of the 12.5 puzzles. It will be either a very difficult monster puzzle or there is some bug in the code
Code: Select all
......................1.2.............1.....3......1..45.........32.6........78..


tarek

[Edit: the above post is incorrect as v1.14.2 had a bug. Disregard all the ratings]
Last edited by tarek on Thu Dec 12, 2019 10:36 pm, edited 1 time in total.
User avatar
tarek
 
Posts: 3762
Joined: 05 January 2006

Re: SudokuX (Diagonal Sudoku)

Postby creint » Thu Dec 12, 2019 9:23 pm

Tarek you probably have bugs, because those are all lower than SE 10.0. (or 10.4)

Here are my timings:
Code: Select all
.......................1..2.......3.1..2.4.........56.7.......8.3.....5..........;0.7 seconds easy chains
......................1................2.3.45..1..6...34......2....78............;0.7 seconds easy chains
......................1...2....2.....3.....4..5..6.......4.7...8.......1.....3...;3.9 seconds harder chains
......................1...2....3.....4.....5.....6.......5.7.8.2.......1.....4...;3.4 seconds harder chains
......................1...2....3.....4.....5..6..7.......8.5...2.......1...4.....;2.5 seconds hard chains
......................1..23....4.....5.....6.....7.......6.8...3.......7.....5...;0.6 seconds easy chains

......................1.2.............1.....3......1..45.........32.6........78..;2.1 seconds hard chains
creint
 
Posts: 393
Joined: 20 January 2018

Re: SudokuX (Diagonal Sudoku)

Postby tarek » Thu Dec 12, 2019 10:35 pm

creint wrote:Tarek you probably have bugs, because those are all lower than SE 10.0. (or 10.4)

Here are my timings:
Code: Select all
.......................1..2.......3.1..2.4.........56.7.......8.3.....5..........;0.7 seconds easy chains
......................1................2.3.45..1..6...34......2....78............;0.7 seconds easy chains
......................1...2....2.....3.....4..5..6.......4.7...8.......1.....3...;3.9 seconds harder chains
......................1...2....3.....4.....5.....6.......5.7.8.2.......1.....4...;3.4 seconds harder chains
......................1...2....3.....4.....5..6..7.......8.5...2.......1...4.....;2.5 seconds hard chains
......................1..23....4.....5.....6.....7.......6.8...3.......7.....5...;0.6 seconds easy chains

......................1.2.............1.....3......1..45.........32.6........78..;2.1 seconds hard chains

Thanks creint,
The bug was found in the chain search algorithm which is fixed ….
This now means I can re-rate the entire collection hopefully.
I'll report back here

Tarek
User avatar
tarek
 
Posts: 3762
Joined: 05 January 2006

Re: SudokuX (Diagonal Sudoku)

Postby tarek » Fri Dec 13, 2019 1:44 pm

Problem fixed and the entire collection (422969 puzzles) successfully rated. The hardest was rated at 10.6

The following are the puzzles that had a first elimination rated at >=3.0
Code: Select all
................1.23..................4.5..........3....6.45.........7...2....8..   9.5/9.5/4.3   DCFC+/DCFC+/g2SK11
....................1.234.......5...67.8...........2...6.....7.....1.............   8.4/8.4/4.3   RFC/RFC/g2SK11
...............1....2.3............41.3..5..........2...6..3.........7..........8   7.9/7.9/4.3   NFC/NFC/g2SK11
...............1....2.3............4..3.5...........6776.....4.......8...........   9.0/9.0/4.0   DCFC/DCFC/HT
...............1....2.3............4..3.5...........6746.....7.......8...........   8.9/8.9/4.0   DLFC/DLFC/HT
................1..2.1.3....3......45...6............7.7..............8..4.......   7.6/7.6/3.0   NFC/NFC/gNP


The following are the only 10.0+ rated puzzles
Code: Select all
.................1.2.............34.1.5........6.........5......3........7..8...5 10.6/1.2/1.2 DCFC+FC/HS/HS
....................1.2.3........4.....5.6..7....8....65.....8.....3............. 10.4/1.5/1.5 DCFC+FC/HS/HS
................1...2...3.....4..5...1........6.....7........6...8........43..... 10.4/1.5/1.5 DCFC+FC/HS/HS
.................1....2.3..14...........5....67.........2...8.......6......4..... 10.3/10.3/2.9 DCFC+FC/DCFC+FC/gI
................1.2....3....4........1..5..4..................67.....3.2....8.... 10.1/2.9/2.9 DCFC+/gI/gI
....................1.2...3.....4..5........16..7..2.............3............87. 10.1/1.5/1.5 DCFC+/HS/HS
....................1.2.3..............4.5.67..2........8.........7.4....6....... 10.1/1.5/1.5 DCFC+/HS/HS
....................1..23......4......3...1......5...65..................7..8..4. 10.0/4.3/2.9 DCFC+/g2SL00/gI
....................1..23......4......3...1......5..........6......7.....4..8..7. 10.0/2.9/2.9 DCFC+/gI/gI
................1.....2..........3.....4.5..67.8......64..............2..5....... 10.0/2.9/2.9 DCFC+/gI/gI
.............1....2.......3.............456....7......3......52.1..8............. 10.0/2.9/2.9 DCFC+/gI/gI
.......1.........2..3..4....5.............3..2...6......74.............6...8..... 10.0/2.9/2.9 DCFC+/gI/gI
....................1.2.3............4.5.6.7...3.....1.5.....6..............8.... 10.0/1.5/1.5 DCFC+/HS/HS
.................1........2.......3...41.2..........5.........6.3..57......8..... 10.0/1.5/1.5 DCFC+/HS/HS
.................1..2.3.............1..4.5..6..7........3.8.7..6................. 10.0/1.5/1.5 DCFC+/HS/HS
................1...2.3...4........5......6..17.8............8...3........4...... 10.0/1.5/1.5 DCFC+/HS/HS
...............1....2.3.................4..5.16.........52.....7.....6........8.. 10.0/1.5/1.5 DCFC+/HS/HS


Solver version & options
Code: Select all
v1.14.3 2019-12-12 S t0 T X1 B2
The following techniques where set and unset:
01, Set   Hidden Single
02, Set   Direct Pointing
03, Set   Direct Hidden Pair
04, Set   Naked Single
05, Set   Direct Hidden Triplet
06, Unset Pointing & Claiming
07, Set   Generalized Intersections
08, Unset Naked Pair
09, Set   Generalized Naked Pair
10, Set   X-Wing
11, Set   Hidden Pair
12, Unset Naked Triplet
13, Set   Generalized Naked Triplet
14, Set   Swordfish
15, Set   Hidden Triplet
16, Set   Scraper, Kite, Turbot
17, Set   XY-Wing
18, Set   XYZ-Wing
19, Set   WXYZ-Wing
20, Set   Unique Rectangle / Loop
21, Unset Naked Quad
22, Set   Generalized Naked Quad
23, Set   Jellyfish
24, Set   Hidden Quad
25, Unset 3 Strong-linked Fishes
26, Set   Generalized Naked Quintuplet
27, Set   VWXYZ-Wing
28, Set   Bivalue Universal Grave
29, Unset 4 Strong-Linked Fishes
30, Set   Aligned Pair Exclusion
31, Unset 5 Strong-Linked Fishes
32, Unset 6 Strong-Linked Fishes
33, Set   Forcing Chains & Cycles
34, Set   Aligned Triplet Exclusion
35, Set   Nishio Forcing Chains
36, Set   Multiple Forcing Chains
37, Set   Dynamic Forcing Chains
38, Set   Dynamic Forcing Chains (+)
39, Set   Nested Forcing Chains
 techs111110101110111111110111011101001111111 f%g;%r/%p/%d;%S/%T/%U;%e o-


The 2.9 rating is a new technique (generalized intersections) which will be available in the next release of Sukaku Explainer
User avatar
tarek
 
Posts: 3762
Joined: 05 January 2006

Re: SudokuX (Diagonal Sudoku)

Postby ton » Wed Apr 01, 2020 1:59 pm

My 12-clue list.
https://xfs.jp/RKVKnI
ton
 
Posts: 8
Joined: 26 October 2010

Re: SudokuX (Diagonal Sudoku)

Postby m_b_metcalf » Wed Apr 01, 2020 2:08 pm

ton wrote:My 12-clue list.
https://xfs.jp/RKVKnI

Sorry, but this link is to a site in Japanese with no list in sight.

Mike Metcalf
User avatar
m_b_metcalf
2017 Supporter
 
Posts: 13621
Joined: 15 May 2006
Location: Berlin

Re: SudokuX (Diagonal Sudoku)

Postby 1to9only » Wed Apr 01, 2020 2:30 pm

m_b_metcalf wrote:
ton wrote:My 12-clue list.
https://xfs.jp/RKVKnI

Sorry, but this link is to a site in Japanese with no list in sight.
Mike Metcalf

- Click on ton's link
- Click on the red link
- Click on the button
- Download sudoku-x-12.zip (5.06 MiB)
User avatar
1to9only
 
Posts: 4177
Joined: 04 April 2018

Re: SudokuX (Diagonal Sudoku)

Postby m_b_metcalf » Wed Apr 01, 2020 6:37 pm

Thanks.

I ran the first 10,000 through SukakuExplainer, and here's a selection of the results (in SE order):
Hidden Text: Show
Code: Select all
000000000000000000000000001000000000002034050060700000010600000000800020000000400   ED=1.5/1.5/1.5   
000000000000000000000000001000002300000040560000007000180000002000060000000003000   ED=1.7/1.5/1.5   
000000000000000000000000000000001000023004000000005067000000704080300000600000000   ED=2.0/1.5/1.5   
000000000000000000000000000000001000002000003400005000000000400560000071000280000   ED=2.3/1.5/1.5   
000000000000000000000000001000102000030000450000006000750000000000000000000000382   ED=2.5/2.0/2.0   
000000000000000000000000001000000000002030040056000000710000005000000080900700000   ED=2.9/2.9/2.9   
000000000000000000000000001000000000002030400506000000040070805170000000000000000   ED=3.0/2.3/2.3   
000000000000000000000000001000020000000030000045600700010000000208000090000400000   ED=3.2/1.5/1.5   
000000000000000000000000001002000000003040500060000000010000000000300782500000000   ED=3.4/2.3/2.3   
000000000000000000000000001000002003000000004500060000070000000001000600000500820   ED=3.6/3.6/2.9   
000000000000000000000000001002340000000500000060000780500000004000007800000000000   ED=3.8/1.2/1.2   
000000000000000000000000001000001000000002000030000400050630700200000060800000000   ED=4.0/4.0/2.9   
000000000000000000000000001000010000000002000304000500060000700000030480020000000   ED=4.1/2.9/2.9   
000000000000000000000000001000020000340000005000060000000004000000010200700000680   ED=4.2/2.9/2.9   
000000000000000000000000001000200304000000000567800000000010060000000000700000080   ED=4.3/3.6/2.9   
000000000000000000000000001000002000000103000040000056000000027000000308060000000   ED=4.4/2.9/2.9   
000000000000000000000000001002000000003040050600030000100700006050000080000000000   ED=4.5/2.9/2.9   
000000000000000000000000001000023000405000006000007000006000000001000000000008430   ED=4.6/2.9/2.9   
000000000000000000000000001002000000000030400015000000000000002000000605700480000   ED=5.0/2.0/2.0   
000000000000000000000000001000000000000000203400500000000620000000400007010000058   ED=5.2/1.2/1.2   
000000000000000000000000001000000023400050000600000000000000700000000540032800000   ED=5.4/1.2/1.2   
000000000000000000000000001000020000034050060000070000820000000100000007000006000   ED=5.5/2.9/2.9   
000000000000000000000000001000000000002030040506070000080000200170000005000000000   ED=5.6/2.9/2.9   
000000000000000000000000001000020000034050060000070000100000080000008032000000000   ED=5.7/2.9/2.9   
000000000000000000000000001000100000002000300004005000500000000600000200000030780   ED=6.2/1.2/1.2   
000000000000000000000000000000102003000040000506000007000000000014080000000000650   ED=6.3/3.4/2.9   
000000000000000000000000001000000002003040500067000000010000007500000086000000000   ED=6.4/2.9/2.9   
000000000000000000000000001000000002003040000005000000010000035400000000600078000   ED=6.5/2.3/2.3   
000000000000000000000000001000000002003040000050600000010502000000000030000700080   ED=6.6/1.5/1.5   
000000000000000000000000001000000002000300000000010000004506700200008000030000090   ED=6.7/2.9/2.9   
000000000000000000000000001000000020000000003000400050015000060700000000000830040   ED=6.8/2.9/2.9   
000000000000000000000000001000020304000000500607001000000000000030500800000000006   ED=6.9/1.5/1.5   
000000000000000000000000001000203040005000000006000000100007000000068500400000000   ED=7.0/7.0/2.9   
000000000000000000000000001000000200010304000000050600007020800000000003040000000   ED=7.1/7.1/2.9   
000000000000000000000000001000000000000000230100045000000600004060200700000000008   ED=7.2/3.4/2.9   
000000000000000000000000001000020000000300000400000560010000003000070002800006000   ED=7.3/2.9/2.9   
000000000000000000000000001000200034000000000560700000000000000003080600004000500   ED=7.4/3.4/2.9   
000000000000000000000000001000200000300000450000006000000000070004050800012000000   ED=7.5/2.9/2.9   
000000000000000000000000001000200003004000005006000000100040000500000607000000800   ED=7.6/4.3/2.9   
000000000000000000000000001002000000003040050600000000100000706050300080000000000   ED=7.7/2.9/2.9   
000000000000000000000000001000000000002000340000056000103000000000700480050000000   ED=7.8/2.9/2.9   
000000000000000000000000001000000000002030400005060000000007200800000040010500000   ED=7.9/2.9/2.9   
000000000000000000000000001000010200034000050000060070005087000000000006000000000   ED=8.0/2.9/2.9   
000000000000000000000000001000002030040000000000300050050000000607800000010000020   ED=8.1/1.2/1.2   
000000000000000000000000001000020000030000400000050000000406700502000108000000000   ED=8.2/3.4/2.9   
000000000000000000000000001000000000000020340500000000600030000107000005000840000   ED=8.3/7.7/2.9   
000000000000000000000000001000200000304050600007000000010000070000000000000060840   ED=8.4/2.9/2.9   
000000000000000000000000001000020030004000000005060000010000026000004070000008000   ED=8.5/3.4/2.9   
000000000000000000000000001000002003000000004500600000000700000001000800000009650   ED=8.6/1.5/1.5   
000000000000000000000000001000010020304000050000060000007000000000080006200000030   ED=8.7/1.5/1.5   
000000000000000000000000001000000000000000023450600000000000000007080500000002460   ED=8.8/3.4/2.9   
000000000000000000000000001000200003405000000600001000070300000000000000800000540   ED=8.9/3.4/2.9   
000000000000000000000000001000020000030000004000560000000000000602000700000308009   ED=9.0/4.0/2.9     
000000000000000000000000001000000000230000004500670003000000060000000800041000000   ED=9.1/2.9/2.9   
000000000000000000000000001000002300004000000050600000000506000701000080000003000   ED=9.2/1.5/1.5   
000000000000000000000000001010000000000020030040000000050000000000230060074080000   ED=9.3/1.2/1.2   
000000000000000000000000001000002340100000000500000000020000000030000000006780005   ED=9.4/1.5/1.5   
000000000000000000000000001000020000034000050001060000200000070050000000000810000   ED=9.5/1.2/1.2   

Curiously, there was no 1.2/1.2/1.2.
Last edited by m_b_metcalf on Thu Apr 02, 2020 3:32 pm, edited 2 times in total.
User avatar
m_b_metcalf
2017 Supporter
 
Posts: 13621
Joined: 15 May 2006
Location: Berlin

Re: SudokuX (Diagonal Sudoku)

Postby Leren » Thu Apr 02, 2020 5:59 am

ton's 12 clue puzzle list contains 543,985 ED puzzles, about 110,000 more than my paltry list of about 434,000 puzzles. But it's not all there is. My list contains the following 964 puzzles not in ton's list.

Hidden Text: Show
...................................12...3...........45.4...........6.27..15..8...
...................................12.3..4..........56.......7.85..1..........34.
...................................12.3..4.........5.6........7.6..1......4...83.
................................1..22......34...56....7..3.8.........6........1..
...............................1......2...3.4....5....16.......5....7......2..7.8
...............................1...2..34.5...6.......7..............31..7...2...8
...............................12......3......45....6.........2.......437.8.6....
..............................12...3..4..5....6.....7.1..78..........4........6..
..........................1............1.2...3.....45....5....2....6...75....8...
..........................1............23......4....5.12....6...6...7......5.8...
..........................1...........2.3..4.5......6..34.........7..1.5.8.......
..........................1..........23.4..5....6......1..2........5..3...7...8..
..........................1........2...........345.........2........6.3..78...54.
..........................1........2..2.3.4.....5.............61.6.....78...4....
..........................1........2.2......3...4..........3.......567..1.....48.
..........................1.......123...4....5.........2....5....1.......67..8...
..........................1.......2......3......4.5.6.51..7....3............8..4.
..........................1.......2......3......4.5.6.51..7....3...........8...4.
..........................1.......2....1.3........4.....2.5.......67.3...4......8
..........................1.......2..3...1......2...4...4..........5...67...8...3
..........................1.......23.451.......1.....6...........7...8..2...6....
..........................1......1........23.45...6.....3.7....8...........1....5
..........................1......2....1.........3.45..........6457..2...........8
..........................1......23.4...........3..5..61........2..........78...4
..........................1......23.4.....5.....26......3...7......1...6........8
..........................1......23.4..5.6.......4..7.1.........3..........82....
..........................1.....1...2.3....4.....56..........72......8...6..4....
..........................1.....2.....3......4.....5.3...6........13....78.....2.
..........................1.....2...1.......3.4......5..6....7.....8.1........62.
..........................1.....2.3........4.5..6..........7..5......6....3.18...
..........................1.....2.3..1......456...7..8..3.........4.............6
..........................1.....2.3.14.5..........6.....3.........47........8.5..
..........................1.....23............45.....667.........1...........382.
..........................1.....23..........4...56.......7...2...38......41......
..........................1.....23...45.....6...78........1.............8.....7.5
..........................1.....234........5...1..6......57.....3.8.......6......
..........................1....2......3....4.....51.........6.5........27.84.....
..........................1....2....3.....45.6..71.....4....8.7............3.....
..........................1....23.4.1........56......7..2..4.......1............8
..........................1...2........3.....14....5......167..8.........23......
..........................1...2....34........56........1.........2...6.......758.
..........................1...2....34.......56....3....1.....7...2...8........6..
..........................11..2.....3........4......5..2..61.........78........4.
.........................12..........3..........4..5......23.......1..6.7.....84.
.........................12........34.....5.6.....1....7....3.........2.8...4....
.........................12.....3.............2.....4.41.........5......6...7.3.8
.........................12....1....3.............24.5.......3..6...57........8..
.........................12..3...4.....1...2..5..6......7...3.6.8................
........................1.................2....3.45......2........6.17...48....3.
........................1................2..34....5......41....3..6......78....3.
........................1...............2.3.1.4.......56.....47...78...........5.
........................1..............2...34.5.....6.....517..3.4....8..........
........................1.............23.4.........5........623....5....7...81...
........................1............2....3......45.1.4.6......7...........1..28.
........................1............2...3.1......4.......52...6.......37.8.....6
........................1..........2...3.4.........5..3......4.....1.6.7.84......
........................1..........2...3.4.........5..4......3.....1.6.7.83......
........................1..........2.3.4.5.........6.7.............61....4.....38
........................1..........2.34....5....2...........6........7.1.854.....
........................1..........21.3.4......5......62......7...51.....8.......
........................1........2.......3........45....6.7...8....15.....3....6.
........................1........2.....3.4..5....6....532....4.....7........8....
........................1........2...1...........34...3..5.....6.4....1....7..8..
........................1........2...2....3.4.....5.....2.6....5.6....7.........8
........................1........2...3...4.........5.....1....36.......4..758....
........................1........2.3..4......5....26..62........7..........8...4.
........................1.......2..3..4.....5.167..............3.8...........17..
........................1......2......3.......4..5..6....4.13....7....8..5.......
........................1......2......3...4......5........63..257.......8....1...
........................1......2......3...4......5.6...........5......27.8.1.6...
........................1......2......3..4....1....5.6......7..8.4........2....3.
........................1......2.....31...45.....6....2..17...........3........8.
........................1......2...3...........4...5...2.....67.8...3......1.4...
........................1......2...3.......4.15.67.....2..8......3...........4...
........................1......2...3....4....1...........15.6.7..3.......8.....2.
........................1......2...31.....4.5.6...7......5............8.3..1.....
........................1......2..34......5...6..7..............8.4.6.........27.
........................1......2.3............4..56...........6...7...4.3.81.....
........................1.....2.............3..4....5.2.....6....3.4........5.78.
........................1.....2..........3....4....5.....1.6.....25......78....2.
........................1.....2........3.....45....6.1....5......2....73.....8...
........................1.....2........3....45............15...6...7.....28....4.
........................1.....2.......3...........4.255...6......2...3......7..8.
........................1.....2......1....3..3...45.6..4.7.......5....8..........
........................1.....2.....34............5..66..............4......3178.
........................1.....2...34....3...51.6.......4.........5.........7..8..
........................1.....2...34..5.......61..............74.3..8.........6..
........................1.....2.34..56....3...7...........7......2.....18........
........................1.....21...............3.....4...356...2....7...81.......
........................1.....23.....1....4.54.........3.......6.2............78.
........................1.....23...4.15......6...7...........5.......6.....48....
........................1.....23...4.5.......1.........3.......42............678.
........................1....2.........3......4.5....2.6..........71.....83....6.
........................1...1...............2.3.4.....25.......6.....7.8....3.4..
........................1...2....3.4.....5...3....6......7.....6.8.........31....
........................1.2............2.3..4....5.......6.17....5.......83......
........................1.2..........3.2...........45...5.1........4..67.......8.
........................1.2........34...5....6.......7......4....78...6.......5..
........................1.2.......3.......4...5...6.....3.4....6...7...8........5
........................1.2......31....4..5......6....47......6........8.....3...
........................1.2.....3.....4......5...............3.4.2.5........6.78.
........................1.2.....3....4......5...67...........6.......3...5.28....
........................1.2.....3...4.......5...67...........3.......6.....782...
........................1.2...3......4.2......15...67.....6...........8.....1....
........................1.2..2.3...............4.2.........567...3........8.4....
.......................1.........2........1.34..56.....7.....6......3......1.8...
.......................1.........2...3....4.1.5.67.....1.......8....4..........6.
.......................1........2...........3..3.....4........5..673....8.....12.
.......................1........2....3....4.....54............2.......61.74.8....
.......................1........21............3.4.............41.5.6.......7...38
.......................1......2.3...........4..4....5........63......1...785.....
.......................1....2........3..2.......4....5.....674........3..85......
.......................1....2.....3......4........5.2.......6.1....7....8..23....
.......................1..2.........3...4..5..........67......1....8..3..21......
.......................1..2.........34..5........6...1......47...........821.....
.......................1..2.......3......2.......4.56...6.....7......8.13........
.......................1..2......34..5...3..........6...46.........7...8........5
.......................1..2...2.....3.....4......56......1...........3.74......8.
.......................1..2...2....3..4.....1.5.6............5...7............68.
.......................1..2.32......1...........45.6............7.3..8........5..
.......................1..2.345......5...........2..6.......4.......7...6....8...
.......................1..23...4..........5.......561.........3.....7..86........
.......................1..23.2.4......5...1........67...........7.....8......5...
.......................1..23.45.........62..7.......1...........7.......8.....3..
.......................1..234..........5.6..78............3.4....5.............8.
.......................1.23.............4.....2.......3......12........56.7.8....
.......................1.23...........4.5..............2.....143..678............
.......................1.23...........42.3..1..1......5.6...7...................8
.......................1.23......4..25...3.........6.7..4.5.8....................
.......................1.23..23........4........5..........6........75..8.....4..
.......................1.23..4.........1.25.......3.....6.4.7......8.............
.......................12......3............4.....5...56..........2..1..7.3.8....
.......................12......3......4..........5...6...1.27..6.......8.5.......
.......................12......3...4..25.........6.........71..4.......5.....8...
.......................12......3...45.2...1....6.7......1...6......8.............
.......................12......3.4...5.6........................7..24....8.....61
.......................12......3.4.15....6.......7......1.......4...........8..5.
.......................12......34....5.....6....1.............5..3.7....8.....1..
.......................12......345.............6.........2.7........5....38....6.
.......................12.....3........4..........5.....6.7...45......8...4.2....
.......................12.....3.......1.4.5....6.........7.8..63.........5.......
.......................12.....3....4..........5.......3........4.2..6.......57.8.
.......................12.....34...............5.6...2.....7.....4....8.1.6......
.......................12.....34.......5.......6...7.1.5.8...4....6..............
.......................12.....34......5...1......6....46..7............78........
.......................12.....34...5.21.....6..7......3......................28..
.......................12.....34..56...7.......1........8........76......6.......
.......................12...1.......................34......5.23.6.7.8........1..
.......................12...13...4.....5........67....7..8....5..4...............
.......................12...3....1...........4...5.......123...6.7...........8...
.......................12...3.45....1..............62...........7.......68......5
.......................12...345..........61...2............7........8....5......3
.......................12.3....4...5.6..3.....2.......7.....6..4.8...............
.......................12.3....4...5.6..3.....2.......7.....6..8.4...............
.......................12.312........4...........5...6......4..7.5........8......
.......................1234.................5...6.....45..........74.6...8.......
......................1............2..3..4..........5.......6...2.57....5.....48.
......................1..........1........2......34.....56.7..........8..12....3.
......................1..........2...3..4..........5.......6...1......37...5.28..
......................1..........2.3.4.5.6....7..4......2...3......8...........7.
......................1..........2.3.4.5.6.7.....4......2...3......8.....7.......
......................1.........2.....3.....45..6............56..7.......18.7....
......................1.........2....1....3.....4.5........6.427.3..............8
......................1.........2....3....4......5.......6.7..25.1.8.....4.......
......................1.........2..3..4.........5....6........2.7......5.1..48...
......................1...2...........3...4.......5...25.6.............1....7.83.
......................1...2........3.3....2.4...5.......6....5.....7..8.......6..
......................1...2.......3.......45.2...67.....53.....................18
......................1...2......13.24.5..........6.....3.7.................8...5
......................1...2.....3.....2.....4...5............3...6...7....1...58.
......................1...2.....3....4..........5.6.7........5.....8...1.2..4....
......................1...2.....3....4.5.6.7......8.........13...........7..2....
......................1...2.....3..4..........5...6..........7.6.2.8..........53.
......................1...2.....3.4.......5.6..278...........3..........4......7.
......................1...2.....34...........5..6..7.....3........5......28.....1
......................1...2....2.....3.....4.....5...........51......2.67..8.....
......................1...2....2....34.....5.....6......7...83........4...6......
......................1...2....34.5.62.......7..........5..1.4..................8
......................1...2....34.5.67.......8..........5..1.4..................6
......................1...2...3....4...........56...7.13.......8..............56.
......................1...2...3.4...........5.6......1..7...84...........2.5.....
......................1..23......4....2...5...1..3......6...7...........4....8...
......................1..23.....4.........5...6.7..4..8........2.1............7..
......................1..23.....45.......67....2................7.1..8........4..
......................1.2................3..4.2.5.....6...7....3.4.....5....8....
......................1.2..............3....4.2.5.....6...7....3.4...5......8....
......................1.2..............3...45..2......341..........67.......8....
......................1.2..............3..1..45......6.17..........8..4...3......
......................1.2..............3.4.....2......35......6....7........21.8.
......................1.2..............3.4..5...6.7....7.........5..........2.81.
......................1.2.............1.3............4..5...63........2..7.8.4...
......................1.2.............134...........56....2.1..........7.5.....8.
......................1.2............21.3............44.......56..7.8....5.......
......................1.2............3............4...54...6.....7.....8....2.13.
......................1.2............3.4.5.....1........6....54........37...8....
......................1.2.......3........4..5.2.......6...7....4.5.....3....8....
......................1.2.......3......4....5.2.......6...7....4.5...3......8....
......................1.2......3...1..........1....4.........53........67..8.4...
......................1.2.....3...............14.....25......6.3.7....8.....4....
......................1.2.....3........4.5.....6...1.7....8........2.....3.....4.
......................1.2.....3......4.....5..2...5...6...7......5....6...8......
......................1.2.....3.4.....5.......67......3...6........7...5......8..
......................1.2...1............3....2...4....5.........6.7......3.8...7
......................1.2...2..........3.4.5..61......78.....4.4.................
......................1.2...2....3.....4.5....6...7...4...6....7..............8..
......................1.2...3.............1...4...5...6.7......1..5...3.8........
......................1.2...3...........4.1...5..............367........8...2..5.
......................1.2.3..............4.5..2.3.....5...6.7......8......4......
......................12.............3....4.....5.........6.3..1...74..........58
......................12...........3...4.......1...........5....6..31...7.....48.
......................123...................2.4.5.....67.......2......8....4...5.
......................123..............4...........2......3.1....456.....78......
.....................1.2.34.............5.6............1.....42....7....8...6....
.....................1.23............4....2......5.......63.4....5.......78......
....................1.....2................13...45..6........4...7.18.........5..
....................1.....2...........3.........4..5......36...5...12....7....8..
....................1.....2...........3.4....56.7...8....6...7............2.....3
....................1.....2.........2.3.........45.....5..62........7...8......4.
....................1.....2.........3...4.1.5.6.78.....2.....8...............5...
....................1.....2........3.4......1...5....6....2.....7..4....8.....5..
....................1.....2........31...4.......5.....23.6............7....8..5..
....................1.....2......3..1...........4...5.....26......78....5......4.
....................1.....2......3.4...5......6............7.......42...6.....58.
....................1.....2.....3........4.....5.....6...61.......2....78.....3..
....................1.....2.....3........4.3..567............8...2.1...........7.
....................1.....2.....3....24.........5.........1.........467.8.....3..
....................1.....2.....3....4......1.5.6....4.....7.3......8.....2......
....................1.....2.....3...3....4......5..6.7.......8...2.1...........5.
....................1.....2.....3..1.4.......3..5...6......7.5...8........2......
....................1.....2.....3.4...5.....6.7..8.........6.....2...........43..
....................1.....2.....345...6.......27........................5....483.
....................1.....2...3......4....5....3..6.......78.54...........6......
....................1.....2...3..41.5........2................5........6..478....
....................1....23.................1...4.5...2..........3...6.....7..84.
....................1....23.............24....567..8..3....................8..7..
....................1....23........1.4......2..34...............5..6.7........8..
....................1....23......1.......4....3...2...............5..6..1..7..8..
....................1...2...............3.4.2.5....6...7......8....2.....8.....5.
....................1...2..........1.3.4...........5...4..67.3.8...............8.
....................1...2..........3..4..........3...........56...7..8..4..2.1...
....................1...2..........3.4...5.67...8......7.....5.............2..8..
....................1...2........1......34.......5....46.....5.7...........3....8
....................1...2........3........1...4..5......3.....6...7.8..5..6......
....................1...2........3.......4........51...4.......6...3........2.78.
....................1...2........3.....4.5.....3...1....6.127..................8.
....................1...2.......3....4...5.....26..7..58.....4.....1.............
....................1...2.......3..1.4.5.......6...7...5.8...4.1.................
....................1...2......3.........4.....5...1...3.....67.....8......2.5...
....................1...2......3........45.....2...6.735......8...4..............
....................1...2......3.....2....41.....5.....6......3..7.....5.....8...
....................1...2......3.....2....41.....5.....6......37.......5...8.....
....................1...2......3.....2....41.....5....6.......3..7.....5...8.....
....................1...2....3.....2...4.5.....6...7...5.....8.....7.....4.......
....................1...2.3..........4.5.6.........3..6.7....5......8..........4.
....................1...2.3......4..5....6.7.......8..67......5.............3....
....................1..2..3...........3.........4..56........4...7..1...6.....8..
....................1..2..3........1........24..5...6...................6..7..58.
....................1..2..3........2.....4.3156.7...............8.............7..
....................1..2..3........4...5........6..........1....7....5......836..
....................1..2..3......4...2....56....73............8.6.......8........
....................1..2..3......45........6......3.........2..74.6.............8
....................1..2..3.....4...........15..6.....7..........3..7..........68
....................1..2..3....3...........4.....5......6..1.........5.72..8.....
....................1..2..3....4....5............3......2..1.6.7.....4.......8...
....................1..2..3....4.3.........2....5.............6.....1...7.....85.
....................1..2..3...31.............4...5......2..6.........5.......78..
....................1..2..3...4....1........256.7....................6.....6....8
....................1..2..3..3.....1........4...5...6........5......4...7......8.
....................1..2..3..3.....4..............4..2.......5..5....6..7......8.
....................1..2..3..3.4...........5....6.7.8...................8..5...6.
....................1..2..3..4.........5........6.....1........7.3.........8..65.
....................1..2..3.23.....4...5........67...........5.............8...6.
....................1..2..3.4......2......3.156.7...............8.............7..
....................1..2..31.....4.5...6........7............7.....1..........68.
....................1..2.34......5.6..............3......76.............7..85....
....................1..2.34.5............3.6......7......6......8.5.......7......
....................1..23..............4..........51...4.67..8............5..3...
....................1..23..............4..........51...4.67..8...........5...3...
....................1..23..............4.......5.6...7..2..1...8.....4.6.........
....................1..23..............4......12...5..67..8...4..............1...
....................1..23.............24.........5...15.........6..7..8......1...
....................1..23........2...4.5..1......6............5.....7..6.....8...
....................1..23......4............5..6.7.....4.......8.7.........6....3
....................1..23......4..........1...4..5......621.7............8.......
....................1..23......4..........2.3.5.6............47..8.............6.
....................1..23......4......3..........5...4..623.7..........8.........
....................1..23....3.4.5...............6.7..46......8............2.....
....................1..23....4.......12.........56...76..........8..1............
....................1..23....4.....1..25........67..8........7...........5.......
....................1.2...3............4........5....1....6........3.7...1....84.
....................1.2...3............4.3.....56...2..78.1...................4..
....................1.2...3...........4.....5...6...........7......3....8.2...61.
....................1.2...3.........4.......5....67..........8.....31...8.....6..
....................1.2...3.........45...6.....2....31..........7....6..8........
....................1.2...3........14....5......6.......3....6...7...8........4..
....................1.2...3.....4......3....2.....5..........46.........6...78...
....................1.2...3.....4......5.6....1.........3............6........748
....................1.2...3.....4.....2.....5.6.7............6...3.........8..7..
....................1.2...3.....4.....2....5....6..7..........8.8..5.4...........
....................1.2...3.....4.....5........26...........7......1....8..4...6.
....................1.2...3.....4.....5........36..7......3..........8.48........
....................1.2...3.....4....5.6.......7...3......8..5................64.
....................1.2...3....14.............5.6...7....5...8............3....6.
....................1.2..34...........2.........5..6..........7..3...5....4.8....
....................1.2..34..........4.........25.........3....6........7.....5.8
....................1.2.3.............4...5.6...78...........83.....1....2.......
....................1.2.3............4...5..6......1........7.........8..5.6...4.
....................1.2.3.4....56...........7.7........8..1..........5.........6.
....................1.2.3.4..2...........56....3......75..8..............6.......
....................1.23..4.....56.73.4.....................85..7................
....................1.23..4...4........5................2.1....6..7..5......8....
....................1.234..........456.7...........2..7......5........8..........
...................1......2.............3.........45...2.6...1.......7.84.5......
...................1......2.............3.4....56..7..2.......1....4.6.........8.
...................1......2............1.3..4..5.......6...........7.85.14.......
...................1......2...........1...3.4...56..........7......2....8.....56.
...................1......2.........3...4..5.........1..6...3.......7....2.1.8...
...................1......2........34...5.6..........7.2......1.........6....48..
...................1......2.......3.......45......6.....3.4....2.......1......7.8
...................1......2......3.1......45.6.7.............1..3.4.............8
...................1......2.....1...3.......4...56...........5..7....6.......2..8
...................1......2.....1...3.......4...56...........5..7....6.....2....8
...................1......2.....3...4.5....6....12...7..............5.4.........8
...................1.....2................3.4.....5...6.......75.2.3..........81.
...................1.....2...............3..14....2...........3....1...56.7.8....
...................1.....2............3.4..5....6......27....13....5..8..........
...................1.....2.........34.......5....6....7.3.........81.6..5........
...................1.....2........3.4.5.6.......7.....2......1.8..45.............
...................1.....2......3.......4....52.....1.......6.37.1.8.............
...................1.....2......3.......4....52.....1.......6.37.18..............
...................1.....23.............4.5...2.......5.6.........1.3...7..8.....
...................1.....23.............4.56..7........3....8............2.1.7...
...................1.....23............1....24.5...6....7.5.8.................5..
...................1.....23...........4..........2..1.......45........6..3.7.8...
...................1.....23..........2.1.......4...5....6...7.........8.5....3...
...................1.....23.........4...5.6...7........3......1....4.....2..8....
...................1.....23......3......4......5........6.7..........2.17...8....
...................1....2.3.......4...........2.5.....6.......14...7......8...5..
...................1....2.3...34..........5...2....61.7..........4........8......
...................1....23........4..5............4.676.........8....5........7..
...................1....23....4......2.5......6.........7......4.3...........81..
...................1....23....4...5..6.........7..3..................1.2..4...8..
...................1...2..3.............4.56...........35.....1....6...7........8
...................1...2..3...........4.5.6........4....7...8.....23..........7..
...................1...2..3.........4...5.67..........13...8..2....7.............
...................1...2..3........4..5.6.7..........2........1..8......5...7....
...................1...2..3........45...6............1..7.8..........1.28........
...................1...2..3........45...6............7......53........8..71......
...................1...2..3......4...3...15......2......46..7...........8........
...................1...2..3......4...3...15......2......67..8...........4........
...................1...2.3..............4.5.6...........5.......6...7......2.8.1.
...................1...2.3............4.5......6........758............3......21.
...................1...2.3........1...4.5..2........63..7...8........5...........
...................1...2.3...453..........62........1............7........8.....6
...................1...2.34.............5.........6...34...1..2....7........8....
...................1...2.34...4........2.1...5.....6....6...7...........8........
...................1..2...3............2.1.....4...5....6.7.8.........5........3.
...................1..2..3......4..........2..5.6..........1..73......8....5.....
...................1..2..34......................4..1...5...6......1....7...3...8
...................1..2..34...............56.....14.....7.......8....7......3....
...................1..2.3.4..............5.6..4...7...7........6.8........5......
...................1.2.3.45.2...........6.7...3....8..........1.........7........
..................12...3.45.............617...............7.8............31......
.................1........2.........3.......4..56..........7.5.2...3......8...6..
.................1........2........3.....4.2..567.......8....7.......5.......1...
.................1........2........3....4....1...5......6...74.2.8..3............
.................1........2........31........4..5...6...7....8.......4....3..2...
.................1........2.......3.......45..67.......1..........8....64...5....
.................1........2.......3...2.........4..........3.5.2.6.......17.....8
.................1........2.......34......5..6....2....35........4...7.8.........
.................1........2......34........5..1............4..6.5...3.....7....8.
.................1........2......34.....5..6..7..........7......6.8....74........
.................1........2......34...15.6........7.......8....3.......5....4....
.................1........2.....3...........4..56.....7......6.8...1........4.5..
.................1........2.....3.4...1.........4...5......6.......7....5.2...8..
.................1........2.....3.4..1....5..2..........56.4...........7......8..
.................1........2....3.45...2......1.....6.....1....7.......8.4........
.................1........2...1....3....4......5.........2.6...2.....5.......78..
.................1........2...3....45........6....4....1.....7...3...8........6..
.................1........2..3..4...........52.......6.........6.1...........378.
.................1.......2................3..4..5..6..123.........7.85...........
.................1.......2.........2....3............4.....563...4.......27..8...
.................1.......2........3.....4.....3.....56......7......7.....283.....
.................1.......2.......3..4.51...........6.7.......8..6...3....7.......
.................1.......2.......3.4......5..6...21.....34...........7.........8.
.................1.......2......3....1.....4.....2......5...6.3.7....5...8.......
.................1.......2.....1.3.4..........52..6.................7.583........
.................1.......2...3........4...5....5..1...62.............4........78.
.................1.......23.........32.............4....4.5.6..1............7.8..
.................1.......23......4........53...6.1.......3....7.5.8..............
.................1.......23......4..1.2.........5..6............7....5....8..3...
.................1.......23...1..............4..5..6..32..7..........5.........8.
.................1.......23...2..4...........4.....5...................456..7.8..
.................1......2...........2...3.....1......4..5.....6........78.3.2....
.................1......2.........34....5...67..8................1...8....6..3...
.................1......2........3.......4........5......36......17...8.7..2.....
.................1......2........3..1....4.........5..67.....4....28.......5.....
.................1......2........3..4...........3..5...6........5....6.....7.1..8
.................1......2......3....4.....5......67...71..........5.28...........
.................1......2.....3..4.......5.....5..6...75...........2........4..8.
.................1......2.3.............4.5....6.........1..7........4...18..3...
.................1......2.3........4....5......2......13...4...6..7...8..........
.................1.....1.........23..1.4.5..........6.......6...........7..83....
.................1.....1.........23..1.4.5........3.6.......6...........7..8.....
.................1.....2..............3..45.......1.....67......7.6..8........4..
.................1.....2.........3.....1..45.6...78.............3.5.............2
.................1.....2........3....1.2....4...5.......6.7..5........8.....4....
.................1.....2........3...4..............5..........36.5.4.....1....78.
.................1.....2........3.2..4..........5...6.........7.....8....18.....4
.................1.....2.......3..4...4...56.....71.....5.......2......8.........
.................1.....2...1..34...........5.......6.....2.7..8.5.......6........
.................1.....2...3.....45.......2..67...1.....5.8.............4........
.................1.....2..3.............4.5.....6..7...4..5.8..3.........2.......
.................1.....2..3.........4.....56......1....2............7..85...6....
.................1.....2..3......4....1............56..2....7.8.4.......5........
.................1.....2..3.....42.......5.........6......7..........8..25....4..
.................1.....2..3...3....4............5...........67.17............8.5.
.................1.....2..3.2........45..6.......3..78......5...........7........
.................1.....2..3.3.............45.......6..2.6..........1...7....8....
.................1.....2..32........3.4.........56..7...........7.8.............4
.................1.....2..32..1............4.......5.6.......78.45...............
.................1.....2.3..43..........5.6........7.....8..4....1........2......
.................1.....2.34..2........1.5.................678....4.......3.......
.................1.....23.....4.....................5..1...5...4.6..........73.8.
.................1.....23.....4...56..37.....2.8............8............4.......
.................1....2....................3..2.4.........5..426.7.1...........8.
.................1....2...............3...45.....1.6..........2.6.4.....7.......8
.................1....2..............3..45....6.......17.6...........48.......2..
.................1....2.............3...4........5...2.1..........6...3..728.....
.................1....2............3........4.567.....2...8..........5.....6..7..
.................1....2.......3...........2..4.........3.....5.6.1..........7.82.
.................1....2.......3.45..6....1...2.....7............7..5....8........
.................1....2.....3...........4...526.........5.............2..7.8...3.
.................1....2....2..1.....3.4......5....6....7...........4.8...6.......
.................1....2....3......4.......56.72...1.....5.......4...............8
.................1....2...3...............4....56..78.1.48.............2.........
.................1....2...3.......4..1...5.........67.........8..1......4..6.....
.................1....2...3.......4.2...5..6........7...2...........1..86........
.................1....2...3......45.......6...1.........6..5...........7.2....8..
.................1....2...3.....4...........2..5......67..3..........5.4......8..
.................1....2...3.....45....6........2.1........3.....7....4........8..
.................1....2...3.....456.......4....1..............2.5.7.8............
.................1....2...3....3..4........5..2.....6.....1...7.......8.6........
.................1....2...3.1............4.5678.............1.........8...4......
.................1....2..3.......4..3..1.2.........5......6.7...4.........8......
.................1....2..34.....5...1.2.....6...7..5.....1....................8..
.................1....2.3...............4............5....642..5..7......81......
.................1....2.3..............45............6....752..6.........81......
.................1....2.3...........45.1.6........72....8.3....................1.
.................1....23......1....4...5...6...2........7.........8...4.3........
.................1...2.3.......1.2...............45..........6..7..8...43........
.................1...2.3..4.......5........6.17...4.....6.......5.8..............
.................1...2.3..4.....5............6.......7..3...85....6........1.....
.................1..1...........2.....3.....4...5...6......7.8..6.............25.
.................1..1...........2...1.......3..4.....5....6.....7....8........27.
.................1..1..........2..........34.....5..6......7..2.3.......6......8.
.................1..1..........2..........34.....5..6......7..5.3.......6......8.
.................1..1.....2...........3.....4...5....6....4..5.....3....7......8.
.................1..1.....2.........3.....45.....62.......7...6.5..8.............
.................1..1.....2.........34.....5......2.........6..7...8....8......3.
.................1..1....23.....45...........6........7........1...3....8.....4..
.................1..1....23.1...45...........6........7............3....8.....4..
.................1..1...2........3...4...........5.....6.....7....1.3......2.8...
.................1..1...2.3.....4....5......2....6.....7.8...6.................4.
.................1..1...2.3....4..........56.....7....4......8.............1.2...
.................1..1.2...3.....4........5....6.........2....4..7..8.5...........
.................1..1.2.3............4.5.6.7................85...........7.1.....
.................1..2...................34..2...5..6..1..7.............4..86.....
.................1..2..................3.1.....4...25.36............7.4..8.......
.................1..2.................3.....4...5..6......78.......1....4.....56.
.................1..2.................3...2.......4...41......5....3.6.7.8.......
.................1..2.................3...2......4....5......146....7..5...8.....
.................1..2................3.4....5..6.......4.....78....32........6...
.................1..2............2...1.......34..1......5..........6...4..7.8....
.................1..2............23.4..5.1........6.....7.....8.........3...7....
.................1..2............3.4..........56........7...65.....14........8...
.................1..2...........3.....4......2.5.....6....1.....7....3...8.....7.
.................1..2...........3.4.........25..67.......8........5..7....1......
.................1..2...........3.4.........5.....1..6....5.............4.7...38.
.................1..2...........3.4.........5.6...1..7....5.............4.....38.
.................1..2...........3.4.........5.6...1..7....5..........3..4......8.
.................1..2...........34..............5..6.72..7............8..1.....3.
.................1..2...........34...........2.1......5..67.......8.....4.....3..
.................1..2..........1.....3....24.....5............6.6.....7..4..8....
.................1..2..........1....3...........45....6........7.....4.58....6...
.................1..2..........3.........1.....4...567....2.4............3....8..
.................1..2..........3..4....5.6...7.....2.8.3.....6..........8........
.................1..2.........1....2.........3..4..........5.......3.....67....48
.................1..2.........1....3........245.6..........7........84.........6.
.................1..2.........3.....4..1.....56...........5.7.4..3........8......
.................1..2........3...........4....25...67.81.......4...........5.....
.................1..2........3...........4..51........45..........36..7..8.......
.................1..2........3.........4.15....6.....7....2...........6..4....8..
.................1..2.......3......4....5..2.67..8.............1.........8...4...
.................1..2.......3..4........5.67.1.....2......8............3......4..
.................1..2.....3.................2...45.........6....7.8..4........75.
.................1..2.....3.............4.....5.6.........31......78....5......6.
.................1..2.....3............1.4.......3.56...........6...7.8....6.....
.................1..2.....3...........1.....4...5.........1.....6.7.85........6..
.................1..2.....3..........23.........4.....5..........6.1....7.....84.
.................1..2.....3........2.........4..56..7...........6.............458
.................1..2.....3........2..1.....45..6............7.....1......3....8.
.................1..2.....3........2..3.....45..6............7.....3......1....8.
.................1..2.....3.......4.......5...1..67.....58......8......6.........
.................1..2.....3.....4..2..........5..............5.6.3.71.........8..
.................1..2.....3....3...........4.52..16............7........8..4.....
.................1..2.....3....3..........4....5.1...........6......2...7..8.4...
.................1..2.....3...1....4.5.......67.8..........5.........6....4......
.................1..2.....3...1....45.............6....6...2................7.58.
.................1..2.....3...1...4..........5...2.67...6..4....................8
.................1..2.....3...3...........45...1...6............5..6...........78
.................1..2.....3...4....2.5..6.....7..................4...8.......57..
.................1..2.....3..4.........1.3.56..7..........7.8..5.................
.................1..2....3...........4.5........6...7.....8..........5.6....2.1..
.................1..2....3.......42..........56.7........6....7..4........8......
.................1..2....3....14.......3.......5...6............5...74......8....
.................1..2...3.................2...1..4...........56.....7.1....8.3...
.................1..2...3..............4.1..5..3.......5......4...6...7..8.......
.................1..2...3..............4.5...263...7....6...........8..5.........
.................1..2...3..............4.5...623....7.1.6..............8.........
.................1..2...3........1....4..........5.....6.....75.....1...87.......
.................1..2...3........4........2......5.....1.....5.6......78.....4...
.................1..2...3......4......56.1....73............7.4.............8....
.................1..2..1..3........4..........5.6...7.3..............58........6.
.................1..2..3...............4.1.....5...26...3.........7.....6...8....
.................1..2..3..........3........2.14..5.......3.6..........7......8...
.................1..2..3..........3....4...2.15..6.........7..........8..7.......
.................1..2..3..........4.......2..5..61...........67.....4........8...
.................1..2..3........2....4...5..6...........3...7.....71........8....
.................1..2..3.......34.....5...6.....7.........1.8..3.......4.........
.................1..2..3.......4..........2...5..1.......3.67.......8........7...
.................1..2..3.......4..........52...67...8.3.......4..........1.......
.................1..2..3.......4.......5.......6......41.....5....2.7........8...
.................1..2..3.......4.2.....5..6......1......3...7......8....7........
.................1..2..3......3....2.................4.1.....5...3......6.....78.
.................1..2..3......45.....................31.6..7.........58.7........
.................1..2..3....4..5...6........2....7.....1.3............8..7.......
.................1..2..3..4........5..........6..7.........8.7..........7.....69.
.................1..2..3..4..3....5................67..8......3.5.6..............
.................1..2..3.4.......5.........2..6.7.................6....7..8..5...
.................1..2..3.45.....6..........2....7..8......1.....8..7.............
.................1..2..3.45.....6..........5....7..8......1.....8..7.............
.................1..2.3.................4.2...5..6.....7...8.15..........1.......
.................1..2.3...............3.4...........56.........1..6..7.8.....5...
.................1..2.3............4....5.6.......3...14.7...8.3.................
.................1..2.3...........3.14...5.........26..1......7.........8........
.................1..2.3..........2..4..5.1.......4......6.7.8..................4.
.................1..2.3..........4......5.......6......5.17...........3.......86.
.................1..2.3.........4...........2...5..6...3..7..........8....1....5.
.................1..2.3......4................5..6......71.8.........6........53.
.................1..2.3...4......5..............6.....3........2.1.7..........85.
.................1..2.3...4.....5.....1....4...4......................3.6..7..8..
.................1..2.3...4.....5..2.........1............4..5.......6.....7..8..
.................1..2.3...4.5...............3.6.7....2....8..........5........8..
.................1..2.3..45......6..5...........6..7..3.............4.........8..
.................1..2.3.4............516...........2........3...7....8...6.......
.................1..2.3.4......1.5....6..........78...7........8..............3..
.................1..2.345.........5....6.1..........4..6......7.......8..........
.................1..23.45.................32..6..7............8........7.......6.
.................1.1......2.3.............45..6....7....5..............34.8......
.................1.1.....2..........3...........4......5....64.......3...2..78...
.................1.1...2..3...........4.5..6..........3.7.....2....6..8..........
.................1.1...2..3........4..5.6............7.......5........8..47......
.................1.2..................3.4.5..............1.6.27......8..5.....3..
.................1.2...............31.........4.2.....3..........5..1.......67.8.
.................1.2...............34.......25..6......7.....8.......5...31......
.................1.2............2...3.4......1..5.....6..........7.6..........85.
.................1.2...........2...34.5.6.7....6............5............1.8.....
.................1.2...........3.2..4.5......1...6.....7........8...7...........4
.................1.2..........3..........1....4...5..63......7...8..........7.2..
.................1.2......3.............4.....1.........5...6.....1.7...8..2.3...
.................1.2......3..........31..2.....4...5....67..8.........5..........
.................1.2......3.........4...5.......6.............27...13.........68.
.................1.2......3.....4............3....1..5.......4.......2....6.7.8..
.................1.2......3.1...........4..5..3......................6.27.8.5....
.................1.2.....3....3......4....5.6....78.............5....2....4......
.................1.2.....34...........5...6......3......6...5........7...8...1...
.................1.2....3.....4......5........6...7..8.........4.....2....8.5....
.................1.2....3.....4......53...........6.7..........7.1..........8.2..
.................1.2...1..3.......4.....5.6........7.......3.2.5........8........
.................1.2...3......14.......5..2..6..........7.......3..........8....6
.................1.2...3...1..45............64....7.............6....2.......8...
.................1.2...3..4........3..1.5...............6.7.85........9..........
.................1.2...3..4.....4..25.6........7...........8.....1....6..........
.................1.2...3..4...1.........5.6......4......6..7......8....2.........
.................1.2...3..44.5.1........6..7........6...7....................8...
.................1.2...34.....5....6.7................5..............2....6..83..
.................1.2...34.....5....6.7................5..........6...2.......83..
.................1.2..1..34.3.............5...6..2.7........8...........8........
.................1.2..3...............1....45...2.............3.6..4..........7.8
.................1.2..3.......4......3............5..61..6...........27....8.....
.................1.2..3......4.........5.....6.7.......5.1............7.81.......
.................1.2..3.....1...........4.5...6.........5.........7.8..2..3......
.................1.2..3.4............3....2.......5...6..............7....58....6
.................1.2..34......1....5...........6.....7...........5..........8.42.
.................1.23...............14..........2.............53.6..7.........48.
.................1.23............42....5.63.....1.....4.......7........8.........
.................1.23...........2............4.......51.6.3...........47.......8.
.................1.23...4..............5.6.7...4..........8.2..5.........6.......
.................1.23...4..............5.6.7...4..........8.2..6.........5.......
.................1.23..45........3..1..6.7..............8......6.......7.........
.................1.23.4...5............1..6.......5.............7..8..2........7.
.................11.......2...........3.4.5....6......72...8.........43..........
.................11.......2......3.....2.4.....3....5...67..8.........6..........
.................11.......22...3........4.5..3..........6.............5....7..8..
.................11....2.3...4...5.....2.3..............67..4...8................
.................11.2..........3.......1.4..5.26...7..........3..............8...
.................112...3..4.............5....6.......74....1..........8........5.
.................12...................3.....4...5.1.....6.3..7.....4..........28.
.................12................3......4.....5.....61..7..........2.....2..85.
.................12............3.......2.4....56...7..4..7............5..8.......
.................12.......3...........4.5.6.......17...3.....2.......8......4....
.................12.......3..........34.5............2...67.4..8.1...............
.................12.......3..........4..5..6.3........1.......7....6..8.....8....
.................12.......3.......4......2.......35.....4....6.......78......1...
.................12.......3......45.......6..78..2......4...8..............5.....
.................12.......3.....42..........5.6........1.7...........4........82.
.................12.......3...4...........5.......6..773................8...5.2..
.................12......34..........5..6.7.......1............4.......2...87....
.................12......34.........1.4..2.........5........6...5...........7...8
.................12......34.........1.42...........5........6...5...........7...8
.................12......34.5....6.......4...1...2......7...8...8................
.................12.....3.................4.3.5..61........7........8...4.....2..
.................12.....3.................4.3.5..67........8........5...4.....2..
.................12.....3..........4.....5..67..2........3........7......48......
.................12.....3........2...1.4.5.........67..................86..7.....
.................12.....3.......42...........5..2...6..1.5...............7..8....
.................12....3........4.........5.67..2......5........6.8......1.......
.................12....3.....41....5......6.......78....6............2..8........
.................12....3....3...2.....4.....5....6..7..........6..........1.....8
.................12....3.4......5.........6.47..2......6..........8......1.......
.................12....34...5.6....71..................6.............2...7...8...
.................12...1...3.........4.....5....1.3..........6...7.......8...2....
.................12...3.4.................2...5..16.......7.........1...8.......5
.................12...3.4.....5................6..7................8.3....1.4.2..
.................12.3........4.........5.16...78......5...............7....3.....
.................12.3..4...4............5.3...6.........7................8.4...6.
.................123..................4.....5...6.......7.4..2.....5..........38.
.................123..................4.....5...6....7....4..8.....5..........23.
.................123..................4...5.....3..6....7............8....1.4...7
.................123............2.....4........56............3.1...4......7...8..
.................123.......4.................5.1..6.............78.3..........2.5
.................123....4.............5.....6...3.....7..2..........4.....1.....8
.................123...4.....51....6......3.......7.8......8.........2...........
.................123...45....67................1..8...................1...86.....
.................123..4.............5..........6..7..8.............5.2....1...3..
.................123.4.5................6...7...........8.71.........3.........2.
.................1234...................5.4...........1.6.4...2.............78...
................1........2..........3.....4.....5.....46....3....2..1.........78.
................1........2....3........4.....52............2....6..75.........4.8
................1.......2.............2.3............44...2......5.....6...718...
................1.......2.............3....4......5...5......6.7..3.........2.38.
................1.......2..........3..1.4......5.....6.3.....67........8.8.......
................1.......2..........3..2.4............53..6.5..7.........8......4.
................1.......2......3...4..5.6.....2.......3......784.............5...
................1.......2......3..45.......6.1.2.......5.........3..........7..8.
................1.......2.....1.....3.....4...5..6....76..5......8....3..........
................1.......2.....34..5.1.2......6.........3.7....8.................6
................1.......2....3.........4.5..62.7.......6............8.......21...
................1.......2....3...4.5...6.7...2.........6.....8.............2...7.
................1.......2.3............4.....15.6......4.....6...7........8.3....
................1......1...........23.4......5..............3..62..7..........48.
................1......1.......2..3..4..5..61....7..........8....2...7...........
................1......1.....234............5....6..7.......2.4.7...............8
................1......1...23.......4............5.67...7....8......4...........2
................1......1..2...3..........2.....4...5....6.5.4.....7.....8........
................1......12...........3.......4....5.6.......37....5.8......8......
................1......12.....34...5.........2...........6.7.....5.......84......
................1......12...3..4.......5....6......7.......78....4.............3.
................1......2.............3..4..5..5..1....2........6.7.....8...2.....
................1......2...........3........4.5.6......7.........2....5....1..86.
................1......2...........3..4..5........6......13.....27....4.....8....
................1......2...........31..45........6.........3...2.......7.48......
................1......2.........34............1.56...6........2...........1..78.
................1......2........3........4...5.6...7.....87.5...6..........6.....
................1......2.......3....4..5.6..3.1.................7.....8...3..7...
................1......2.......3..4......56...52..7.........8..............14....
................1......2.......3..4.53...6..........71..7...........8.........6..
................1......2......3................4....5.63...7.....8..........5..41
................1......2......3..............1....4.5....5.........61....73.....8
................1......2......3........4......56...............72..61.........4.8
................1......2......3..4...2.........5..4..........567....8..........2.
................1......2......34......5.....1.........36..............57.4.....8.
................1......2.....1........3.4........56..72......3.....8.....5.......
................1......2.....3...2.4...56........7...87......5............4......
................1......2.....3.4.........5......6.....34...7.................8.61
................1......2....3.....4......5............65.......2.......5...7..89.
................1......2....3....4.5....6.7.....8......2.........3..........1..8.
................1......2....31...........4......5........1.....6.2.....7...38....
................1......2....34..........5.......6.1.5...7........8...4.2.........
................1......2....34.5............6.......72......8.....8..5..6........
................1......2...1.3....4......5......6......5.7.........8..3.........2
................1......2...13.....................45.6.......7..84..6..........3.
................1......2...13...4................56....5.........78..5....6......
................1......2...3........41......2....567........8....6.............3.
................1......2...3.....1....4..5........6....6.1........73............8
................1......2...3....4................5...6....7..4.......2...768.....
................1......23......4....3.2.5......6.......7...8..........3..4.......
................1......23.....45....2.6...1...........7...........1......48......
................1......23...1.......4..5.6........7......1..8......5............2
................1......23...4..........5.....1...6.........72..6.4...........8...
................1.....1......2.........3.4..56.1......75..........82.............
................1.....1.2.3............4.5..6.1...........2....6............7.8..
................1.....2....................3..4..5......6..7...2.....5.48..1.....
................1.....2.........3............4.....5.........31..6.7.....2.68....
................1.....2.........3...2.......4.....5...6........4.5.......7...8.3.
................1.....2........3.......24.....5.....6.........23.....7.8.....5...
................1.....2........3......1..........4....35..........1.6.78...8.....
................1.....2........3.....45..2.6.....7.....8....3.2...1..............
................1.....2........3....1......4.....5..........67.3.5.......82......
................1.....2........3.4.2.5.......61........7..8..........3.........6.
................1.....2.......3.....1.4....5.....67...7......2.........4.......8.
................1.....2.......3...4.......1..25...6.....4........7.......8......5
................1.....2......2...........3.4.5..............5.3..671..........8..
................1.....2..3......4....56...7...8........3..7........3...........4.
................1.....2.3.............4..5............6...7.8.......4....45.....6
................1.....2.3.............4..5............6...7.8.....4......45.....6
................1.....2.3......4.5.67........31........4.........6..........8....
................1.....23.....................3.45.....62..7.3..8...........4.....
................1.....23.............1....3.....45.......6..74.2.........8.......
................1.....23...2..........1...........4...43....5.....67...8.........
................1.....234...5..6............4....7.......1.8...2...........4.....
................1....2.3..4..........15.6.7...........4........2.....8....8......
................1...1.............2....3.4.....2...5..3.6......4............78...
................1...1...2..............3.....4...5...65.......3.....7..8.....2...
................1...1...2......3......4..........5......62.4.........7..8..1.....
................1...2.....................13.....24...2......5.6.78.......4......
................1...2.....................3..4..51...........65........4.78.3....
................1...2..............1.3.4.5..........6..47..1.......6..8..........
................1...2..............3...2.4..........5.....1..674.........28......
................1...2..............3...4.2..........5.....1..674.........28......
................1...2..............3.4...5..........2........54........67.81.....
................1...2..............34......215..67............7......6..8........
................1...2.............2...........3..4......5.1....4..2..6.7.......8.
................1...2.............2.3....4.....15...6...6............4......7.8..
................1...2............1.3.4.5.6.........7........3..........8.6.....4.
................1...2............2..........31..45...........463...7...8.........
................1...2...........3........4....56...7.........3....27..........84.
................1...2...........3....45.........6..7..........84...7...........61
................1...2...........3...2.......45.......6.......3.......1...784.....
................1...2..........3..4.5....6....1.....2.........7........6....4.8..
................1...2..........34..............5....2.67..8............43.......5
................1...2.....3..........435....6...7...............8....5.......2.7.
................1...2.....3......4.......2....5...3.....6.......4..6.7.........8.
................1...2.....3......4....3.........5..16.....7...8.........6..1.....
................1...2.....3...3..4.567........2..............7...5.............8.
................1...2.....3...4............5..6...........5..6...3...7....8.1....
................1...2...3......4......3...2......5.....4....6......7.....7...8...
................1...2...3.....4........2......1.....3....5.6.7.8..........4......
................1...2...3.....4..5..6........73............6..8..........25......
................1...2..3...........24..5.............3.6.7...5.......4.........8.
................1...2..3..........3....4.....3......5...6..5.........4.7......8..
................1...2..3........4......5.6.....7......5.......6.8..1............7
................1...2..3.......4..2..........2......3...56....7........8......4..
................1...2..3......4........5.......6...7..48......5.....2........7...
................1...2..3......45......6....7...........4.........5.....7......18.
................1...2..3....2.....4....5..........6..26.....7......8........4....
................1...2..3....24...5.6...7.............3.......7...6.............8.
................1...2..3...14...5.........6.....4...........7.....7.....8......4.
................1...2..3..4................5......4.........3...6..1....5......78
................1...2..3..4..........5....6...7.8.....4......8...3..2............
................1...2..3..4.........5...........6......2..74..........8.......16.
................1...2..3..4........2.2......5...6..7........86............4......
................1...2..3..4........5..3.........67.........2.........8..7......6.
................1...2..3..4.....5...........67..8......2.....7...4.............5.
................1...2..3..4...56....7............................6...7..8...1.6..
................1...2..3.4...........5............4...........6.6....7.5....18...
................1...2..34........34..............5....5...6..78..............4...
................1...2..34..1.............5......6.7..............5..4.......8..6.
................1...2.1......3.........4..5.6.78......6............2.....4.......
................1...2.1.3..............4.5....16......7.......5........4....8....
................1...2.3....................4......51...5.......6.7..........4.83.
................1...2.3...................1..4..5........4....53.........67.8....
................1...2.3...................4.....5......26....5.........1..3.78...
................1...2.3...................4.....5.6......7.1.5.2..8......4.......
................1...2.3...........4..........3...56......1........7......5....8.3
................1...2.3.........4........5....6....3..........57.3.8...........4.
................1...2.3....4............5..6.7............2...48.......7....6....
................1...2.3...4...........4..5....2...........6.7........5.....18....
................1...2.3..4........3..5.6............7.....2....5.....6.8.........
................1..2.........................3...45..65........4.7...........128.
................1..2.............3..4...........25.6...5.......7....4.8.........1
................1..2............2.....3....4......5....5....6.....24.....7....8..
................1..2............3........4..25.........42....6....75.......8.....
................1..2...........3..........4....5.26...............4...7..36.....8
................1..2...........3.4.25........1.6..7.......4........8..6..........
................1..2..........3......45...........6..7.........7.3.....8.....12..
................1..2.........3...4.....5......6..2...7......1.....4.....8.......4
................1..2.......3......4.......2..5.1.6.....4..........4......7......8
................1..2......3............3...2.4.1........5.6.4...........7...8....
................1..2.....3........2....3.........45.....6.........7....42.......8
................1..2....3................4..51.6.......5..7......4.......8....4..
................1..2....3....4.........5....61.........6..7......5.......8....5..
................1..2...1............3.....4.5...6.....4.7...8...............2..6.
................1..2...3...4.5......6...........2...........4..78.1...........5..
................1..2...3..45.....6..............7......3.........4.........81..2.
................1..2...34...5.............3.......6......51....6.....7.8.........
................1..2..3..........3.....4..2.5..1..............2....6..7........8.
................1..2..3........2.............4.56.....7...........1..68..3.......
................1..2..3.4.......5....6....2.......7..........5.7......8...8......
................1.2......3.....4......1...2....5......3........4.....6.7.8.......
................1.2.....3..............2.45...13...6..4...............7.....8....
................1.2.....3.............1...42.....5....5........6.........7.8.3...
................1.2.....3......4.....51...........6..46........7...........5..8..
................1.2.....3.....4.......5..6....7...8...........16............1..3.
................1.2.....3....45...6.3........1....7..........8...5............2..
................1.2....3...................4.......5.....16....3.7...2...8..4....
................1.2....3.......4.....1.....56....7....4............2.4..........8
................1.2....3......4...56.........1..7......4.........6..8.....5......
................1.2....3......45..6.1..................4.........7..82...6.......
................1.2....3....4........1..5..6....4.....7.......83.....7...........
................1.2....3..45............6..7..................5........2..3.87...
................1.2....34......2.....1....54.....6....6.........7.............8..
................1.2...34........5.......6....4.....7......1..........2....7...8..
................1.2.3.............4.........5....2.......6.1...7.2...8.......4...
................1.2.3........4.........5.......67...8..2.............5......34...
................1.23.............4...1......2.5.6.....4....2...7....8............
................1.23......4........2....5..6.7........48...2..............7......
................1.23...4......5......14.....6.........7....8...5......6..........
................1.23..4.........3......56......7..1.8.........2...............7..
................1.23.4.5..............1.6.........7......68..........7.5.........
................12.....3.....4.......5...67.......8.........3......2.......14....
................12.....3.45...4...............6....7..5...8....3..............6..
................12..3.4...............5...3...6..7.....7....4............8...1...
................12..3.4.......5..3..6.........7........2....87...............1...
................12.3...................4....5..67.....5...8...........6.......37.
................12.3..................4.5..6......7...7.......3..2..........6..8.
................12.3................4.......5.....6....5...........4.3...72..8...
................1223.....45....3..........6....4........7...8.......4............
...............1.........2...2...............3..4....5........3..6....7..28.1....
...............1.........23.........4.5.6...7.....8........3..5..........21......
...............1.......2........3...4...5....................4...2....63..178....
...............1.......2....1...........3..4..56........7.8..........5.28........
...............1.......2..3...3.......4....2.....5...........6..4..1....7......8.
...............1.......2..31............4.....5.6............6.7.3..8..........5.
...............1.......2..33...4......2.........51.6.......7....8....4...........
...............1......2..............23.4..........52.6..7.8...1................3
...............1......2...........2..23.4..........5..6..7.8...1................3
...............1......2..........3....4...2....56.....13...............5.7.....8.
...............1......2.........3..4.5..4...67.8..............3...........1...8..
...............1......2.....2..3.............4....5.6...7..6.........2.3......8..
...............1......2...3.......2.41..............3...........5....6..7...8.4..
...............1......2.3.............3.4...........5.65.....7....1......1.....8.
...............1......2.3.........4...3.5.......1.....64.....7......1..........8.
...............1....1.2.............3..4.5.........2.........53.6..7........8....
...............1....2.....3.............4..........5..31..........42...67...8....
...............1....2.....3..........1......24..56...............3...7........85.
...............1....2....3.......4...3..5.....4...........67...1.....8.....3.....
...............1....2...3.............1.4.2...5.......67.....5..8...1............
...............1....2...3........2.3......4...5.......67......8....1...........5.
...............1....2..3............41....5...3.6.......7.....6....8............3
...............1....2.3..............1...45.....6...3...73...........4...8.......
...............1....2.3..............4.1.5..6.............7......8.2..........45.
...............1....2.3........4.......1..5..6.......7..87..............7.......6
...............1...2....3.....4...56..7.......31..........8....4......6..........
...............1...2..3...............4.5...3......6......4....1.....7...7....8..
...............1...2..3............4....5...6.17.2......8............7..4........
...............1...2..34......1.5......6....7..............3.....6....7...8......
...............1...23.........4......3.....5......6.7..........8.1..........5.2..
...............1..1......2..3...4......15....6........7..........2...........3..8
...............1..2.......3.....3....4..........5......1.....5.......2...67..8...
...............1..2.......3.....3....4..........5......1.....5.......2...67.8....
...............1..2......3.....4..5617.......3..........6...7...............8....
...............1..2....3..............2....4......1.........52..3..6...........78
...............1..2....3........2.34..5.......61.......7....6................8...
...............1..2....3.....1..........4..5...........3......2.6..5..7..8.......
...............1..2....3.....1..2....................4......3........5...46.7...8
...............1..2....3...4...56.........7.8...2......7........8........1.......
...............1..2...3...4....1......5....6.1...4..........7...8...........2....
...............1..2.3..4......2.....5.............1..........67........5..4..8...
...............1..2.3.4.................5...4........2.....2.............1.6..78.
...............1..23....................4..5....6.....1.7.....2...45........8....
...............1..23...4......5............6........2.....2.7...4.1.....8........
...............1.2.....1..3..........1.3............4...5.6..7..........4...8....
...............1.2.....3......24......5....3.................6..5..1....7......8.
...............1.2....3...............1..2.....4.............4.5...6..3.78.......
...............1.2....3.............3.4.5..........6...2.7.8..........4..1.......
...............1.2..1...3............4.2..........35.........46.........7...8....
...............1.2..3.....4..........5...........4..3.....2........1..6.7......8.
...............1.2..3.....4.......3..5...........4........2........1..6.7......8.
...............1.2.3......4...............56..1.......5.6....7.........38........
...............1.23....4...4....3.5....................1.........2....6..7.....8.
...............1.23.4.56......7............5..........72.......8...............3.
...............1.234...5...6....4...................6.....1.7...6...........8....
...............1.234...5...6....4.7....................2.........8.......1.....3.
...............1.234..5........................6..1..7.............8.3....2...4..
...............1.2345.6.......2..............7...............4..5....3.....8.....
...............12......3...................4.......5.....26....3.1...7...8..4....
...............12......3..........4......5............67......1....28.....1..4...
...............12.....3.........4...1..........3...5.....56...........78.......4.
...............12...3..4............51.6............73.......6..8.............5..
...............12..3..4.......5............6......7...5..8.......1.....6.......3.
...............1234....5....6...4...............7......3....8.............8.....1
...............1234...2..............2..5......................3......64....78...
..............1.......2.1.....34......5...67......8...8............5.....3.......
..............1.....2.....3.....4.............5............67142.3.8.............
..............1.....2.....3.....4....56........7.....2.......1...3...4........8..
..............1.....2.....345...6..................7.2.....4.6...8........7......
..............1.....2....3..................3..4......5..3........26....17....8..
..............1.....2....3......4.....5.........6.....78.............6.1..3.5....
..............1.....2....3....3......4....1........5....6..7...25.....8..........
..............1.....2...3..........4..3.56..........7.17.......6............8....
..............1.....2...3......4......5............1..6......4778..........1.....
..............1.....2...3.4........5.67..............8....3.7........6.........1.
..............1....2......3..45................6..3......7......8....2........81.
..............1....2.....3............4..............2....4....56....1...7.38....
..............1...2.......3...........4.5.6.....1..7...3......2.............4.8..
..............1..2...3..4..5......................6....1.......2.6..........7.38.
..............1..2..3...1...2.............4..56.........4.1....7.......8.........
..............1..2..3...4...5...........6.3...........67......8..........1.....7.
..............1..2.3..4.......5......4............6...2.6............3......7..8.
..............1..234....5..............3....6...4.....7..............7....6.....8
..............1.2...3.....4.....5.........6..................176.8.......4..3....
..............1.2..34.2.5..6............7.............16.......8................2
..............1.2.3.4.5.................6.5.3......4...............3.....7.....8.
..............1.2.34......5....6.......5.3.....7........8.............1.......5..
..............12....3.4.........5..1..................6...............75.286.....
.............1........2...1.........2.......3...4.....53.......6.1............78.
.............1......2.....3.....4....1......56.........7.....8...3............64.
.............1......2....3...........4....5.....6.3.....7.8...1.8..............4.
.............1......2....3........2..45...........367.........1.8.............8..
.............1......2....34..........5....6.......3.......7...82......4.......1..
.............1......2.3..1..........4......3....5.....6..............5.7.38......
.............1.....2......3...........43.2.....5............6......6.71..8.......
.............1.....2.....34................1......5...5...........67...2..3.....8
.............1.....2.....34...............1.......5...5...........67...2..3.....8
.............1...2....34......5.......12.....................6..1....4..7..8.....
.............1...23.......4..........................3...3..5....6....1..4.7.8...
.............1...23.......4...........1.5.6........7...4.....3.......8.........2.
.............1..2.3.....4...............5...6........7.6.........7.....3....2.8..
.............1..2.3.....4...............5...6........7.6.......7.8.....3....2....
.............1.2.3..3.....4................5.1...2...........6..7.....8.....4....
.............12....3....45............6.7....................386.........5..6....
.............12...3.......4............56.....7..........4..7.........2...83.....
.............12...3.....4.5................1....6..7...8..............24..5......
.............12..34.....5....2.....1..........6.4...........7.....6.............8
...........1.....23......41..........5..6.7................2....8.....7..4.......
...........1....2...3...4.....5...................6..7.7.......8..7.........4.1..
..........1......2..3..1..4........5...6........7............8......5...6.....7..
..........1......23.......1.........4................5.2.....6...5...7......8..4.
........1........2....1.......3...........1..4.........3.....5.6.2..........7.8..
........1........2..3..4..................5.....6......3..78...5.......6.......1.
........1........2.3..........2.....4.............5..6..7.8..3.......7.........4.
........1.......2.....1.............3.45..6........7......8.....7..2..........4..
........1.......2...3.....4.....5............3.....6.......1.75..6.......8.......
........1.......2..1...........3....4.5......6...7.........1......2.8.........3..
........1.......2..3...4.....5...6....................56...7...........8.......12
........1......2.......3..........................4.5..2.......4.3....6....71..8.
........1......2.3.4....................3....5......4...5.6..7.....1...........8.
........1.....1........2.......3.4..5..6...............4..............5..67..8...
........1.....1........2....31.......4............56..6...............7..8.4.....
........1.....1....2..........3.......1.4..........5..6......2.....7...8.5.......
........1.....2.....3.....4.............5.......6.....78.....2.....1.5......3....
........1.....2.....3....4..............5...6.........7............6.....48...72.
........1.....2...3........4.1....................567.........4........8.5...6...
........1....2......3....................456...1.......4.......5...3..7.....8....
........1....2......3..................4..56...1.......4.......5...3..7.....8....
........1....2......3.............2.......45....6.............64.........3.7....8
........1..1......2...3...4...............56.....4......7...8......2.....2.......
........1..1....2.3...........4.......5.......6........4............72.3.8.......
.......1.........2..3..4....5..1..........6......2......4...7..5............8....
.......1.........2..3..4....5..2..........6......1......4...7..5............8....
.......1........2345..6........................7..1...........4....8.5....3......
.......1.......2..3......4............5.....3.....4....5......6......7.8.....1...

I'll add these to ton's list, to make a combined total of 544,949 and continue on.

Leren
Leren
 
Posts: 5117
Joined: 03 June 2012

Re: SudokuX (Diagonal Sudoku)

Postby m_b_metcalf » Thu Apr 02, 2020 3:30 pm

m_b_metcalf wrote:I ran the first 10,000 through SukakuExplainer, and here's a selection of the results (in SE order):
[snip]
Curiously, there was no 1.2/1.2/1.2.

To obtain a more representitve selection, I processed every fiftieth puzzle instead of the first 10,000 (up to 500,000). That took longer, and any higher ratings for a given ER are shown here:
Code: Select all
000000000000000000000000100000000200003000004005600000000000040000000530780002000   ED=2.0/2.0/2.0   
000000000000000000001000002000030040050000060000007000000000003080020005000060000   ED=2.3/2.3/2.3   
000000000000000010023000000000000000450000000000006300000000000076008000000000450   ED=4.3/4.3/2.9   
000000000000000000000001200020003000000000040005006007000470000000000300000000800   ED=4.5/4.5/2.9   
000000000000000000001000200000000300000400500000060000470000006000000080002001000   ED=4.7/1.5/1.5   
000000000000000001000002003000000040001503000000000600000000700000000080020001000   ED=5.6/4.3/2.9   
000000000000000001200000300000004005600000000000007000010000000008060200005000000   ED=5.7/4.0/2.9   
000000000000000010002003000000000000040000000000005006300000000000046000007000180   ED=6.2/2.9/2.9   
000000000000000102340000000000000000000000005001006000000000070806001000000000030   ED=6.5/2.9/2.9   
000000000000000000001000200000000304000000500000670000070000080000000006004050000   ED=6.6/2.9/2.9   
000000000000000100002003000000004030000000000000000050000506020100000000078000000   ED=6.7/4.1/2.9   
000000000000000001000002003001000000000040500000000600000070800000900000032000000   ED=7.2/7.2/2.9   
000000000000000100002000300000000200000405000006000000150000074000080000000000000   ED=7.7/7.7/2.9     
000000000000000010002003000000000042053000060000000000000030000070000500000000008   ED=7.8/7.8/2.9   
000000000000000010000000230000000000000000400030500000600030000700000500000001080   ED=7.9/4.3/2.9   
000000000000000010002030000000000000004000000000500060000040300000070000650000800   ED=8.0/4.3/2.9   
000000000000000000001020003000000000040005600000003000007080000000000507000010000   ED=8.1/2.3/2.3   
000000000000001020000345000000000000060070000000000000000060800503000001000000000   ED=8.3/8.3/2.9   
000000000000000100002000300000004200000506000007000000050000086000010000000000000   ED=8.4/7.7/2.9   
000000000000000000000001200000000003405000000000060070000100500308000000070000000   ED=8.6/2.9/2.9
000000000000000000010203004000000000000050600000000000040000023000000000700068000   ED=8.8/7.9/2.9     
000000000000010000002000300000000000004000500000060007760000010000400000000008000   ED=8.9/4.3/2.9 
000000000000000010020030000000020000004000000000050600030700000000400080050000000   ED=9.0/8.3/2.9   
000000000000000010000020000000030000100000045000060070060000000000005080020000000   ED=9.1/9.1/2.9   
000000000000000001002003400000050000003000000000060000170000056000800000000000000   ED=9.2/9.2/2.9   
000000000000000010020003004005000000006070800000000000040000023000000000000060000   ED=9.3/7.1/2.9   
000000000000000000010002034000000000005060000000000000030000021400070000000080000   ED=9.4/2.9/2.9   
000000000000000000001002300000000400000000000000050000050000006000004070002001080   ED=9.5/2.9/2.9   
000000000000000000001020300000000000000405006002000000040000067500000000000080000   ED=9.6/9.6/2.9   
000000000000000001002304000000010000000000300000050000060000078000000000400002000   ED=9.7/1.5/1.5   
000000000000000001230000400000000000000000200000005000000006000070000300005008006   ED=9.8/2.9/2.9   

Still no 1.2/1.2/1.2.
User avatar
m_b_metcalf
2017 Supporter
 
Posts: 13621
Joined: 15 May 2006
Location: Berlin

Re: SudokuX (Diagonal Sudoku)

Postby Hajime » Sat Apr 04, 2020 11:30 am

These are all proper SudokuX with a single solution, I presume.
Hidden Singles in boxes have SE 1.2 . In rows or columns SE 1.5.
Probable you cannot solve the puzzle without the use of the diagonals constraint.
Presumably the diagonals have SE 1.5 also.
So you won't find 1.2/1.2/1.2.

PS what are "Generalized Intersections/Naked Pairs/... etc"
User avatar
Hajime
 
Posts: 1375
Joined: 20 April 2018
Location: Fryslân

Re: SudokuX (Diagonal Sudoku)

Postby HATMAN » Sat Apr 04, 2020 5:42 pm

Hajime

Not quite sure, but in my memory tarek tries to include all sorts of things in one big box. It is natural for him to be inclusive; as he comes from the east and also the west (of the English Midlands of course).


Maurice
HATMAN
 
Posts: 315
Joined: 25 February 2006
Location: Saudi Arabia

Re: SudokuX (Diagonal Sudoku)

Postby tarek » Sat Apr 04, 2020 7:24 pm

Hajime wrote:PS what are "Generalized Intersections/Naked Pairs/... etc"

All candidates that are peers to the locked candidates are eliminated … In Sudoku variants, the eliminations may go beyond the interacting 2 houses (sectors)

HATMAN wrote:Not quite sure, but in my memory tarek tries to include all sorts of things in one big box. It is natural for him to be inclusive; as he comes from the east and also the west (of the English Midlands of course).
:lol: That too
User avatar
tarek
 
Posts: 3762
Joined: 05 January 2006

Re: SudokuX (Diagonal Sudoku)

Postby creint » Sat Apr 04, 2020 9:03 pm

My solver solved all except these 62:
Hidden Text: Show
000000000000000000000001200000030004000000000500000000000520100003000000067000080;1.815 seconds
000000000000000000000010002000020000000000340000050000100000000040607080000009000;1.383 seconds
000000000000000000001000200000000102030405000000060000007000800000008003000000000;3.136 seconds
000000000000000000001002300000040000000000201000050000450000067000000000000008000;12.659 seconds
000000000000000000001002300000040000003000100000050000000000600000070000040080070;7.395 seconds
000000000000000000001002300000040000003000100000050000006000004500780000000000000;1.835 seconds
000000000000000000001002300000040000003000100000050006500000000000000000070080040;3.73 seconds
000000000000000000001002300000040000003000100005060000060000000700080004000000000;3.25 seconds
000000000000000000001020003000004000002000000050600070000500060000000000003080000;1.684 seconds
000000000000000000001020003000004005000000001600700200000000000003000000000000870;2.305 seconds
000000000000000000001020300000000000000405067000000000060000074000000000050080000;3.846 seconds
000000000000000000001020300000000000000405067000000000060000074500000000000080000;3.063 seconds
000000000000000000001020300000000000000405067002000000008000000000704000060000000;4.832 seconds
000000000000000000001020300000000400000506007000080000650000080000030000000000000;1.562 seconds
000000000000000000010000020000000003000000041506070000000000500040300000000000800;0.725 seconds
000000000000000000120000034000000000000003042005000000006070800000000001000000000;6.956 seconds
000000000000000001000000002000000030004102000000000050000000006030057000000800000;3.684 seconds
000000000000000001000000200000013004005000000062000000000000500400708000000000000;2.944 seconds
000000000000000001000000200030000400000500000600000000000023000708000005000004000;1.897 seconds
000000000000000001000020300140000000000050000670000000002000800000006000000400000;2.171 seconds
000000000000000001002030000000000000100405006007000000003080700600000000000000000;3.511 seconds
000000000000000001002030000000000040000560000040000000007000000000801000300009000;2.956 seconds
000000000000000001002030000000004200000100000050000630000000007000000004008000000;2.473 seconds
000000000000000001002304500000010000005000200000060000070000000000000006000080000;3.591 seconds
000000000000000001020000000000000340105000000006000000000500000030000000070080005;0.508 seconds
000000000000000001020000003000000040000000520607001000000000000040500000000080000;0.601 seconds
000000000000000001020003000000400000000000000030000200405000000001000000000006780;1.464 seconds
000000000000000001023040000000000000000501006007000000004030000000000008000000500;2.805 seconds
000000000000000001200000003000000000004056700000000008300000002800000000000090000;2.193 seconds
000000000000000001200000030004000000000050670800000000300000002100000009000000000;3.026 seconds
000000000000000001200000300000002000040000000300500000060000000000000700010080004;2.741 seconds
000000000000000001200030000000040520060000000017008000000000000000800000000000007;1.661 seconds
000000000000000001200030000000040520060000000017008000000000000800000000000000007;1.53 seconds
000000000000000001203040000000000005000600000000200007000030400000080000060000000;1.496 seconds
000000000000000001230000000004000005000000000600001000000700000080000200000000380;1.369 seconds
000000000000000010000000200002000000000304005600070000008060000009000080000000000;1.972 seconds
000000000000000010000002000000030405060000000170000000000000000000000370004080000;3.014 seconds
000000000000000010000002000034000005000067000000000070000000002080010000000000900;2.241 seconds
000000000000000010000020000000000300000405006708000000640000000000000020050000000;1.418 seconds
000000000000000010002000300000400500010000000060000070000000060008000000004300000;0.996 seconds
000000000000000010002003000000000002000000403560700000000800060000006000000000000;0.751 seconds
000000000000000010002003000140050000000000206000000003000700080006000000000000000;1.064 seconds
000000000000000010002030000000000400000000200050600000000000067408000000009000000;4.792 seconds
000000000000000010002030000020000000000000400500000000000100056708000000009000000;1.872 seconds
000000000000000010002030004000000005000000600170800000000000080003000000004000000;1.981 seconds
000000000000000010020000000000300405060000000070060000300000800000000002008000000;3.857 seconds
000000000000000010020003000000000004001030000000000000005000002000670080040000000;0.776 seconds
000000000000000010023000000000000400000000305160020000000070000000000000005000800;2.037 seconds
000000000000000010200000003000000000004050060000000000300004002000000708000010000;2.586 seconds
000000000000000010200003000040000000010050040000000000000000006700000302000080000;3.353 seconds
000000000000000010200003000040000000013000000000050600700000000800000000000000084;1.945 seconds
000000000000000010230000004000000000001050060000000000400037002000000000000080000;3.274 seconds
000000000000000010230000004000000000005060000000000000007050100000000070040008000;4.085 seconds
000000000000000010230004000000000000506070800000000000040000002000000000000710000;1.917 seconds
000000000000000100002030000000000000400100500000000006007083000000090400000000000;1.512 seconds
000000000000000100203040000000000005000106000000000007000030400000080000060000000;2.288 seconds
000000000000000102300004000000005000000000006700000000020060000001000800000000030;2.338 seconds
000000000000000102340050000000000000000000030000006000000000078006001000000002000;1.133 seconds
000000000000001000002000003040056000000000002000070008000000010008000000000000600;3.177 seconds
000000000000001000002000034000400020560000000700000000000800000010000000000000700;1.633 seconds
000000000000010000020003040000000000005060700000080000040000020000000500700000000;2.798 seconds
000000000000010000200000034000000000005060700000000000001000800400003009000000000;4.233 seconds

Adding one more dynamic part would solve them all.
creint
 
Posts: 393
Joined: 20 January 2018

PreviousNext

Return to Sudoku variants