## Domino Loops (SK Loops & Beyond)

Advanced methods and approaches for solving Sudoku puzzles

### Domino Loops (SK Loops & Beyond)

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1. Introduction
This thread summarises the contributions of Blue, Champagne, Leren, Obi-Wahn, & Ronk on SK Loop type structures in the <Exotic Patterns a Resumé> thread, and suggests the Domino Loop (DLoop) term as a collective name for them. This relates to the pattern of the containing cells which is constant regardless of the other differences that have been discovered.

These loops are not AICs and have led to several quite experienced players stumbling when trying to apply strong and weak links to terms that aren't Boolean. Attempting to overcome this, their workings are explained considering them as locked set terms.

The patterns can also be expressed as XSudo rank 0 structures or as multi-sector locked sets which allows them to be formulated in a large variety of ways. However the only formulation used here is that given by the DLoop structure because, not only is it simpler, but it also provides inferences that are potentially useful in later solution steps.

I hope that I've been fair to everyone and that the Domino Loop term wins general approval. Feedback on this and any bludners or omissions I've made would be welcomed.

DPB
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TAGdpbDominoLoops
David P Bird
2010 Supporter

Posts: 1043
Joined: 16 September 2008
Location: Middle England

### Re: Domino Loops (SK Loops & Beyond)

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2. Domino Loop Definition
Domino Loops consist entirely of arguments for the digits that can occupy a loop of cell pairs. When they occur the grid can be rearranged so that all the cells pairs resemble dominos with pivot cells containing givens.
Code: Select all
` *--------*--------*--------* | * H H  | . . .  | G G *  |   | A . .  | . . .  | . . F  |  Lettered cell pairs all   | A . .  | . . .  | . . F  |  contain four candidates.   *--------*--------*--------* | . . .  | . . .  | . . .  |  * = Pivot Givens | . . .  | . . .  | . . .  |  | . . .  | . . .  | . . .  |  *--------*--------*--------* | B . .  | . . .  | . . E  |  | B . .  | . . .  | . . E  |  | * C C  | . . .  | D D *  |  *--------*--------*--------*`

SK Loops, discovered by Stephen Kurzhals, were the first form of domino loop to be identified. They are recognisable when a 'signature' pattern of givens exist in the boxes holding the loop when the domino cells are linked by digit pairs. These digits pairs are then locked in the two linked dominos and can be eliminated from the other cells in the same house. These have been described as either hidden or naked pair loops according to the viewpoint taken.

However, later discoveries have shown that
1) Domino loops can exist with other distributions of givens
2) The number of common digits between adjacent dominos can alternate between 1 and 3.
The confusion in terminology that these findings have caused is the reason for suggesting the adoption of the 'domino loop' term.

In notating a loop, all the candidates in the domino nodes must be listed, split into two groups. This usually will be in pairs with two linking digits between adjacent dominos in the loop giving a pattern which is customarily shown as:
(ab=cd)domino1 - (cd=ef)domino2 - (ef=gh)domino3 – etc
However there are no strong links here and it would be more accurate to use:
(ab-cd)domino1 - (cd-ef)domino2 - (ef-gh)domino3 – etc
With every node containing 4 candidates, provided it's possible to complete this loop using this pattern, it demonstrates how the truths may be distributed in the 16 cells. Each domino must hold two truths so if (ab)domino1 holds x truths, (cd) must hold (2-x) truths leaving (cd)domino2 only capable of holding x truths - and so on until the loop returns to domino 1 again. Hence the truth splits between the left- and right- hand terms throughout the loop will always be same and could be 0/1, 1/1, or 0/2. Now, considering digits (c) & (d), either one will be true on domino1 and the other in domino2 or they will both be true in one of them. Therefore (c) & (d) can be eliminated from the other cells in the same house, and likewise for all the other linking pairs in the loop.

The loop still operates when it is partially solved, eg when one candidate is locked in one domino and absent from the neighbouring one. The only effect this has is that one of the ways of splitting the truths in a node will no longer be possible.

When the groups consist of triplets and singles the pattern becomes:
(a=bcd)domino1 - (bcd=e)domino2 - (e=fgh)domino3 -.
The single terms will hold 0 or 1 truths and the triplets will hold 1 or 2 truths which creates a repeating pattern of the number of truths in the left and right hand terms. Again provided the loop can be completed, regardless of how the triplet is split between them, dominos 1&2 must contain (bcd) and similarly dominos 2&3 must contain (e) giving equivalent eliminations.

In practice Domino Loops prove to be a useful way to recognise and present a Multi-sector Locked Sets or XSudo 0 Rank patterns. The 16 cells in the loop are all truths to be satisfied by the cover sets formed using the linking digits in each house. However the loop notation provides additional derived inferences which can sometimes be useful later in a solution as will be shown.
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David P Bird
2010 Supporter

Posts: 1043
Joined: 16 September 2008
Location: Middle England

### Re: Domino Loops (SK Loops & Beyond)

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3. The Classic SK Loop
1.....9...3...7.4...5.....2.....6.7.....1.....4.3.8...9.......5.7.8...3...2...1..#8668;tax
Code: Select all
` *--------*--------*--------* | 1 . .  | . . .  | 9 . .  |  | . 3*.  | . . .  | . 4*.  |  | . . 5  | . . .  | . . 2  |  *--------*--------*--------* | . . .  | . . .  | . . .  |  | . . .  | . . .  | . . .  |  | . . .  | . . .  | . . .  |  *--------*--------*--------* | 9 . .  | . . .  | . . 5  |  | . 7*.  | . . .  | . 3*.  |  | . . 2  | . . .  | 1 . .  |  *--------*--------*--------*`

Inspecting the givens for this puzzle, four of them in different boxes at r28c28 make a rectangle and are accompanied by two other diagonal givens to fill in each box (known as base cells). The four cells are potential pivot cells for the domino cell pairs in a possible loop. The two sets of digits held by the four pivot cells (347) and the other eight base* givens (1259) should have no digits in common. If these conditions are satisfied then an SK loop will probably exist, and should be investigated further by checking that there are no more than four candidates in the cell pairs in the loop. If this is so then the linking digits in the boxes will be drawn from the base set and those in the rows and columns from the complementary digits.

* This term is used because if an Exocet pattern exists in combination with the loop, these digits will occupy its base cells.

Code: Select all
`       *-----------------------*-----------------------*-----------------------*       | <1>    268    47      | 2456   234568 2345    | <9>    568    37      | (29)b1 (68)r2 | 268    <3*>   689     | 1259-6 259-68 <7>     | 568    <4*>   168     |        | 47     689    <5>     | 1469   34689  1349    | 37     168    <2>     | (15)b3       *-----------------------*-----------------------*-----------------------*       | 2358   1259-8 1389    | 2459   2459   <6>     | 23458  <7>    13489   |        | 235678 259-68 36789   | 24579  <1>    2459    | 234568 259-68 34689   |        | 2567   <4>    1679    | <3>    2579   <8>     | 256    1259-6 169     |        *-----------------------*-----------------------*-----------------------*       | <9>    168    3468-1  | 12467  23467  1234    | 4678-2 268    <5>     | (15)b7(46)r8 | 456    <7*>   146     | <8>    259-46 1259-4  | 246    <3*>   469     |         | 3468-5 568    <2>     | 45679  345679 3459    | <1>    689    4678-9  | (29)b9        *-----------------------*-----------------------*-----------------------*                (68)c2                                          (68)c8`

This is the SK loop with the pivot givens asterisked. There are of 8 nodes of two cells each contained by four boxes, two rows, and two columns and their linking pairs in each house are shown alongside the grid:
(29=68)r13c2 - (68=15)r79c2 - (15=46)r8c13 - (46=29)r8c79 -
(29=68)r79c8 - (68=15)r13c8 - (15=68)r2c79 - (68=29)r2c13 - SK Loop
=> 16 Elims: 8r4c2, 68r5c2, 1r7c3, 5r9c1, 46r8c5, 4r8c6, 2r7c7, 9r9c9, 68r5c8, 6r6c8, 6r2c4, 68r2c5
Each repeating digit pair will be locked in the four pattern cells so eliminating them from the other cells in the same house.

Unusually, because of the distribution of the (6)s, there is a second way to build a loop through these cells using a 1/3 digit split:
(296=8)r13c2 - (8=156)r79c2 - (156=4)r8c13 - (4=269)r8c79 -
(296=8)r79c8 - (8=156)r13c8 - (156=8)r2c79 - (8=269)r2c13 - DLoop
=> 14 Elims: 8r45c2, 16r7c3, 56r9c1, 4r8c56, 26r7c7, 69r9c9, 8r5c8, 8r2c5

Depending on the loop used, one or other (6)4-Fish will be exposed and, once those eliminations are made, the same grid is produced.

The reduced grid contains this sub-pattern which derives an extra inference using the SK Loop.
Code: Select all
` *----------*----------*----------* | 1  .  .  | 5  5  5  | .  5  .  |  | .  .  .  | 15 5  .  | 5  .  1  |  | .  .  5  | 1  .  1  | .  1  .  |  *----------*----------*----------* | 5  15 1  | 5  5  .  | 5  .  1  |  | 5  5  .  | 5  1  5  | 5  5  .  |  | 5  .  1  | .  5  .  | 5  15 1  |  *----------*----------*----------* | .  1  .  | 1  .  1  | .  .  5  |  | 5  .  1  | .  5  15 | .  .  .  |  | .  5  .  | 5  5  5  | 1  .  .  |  *----------*----------*----------*`

Consider the right hand pairs (15)r79c2 & (15)13c8 in the SK loop which must hold the same number of truths.
a) If they both hold no truths then (1,5)r4,5c2 & (5,1)r5,6c8 will be forced producing a contradiction for (5).
b) If they both hold two truths then (1,5)r2c4,5 & (5,1)r8c5,6 will be forced giving a second (5) contradiction.
The derived inference is that each digit pair in the loop is therefore a Single Truth Pair (STP) with a conjugate inference between the two digits.

This sub-pattern exists when the same base digits occur in diagonal boxes in the loop, digits (1&5), and one of them is a given, (1), in a cell, r5c5, where it restricts how the non-loop cells in the rows and columns can be occupied. When only one of the restrictions a) & b) operates, the derived inference will be weaker; the left hand digit pairs in each node must hold 0 or 1 truths and the right hand one must hold 1 or 2 truths or vice versa.
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David P Bird
2010 Supporter

Posts: 1043
Joined: 16 September 2008
Location: Middle England

### Re: Domino Loops (SK Loops & Beyond)

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4. SK Loop Close Relation?
This loop is very close to a SK loop but all four diagonals of givens in the boxes hold the same digit, digit (1). Consequently (1) is never a member of a linking pair which makes it different. It's therefore an open question whether it should be considered an SK loop or not.

..6...3...97..5...18..........2....5....67.1..4...169.2...781......1.98....3....4 Blue 2011b
Code: Select all
`       *----------------------*----------------------*----------------------*       | 45     25     <6>    | 1789-4 248    249    | <3>    2457   1789-2 |         | 34     <9>    <7>    | 168-4  2348   <5>    | 248    246    168-2  |        | <1>    <8>    2345   | 679-4  2349   23469  | 2457   24567  679-2  |        *----------------------*----------------------*----------------------*(34)r4 | 6789-3 167-3  189-3  | <2*>   3489   349    | 478    347    <5*>   |        | 3589   235    23589  | 4589   <6>    <7>    | 24-8   <1>    238    |(89)b5       | 3578   <4>    2358   | 58     35-8   <1>    | <6>    <9>    2378   |(78)b6       *----------------------*----------------------*----------------------*       | <2>    356    49     | 49     <7>    <8>    | <1>    35-6   36     |        | 34567  3567   345    | 456    <1>    24-6   | <9>    <8>    2367   |(69)b8(25)r9 | 6789-5 167-5  189-5  | <3*>   259    269    | 257    2567   <4*>   |(67)b9       *----------------------*----------------------*----------------------*                                 (45)c4                               (23)c9`

(89=45)r56c4 - (45=69)r78c4 - (69=25)r9c56 - (25=67)r9c78 -
(67=23)r78c9 - (23=78)r56c9 - (78=34)r4c78 - (34=89)r4c56 – SK Loop or DLoop?
=> 16 Elims: 4r123c4, 6r8c6, 5r9c123, 6r7c8, 2r123c9, 8r5c7, 3r4c123, 8r6c5

The pivot set is (2345) and the apparent base set is (168) but with (1) is a given in all four boxes it won't appear in the loop which actually uses the base set (6789). This requires (7) & (9) to be givens in the other diagonals through the pivot digits in each box.

In this case the base digits can be deduced by inspecting the other digits in the cell pairs when the 4 digits in the pivot set are ignored.

Using the Derived Inferences
Opening 5 Chains: Show
01: (2)r1c2 = r5c2 - r6c3 = r6c9 - r8c9 = r8c6 => r1c6 <> 2
02: (3)r3c6 = (3)r4c6 - (3=5)r6c5 - (5)r6c4 = (5-6)r8c4 = (6)r9c6 => r3c6 <> 6 (Single(6)r9c6)
. . . .(167)HiddenTriple:r123c4 = r12c4 <> 8, r13c4 <> 9
. . . .(6)boxline:b9c9 => r23c9 <> 6
. . . .(8)boxline:b2c5 => r4c5 <> 8
Now r9c56 as can't hold (25) the left hand DLoop pairs can't hold two truths, but before this can be used some further eliminations are needed.
03: (3)r4c6 = (3)r3c6 - (3)r2c3 = (3-4)r2c1 = (4)r1c1 - (4=9)r1c6 => r4c6 <> 9
. . . .(9)linebox:c6b2 => r3c5 <> 9
04: (9)r3c9 = (9-1)r1c9 = (1-7)r1c4 = (7)r3c4 => r3c9 <> 7
. . . . Singles (9)r3c9,r1c6
05: (9)r4c5 = r9c5 - r7c4 = r7c3 => r4c3 <> 9

Code: Select all
`       *----------------------------*----------------------------*----------------------------*       | 45       25       <6>      | 17       248      9        | <3>      2457     178      |        | 34       <9>      <7>      | 16       2348     <5>      | 248      246      18       |        | <1>      <8>      2345     | 67       234      234      | 2457     24567    9        |        *----------------------------*----------------------------*----------------------------*(39)r4 | 6789     167  f   18    e  | <2>      349      34       | 478 d    347 d    <5>      |        | 3589     235      23589    | 4589     <6>      <7>      | 24       <1>      238   c  |(89)b5        | 3578     <4>      2358     | 58       35       <1>      | <6>      <9>      2378  c  |(78)b6       *----------------------------*----------------------------*----------------------------*       | <2>      356      49       | 49       <7>      <8>      | <1>      35       36    b  |        | 34567    3567     345      | 45       <1>      24       | <9>      <8>      2367  b  |(69)b8  (25)r9 | 789      17  g    189      | <3>      259      6        | 257 a    257 a    <4>      |(67)b9        *----------------------------*----------------------------*----------------------------*                                  (45)c4                                    (23)c9`

With these reductions in the cell pairs it's now possible to use the loop inference that the left hand terms must hold 1 or 2 truths and the left hand terms must hold 0 or 1 in this segment of the loop:
"(25=67)r9c78 - (67=23)r78c9 - (23=78)r56c9 - (78=34)r4c78 -".

06: (7)r9c78(RH1) = (78)r4c78(LH2) - (8=1)r4c3 - (1)r4c2 = (1)r9c2 => r9c2 <> 7
. . . (or (7)r9c78 = (67-2|3)r78c9 = (23-7|8)r56c9 = (78)r4c78 - (8=1)r4c3 - (1)r4c2 = (1)r9c2 => r9c2 <> 7)
. . . Singles (1)r9c2,r4c3
07: (257)r9c578 = (7)r8c9 – (7)r6c9 = (7)r6c1 – (7)r9c1 = (257)r9c578 => r9c5 <> 9
Finish: Show
. . Singles (9)r4c5,r7c4, (4)r7c3
. . (4)LineBox:c5b2 => r3c6 <> 4
. . (7)LineBox:r9b9 => r8c9 <> 7
08: (3=2)r3c6 - (2=4)r8c6 - (4=5)r8c4 - (5=3)r8c3 => r3c3 <> 3
. . Singles: (3)r2c1, (4)r1c1
. . (5)XWing:r17c28 => r58c2, r29c8 <> 5
09: (5)r8c4 = (5-3)r8c3 = (3)r78c2 - (3=2)r5c2 - (2=4)r5c7 - (4)r5c4 = (4)r8c4 => r8c4 <> 4
. . Singles to the end
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Last edited by David P Bird on Sat Nov 14, 2015 10:06 am, edited 1 time in total.
David P Bird
2010 Supporter

Posts: 1043
Joined: 16 September 2008
Location: Middle England

### Re: Domino Loops (SK Loops & Beyond)

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5. Champagne's Virus Loop
This loop with linking pairs is far away from the SK pattern template and was responsible for bringing the naming issue to light. Expecting that similar loops could exist in other puzzles, Champagne called it a Virus Loop, but so far this is the only example that has been presented. They must surely exist but may be confined to simpler puzzles where they aren't needed.

.........1....72...7..84.6...8....93.6..4..7.93....6...9.73..8...59....2.........
Code: Select all
`       *-----------------------*-----------------------*-----------------------*       | 468-5  458    469     | 156    2      3       | 4789-1 145    4789-1  | (25)b1       | <1>    458    346     | 56     9      <7>     | <2>    345    48      | (39)r3 | 235    <7*>   239     | 15     <8>    <4>     | 139    <6*>   19      | (15)b3       *-----------------------*-----------------------*-----------------------*       | 457    15-4   <8>     | 2      1567   156     | 145    <9>    <3>     |        | 25     <6>    12      | 3      <4>    9       | 158    <7>    158     |        | <9>    <3>    147     | 8      157    15      | <6>    2      145     |        *-----------------------*-----------------------*-----------------------*(46)r7 | 46     <9*>   146     | <7>    <3>    2       | 145    <8*>   1456    | (12)b7       | 34678  148    <5>     | <9>    16     168     | 347-1  134    <2>     |        | 3678   2      367-1   | 4      156    1568    | 379-1  13     679-1   | (15)b9       *-----------------------*-----------------------*-----------------------*               (48)c2                                          (34)c8`

(25=48)r12c2 - (48=12)r2c79 - (12=46)r7c13 - (46=15)r7c79 -
(15=34)r89c8 - (34=15)r13c8 - (15=39)r3c79 - (39=25)r3c13 - DLoop or VLoop
=> 8 Elims: 4r4c2, 1r9c3, 1r89c7, 1r9c9, 1r1c79, 5r1c1

After the preliminary eliminations have been made the loop is already degenerate (partially solved) as (2)r9c2 is assigned. Therefore the left hand pairs can only have one truth at most.

The irregular pattern of the givens makes determining the digits in the base and hinge sets more awkward here.
The pivot set of (6789) is extended by adding (3) and (4) as these digits also exist as givens in the pattern rows and columns.
This leaves (125) as the base set, and on checking this combination out, the loop is found to be good as the cell pairs all contain two candidates from each set.

The puzzle is fairly simple but it demonstrates that domino loops can be found with other configurations of givens too. If the pattern had been applied before the preliminary eliminations it would have made 12 more eliminations.
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Last edited by David P Bird on Sat Nov 14, 2015 10:09 am, edited 1 time in total.
David P Bird
2010 Supporter

Posts: 1043
Joined: 16 September 2008
Location: Middle England

### Re: Domino Loops (SK Loops & Beyond)

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6. Domino Loop with 1 & 3 Linking Candidates
The rank 0 pattern in this puzzle was first analysed by Ronk and later by Obi-Wahn

98.7.....7.6...8...54......6..8..3......9..2......4..1.3.6..7......5..9......1..4 # 7658;GP;H1521
Code: Select all
`      *--------------------------*--------------------------*--------------------------*      | <9>     <8>     123      | <7>     12346   2356     | 1245-6  13456   2356     |       | <7>     12      <6>      | 12459-3 1234    2359     | <8>     1345    2359     |       | 123     <5>     <4>      | 129-3   12368   23689    | 129-6   1367    23679    |       *--------------------------*--------------------------*--------------------------*(7)r4 | <6>     1249-7  1259-7   | <8*>     127     257     | <3*>    457     579      |       | 13458   147     13578    | 135     <9>     367-5    | 456     <2>     678-5    | (125)b5      | 2358    279     235789   | 235     367-2   <4>      | 569     678-5   <1>      | (459)b6      *--------------------------*--------------------------*--------------------------*(8)r7 | 1245-8  <3>     1259-8   | <6*>     248     289     | <7*>    158     258      |       | 1248    12467   1278     | 234     <5>     378-2    | 126     <9>     368-2    | (249)b8       | 258     2679    25789    | 239     378-2   <1>      | 256     368-5   <4>      | (125)b9      *--------------------------*--------------------------*--------------------------*                                   (3)c4                     (6)c7`

(125=3)r56c4 - (3=249)r89c4 - (249=8)r7c56 - (8=125)r7c89 -
(125=6)r89c7 - (6=459)r56c7 - (459=7)r4c89 - (7=125)r4c56 – DLoop
=> 16 Elims: 3r23c4, 2r8c6, 2r9c5, 8r7c13, 2r8c9, 5r9c8, 6r13c7, 5r5c6, 5r5c9, 7r4c23, 2r6c5, 5r6c8

Blue ingeniously found this way to express this loop as an AIC:
(3)r56c4 = (1257-3125)r56c4,r4c56 = (7)r4c56 -
(7)r4c89 = (4596-7459)r4c89,r56c7 = (6)r56c7 -
(6)r89c7 = (1258-6125)r89c7,r7c89 = (8)r7c89 -
(8)r7c56 = (2493-8249)r7c56,r89c4 = (3)r89c4 – Loop

Although there are three sets of possible pivot digits in boxes5689 two of them can be eliminated as the cell pairs would hold 5 digits. The pivot set that is left is (3678) which has no digits in common with the base set (12549) using the other givens in the boxes. On forming the loop, the candidates are found to split into singles and triplets, with no way to convert it to a loop with two linking digits.

For 0 Rank pattern enthusiasts
A <similar puzzle> also presented by Ronk is
1.......9.4...2.8...6...3.....4.3.7.....6.....2.5.8.....9...1...8.7...4.5.......6 # Pattern Game 119/gsf/11.3/11.3/3.4
which has a base digit set of (13569). Using a Multi-fish approach Leren then discovered a second Multi-sector Locked set using a base digit set (169).

Another puzzle of potential interest is
........4.1...3.9.6...9.8.....3.5.....71......3...2.1...4...7...9.5...2.8.......6 # tarek_Pearly_#3298
which could be classed as an Almost SK Loop embedded in a rank 0 pattern.

The inferences available when SK Loops and JExocets are found in combination are covered in the JExocet Compendium thread.
.
Last edited by David P Bird on Sat Nov 14, 2015 10:19 am, edited 2 times in total.
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### Re: Domino Loops (SK Loops & Beyond)

.
7. AIC Proof of Domino Loops
The AIC loop to make the DLoop eliminations with 1 and 3 linking digits has been given in the previous post. With 2 linking digits two alternative AIC loops must be used, described here only for those who would like to delve further.

Here is the original SK Loop example repeated:
Code: Select all
`       *-----------------------*-----------------------*-----------------------*       | <1>    268    47      | 2456   234568 2345    | <9>    568    37      | (29)b1 (68)r2 | 268    <3*>   689     | 1259-6 259-68 <7>     | 568    <4*>   168     |        | 47     689    <5>     | 1469   34689  1349    | 37     168    <2>     | (15)b3       *-----------------------*-----------------------*-----------------------*       | 2358   1259-8 1389    | 2459   2459   <6>     | 23458  <7>    13489   |        | 235678 259-68 36789   | 24579  <1>    2459    | 234568 259-68 34689   |        | 2567   <4>    1679    | <3>    2579   <8>     | 256    1259-6 169     |        *-----------------------*-----------------------*-----------------------*       | <9>    168    3468-1  | 12467  23467  1234    | 4678-2 268    <5>     | (15)b7(46)r8 | 456    <7*>   146     | <8>    259-46 1259-4  | 246    <3*>   469     |         | 3468-5 568    <2>     | 45679  345679 3459    | <1>    689    4678-9  | (29)b9        *-----------------------*-----------------------*-----------------------*                (68)c2                                          (68)c8`

(29=68)r13c2 - (68=15)r79c2 - (15=46)r8c13 - (46=29)r8c79 -
(29=68)r79c8 - (68=15)r13c8 - (15=68)r2c79 - (68=29)r2c13 - SK Loop
=> 16 Elims: 8r4c2, 68r5c2, 1r7c3, 5r9c1, 46r8c5, 4r8c6, 2r7c7, 9r9c9, 68r5c8, 6r6c8, 6r2c4, 68r2c5

As notated the strong links in each two-cell node are invalid and are weak as there are 6 combinations of the 4 digits that could be true together. Using the pairs shown there are three ways the true digits could be split; both true on the left, both true on the right, and one true on each side.
It therefore needs two AIC loops to express DLoops using weak and strong inferences:
A: (2&9 = 6|8)r13c2 - (6&8 = 1|5)r79c2 – etc
B: (2|9 = 6&8)r13c2 - (6|8 = 1&5)r79c2 – etc
Where to be true (2&9) would require the cells to hold both (2) AND (9), and (6|8) would require them to hold (6) OR (8) either individually or together.

Provided these chains close to become conjugate loops, all the left hand terms will be true or all the right hand ones will be.
Now there are three valid ways these two loops can combine and one further way which is impossible.
1. All left hand terms are true in both loops = left hand terms hold two truths
2. All right had terms are true in both loops = right hand terms hold two truths
3. All right hand terms in loop A are true and all left hand terms in loop B are true = all terms hold exactly one truth
4. All left hand terms in loop A are true and all right hand terms in loop B = impossible (as (68) would be true in both r13c2 and r79c2)

Cases 1 and 2: (6) and (8) can be eliminated from the other cells in c2 as they must both be true in one or other of the cell pairs.
Case 3: one of the digits must be true in the first cell pair and the other one must be true in the second cell pair to make the same eliminations.

A more formal proof for case 3 is given by these chains
(6=8)r13c2 – (8=6)r79c2 to make any (6) eliminations.
(8=6)r13c2 – (6=8)r79c2 to make any (8) eliminations.

Therefore without knowing which of these three cases applies, it is known that the two linked digits must occupy the four cells in the two nodes. Only later in the solution will it be known which of these cases it is.

Notating DLoops with strong and weak link symbols is merely a matter of convenience as they are not AICs, and this must therefore be remembered.
.
David P Bird
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### Re: Domino Loops (SK Loops & Beyond)

David P Bird wrote:.
5. Champagne's Virus Loop
This loop with linking pairs is far away from the SK pattern template and was responsible for bringing the naming issue to light. Expecting that similar loops could exist in other puzzles, Champagne called it a Virus Loop, but so far this is the only example that has been presented. They must surely exist but may be confined to simpler puzzles where they aren't needed.

.

The virus chain has properties that a solver can use even if the "virus chain" does not loop.

My first solver used that property. Meantime appeared Exocet and multi fish pattern, more exiting as solving tools. I did not work again for long on virus chains.

When the SK loop was shown, I immediately noticed that it was one application of the virus chain. This explain why I always considered the proof through the virus chain loop
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### Re: Domino Loops (SK Loops & Beyond)

champagne wrote:The virus chain has properties that a solver can use even if the "virus chain" does not loop.

Thanks for pointing this out and I'm sorry if there is something wrong or missing in my description of virus loops.

Your use of a virus chain that doesn't complete a loop a loop is new to me, but I'm not sure if would fit into this summary or not. Would you say that these chains are suitable for use by a human solver?

I'd like to study a solution that uses one and would be grateful if you would either provide an example or a pointer to one in another post.

When I understand them properly I will know how to correct that post.

In trying to find a collective name for these patterns those with alternately 1 and 3 linking digits were a problem. They weren't SK loops and when I asked you, you said they weren't virus loops either! That's why I coined the domino loop term to include all of them.

David.
David P Bird
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### Re: Domino Loops (SK Loops & Beyond)

David P Bird wrote:
Your use of a virus chain that doesn't complete a loop a loop is new to me, but I'm not sure if would fit into this summary or not. Would you say that these chains are suitable for use by a human solver?

David.

Hi David,

This is a side "historical" remark. I used the "virus chain" property in hardest puzzle, but the solution remained a long and boring one, not something players are looking for.

I did not try to use it in middle hardness puzzles. Meantime, the exotic properties appeared much more attractive.

In that thread focusing on loops (if I catch your point), just forget it
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### Re: Domino Loops (SK Loops & Beyond)

David P Bird wrote:.
6. Domino Loop with 1 & 3 Linking Candidates
The rank 0 pattern in this puzzle was first analysed by Ronk and later by Obi-Wahn

98.7.....7.6...8...54......6..8..3......9..2......4..1.3.6..7......5..9......1..4 # 7658;GP;H1521

Blue ingeniously found this way to express this loop as an AIC:
(3)r56c4 = (1257-3125)r56c4,r4c56 = (7)r4c56 -
(7)r4c89 = (4596-7459)r4c89,r56c7 = (6)r56c7 -
(6)r89c7 = (1258-6125)r89c7,r7c89 = (8)r7c89 -
(8)r7c56 = (2493-8249)r7c56,r89c4 = (3)r89c4 - Loop

A chain (or loop) of four ALS nodes is usually expressed with four strong inferences. What is the purpose in using eight?
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### Re: Domino Loops (SK Loops & Beyond)

ronk wrote:
David P Bird wrote:.
6. Domino Loop with 1 & 3 Linking Candidates
The rank 0 pattern in this puzzle was first analysed by Ronk and later by Obi-Wahn

98.7.....7.6...8...54......6..8..3......9..2......4..1.3.6..7......5..9......1..4 # 7658;GP;H1521

Blue ingeniously found this way to express this loop as an AIC:
(3)r56c4 = (1257-3125)r56c4,r4c56 = (7)r4c56 -
(7)r4c89 = (4596-7459)r4c89,r56c7 = (6)r56c7 -
(6)r89c7 = (1258-6125)r89c7,r7c89 = (8)r7c89 -
(8)r7c56 = (2493-8249)r7c56,r89c4 = (3)r89c4 - Loop

A chain (or loop) of four ALS nodes is usually expressed with four strong inferences. What is the purpose in using eight?

When I first read this I was reminded of a reply given by Victor Borge when he was asked if he was going to continue performing his one-man show:
"Well I don't see how I can do it with less."

You seem to overlook that this loop is directly comparable to the standard SK Loop and has the same number of links.
By using overlapping nodes Blue managed to compose it so it uses just the default notation conventions for when each term will be considered true and false. It justifies all the eliminations that are proved when the pattern is analysed as a MSLS or being 0 Rank.

In comparison my best effort was
(125#2=3)r56c4 - (3=249#2)r89c4 - (249#2=8)r7c56 - (8=125#2)r7c89 -
(125#2=6)r89c7 - (6=459#2)r56c7 - (459#2=7)r4c89 - (7=125#2)r4c56 – Loop
Here firstly #2 has to be specified as the requirement for the triplet terms to be true (which no-one likes) and secondly the eliminations made by the linked triplets aren't obvious and have to be justified.
.
David P Bird
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### Re: Domino Loops (SK Loops & Beyond)

David P Bird wrote:.
Blue ingeniously found this way to express this loop as an AIC:
(3)r56c4 = (1257-3125)r56c4,r4c56 = (7)r4c56 -
(7)r4c89 = (4596-7459)r4c89,r56c7 = (6)r56c7 -
(6)r89c7 = (1258-6125)r89c7,r7c89 = (8)r7c89 -
(8)r7c56 = (2493-8249)r7c56,r89c4 = (3)r89c4 -
Loop

ronk wrote:A chain (or loop) of four ALS nodes is usually expressed with four strong inferences. What is the purpose in using eight?

David P Bird wrote:You seem to overlook that this loop is directly comparable to the standard SK Loop and has the same number of links.

You seem to overlook that the sk-loop, i.e., Steve Kurzhal's hidden-pair-loop, is comprised of eight aahs (or ahs) elements, so use of eight derived strong inferences makes perfect sense. In the GP-H1521, a v-loop, naked-pair-loop "1/3 loop", we have four derived strong inferences. Hence, when omitting als weak-link values, the loop can be written as ...

(3=7)r56c4,r4c56 - (7=6)r4c89,r56c7 - (6=8)r89c7,r7c89 - (8=3)r7c56,r89c4 - loop

Surely there must be some way to add the weak link values internal to an als without adding more strong inferences to the expression, perhaps something like:

(3=7,125)als:r56c4,r4c56 - (7=6,459)als:r4c89,r56c7 - (6=8,125)als:r89c7,r7c89 - (8=3,249)als:r7c56,r89c4 - loop
Last edited by ronk on Sun Nov 22, 2015 7:38 pm, edited 1 time in total.
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### Re: Domino Loops (SK Loops & Beyond)

ronk wrote: You seem to overlook that the sk-loop, i.e., Steve Kurzhal's hidden-pair-loop, is comprised of eight aahs (or ahs) elements, so use of eight strong inferences makes perfect sense.

In the GP-H1521 v-loop, a naked-pair-loop, we have four strong inferences. Hence, when omitting als weak-link values, the loop can be written as ...
(3=7)r56c4,r4c56 - (7=6)r4c89,r56c7 - (6=8)r89c7,r7c89 - (8=3)r7c56,r89c4 - loop

Surely there must be some way to add als weak link values without adding more strong inferences to the expression, perhaps something like:
(3=7,125)als:r56c4,r4c56 - (7=6,459)als:r4c89,r56c7 - (6=8,125)als:r89c7,r7c89 - (8=3,249)als:r7c56,r89c4 – loop

In the analysis I did which stems mainly from pages 52 – 54 of the "Exotic Patterns a Resumé" thread I stepped back and realised that I only had the vaguest idea of what you meant by your hidden-pair-loop and naked-pair-loop terms that I've marked in your response. I would grateful if you would define them for me. (This hasn't been an issue before as all I've been concerned about has been the end results of the pattern.)

You also say that GP-H1521 is VLoop, but <here> Champagne expressly denied that. This is what caused me to try to find a new collective name for SK-loops, VLoops, and these 1/3 Loops.

Now we are at odds about the strong inferences in SK loops. As written they contain no strong inferences, but the pattern proves that all 16 links are strong 8 of which make the eliminations. In Blue's loop as written there are 8 genuinely strong links and 8 weak ones. When the loop is closed the weak inferences become strong and these make the eliminations.

You could be right about there being another AIC loop to cover the 1/3 case, but I suspect that, like my effort I gave, it would still need to have special rules for interpreting the logic it employs. Either that or two separate chains may be needed.

In my opinion AICs are good at proving some methods but not all of them (eg fish) and SK loops are better proved without them. I therefore only provided the AIC proofs as a matter of interest, and was pleased when Blue's loop covered the 1/3 linking digits case for me.
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### Re: Domino Loops (SK Loops & Beyond)

Oops, I lost sight of H1521 being a "1/3 loop" rather than a "2/2 loop." I'll tweak my post tomorrow (and make an email generating post as well.)
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