## SK and related loops

Advanced methods and approaches for solving Sudoku puzzles

### SK and related loops

SK and Related Loops

Introduction

The purpose of this post is to document the rich variety of SK-type loops that I have discovered. By locating numerous examples of variant SK loops,
I have been able to describe patterns that allow one to propose some rules for these loops to be valid. The first described SK loop (SK for Steve
Kurtzal) occurs in the "Easter Monster" puzzle shown here after basics. It has 8 double links: 1 in rows 2 and 8, 1 in columns 2 and 8 and 1 in
boxes 1,3,7,9.

1.......2.9.4...5...6...7...5.9.3.......7.......85..4.7.....6...3...9.8...2.....1

Code: Select all
` 1       7(48)   3458-7 | 3567   3689   5678   | 3489   6(39)  2       2(38)   9       7(38)  | 4      126-38 1267-8 | 1(38)  5      6(38)     3458-2  2(48)   6      | 1235   12389  1258   | 7      1(39)  3489   ------------------------+----------------------+--------------------- 2468    5       1478   | 9      1246   3      | 128    1267   678     234689  126-48  13489  | 126    7      1246   | 123589 126-39 35689   2369    1267    1379   | 8      5      126    | 1239   4      3679   ------------------------+----------------------+--------------------- 7       1(48)   4589-1 | 1235   12348  12458  | 6      2(39)  3459    6(45)   3       1(45)  | 1267-5 126-4  9      | 2(45)  8      7(45)     4589-6  6(48)   2      | 3567   3468   45678  | 3459   7(39)  1  `

The loop is show here:
(38=27)r2c13 - (27=48)r13c2 - (48=16)r79c2 - (16=45)r8c13 - (45=27)r8c79 - (27=39)r79c8 - (39=16)r13c8 - (16=38)r2c79 - loop

SK Loops such as this with a (2-2-2-2-2-2-2-2-) pattern are exceeding common amongst very hard puzzles, especially those with symmetry.
A selection of puzzles containing these "normal" SK loops can be seen on my web site (http://www.philsfolly.net.au). The rest of this post will focus
on "variant" loops. In each different type of variant loop, I will present one example, again pointing out there are many more on my web site.

Proposed Rules

1) Loops can include single, double or triple links, as long as the sum of the links is less than or equal to 16.
2) The loop requires 4 boxes in 2 bands and and 2 stacks, and 8 links, one link in each box, one link in each row and one link in each column.
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, but not a given.

Categories of variant loops:
Note that I am presenting one example in each case to keep the size of the post reasonable, but there are many more examples on my web site.

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

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

Type C: One single link in a box, one triple link in an adjacent row or column: (B1-3-2-2-2-2-2-2-):
1..2....3.....145.....6..1......4.6.....5.1...2.7....86.7......4..8......58.....9 Patterns game joel64 jul 22, 2010

Type D: Two triples, two singles, and four doubles. There are several possibilities for this, and the example has (B2-3-1-2-3-1-2-2-2-):
..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

Type E: There can be three singles and three triples, and two doubles (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

Type F
The following puzzle contains an SK loop which can be coded as a normal SK, all links pairs, or as a Type A variant.
The same 16 cells are used in each loop.

Please note that in the following types, the software gives full details for both loops in the output box, but displays only the second
of the two in the grid. If the first has more eliminations than the second, run the program again.
.5......49......1...6..37......156.....3.6.....289......37..2...9......14......5. patterns game gsf, Mar 13, 2009

Type G
Alternatively, the following puzzle contains an SK loop which can be coded as a normal SK, all links pairs, or as a variant Type B.
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

Type H
Aditionally, some puzzles have two different variant loops: A type A loop and a type C loop:
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

Type A sub-types
One of the cells in one (or more) of the cell pairs can be a solved cell (not given). The link within the box in this case can be double.

First example has three (B2-1-2-1-3-1-2-1-) with only 13 links:
..3...5...8...1.4.5.......2.....8.1.....2.....7.3.5...6.......5.1.4...7...2...3..

The second example has two (B3-1-2-1-3-1-2-1-) with 14 links:
2.......4.8.5...7...1.2.3.....7...9.....6.....7...8.....3...1...9...7.5.4....1..2

The third has only one (B3-1-3-1-2-1-3-1-) with 15 links:
2.......4.9.6...5...17..3.....8...9.....5...3.5...9.....4...1...6...5.7.3.......2

Type E sub-type
This one has a given at r4c3, 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

-----------------------------------------------------------------------------------------------------------------------------------------

Almost SK (+1) Loops
This section is added out of interest, as trial and error is employed.

These are every bit as common as SK loops. They have 17 links rather than the 16 of normal 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 simply 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 candidate eliminations can be carried out. I would be delighted if someone could discover a theoretic framework to establish this
approach to be valid.

Almost SK (+1) Loops

Type A: The following puzzles have a single triple box link (B3-2-2-2-2-2-2-2-). It occurs typically in very hard puzzles:
.......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

Type B: 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

Type C: The next are examples of a triple link being in a column (B2-3-2-2-2-2-2-2-):
.12......3.4...2..56.7.......18..9......6...7.....7.5..8.2..1.......5..3....7..6. patterns game Dec 2017, papy999

Variant Almost (+1) 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.

..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

Type E: 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

98.7.....6.....7....7.5.....4.....3...98..4......24..1..89..6......1...2.....3.4.

Type G: This has 4 triple links, 3 single links and one double (but also a type B loop).
9..3....6.....597.....9..3......1.9.....7.5...6.4....35.1......7..6......36.....8 patterns game Jul 2010, champagne

Type H: This one has 3 triple and 2 single and 3 double links, (or a type D loop)
98.7.....6.....7....7.5.....4.....3...98..4......24..1..89..6......1...2.....3.4.;104;GP;H13; ; ;E; ;;

----------------------------------------------------------------------------------------------------------------------------------------------

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
of the many examples tested. The famous "Fata Morgana" puzzle has this pattern, and subsequently solves with basic moves (fish & ALS).

Type A: The following have 2 triple links and 6 double links:
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

3....84...5..3..9...21....7..85.6...............3.16..4....39...9..1..5...79....2

Type C: These has 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

Type D: The next example is specially interesting in that it has 3 loops. First a normal SK loop, second a normal type A Variant, and third an
SK (+2) loop. The first 2 occupy the same loop of cells, but the +2 loop uses a different set of cells.
..1..23...6..8..4.7.......9...5.......2.3.5.......4...9.......7.4..5..6...31..2.. Patterns game jsf Jan 06, 2009

1..2......3..4..5...6..7...9.....6...4..5..2...7.....9...4..9...2..1..3......6..7
pjb
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### Re: SK and related loops

I wrote:Good stuff, Phil. I hope I'm not polluting this thread by asking about a somewhat minor detail that has been bothering me about SK-Loops.

pjb wrote:The loop is show here:
(38=27)r2c13 - (27=48)r13c2 - (48=16)r79c2 - (16=45)r8c13 - (45=27)r8c79 - (27=39)r79c8 - (39=16)r13c8 - (16=38)r2c79 - loop

As is obvious and well-known, that common style of writing SK-Loops is not a correct AIC. Furthermore, I've seen people like David claim that it's not even possible to write it correctly as a single AIC. Why not?

Edit: Sorry about that stupid question. Having put some more thought into this, I think I've finally understood David's arguments and have to agree with them. I can't find a way to write that as a valid AIC-loop at all (or I can, but it doesn't produce any eliminations). To get all the eliminations I have to write six AICs (or eight in the general case, but two don't produce any eliminations here), but they're no loops. Here's what I came up with:

SK-Loop => AICs: Show
(38=1267#3)r2c1379 - (1|6)r13c8&(2|7)r13c2 = (3927)r1379c8|(4816)r1379c2 - (2|7)r8c79&(1|6)r8c13 = (4516|4527)r8c1379 - (1|6)r79c2&(2|7)r79c8 = (4827)r7913c2|(3916)r7913c8 - (2|7)r2c13&(1|6)r2c79 = (38)r2c1379 => -38 r2c456

(45=1267#3)r8c1379 - (1|6)r79c2&(2|7)r79c8 = (4827)r7913c2|(3916)r7913c8 - (2|7)r2c13&(1|6)r2c79 = (3816|3827)r2c1379 - (1|6)r13c8&(2|7)r13c2 = (3927)r1379c8|(4816)r1379c2 - (2|7)r8c79&(1|6)r8c13 = (45)r8c1379 => -45 r8c456

(48=1267#3)r1379c2 - (1|6)r8c13&(2|7)r2c13 = (4527)r8c1379|(3816)r2c1379 - (2|7)r79c8&(1|6)r13c8 = (3916|3927)r1379c8 - (1|6)r2c79&(2|7)r8c79 = (3827)r2c7913|(4516)r8c7913 - (2|7)r13c2|(1|6)r79c2 = (48)r1379c2 => -48 r456c2

(39=1267#3)r1379c8 - (1|6)r2c79&(2|7)r8c79 = (3827)r2c7913|(4516)r8c7913 - (2|7)r13c2&(1|6)r79c2 = (4816|4827)r1379c2 - (1|6)r8c13&(2|7)r2c13 = (4527)r8c1379|(3816)r2c1379 - (2|7)r79c8&(1|6)r13c8 = (39)r1379c8 => -39 r456c8

(27)b1p2468 = (3816)r2c1379|(4816)r1379c2 - (1|6)r13c8&r8c13 = (3927)r1379c8|(4527)r8c1379 - (2|7)r8c79&r79c8 = (4516)r8c7913|(3916)r7913c8 - (1|6)r79c2&r2c79 = (3827)r2c7913|(4827)r7913c2 => -27 b1p37

(16)b3p2468 = (3827)r2c7913|(3927)r1379c8 - (2|7)r13c2&r8c79 = (4816)r1379c2|(4516)r8c7913 - (1|6)r8c13&r79c2 = (4527)r8c1379|(4827)r7913c2 - (2|7)r79c8&r2c13 = (3916)r7913c8|(3816)r2c1379 => -16 b3p19 (n/a)

(16)b7p2468 = (4527)r8c1379|(4827)r7913c2 - (2|7)r79c8&r2c13 = (3916)r7913c8|(3816)r2c1379 - (1|6)r2c79&r13c8 = (3827)r2c7913|(3927)r1379c8 - (2|7)r13c2&r8c79 = (4816)r1379c2|(4516)r8c7913 => -16 b7p37

(27)b9p2468 = (4516)r8c7913|(3916)r7913c8 - (1|6)r79c2&r2c79 = (4827)r7913c2|(3827)r2c7913 - (2|7)r2c13&r13c2 = (3816)r2c1379|(4816)r1379c2 - (1|6)r13c8&r8c13 = (3927)r1379c8|(4527)r8c1379 => -27 b9p37 (n/a)

Pretty horrible (if even correct). Perhaps there are simpler ways to do that with chains, but I can't see them. Because I still don't like the convention of writing SK-Loops as incorrect AIC-loops, no matter how convenient that is, I think I'd prefer the unambiguity and conciseness of set logic:

Alien 16-Fish (Rank 0); Pattern: SK-Loop

{28N1379 1379N28} \ {38r2 45r8 48c2 39c8 27b19 16b37}

(Or alternatively some other-than-AIC syntax for writing the "loop".)
Last edited by SpAce on Mon Apr 01, 2019 12:37 am, edited 2 times in total.

SpAce

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### Re: SK and related loops

Hi phill,

A short post to show interest in your work.
As I have closed the coding on the 17 clues search, I was restarting thoughts on the "exotic patterns" analysis.

You start from the sk loop, but IMO, the rest is more in the sets/linksets general field.
Anyway, digging in your examples is not possible for me in the next 2 weeks,

I had in mind, one day, to investigate "potential hardest" with many eliminations in multi floors to try to extract new "exotic patterns". Your example can give some solutions and the time to do it is there for me this year.
champagne
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### Re: SK and related loops

Hi Phil, tested all of your sample puzzles and got a lot of hits with MSLS plus a few with SK Loop and Multifish hits, so there is obviously a lot of overlap there.

Had a chuckle over Mauricio's puzzle. It solves with basics and an X Wing but ignoring that I can see a MSLS with 10 eliminations.

Leren
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Joined: 03 June 2012

### Re: SK and related loops

Hi again phil,

To go in leren's direction, we could see what you have on fata morgana.

It has been one of the 2 sudokus helping to discover the exocet property.

The multifloors 136 has a good potential, but the exocet property itself requires only, if I am right a 11 truths 12 links
the pm

Code: Select all
`2458  2467  245678 |126789 16789 1678  |24589  1248  3     2348  2347  1      |23789  3789  5     |6      248   249   2358  9     2568   |12368  4     1368  |258    7     125   ----------------------------------------------------------12348 12346 2468   |13678  13678 9     |2347   5     12467 7     12346 2469   |136    5     136   |2349   12346 8     1389  5     689    |4      13678 2     |379    136   1679  ----------------------------------------------------------145   8     457    |1367   2     13467 |3457   9     4567  249   247   3      |5      6789  4678  |1      2468  2467  6     1247  24579  |13789  13789 13478 |234578 2348  2457    `

the 136 multi floors having the exocet

Code: Select all
`-   6+   6+ |16+  16+  16+  |-  1+       3+  3+      |3+   3+   -    |   -    -   3+  -    6+ |136+ -    136+ |-  -    1+  ----------------------------------------13+ 136+ 6+ |136+ 136+ -    |3+ -    16+ -   136+ 6+ |136  -    136  |3+ 136+ -   13+ -    6+ |-    136+ -    |3+ 136  16+ ----------------------------------------1+  -    -  |136+ -    136+ |3+ -    6+  -   -       |-    6+   6+   |   6+   6+      1+   -  |13+  13+  13+  |3+ 3+   -     `

we have here a classical JE base r5c46 r4c2 r6c8

the truths are columns 2,5,8 and cells r5c46

the links row 5 and box 5 plus rows having candidates in columns 2,5,8

what is your clearing group in this specific case
champagne
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### Re: SK and related loops

Fata Morgana was interesting from an Exocet point of view. Since the last time I looked at it (many years ago), I had coded David P Bird's pattern based exclusion of base digits from his JExocet compendium masterpiece.

Surprisingly it is able to exclude 1 from the base cells, reducing them to 3 & 6 and enabling the elimination of 3 & 6 orthogonal to the cross lines, for a total of 28 Exocet eliminations.

I have a vague memory that the previously used method to achieve this was an abi loop. If so, well done David.

Leren
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### Re: SK and related loops

Leren wrote:Fata Morgana..

I have a vague memory that the previously used method to achieve this was an abi loop. If so, well done David.

Leren

yes it is and it remains a valid (and general) way to do it.
But the solution of phil can do it as well in a different way. Let's see

EDIT BTW, the use by "abi" of the same loop in platinum blonde helped to understand the logic of Exocets with a locked digit
champagne
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### Re: SK and related loops

Hi Champagne,

Just checked Phil's platinum and David's UR threat method proves 6 false in the base and target cells. Similar to Fata Morgana the base cells are reduced to two digits, with a total of 34 Exocet eliminations.

Looks like David's UR threat method may be an alternative to the abi loop. Perhaps you could provide a list of Exocet puzzles where the abi loop is used to reduce the base digits to two and I can check them with David's method.

Leren
Leren

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Joined: 03 June 2012

### Remembering what is the abi loop

Hi leren,
As soon as you mention a UR threat linked to an exocet, it is an abi loop.

Let's restart from the fata morgana example

Code: Select all
`-   6+   6+ |16+  16+  16+  |-  1+   3   3+  3+   1  |3+   3+   -    |6  -    -   3+  -    6+ |136+ -    136+ |-  -    1+  ----------------------------------------13+ 136+ 6+ |136+ 136+ -    |3+ -    16+ -   136+ 6+ |136  -    136  |3+ 136+ -   13+ -    6+ |-    136+ -    |3+ 136  16+ ----------------------------------------1+  -    -  |136+ -    136+ |3+ -    6+  -   -    3  |-    6+   6+   |1  6+   6+  6   1+   -  |13+  13+  13+  |3+ 3+   -   `

Trying pair 13 and doing exocet linked clearing, we are there

Code: Select all
`-   6+   6+ |16+  16+  16+  |-  1+   3   3+  3+   1  |3+   3+   -    |6  -    -   3+  -    6+ |136+ -    136+ |-  -    1+  ----------------------------------------13+ 13   6+ |6+   6+   -    |+  -    6+ -   6+   6+ |13   -    13   |+  6+   -   +   -    6+ |-    6+   -    |3+ 13   16+ ----------------------------------------1+  -    -  |136+ -    136+ |3+ -    6+  -   -    3  |-    6+   6+   |1  6+   6+  6   1+   -  |13+  13+  13+  |3+ 3+   -   `

we have here 2 UR threats R35C46 and R37C46.
Note : using an exocet threat, we introduce an external condition (unique solution).
This changes the clearing potential of the truths/link approach.
Allan Barker added this property in the Xsudo after the discussions on this loop.

Consequence of the threat, we must have
a) 3r3c1 or 1r3c9
b) 1r7c1 or 3r7c6

if we have 1r4c2 3r7c8 (exocet sub solution 1) then a) is not verified
if we have 3r4c2 1r7c8 (exocet sub solution 2) then b) is not verified

To have this working, several conditions must be satisfied eg: no 1;3 in r3c5 r7c5
David looked for patterns having the "necessary and sufficient" conditions to have an abi loop.
As said earlier, Allan Barker just introduced the threat in Xsudo.
champagne
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### Re: SK and related loops

Something slightly different, still with fata morgana.
In very hard sudokus having an exocet, in a multi floors analysis only the exocet effect comes.

I run on fata morgana my last version of code for the multi floors analysis (so the results are subject to..).
I got the following eliminations that seem correct

Code: Select all
`-   6+   6+ |16+  16+  16+  |-  1+   3   3+  3+   1  |3+   3+   -    |6  -    -   3+  -    6+ |136+ -    136+ |-  -    1+  ----------------------------------------13+ 136+ 6+ |136+ 136+ -    |3+ -    16+ -   136+ 6+ |136  -    136  |3+ 136+ -   13+ -    6+ |-    136+ -    |3+ 136  16+ ----------------------------------------1+  -    -  |136+ -    136+ |3+ -    6+  -   -    3  |-    6+   6+   |1  6+   6+  6   1+   -  |13+  13+  13+  |3+ 3+   -`

elims
24R4C2
3: R2C1 R5C78 R6C1 R9C467
6: R1C346 R4C9 R5C23 R8C9

Assuming that this is correct, I was expecting a truths/links group from phil covering this.
champagne
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### Re: SK and related loops

I get 32 eliminations for Fata Morgana:

24 r4c2 : Non Base digit in Target cell.

1 r4c2, r5c46, r6c8 : Incompatible Base digit.

3 r4c457, r5c278, r6c15 : Established Base digit sees both Base or both Target cells.

6 r4c459, r5c238, r6c35 : Established Base digit sees both Base or both Target cells.

3 r2c14, r9c467 : S cell eliminations.

6 r1c346, r8c69 : S cell eliminations.

Leren
Leren

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Joined: 03 June 2012

### Re: SK and related loops

Thank you all for your responses.
Firstly, SpAce, horrible is an understatement. We'll have to agree to disagree on this one. The "shorthand" way of representing the loops is utterly unambiguous, so why create such a nightmare?
Secondly, champagne, I didn't put in the loop and its eliminations for each different case as it would make it a massive document. One can simply put the puzzle into my solver to see all the output. It is of interest that some, but not all, of these "exotic" loops can be easily expressed as MSLS or multifish. My interest has been solely on detecting these loops, and I have not been trying to find truth/link sets that underlie them. Champagne, by "clearing group" do you mean eliminations? My output for fata morgana is:

Almost (+2) SK loop: (258=36)r3c13 - (36=247)r12c2 - (247=1)r89c2 - (1=457)r7c13 - (457=36)r7c79 - (36=248)r89c8 - (248=1)r12c8 - (1=258)r3c79 - loop
No contradiction when 4 at 7,6 is true and 2 at 5,8 is true and others are all false
Eliminations: r3c4 <> 2, r3c4 <> 8, r3c6 <> 8, r7c4 <> 7, r7c6 <> 7, r4c2 <> 2, r4c2 <> 4, r5c2 <> 2, r5c2 <> 4, r5c8 <> 4, r1c3 <> 6, r2c1 <> 3, r8c9 <> 6, r9c7 <> 3

As I discuss in my post, the proposition for "Almost (+1) loops" that one of the predicted eliminations based on the loop must be actually not false but true, stands up to every one of a huge number tested. Similarly for "Almost (+2) loops". Maybe there is a way to link this observation with approaches such as exocets and truth/link sets.

Phil
pjb
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### Re: SK and related loops

pjb wrote: Firstly, SpAce, horrible is an understatement. We'll have to agree to disagree on this one.

I don't think we disagree on that one at all Horrible is an understatement indeed. I hope it's pretty clear that I'm not suggesting representing SK-Loops with those chains for any practical purposes. My point is that if one really wants to use AIC notation with SK-Loops and remain on the correct side, then something like that is probably needed -- unless someone can convincingly argue otherwise. In other words AICs aren't a practical tool for notating SK-Loops, and nothing else should be implied by using incorrect AIC notation in that context.

The "shorthand" way of representing the loops is utterly unambiguous, so why create such a nightmare?

Okay, here we seem to disagree. I truly wish you were right, because that would make things really simple, elegant, and intuitive. It also happens to produce correct eliminations, so it works for practical purposes just fine. I just don't think it's conceptually correct -- or even close to correct -- when interpreted as normal AIC syntax, which makes me cringe when I see it. It's pretty easy to see why if we look at your loop:

(38=27)r2c13 - (27=48)r13c2 - (48=16)r79c2 - (16=45)r8c13 - (45=27)r8c79 - (27=39)r79c8 - (39=16)r13c8 - (16=38)r2c79 - loop

In the AIC notation '=' means 'OR' which implies that at least one of the terms connected by it must be true. That simple rule doesn't hold here. If it did, either all of the left hand terms or all of the right hand terms would have to be true in the solution grid. They could be, but they're not the only alternatives for the digit distribution, as explained by David (and which should be pretty obvious anyway -- if it wasn't for the misleading notation). Luckily for this demonstration, one of those alternatives happens to occur here:

Code: Select all
`.-------------.---------.-------------.|  1  (7)  4  | 3  8  5 |  9  (6)  2  || (2)  9  (3) | 4  6  7 | (1)  5  (8) ||  5  (8)  6  | 1  9  2 |  7  (3)  4  |:-------------+---------+-------------:|  4   5   1  | 9  2  3 |  8   7   6  ||  9   2   8  | 6  7  4 |  3   1   5  ||  3   6   7  | 8  5  1 |  2   4   9  |:-------------+---------+-------------:|  7  (1)  9  | 5  4  8 |  6  (2)  3  || (6)  3  (5) | 2  1  9 | (4)  8  (7) ||  8  (4)  2  | 7  3  6 |  5  (9)  1  |'-------------'---------'-------------'`

As you can see, neither (38) nor (27) is true in r2c13, and the same happens for all the other "strongly linked" pairs in your loop. So, unless I've seriously misunderstood something, that simple counter-example proves that all of your loop terms are logically incorrect because they really aren't strongly linked at all (they're possible cases, but not the only ones -- you'd need a larger SIS to account for them all). To write the loop correctly, you should do something like this (also explained by David):

(3|8=27)r2c13 - (2|7=48)r13c2 - (4|8=16)r79c2 - (1|6=45)r8c13 - (4|5=27)r8c79 - (2|7=39)r79c8 - (3|9=16)r13c8 - (1|6=38)r2c79 - loop

(38=2|7)r2c13 - (27=4|8)r13c2 - (48=1|6)r79c2 - (16=4|5)r8c13 - (45=2|7)r8c79 - (27=3|9)r79c8 - (39=1|6)r13c8 - (16=3|8)r2c79 - loop

They are indeed valid AIC-loops, unlike the original, but by themselves they don't produce any eliminations using the normal AIC-loop logic. David explains how both of those loops need to be considered together, which produces three valid combinations of the left and right hand terms. Even that doesn't produce a simple way to prove the eliminations, because the third case is a mix of the OR-terms from both, which complicates things a lot. Basically you have to prove them pair-by-pair, just like I did with my horrible AICs.

So, to me it seems that there's no simple way to write SK-Loops as correct AICs, as David already proved a long time ago (I never really doubted him -- it just took a while to understand his argument). If you can really disagree with that, please show me how!

--

From a practical point of view it's of course great to look at the SK-Loop as conventionally presented, because it's intuitive and it works, but it's not a correct AIC-loop (unless you pull a miracle out of your hat and prove otherwise) -- and it's misleading and confusing to present it as such, imho. Therefore, I would prefer a couple of changes. At the very least, the "- loop" ending should rather be "- SK-Loop" to demonstrate that it's not a real AIC-loop but uses the SK-Loop logic. Second, I would very much like to change the '=' to something else because those links are not strong links. I'm pretty sure something like this has already been suggested elsewhere (probably by David):

(38/27)r2c13 - (27/48)r13c2 - (48/16)r79c2 - (16/45)r8c13 - (45/27)r8c79 - (27/39)r79c8 - (39/16)r13c8 - (16/38)r2c79 - SK-Loop

That, or something similar, I could easily accept. Just not AIC syntax. [Added: As I kind of remembered, it has been suggested by David here:

David P Bird wrote:On notating the loops my preference would be not to use weak and strong link symbols as they are liable to be misinterpreted. I would prefer something like (29/68)r13c2 - (68/15)r79c2 - (15/46)r8c13 - but this isn't a major issue.

[Added: As an afterthought, I think I might actually see a way to get (at least) the row and column eliminations with a single loop if the terms are written as almost-almost-quads instead of almost-almost-pairs:

(2|7=48'16)r1379c2 - (1|6=45'27)r8c1379 - (2|7=39'16)r7913c8 - (1|6=38'27)r2c7913 - loop

=> -48 r456c2, -45 r8c456, -39 r456c8, -38 r2c456

Written like that we can see easily which digits in the AALS-nodes are bystanders and thus locked and providing eliminations, no matter which way you read the loop. That's standard behavior with ALS-loops, so why not with an AALS-loop as well? Does that loop also prove the box eliminations using the weak links, as in normal loops? I'm not so sure about that. If it does, then the pair-based loops should too, which actually was my original intuition before reading David's thoughts. I'm confusing myself here again. Did David over-complicate things after all?]
Last edited by SpAce on Thu Apr 04, 2019 1:55 am, edited 2 times in total.

SpAce

Posts: 2016
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### Re: SK and related loops

Hi phil,

No problem to see where is your loop. This is a typical SK loop pattern if we just consider the empty/assigned cells r3c28 r8c28.
As SpAce, i would try to find another way to express the loop, but more important, I am not sure to catch your point

No contradiction when 4 at 7,6 is true and 2 at 5,8 is true and others are all false

First of all, this has nothing in common with the exocet.
Second quick remark, the loop has 8 digits active. Not very exciting for a multi floors approach.

I tried in my own way to express your loop, showing deviations to a classical SK loop.

My result is the following

36r3c13 => 24(_7)r12c2=>17r89c2=>45r7c13=>36(_7)r7c79=>24(_8)r79c8=>18r12c8=>25r3c79=>36(_8)r3c13 loop

so the loop is possible if if if if (four extra digits)

having a look at your eliminations, one of these extra digits is true, so, no chance to apply the loop.
At the end, in a first approach, I don't see what to do with your result.
champagne
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Location: France Brittany

### Re: SK and related loops

pjb wrote:My output for fata morgana is:

Almost (+2) SK loop: (258=36)r3c13 - (36=247)r12c2 - (247=1)r89c2 - (1=457)r7c13 - (457=36)r7c79 - (36=248)r89c8 - (248=1)r12c8 - (1=258)r3c79 - loop
No contradiction when 4 at 7,6 is true and 2 at 5,8 is true and others are all false
Eliminations: r3c4 <> 2, r3c4 <> 8, r3c6 <> 8, r7c4 <> 7, r7c6 <> 7, r4c2 <> 2, r4c2 <> 4, r5c2 <> 2, r5c2 <> 4, r5c8 <> 4, r1c3 <> 6, r2c1 <> 3, r8c9 <> 6, r9c7 <> 3

Hi Phil,

interesting stuff.

Have no time now to look at it more careful, but i doubt that this is correct. But maybe i just made a manual mistake.

Grid:
Code: Select all
`    2458   6/24+7 245678 |126789 16789 1678  |24589  24/1+8 3         2348   3/24+7 1      |23789  3789  5     |6      24+8   249       28/3+5 9      28/6+5 |12368  4     1368  |/28+5  7      1/2+5      -------------------------------------------------------------    12348  12346  2468   |13678  13678 9     |2347   5      12467    7      12346  2469   |136    5     136   |2349   12346  8         1389   5      689    |4      13678 2     |379    136    1679     ------------------------------------------------------------    1/5+4  8      /57+4  |1367   2     13467 |57/3+4 9      57/6+4     249    24+7   3      |5      6789  4678  |1      6/24+8 2467     6      24/1+7 24579  |13789  13789 13478 |234578 3/24+8 2457 `

Assigned:
Code: Select all
`    2458   7      245678 |126789 16789 1678  |24589  24    3         2348   3      1      |23789  3789  5     |6      24    249       2      9      5      | 12368  4     1368 |8      7     1       -------------------------------------------------------------    12348  12346  2468   |13678  13678 9     |2347   5      12467    7      12346  2469   |136    5     136   |2349   12346  8         1389   5      689    |4      13678 2     |379    136    1679     ------------------------------------------------------------    1      8      5      |1367   2     13467 |7      9      4     249    24     3      |5      6789  4678  |1      6      2467     6      24     24579  |13789  13789 13478 |234578 8      2457 `

This way there is no 6 in the loop cells in box 1.

btw: I remember Steve K. as Steve Kurzhals, my favorite manual solver (together with RW and Myth).
eleven

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