Help for Loops
 

Normal SK Loops

SK loops are continuous chains of hidden pairs. They were discovered during attemps to find a solution to the 'Easter Monster' puzzle. The loop involves 4 boxes arranged in 2 shutes and 2 stacks to form a rectangle. They involve 16 cells, or 8 pairs of cells, with each pair of cells having implications for the next pair of cells in the loop.

The following is a small selection of puzzles have this "normal" pattern (2-2-2-2-2-2-2-2-):

1.......2.9.4...5...6...7...5.9.3.......7.......85..4.7.....6...3...9.8...2.....1    // Easter Monster
12.3.....34....1....5......6.24..5......6..7......8..6..42..3......7...9.....9.8.    // Cigarette
1.........2.4...6...3...5.1.4..86.......49.8....2.....5.7...3...6.9...4.........7    // Cloudy Bay
.....678........327.....5....1.4.....9.2.....6....7.5.3....58....29......4..1....    // Double magic
1.......2.2.....6...34..5.....8.5.....8.3.9.....9.4.....5..34...7.....1.6.......7    // Dukdiamond1
..3.8....7..2......6...9.1.........3.....596..9.....54.1...45..8...3......27.....    // tax;tarek071223170000-;23;21;
1.......2.9.4...5...6...7...5.3.4.......6........58.4...2...6...3...9.8.7.......1    // coloin coloin-04/13-1414
1......89.....91.2......4....76......3..4....9....2..5..4.7....5....8.1..6.3.....    // tarek 04/08
5.......9.2.1...7...8...3...4...2.......5.......7.6.1...3...8...6...4.2.9.......5    // tax StrmCkr   ED=11.6/11.6/3.4
..4.....9.1...3.7.6.....2.......8....5..3..8....1.5.....2.....4.7.3...1.9...7.6..    // tarx0037, tarek
12.3.....3.4...1...56....3.2..5...7.....8.9.......6..4.7..9.8....21...5......4..6    // Patterns game, lost reference
..7..1...6...9.2...3.5.....9.....6.8.......3.....8..922...4...9..13......5...7...    // tax;tarek071223170000
..1..23...4..5..6.7.......8...3.......5.1.2.......9...8.......4.6..7..8...31..9..    // Patterns game JPF, Jan 04, 2009

Taking 'cigarette' as an example, the following result occurs (Loop - red cells, eliminations -green cells):

Starting with the pair of candidates 7 & 9 in r5c4 and r6c4, if these are both true, then the cells r4c5 and r4c6 contain the pair of numbers 1 & 3. Therefore, the 1 & 3 in r4c8 and r4c9 are false and 8 & 9 are true. This makes the 8 & 9 in r5c7 and r6c7 false, and the 2 & 4 to be true. Therefore the 2 & 4 in r8c7 and r9c7 are false and the 6 & 7 true. This makes the 6 & 7 in r7c8 and r7c9 false and the 1 & 5 true. Therefore the 1 & 5 in r7c5 and r7c6 are false and the 6 & 8 true. This makes the 6 & 8 in r8c4 and r9c4 false and the 1 & 5 true. Therefore the 1 & 5 in r5c4 and r6c4 are false and the 7 & 9 true, thus completing the loop.

The corresponding chain is:
(79=15)r56c4 - (15=68)r89c4 - (68=15)r7c56 - (15=67)r7c89 - (67=24)r89c7 - (24=89)r56c7 - (89=13)r4c89 - (13=79)r4c56 - (79)r56c4: loop

The eliminations are:
-4 r1c7, -5 r2c4, -1 r3c4, -13 r4c2, -15 r7c2, -24 r3c7, -8 r5c9, -9 r6c5, -9 r6c8, -5 r7c1, -6 r8c8, -7 r9c9;

The last two of the above examples are of interest because of the different configurations in all four boxes.


Variant Loops

The links in the classic SK loop equal 8 x 2 = 16. However, there can be triple or single links, and if the sum of the links add up to 16, the logic still holds. In each box the cell at the intersection of the 2 pairs of cells must be a given. Of the sixteen cells comprising the loop, one or more of the cells can be solved, but NOT given. For each such solved cell, the link count can be increased by one to keep the link count to 16.

This can be summarized with the following Rules:

1) The loop requires 16 cells in 4 boxes in 2 bands and and 2 stacks.
2) Loops can include single, double or triple links, as long as the sum of the links is 16.
3) In each box the cell at the intersection of the row cells and column cells is a given.
4) Of the 2 cells involved in a row or column within a box, one of the two can be solved or given.
5) For each solved or given cell, the link count should be increased by one.

Type A

These loops (B3-1-3-1-3-1-3-1-) have four triple digit links within the boxes, and four single digit link between the boxes. Here are some examples:

....1.......234.....15.62....2.7.8...6.....9.4.......5..8.2.7...5.....6.3.......4  Patterns Game m_b_metcalf Feb 06, 2011
1...2...3.4.....5...6...7.....5.6...8...9...1...3.......7...6...5.....9.2...3...8  Patterns Game 116/m_b_metcalf/9.2/9.2/3.8
1.......9.4...2.8...6...3.....4.3.7.....6.....2.5.8.....9...1...8.7...4.5.......6  Patterns Game 119/gsf/11.3/11.3/3.4
..1.....2.3...4.5.6.....7.....68.......4.9.3..9..31.....8.....1.4..5..9.7.....6.. # 7658;GP;H1521
98.7.....7.6...8...54......6..8..3......9..2......4..1.3.6..7......5..9......1..4
..9...2...4...8.7.7.5.....6.....7.3.....5.....3.9.4...6.....7.9.9.8...4...2...1..
6...4...3.1.....7...5...8.....5.2...3...9...2...1.3.....8...9...7.....5.2...3...4
..1..23...4..5..6.7.......8...4.......3.2.5.......6...6.......7.9..8..4...53..1..  Patterns Game m_b_metcalf Jan 05, 2009
...1.9......6.5.....6.7.5...1.....4...57.18..3.......14.......3..7.8.6...2.....9.  Patterns Game tarek Feb 19, 2009
1....23...4..5..6...78....9..5.......6...1...9...3..8.2.....7...8...4.5...3.....1  Patterns Game m_b_metcalf Aug 29, 2009
..1.....2.3...4.5.6.....7.......1.3.....2...8.8.3.5.....6...9...5.4...8.2...3...6  Patterns Game JPF May 30, 2009
.....9..5.3.....4...567.8....2..67....8.2....4..1.......67..2...9.....1.5.......3  Patterns Game gsf Jun 21, 2009
..1.....2.3...4.5.6...7.8.......7....5..1..7....8.......9.5...6.7.3...4.2.....1..  Patterns Game m_b_metcalf Jan 04, 2010
..1...2...3...4.5.6.......7...4...8.....2.1...5...3...2...1...9.8.5...3...7...6..  Patterns Game joel64, Jun 27, 2010
1.....2.3.4.5...6...3.......6...5.7.....12......67....2.......8.5.7...4.9.....3..  Patterns Game JPF, Sep 02, 2010
1.2...3...43......56.3.......78..2.......1.5.....6...98..2..7......5..24.....9.1.  Patterns Game m_b_metcalf May 18, 2017
.12......3.4...5..65.2.......75..8......9...4.....3.1..8.7..2.......6..3....1..9.  Patterns Game m_b_metcalf Dec 17, 2017
3...5...9.8.1...6...7...4....5..8....1..6..2....2..8....4...5...6...2.1.7...9...3  Patterns Game Pat Oct 02, 2017
12.......3.4...5...657.......35..8......2...4.....4.6..8.3..7.......1..9....6..2.  Patterns Game Champagne Feb 27, 2018
.123.....43.......5.6...1..3..1..7......4...5.....8.9...72..3.......5.4.....8...6  Patterns Game Dobrichev May 19, 2018

The chain for the first example is: (359=7)r13c2 - (7=136)r79c2 - (136=2)r8c13 - (2=359)r8c79 - (359=2)r79c8 - (2=156)r13c8 - (156=7)r2c79 - (7=359)r2c13 - loop

Type A sub-type

One of the cells in one (or more) of the cell pairs can be a solved cell (not given). First example has 13 links and 3 solved cells, so its corrected link count is 16.
It has 2 alternate loops: (B2-1-2-1-3-1-2-1-) or (B1-2-1-2-2-2-1-2-)

..3...5...8...1.4.5.......2.....8.1.....2.....7.3.5...6.......5.1.4...7...2...3..

It's chain is (69=4)r13c2 - (4=39)r7c2 - (39=8)r8c13 - (8=269)r8c79 - (269=8)r79c8 - (8=69)r1c8 - (69=7)r2c79 - (7=69)r2c79 - loop

The second example has two (B3-1-2-1-3-1-2-1-):

2.......4.8.5...7...1.2.3.....7...9.....6.....7...8.....3...1...9...7.5.4....1..2

These examples have one (B3-1-3-1-2-1-3-1-):

1.......4.5.7...9...26..3.....8..45.....5.....9...7.....3...2...8...5.7.4.......1
2.......4.9.6...5...17..3.....8...9.....5...3.5...9.....4...1...6...5.7.3.......2
.1......22......3...4..56......217.....4.6.....783......67..4...8......13......2.
..1..23...6..1..5.7.......4...2.......3.7.1.......3...5.......7.4..5..6...81..2..

Type B

Alternatively, a loop can have four triple digit links between the boxes, and four single digit link within the boxes (B1-3-1-3-1-3-1-). These are much less common than type A's

1....23...4..5..6...74....1..4.......2...8...5...2..9.3.....1...8...6.2...1.....7  Patterns Game m_b_metcalf Aug, 2009
1....23...4..5..6...34....7..4.......2...5...5...6..2.3.....7...6...8.9...7.....1  Patterns Game m_b_metcalf Aug, 2009
1....2..3..45..6...3..7..1.8......2...5...9...7......4.1..4..3...8..62..3..2....7  Patterns Game m_b_metcalf Jan 02, 2011

Type C

There can be one single box link and one triple row or column link (B1-3-2-2-2-2-2-2-)

1..2....3.....415.....1..6......6.4.....5.6...2.3....74.5......8..9......13.....8  Patterns game m_b_metcalf jul 21, 2010
1..2....3.....145.....6..1......4.6.....5.1...2.7....86.7......4..8......58.....9  Patterns game joel64  jul 22, 2010
98.7.....7.3...8...41......6..8..9.......5..4....3..5..1.6..2.......4.15....1..3.  Patterns game Pat  Mar 01, 2017
1.2...4...43......79.3.......56..2.......7.8.....9...46..5..1......7..49.....8.7.
..1...2...3...4.5.6.....1.3...7...2.....2.4...7...6...8.3.7...2.4.5...9...6...8..  Patterns game joel64  Aug 04, 2010

Type D

More complicated, there can be 2 triple, 4 double, and 2 single links (B3-2-2-1-3-1-2-2-):

..1...2...2...3.1.4.....5.6...7...8.....5.3...1...6...5.2.7...9.3.8...7...6...4..
..1...2...2...3.1.4.....5.6...2...7.....4.1...3...8...5.6.8...1.7.9...8...2...4..
..1...2...2...3.4.5.4.....6.....4.3.....5.....7.6.8...1.....9.4.8.7...2...9...5..
..5...7...9...1.3.6.8.....2.....6.1.....8.....4.3.7...7.....8.6.5.9...2...2...5..
..1...2...2...3.4.4.....5.6.....7.8.....2.....3.1.6...6.5.....1.7.8...9...2...4..  Patterns game m_b_metcalf May 25, 2012
..1...2...2...3.1.4.....5.6...5...7.....1.8...7...6...2.4.9...5.9.3...8...6...1..  Patterns game joel64  Aug 04, 2010

Or slightly different (B3-2-2-2-1-3-2-1-):

..1...2...2...3.4.4.....5.6.....7.8.....5.....3.2.9...6.4.....1.7.8...9...2...4..  Patterns game gsf May 25, 2012

Type E

There can be 3 triple and 3 single links, and 2 double links (B3-1-3-1-3-1-2-2-)

...1.2......3.4.....1.5.6...7.....8...32.64..9.......33.......9..2.4.1...8.....7.  patterns game JPF, Feb 19, 2009
...1.2......3.4.....4.5.2...6.....7...72.51..8.......99.......8..2.1.4...7.....6.  patterns game JPF, Feb 19, 2009
..1...2...3...1.4.5.....1.6...4...7.....5.8...7...3...9.6.8...2.4.3...8...3...9..  Patterns game joel64  Aug 04, 2010
...1.2......4.7.....4.5.2...6.....7...72.51..8.......99.......8..2.1.4...7.....6.  patterns game Pat, Feb 20, 2009

Type E sub-type

These have a solved cell in place of one of the usual occupied cells, leaving 15 links: (B3-1-3-1-3-1-2-1-):

...1.2......3.4.....1.5.2...6.....4...57.18..9.......77.......6..2.8.5...9.....3.  patterns game JPF, Feb 20, 2009
..1..23...6..1..5.7.......4...2.......3.7.1.......3...5.......7.4..5..6...81..2..  patterns game jsf, Jan 07, 2009

Type F

The following puzzles contain an SK loop which can be coded as a normal SK, all links pairs, or as a variant with a single link between the boxes and a triple link within the boxes.

Please note that in the following types, the software gives full details for both loops, but displays only the second of the two. If the first has more eliminations than the second, run the program again.

..1..23...4..5..6.7.......8...8.......5.3.2.......4...8.......7.6..9..4...23..1..  patterns game m_b_metcalf, Jan 05, 2009
....8.......621.....83.56....6.5.7...1.....4.4.......2..7.6.5...4.....3.2.......1  patterns game ronk, Feb 06, 2011
.123.....45.......3.6...7..6..7..4......2...8.....5.9...46..3.......9.5.....8...2  patterns game Dobrichev, May 19, 2018
.123.....34.......5.6...3..7..8..9......1...5.....2.4...89..7.......4.2.....2...1  patterns game Papy999, May 20, 2018
.123.....43.......5.6...2..6..7..3......8...1.....9.4...72..6.......1.8.....4...9  patterns game Champagne, May 19, 2018
.123.....45.......3.6...2..2..7..3......5...8.....8.1...32..9.......1.6.....8...5  patterns game Dobrichev, May 19, 2018
12.......3.4...2...567.......26..8......3...9.....4.1..8.2..6.......1..4....9..3.  patterns game Champagne, Mar 01, 2018
12.......3.4...5...672.......23..8......4...1.....6.7..5.8..3.......1..4....7..6.  patterns game Patrice, Mar 02, 2018
12.......3.4...5...672.......85..9......4...1.....6.7..5.9..8.......1..4....7..6.  patterns game Champagne, Mar 02, 2018
12.......3.4...1...561.......75..8......2...4.....3.9..8.7..5.......4..2....9..3.  patterns game Patrice, Feb 28, 2018
.12......3.4...5..65.7.......54..7......1...6.....8.2..9.5..4.......2..1....6..8.  patterns game Patrice, Dec 19, 2017
.12......3.4...1..56.1.......65..4......7...3.....8.9..5.4..6.......9..7....3..8.  patterns game Patrice, Dec 19, 2017
..2.....7.4...3.5.7..8..2....63...4.....1.....3...58....7..6..2.5.4...3.2.....1..  patterns game Ronk, Apr 17, 2011
..1..23...4..5..6.6.......7...4.......5.8.1.......1...7.......4.6..1..8...32..5..  patterns game Mauricio, Jan 03, 2009
...1.2......3.4.....1.5.2...6.....7...52.73..7.......48.......7..2.3.1...4.....6.  patterns game JPF, Feb 19, 2009
.5......49......1...6..37......156.....3.6.....289......37..2...9......14......5.  patterns game gsf, Mar 13, 2009
.1......23......4...5..67......685.....9.3.....872......76..8...4......32......1.  patterns game JPF, Mar 14, 2009
..3.....7.9...2.1.8.....4.......6.5.....7...2.2.9.5.....4...8...1.5...9.7...1...3  patterns game Mario133, May 29, 2009
..1.....2.3...4.5.6...7.8.......7....4..6..3....2.......8.9...6.5.3...9.2.....1..  patterns game m_b_metcalf, Jan 04, 2010
1..2....3.....451.....6..7......7.4.....5.6...2.3....18.4......7..8......93.....8  patterns game m_b_metcalf, Jul 21, 2010
1..2....3.....415.....6..7......5.4.....7.6...2.8....14.8......9..3......56.....8  patterns game joel64, Jul 21, 2010
..1...2...3..4..5.6.......7...1.3....8.....3....6.4...2.......6.4..5..8...7...1..  from http://diuf.unifr.ch/people/juillera/Sudoku/InterestingSudokus.html
1.....2.3.4.5...6...7.......5...1.8.....53......89....2.......7.8.6...4.3.....1..  patterns game joel64, Sep 2, 2010
..1...2...4...8.9.6.......7.....3.4.....2.....8.9.5...7.......6.9.4...3...2...1..  38021;ig;Patrice;K; ; ;R;;
1.....2...3...4.5...6.....7.....5.3.....6.....4.1.8...7.......1.5.3...9...2...6..  patterns game m_b_metcalf, Apr 08, 2012
1.....7...2...3.6...4.....5.....9.2.....4.....6.8.2...5.......1.8.6...3...7...4..  patterns game Patrice, Apr 08, 2012
1.2...3...43......56.7.......63..8.......5.2.....1...98..6..7......9..18.....2.9.  patterns game 2252, may 18, 2017

Type G

Alternatively, the following puzzles contain an SK loop which can be coded as a normal SK, all links pairs, or as a variant with a single link within the boxes and triple links between the boxes.

9....21...8..4..5...67....9..8.......3...1...5...8..4.1.....9...4...3.2...9.....6  Patterns game gsf Aug 2009
1....23...4..5..6...74....1..4.......8...9...9...2..8.3.....1...6...5.2...1.....7  Patterns game JPF Aug 2009
..1...2...3...4.5.2.......6...7...3.....8.5...4...9...1...5...2.9.3...7...6...1..  Patterns game JPF Jun 28, 2010
..1...2...3...4.5.6.......1...6...3.....7.8...4...1...1...8...2.8.5...9...2...6..  Patterns game JPF Jun 28, 2010
..3.4.1...4...9.2.6.....3.4.......5.7...1.2...9.......1.6.9...3.7.5...1...4...6..  patterns game ronk Sep 2010
93.8.....1.6...8...84......6..2..9.......1..4....5..3..6.9..2.......3.91....4..5.  Patterns game JPF Mar 2017
7....2..3..68..4...3..9..7.5......2...8...5...1......6.7..6..3...5..42..3..2....1  Patterns Game ronk Jan 01, 2011

Type H

Aditionally, some puzzles have two different variant loops:

.1......22......3...4..56......217.....4.6.....783......67..4...8......13......2.  Patterns game.m_b_metcalf Mar 18, 2009
...1.2......3.4.....4.5.2...6.....7...32.51..1.......88.......1..2.3.4...7.....6.
...1.2......4.7.....4.5.2...6.....7...72.51..8.......99.......8..2.1.4...7.....6.
...1.2......3.4.....1.5.6...7.....8...32.64..9.......33.......9..2.4.1...8.....7.
12.3.....4.5...6...76......8..1..3.......9..5....4..7..9.8..1.......7.59....9..4.  Patterns game m_b_metcalf Mar 05, 2017

This example has within the same 16 cells a normal SK loop, a variant type F and also a variant type G

2...8...7.3.1...5...7...2....6..9....9..6..3....4..9....2...7...5...3.1.7...1...8.  Patterns game Pat Oct 03, 2017

Related Loops

The following are added out of interest, as they share the the same 4 box, 8 cell pair configuration, but are not actually SK loops.

Single digit loops

The Variant SK code also detects single digit loops.
The first type has strong links in rows and columns, with eliminations within the boxes:

1.......9.4..7..1...29.16....36.91......8......57.42....43.27...3..6..4.6.......5  Patterns game.Tarek Sep 29, 2008
1.......2.3..1..4...56.47....24.15......8......37.64....19.52...2..6..7.9.......6  Patterns game.m_b_metcalf Sep 30, 2008
The chain for this example is (8)r3c12 = r3c89 - r12c7 = r89c7 - r7c89 = r7c12 - r89c3 = r12c3 - r3c12: loop, and eliminations are:
r1c2 <> 8; r2c1 <> 8; r1c8 <> 8; r2c9 <> 8; r8c1 <> 8; r9c2 <> 8; r8c9 <> 8; r9c8 <> 8;

The second type has strong links within the boxes, and eliminations in rows and columns:

2...9...6.3.1...5...7...8....6..5....1..8..3....4..9....2...7...5...3.1.7...2...8
.....1..2..12..4...7..3..5..5......4..9.7.8..8......6..8..5..7...6..29..3..8.....
....3......74.69...8.....5.4.......9.2.7.5.8...1...6...5..2..7...6...1......7....
....3......74.69...8.....5.4.......9.2.7.5.8...1...6...5..2..7...6...1......7....
1..2......3..4......5..67..2..7..1...8..5..4...6..4..9..71..2......9..6......8..5
.12...3..45.2.....3...6.....6.4..2....5.7...8.....9.5.2..3..4.......7..9....8..7.
.14...2..37.9.....5.9.....1.9.7....4.....95......8..6.2...6.8.......5.1...14....7
.12.3....4.5...1..67.5.......18..3..5....9.7.....6...4.8.1..5......9..4......7..9
8..6....2.....576.....9..1......1.9.....7.5...6.2....35.9......7..3......16.....8
1.2...3...4.....1.5....67.2...3.14.............87.2...3.74....5.9.....8...5...1.7
..1...2...3...4.5.2.....3.6.....3.7.....8.....5.7.9...5.2.....8.7.4...3...8...1..
..1..23...9..8..4.7.......6...9.......2.7.8.......1...9.......7.4..2..6...83..2..
...1.2......3.4.....1.5.2...3.....6...52.17..1.......88.......6..2.7.5...4.....3.
2.6...3...5.....1.9....56.2...2.79......3......98.1...7.34....9.1.....5...2...7.4
..9.....3.8...1.6.3.....7.......6.8.....4...5.1.2.5.....7...1...6.8...5.4...9...7
1..2....3.4..5..6...3...7..6....5.8..8..4.6.....9.......2.7.3...5.1...4.3.......2
.1...9..37..5...6...2.......2...3..1.........6..7...5.......5...3...1..89..6...7.

With the next one, if one runs SK loop first, both variant and normal SK loops are found. However, if one runs SK loop again, one gets a single digit loop with strong links in the boxes:

1.2...3...43......56.7.......63..8.......5.2.....1...98..6..7......9..18.....2.9.  Patterns game 2252 May 18, 2017

XY-type loops

These are degenerate SK loops, which can also be viewed as XY loops containing various numbers of ALSs

These have eliminations within the boxes. They can alternately be viewed as x-cycles as above:

.12.3....45.6.....7.....8...4...1.9.3...7.2.....9....4..6.8.3.....1....9.....9.2.
1.2...3...43......56.2.......74..2.......8.3.....5...64..1..7......8..53.....5.8.
2....18...8..2..7...57....4..42.6...............9.53..6....27...2..6..8...94....5
12.......3.4...5...653.......78..3......2...4.....6.2..8.5..7.......2..1....1..6.
12.......3.4...5...672.......26..8......3...7.....9.4..8.5..2.......7..9....9..3.
12.3.....3.4...5...67......7..4..8.......2..9....9..1..8.7..3.......9.28....1..5.
1.2...3...45......63.7.......86..7.......9.2.....5...13..8..6......9..35.....1.9.
1.2...3...45......67.8.......14..8.......9.3.....6...74..1..2......9..63.....6.9.
1.2...3...43......56.2.......74..2.......8.3.....5...64..1..7......8..53.....5.8.
1....23...4..5..6...31....7..4.......8...9...7...2..4.3.....7...6...8.5...2.....3
1.2...3...45......67.2.......18..2.......6.3.....9...74..1..8......6..93.....9.6.
..1...7...3...4.1.5.6.....9.....1.3.....7.....8.3.2...7.....9.5.9.4...8...5...6..
..1...2...3...4.5.2.....6.7.....5.8.....7.....5.3.9...3.7.....6.4.8...2...6...1..
...1.2......3.4.....3.5.2...2.....6...42.73..6.......82.......1..7.9.4...1.....8.
1..2....3.4..5..6...3...1..7....8.2..6..4.5.....9.......7.9.3...5.6...4.3.......1
.1......22......3...4..56......378.....2.1.....684......85..4...9......13......2.
.123.....45.......6.7...1..8..1..5......6...2.....9.4...15..8.......4.9.....9...6

While in this example the eliminations are in the rows and/or columns:

12.3.....4.....3....5.4..1.6...1..2...37..6.......8..9.6..2..4...78..9.......9..8

Almost SK Loops

This section is added out of interest, as trial and error is employed.

These are as common as SK loops. They differ by having a linking triple rather than a linking pair in one of the boxes or rows/columns. The bidirectional logic is still there, but an absurdity is introduced by suggesting 3 numbers to be true in the 2 linking cells. The result of this is that in the final solution one is trying to fit 17 numbers into the 16 cells of the SK pattern. The one that is squeezed out leads to just one of the eliminations being false, ie the candidate elimination must be true. One can test this by in turn making each candidate elimination true and the rest false, and then testing to see if it creates a contradiction. In all of the vast number of cases I've tested, this theory holds up. Thus this number can be assigned to the cell, and the remaining andidate eliminations can be carried out.

Almost SK (+1) Loops

The following puzzles have a single triple box link (B3-2-2-2-2-2-2-2-)

.......39.....1..5..3.5.8....8.9...6.7...2...1..4.......9.8..5..2....6..4..7.....; Golden Nugget
98.7.....7.....6....6.5.....4...5.3...79..5......2...1..85..9......1...4.....3.2.; Champagne Dry
1.......7.2.4...6...3...5...9..4........62.4....9..8....5.....3.6.2...8.7....1...; Silver Plate
4....3..2..31..9...7..8..3.2......4...5...2...8......6.3..2..7...9..48..1..6....5; Patterns game Jan 02 2011, gsf
98.7.....6...5......4..93..5......6..7..8......9..24....1.....4....9.1.......1.32; 11;GP;H1;;;2BN H8I8;;
98.7.....76....9....5......4...6..3...75..6.......2..1..86..5......1...2.....3.4.; 36;GP;H5;;;2BN A3B3;;
98.7..6..7..........5.4..7.3.....9...6......8..7..4.5...24...1.....2...3....15...; 154;GP;H19;;;2BN D8D9;;
9876.....65....7...........5...4..3..2......1..68..5....59..8......3..2......1..4; 153;GP;H21;;;2BN A3B3;;G1
........1.6...9.3...2.3.7..4..8.5........6....3.9...8...1.....4.5...8.9.7.....2..; cola199 (gsf morph)ED=10.6/1.2/1.2
........6..3.8..9..4.....85..5.9...8.7...2...1..4.......6.3..5.2....73.....1.....; 11.10;1.20;1.20;col;H300;5010;21
....5..8...67..1.......3..426.9.............19.7...2...3..4..1......8..5..21..6..; 92;elev;41;;;2BN A5B5;;G13
1....6.8....7....3....2.4....5.4....6....8.5....3..2....8..1.9..1......796.......; 182;elev;L14;1;2;r8c3 r9c3 r1c2 r5c2
1....6.8..5....2....9.....4.....86.33...7........1..7..4.9.....8....7.6...2...5..; 67;elev;L6;;;2BN D4E4;;
1....6....571.......9...51..4....3.......8..29..7...6......24..5..6...9.....3...8; 155;elev;129;;;2BN B1C1;;
..1.....9.4...1.3.7...8.6.....5.......21.4....1..3..5...6...2...3.8...4.9.......7; 9671;TkP;3242;S
..34......5...9...7...2...62...7..1..9...5......3....8..1.6...78......21......4..; 4024;elev;L406;11.10;11.10;3.40;881
..1...5...2.4...6.3....7....6.28........9..2.......4.65.....1...9.8...4...7.....3; 11.70;11.70;2.60;col;H2;54;21
1.......9..67...2..8....4......75.3...5..2....6.3......9....8..6...4...1..25...6.; 7;elev;15;;;2BN E6F6;;
.2...67..4...8......93........9..57..1...7..2......61.3...4..6...8.......6...5.2.; 9;tarekdb;tarx0075;;;2BN I4I6;;
1..2...34......2.1.....156...6..24...5..7....8..1....2..4..7....7..6....9..8....3; Patterns game Sep 2010, joel64 (ED=5.6/3.8/3.4!!)
..1..2..3.4.....5.6...7.4.....7.3..2..3...6..8..6.1.....7.1...8.5.....4.1..2..9..; Patterns game Aug 2008, JPF 

The following are examples of the triple link being in a row (B2-3-2-2-2-2-2-2-):

..1...2...3...4.5.2.......6...7...8.....9.1...5...3...9...1...2.4.9...7...6...9..  Patterns game Jun 2010, joel64
68.2.....2.9...8...35......1..6..9.......5..4....7..3..7.9..1.......7.95....3..7.  patterns game Mar 2017, JPF
12.......3.4...5...653.......76..3......1...2.....8.9..8.5..7.......2..8....8..1.  patterns game Mar 2018, champagne
12.......3.4...5...657.......84..7......1...2.....9.3..9.6..4.......2..9....9..1.  patterns game Mar 2018, Patrice
12.......3.4...5...657.......86..4......2...9.....3.1..9.8..6.......9..2....1..9.  patterns game Mar 2018, champagne
8...9...7.6.1...5...7...2....6..5....1..6..3....4..9....8...7...5...8.9.7...2...8  patterns game Oct 2017, Pat

The next are examples of a triple link being in a column:

.12......3.4...2..56.7.......18..9......6...7.....7.5..8.2..1.......5..3....7..6.  patterns game Dec 2017, papy999
1.2...3...43......56.7.......18..5.......7.9.....4...68..5..1......7..54.....6.7.  patterns game May 2017, m_b_metcalf
1.2...3...43......56.7.......48..2.......7.1.....5...68..2..4......7..25.....6.7.  patterns game May 2017, m_b_metcalf

Variant Almost SK Loops

If the total number of links equals 17, and the number of triple links minus the number of single links = 1, then the logic holds.

In the following examples, there is a single box link straddled by a triple row link and a triple column link (3-B1-3-2-2-2-2-2-)

.2.7...1.7...4...8..6...7..3..5.8.......9.......4.3..5..1...6..8...3...9.7...4.2. patterns game Mar 29, 2015, blue
..1..2..3.4.....5.6...7.1.....8.3..2..3...6..9..6.7.....4.8...1.5.....4.8..2..9.. patterns game Aug 23, 2008, JPF
98.7.....6.....9....5....7..4..3..2...85..4.......4..1..69..5......2...3.....1.4. 35;GP;H4; ; ;E; ;;
98.7.....7.6...8......5.....7.4..6....4..3.2.....1...4.6.9..5......4..3......2..1
98.7.....6.....9....5....7..4..3..2...98..4.......4..1..65..8......2...3.....1.4.
98.........79..6......5....7..4..9....6.3..2......5..1.7.5..8....4.1..5......2..3
1....6.8....7..1........5.6..9.4.....7.2...3.8....76..3....1..5.4.9.......2.7....
9.....4.2.2.8...3...5.......6...4.7.....59......63....4.......5.8.3...1.2.....9..  patterns game Sep 2010, champagne
........94....92......7..45..1.3.....7.6..9..8....7..2.3.7..8....6.1....9....5.2.
9..8..7...8..7..6...7..5..48..9..3......6..5......4..74....2..1.9.1.......3...2..
1.2...3...4.....5.6....72.1...1.37.............85.2...7.38....2.6.....9...1...6.3
..1.....2.3...4.5.6...3.8.......8....5..6..7....1.......8.4...1.7.5...2.2.....6..
.12......3...4....5..1...3...16...5..6..7...8.....96.......69....34...2.....8...7
.12......3...4....5..3...6...41...2..6..7...1.....89.......98....32...5.....1...7
87.1.....9.3...8...54......6..8..9.......5..4....7..3..9.7..2.......4.95....3..7.
1...2...3.3.4...5...6...7....3..8....2..5..8....9..2....7...1...9...5.2.3...4...6
..3..6.8.4..7......9..3.....3.....15..5..18.........6.3..9..2...7..4......2..5..1; 48;elev;L3; ; ;E; ;;	
98.76....75....86...6......4..3....2..8.4.5.......1.4...7.8.9.....2...1......4..3
98.7.....6.....97...7.5.....4..3......94..5.......2.14..85..6......4..3......1..2
98.7.....6...5.8....4..8.9.5...6.....7.8....3..9..3.2...1...4.9...3...1......1..2
1.....2.3.3.4...5...6.......5...7.8.....21......58....3.......1.4.7...9.2.....6..  patterns game Sep 2010, JPF
....1.......234.....56.71....1.7.3...8.....2.4.......6..8.9.7...6.....8.2.......4  patterns game Feb 06, 2011 m_b_metcalf
9876.....65....8......7....4....3.2...94..5......1...4..87..9......4...3.....2.1.  42357;GP;12_07;2;1234;R;C; ; ;;
........1....23.4...2.45.....45...1..1..6...78.....9....53...6..7......89.....1..  35071;GP;12_05;2;1789;R;C;

Slight variation (B3-2-2-2-1-3-2-2-):

12.3.....4.5...6...78......9..8..3.......3..6....6..5..8.1..9.......5.72....7..6.

Another slight variation (B3-2-1-3-2-2-2-2-):

.......12........3..23..4....18....5.6..7.8.......9.....85.....9...4.5..47...6... Platinum Blonde

This example has 3 triple links, 3 double links, and 2 single links:

1....2..3..4....5..6..7.1.......3.7.2.......8.3.1.......5.9..4..1....6..8..7....2 (B3-1-3-1-3-2-2-2-) or (3-B1-3-2-2-2-2-2-)
98.7.....6.....7....7.5.....4.....3...98..4......24..1..89..6......1...2.....3.4. (B3-1-3-1-3-2-2-2-)
8.9...7...2.....9.5....48.6...4.52.............37.1...7.58....9.1.....3...6...5.7 (B3-1-3-1-3-2-2-2-)
.12......3...2....4..5...6...78...5..2..1...4.....38.......89....56...7.....4...3 (B3-1-3-1-3-2-2-2-)
1..2....3.4..5..6...7...4..3....8.5..2..9.8.....1.......4.6.7...8.5...2.9.......1 (B3-1-3-1-3-2-2-2-)
38.6.....5.4...8...69......2..1..6.......5..3....9..4..1.9..4.......4.59....3..7. (B3-1-3-2-2-3-1-2-)

This has 4 triple, three single and one double link

.12......3.4...2..56.7.......18..9......6...7.....7.5..8.2..1.......5..3....7..6.  Patterns game papy999 Dec 20, 2017

These have 4 triple links, 3 single links and one double, but also 7 double links and one triple link in a row.

.1......22......3...4..25......614.....7.3.....658......54..6...2......73......1.
9..3....6.....597.....9..3......1.9.....7.5...6.4....35.1......7..6......36.....8  patterns game Jul 2010, champagne
1...2...3.4.1...5...3...1....6..7....5..8..4....5..9....1...3...7...4.8.3...6...2  Patterns game Patrice Sep 28, 2017
1.2...3...43......56.7.......18..5.......7.9.....4...68..5..1......7..54.....6.7.  Patterns game m_b_metcalf May 20, 2017

These have 3 triple and 2 single and 3 double links, or 2 triple, 1 single, and 5 double links

98.7.....6.....7....7.5.....4.....3...98..4......24..1..89..6......1...2.....3.4.  104;GP;H13; ; ;E; ;;
.7......64......9...5..17......735.....4.9.....286......15..2...6......79......4.  Patterns game jsf Mar 18, 2009

This has 3 triples, 4 singles and one double, but also 2 solved cells:

...1.2......3.4.....4.5.1...1.....6...32.57..5.......88.......5..2.7.3...6.....1.  Patterns game JPF Feb 21, 2009

Almost SK (+2) Loops

This approach can be extended to accomodate loops with 18 links in the loop, ie 2 more than the normal SK loop. In line with the above approach, it is proposed that 2 of the eliminations predicted by the weak links will be untrue, and therefore must be true. This again holds up in all the examples tested. The famous "Fata Morgana" puzzle has this pattern, and subsequently solves with basic moves (fish & ALS)

The following have 2 triple links and 6 double links:

1..8....3.4.....5...2..6...6..1.37......2......36.9..8...5..2...7.....4.8....1..9  Patterns game JPF Mar 28, 2017
1.......2.3..4..5...23.56....37.45......8......75.61....14.82...9..7..4.2.......6  Patterns game Mauricio Oct 08, 2008
..1..2..3.4.....5.6...7.1.....6.3..8..2...6..8..7.9.....9.3...1.5.....4.1..2..7..  Patterns game JPF Aug 23, 2008
..1..2..3.4.....5.3...6.1.....4.3..6..2...7..6..7.1.....3.8...1.5.....4.7..2..9..  Patterns game JPF Aug 26, 2008 ED=4.5/4.5/2.6
....5..8.4....9..3...2..1...6...3.....18...3.93......7..2...5...7...4..66..5.....  98;elev;L8; ; ; ;R;;
...4..78......92..6...7...1.6..1...53..........5..84...3.....2.71.9.......6.3...7  145;elev;118; ; ;E; ;;
1..4..6...2...5.....3....7.8...1...4...8..9...5...3...4...8.1....7....3....9....2
1..2......3..4......2..56..4..5..7...8..3..5...5..2..9..76..5......2..8......9..1
3....49...6..5..3...71....2..13.5...............4.28..8....12...9..2..6...25....7
7....46...9..2..8...31....2..52.1...............5.74..6....52...8..1..7...24....3
.7.3..5..4...9...6..1..8...8..1......9..7.3....6..2.4.2...8.7.......4.2..5......1
.....1..2....3..4...56..7....6.....3.5...248.8..9..5....8.....4.6.8..9..9.7.1....
.....1..2..2....3..4..5.6......7..1..7..685..9..3.......6...8...8.5.6..74.7.8....
98.7..6..75.....9...6...7...4..3......58...6......2..1.3....2....85...7......1..4
.....1..2....3..4...56..7....8.....1.5.9..6..6....2....86..9.3..97.4...85..8..9..
1..2......3..4......2..56..4..5..7...8..3..5...5..2..9..76..5......2..8......9..1  Patterns game Champagne Apr 12, 2017
..12......3...45..4......6.6..1...7.....8.....9...3..8.5......4..27...1......89..
...4..78..5...9..68.........1.....9.3....15..6.5.....1......42..6...5..3..27.....
........1..2..3.4..4..5.6.....7..5.6.6..8..1.9.......8..39......5..6...47....2...  36067;GP;12_05; ; ;E; ;;
.....6.7..9...8.....12..3..7......9...3.2.8...5......4..8..92.....3...5..4.1.....  Patterns game jsf Feb 05, 2009
.....1.2..3...4.....56..1..7......3...8.6.4...2......9..4..65.....8...7..9.5.....  Patterns game JPF Feb 05, 2009
.....1.2..3...4.....56..7..8......3...1.7.5...2......6..4..51.....3...8..6.9.....  Patterns game m_b_metcalf Feb 06, 2009
.....6....5..8..2...41....77.8....9.1...9...6.9....4.34....71...8..5..3....6.....  Patterns game Patrice Aug 14, 2014
.....1....2..3..4...56....77.2....8.8...9...5.9....4.66....57...8..4..3....1.....  Patterns game champagne Aug 12, 2014
...........1.9.6...9.5.6.8...71..4...2...7.5...6.4.7...1.8.2.6...9.7.1...........  Patterns game 1to9only Dec 31, 2018
1..2..3...4..5..6...5..1...5.7....8.....8.....8....4.6...9..7...5..7..4...9..3..2  Patterns game champagne Oct 15, 2015
.....1..2..3.4..5..6.7.......8.....4.5..3..6.1.....9.......21...4..5..8.7..9.....  Patterns game joel64 Jan 28, 2011
.....1..2..3.4..5..6.7.......8.....1.9..5..3.2.....7.......28...5..9..6.7..6.....  Patterns game m_b_metcalf Jan 28, 2011
..1..2..3....6..5.6..9..8....71....4.5.....7.8.....9....2..3..7.6..5....9..8..3..  Patterns game gsf Feb 14, 2011
98.7..6..76..9......5..6...6...4.3....8....5......2..14....1.8..3..7.9..........2  41962;GP;12_07;2;1258;R;C;
1...2.....3...45....67...8...8....7.9.......2.4....3...5...3..4..72...9.....1.6..  Patterns game m_b_metcalf Jan 11, 2013
2.....1...9..5..7....4....3..67.5.8.....6.....7.9.86..4....3....8..9..6...1.....2  Patterns game Patrice Nov 29, 2017
1.......2.3.4..5....6....7..5..8.......9.2..1....36....9....3....7....2.2...1...6  Patterns game champagne May 29, 2013

These have 3 triple links and one single link, and 4 double links. The last also has 2 triple links and 6 double links

1....2..3.4..5.6.....7...8...91....2.2..7.1..3....5....8..6.7....6....4.5..9....6  Patterns game champagne May 31, 2019
3....84...5..3..9...21....7..85.6...............3.16..4....39...9..1..5...79....2
1.2...3...4.....5.6....78.1...3.26......9......34.5...8.79....2.1.....8...6...1.7
.1...2..32..3...4...5...6...2..3...1...7.4...8...2..9...6...5...9...7..44..1...8. Patterns game Robbie May 20, 2019
3....61...9..4..8...51....4..72.1...............7.45..6....37...8..1..9...24....1
98.7.....6.....9....5.6..7.4......3...86..4......24..1.3...6.4...65..7......1...2
5...3...7.4.9...5...6...2....2..6....1..7..3....4..9....8...5...9...8.1.7...4...8 Patterns game Pat Sep 23, 2017
12.......3..4..2....5.2..3..6..3..1...78..6.......5..9.9..1..2...85..1.......7..6 Patterns game joel64 Mar 12, 2014

These have 4 triple links, 2 single and 2 double:

........3..1..56...9..4..7......9.5.7.......8.5.4.2....8..2..9...35..1..6........ Fata Morgana
1..2....3.4..5..6...7...8..2....1.9..6..7.2.....3.......8.4.6...5.9...8.3.......1 Patterns game champagne, Sep 27, 2015
1..2....3.3..4..5...2...6..7....5.8..9..1.7.....4.......6.2.9...5.8...4.3.......1 Patterns game m_b_metcalf, Sep 25, 2015
..72......5...43..1......6.6..1...3.....5.....4...9..8.3......5..27...4......89..
..1..23...6..5..4.7.......9...5.......5.1.8.......7...9.......7.4..8..6...31..2.. Patterns game gsf Jan 03, 2009 
..1..23...4..5..6.7.......8...3.......5.4.9.......5...8.......7.6..9..4...31..2.. Patterns game m_b_metcalf Jan 04, 2009
..1..23...4..5..6.7.......8...8.......5.3.2.......4...8.......7.6..9..4...23..1.. Patterns game m_b_metcalf Jan 05, 2009
..1..23...6..8..4.7.......9...5.......2.3.5.......4...9.......7.4..5..6...31..2.. Patterns game gsf Jan 06, 2009
6....4..9..28..1...9..3..7.5......4...8...2...2......8.1..7..6...5..34..3..2....1 Patterns game ronk Jan 07, 2011
..1.2..3.3.......4.5....6...7.8.......4.3...1.....5...9...4...3.8...67....2....9. Patterns game Robbie May 13, 2019

This has an almost (+2) loop with 4 triple links, 2 single and 2 double links, but also has a separate Type A normal variant loop:

1..4....7.2..5..8...3...9..4....5.6..6..7.2.....1.......9.1.3...8.6...5.7.......4 Patterns game CSC, Sep 21, 2015

These has 5 triple links and 3 single links:

1..2......3..4..5...6..7...9.....6...4..5..2...7.....9...4..9...2..1..3......6..7
4..9......8..2..3...9..7...7.....4...3..6..2...1.....8...4..6...2..3..8......9..7
5..2......3..9..1...6..4...1.....6...4..3..8...2.....9...4..2...7..8..3......6..4 Patterns game gsf Apr 18, 2012
1..3......3..4..5...6..7...7.....6...8..5..1...1.....9...6..9...2..1..4......9..7 Patterns game gsf Apr 18, 2012

The following has only one triple link, but a solved cell that increases the link count to 18,

.....1..6..64..9...8..6..3..2......9..7.8.1..5......4..9..3..5...3..67..4..9..... Patterns game 1to9only Apr 07, 2019
while this next one there are only 15 links but 3 solved cells, raising count to 18:
..1.6..4.6.......3.2....7...8.7.......5.4...1.....9...4...3...6.9...82....3....1. Patterns game 1to9only May 14, 2019

Almost (+3) SK Loops

Not surprisingly, if the total number of links equals 19, then the logic again holds. Three of the "eliminations" are in fact true, in line with the above cases.

The first examples have 3 triple links and 5 double links:

1..2....3.4..3..5...5...6..7....8.2..9..6.3.....1.......3.4.5...5.9...3.8.......7 Patterns game champagne, Sep 20, 2015
..12...3........456...5.2....4..7....3..8.9..9..1....7.....9..1.7..3....2..6..5.. Patterns game patrice, Dec 02, 2011
.1..2.3..2..1...4...6..7..1.2..3...54..2...8...7..6...9.........8..5..1...59....2 Patterns game jouinJP, Dec 23, 2013
.1...2..34..5...7...8.......2...6..1.........9..4...8.......4...6...1..25..8...9. Patterns game jsf, May 02, 2009
..1..23...4..5..6.7.......8...1.......2.4.9.......7...3.......1.6..9..7...58..2.. Patterns game m_b_metcalf Jan 04, 2009
1...2.....3...45....67...1...7....2.3.......8.5....3...8...5..9..12...4.....7.6.. Patterns game m_b_metcalf Jan 13, 2013
..12...3......45..2...6...77..6.8.9.....9.....9.7.2..86...2...9..45......8...61.. Patterns game m_b_metcalf Dec 30, 2015
1..2......3..4.5....6..7.8.8..1...3..9..6...7..5...9...7...43....23.........5...6 Patterns game champagne, Oct 16, 2017
..12..3...4...52..2...1...61...3...7..21..8...5...4...31..8.7.........98..97...3. Patterns game m_b_metcalf Jan 18, 2018
..12...3.4...3...5.6....7..8..9.3.4.....8.....3.4.5..8..7....6.3...4...9.4...12.. Patterns game champagne, Feb 25, 2015
12.3.....4.5...6...67....1.2..1...4.....2.8.......7..9.5..8.4....24...3......9..7 Patterns game m_b_metcalf Oct 24, 2013

These examples have 3 triple links and 5 double links, or 4 triple links and one single link:

98.7.....7...8.6....5..4...6..3..9...3.........4..2.1..7.8..3.......1..2.......5. 38678;GP;12_07;2;1245;R;C;
1..2....3.2..4..5...4...2..3....6.7..8..5.9.....1.......2.8.5...9.7...2.6.......7 Patterns game champagne, Sep 19, 2015
1..2....3.2.....4...5..67..8....95......3......97....6..79..2...4.....1.3....8..7 Patterns game champagne, Feb 01, 2015

These examples have 4 triple links, 3 double links and one single link. The second also has 3 triple and 5 double links. The third similarly has 3 triple and 5 double links, but also a type A variant loop

1..2....3.4..5..6...5...7..5....1.4..2..3.8.....9.......8.7.6...3.4...7.6.......9 Patterns game m_b_metcalf, Sep 18, 2015
1....2..3..4....5..5..6.7.......4.2.8.......9.9.7.......5.8..4..7....9..3..2....1 Patterns game patrice, Nov 21, 2017
..12...3........456...3.2....6..7....1..8.3..9..5....2.....4..7.8..9....2..1..9.. Patterns game champagne, Dec 01, 2011
.1.2.3.4.4...5...6......7..2..4.6.8...........6.9.5..3..8......6...9...4.7.3.2.1. Patterns game jouinJP, Jan 20, 2014
1...2...3.4.5...6...7...8....5..9....2..5..9....4..6....8...7...9...5.3.3...1...6 Patterns game m_b_metcalf Sep 20, 2017
1..2......2..3......4..56..4..6..7...3..4..1...5..3..4..87..9......1..4......8..5 ED=5.0/5.0/3.8 - m_b_metcalf
..1...2...3...4.5.6.......7...6...8.....7.9...4...5...7...9...6.6.8...3...2...1..

This example has 5 triple links, 2 single, and 1 double link:

7..9....6.9..5..3...8...2..6....8.5..1..2.4.....7.......4.9.8...5.6...4.2.......7  Patterns game Pat Sep 24, 2015

These have 4 triple links, 2 single links, and 2 double links, but also one solved cell, so the link count is increased from 18 to 19.

.1...2..94..5...6...9.......2...1..7.........3..8...4.......8...7...9..18..4...3.
...........1.2.3...2.4.5.6...27..6...6...8.7...8.5.9...5.8.7.9...3.9.1...........  Patterns game Robbie Dec 31, 2018
.1...2..37..4...9...6.......2...5..1.........6..7...3.......4...5...1..29..6...7.  Patterns game gsf May 05, 2009

These have 2 triple links, 1 single link and 5 double links, or 3 triple links, 2 single links and 3 double links. There are 2 given cells, so the link count is increased from 17 to 19.

.12.....34.5..2...673.........7...5.....8.9.6.2...4.......9.4.....5...9.3...1...8  Patterns game Robbie Dec 31, 2018
1.......3.6..2..4...56.78....92.64.............41.85....67.29...7..6..2.9.......4  Patterns game gsf Nov 14, 2008
..1...2...2.1.3.4.5.......6.1.3.6.7...........3.4.7.2.1.......7.6.7.4.3...8...9..