The New Sudoku Players' Forum

[ag24ag24]

I have a question about SK-loops and their variations; apologies if it is a terribly newbie one, but I couldn't answer it by searching here.

Everything I have read about SK-loops is unambiguous that in each box, the cell in the same row or column as all four of the cells that are part of the loop must be given/solved. But I don't see why this needs to be so. Let us consider the following partial grid, which is what I understand to be a maximally general SK-loop:

**** abcd abcd | abef abef ****
cdgh [c?] ---- | ---- ---- efij
cdgh ---- ----| ---- ---- efij
----------------------------------
ghkl ---- ----| ---- ---- ijop
ghkl ---- ----| ---- ---- ijop
**** klmn klmn | mnop mnop ****

(sorry if the spacing is confusing - everything except the "[c?]" is intended to be in columns 1 or 6 or rows A or F)

where the letters a...p denote candidates in the indicated cell, and (necessarily) different letters not appearing in a single cell (such as c and p) can specify the same value. It seems to me that we can derive the SK-loop eliminations without saying anything at all about the cells marked with *. Consider whether cell B2 can be c. If it is, then cells B1 and C1 must be two of d,g,h. If they are g and h, we can go around the loop anticlockwise and derive DE1 = kl, F23 = mn, F45 = op, DE6 = ij, BC6 = ef, A45 = ab, A23 = cd, contradicting B2=c. But if BC1 are not g and h, one of them must be d, so A23 is ab, so we can instead go around the loop clockwise and derive A45 = ef, BC6 = ij, DE6 = op, F45 = mn, F23 = kl, DE1 = gh, BC1 = cd, again contradicting B2=c. So we're done, B2=c can be eliminated, and we never looked at AF16. What am I missing? Is this not a bona fide SK-loop, or not a truly generic one?

[denis_berthier]

Everything I have read about SK-loops is unambiguous that in each box, the cell in the same row or column as all four of the cells that are part of the loop must be given/solved.

I don't know where you have read this, but the definition of an sk-loop doesn't involve any given at all. It's based only on candidates.
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[ag24ag24]

I meant, for example, rule (3) below, which is from https://www.philsfolly.net.au/Sudoku/loops_help.htm

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.

[denis_berthier]

I meant, for example, rule (3) below, which is from https://www.philsfolly.net.au/Sudoku/loops_help.htm .

They explicitly call it a variant!

If you want a formal definition and a detailed discussion of it, see mine in chapter 13 of [PBCS] here: https://www.researchgate.net/publication/356313228_Pattern-Based_Constraint_Satisfaction_and_Logic_Puzzles_Third_Edition

Note that most patterns have special cases, extensions and degenerated cases, and it's often very difficult to define all the possible variants.
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[ag24ag24]

Thanks! I didn't think I was resorting to variants (in the five rules that I quoted, I think only rules 4 and 5 describe variants, not rule 3). However, I see that your definition of SK-loops indeed says nothing about the cells I was referring to (the ones you define as the centre of a block), other than "in all the known examples the center of the block is occupied by a clue" - which I suppose is why other (less formal) treatments tend to include that as part of the definition rather than as merely a seemingly-universal property.

[denis_berthier]

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In my formal definition, I tried to generalise as much as possible from the known examples. That didn't require to put any condition on clues.
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[totuan]

I have a question about SK-loops...

You can refer original post by Steve-K: https://sudoku.com.au/The-Easter-Monster-An-Opening-Volley.aspx

totuan

[denis_berthier]

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This an undated web page. It's very far from being the original post.
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[ag24ag24]

I have another question (somewhat related). In getting to grips with SK-loops, I have realised that there are smaller loops which yield to the same logic. For example, if boxes 1 and 2 look like this:

+----------------+----------------*---
| ---- 1234 1234 | ---- 1256 1256 |
| ---- 3478 3478 | ---- 5678 5678 |
| ---- ---- ---- | ---- ---- ---- |
+----------------+----------------+---
| | |

then surely we can use the SK-loop logic to eliminate 1 and 2 from r1c14789, 3 and 4 from r1c1, r2c1 and r3c123, 5 and 6 from r1c4, r2c4 and r3c456, and 7 and 8 from r2c14789, no? And one can also make versions with 12 cells in (for example) r1c2356, r2c2389 and r3c5689. But I have never seen such patterns discussed, nor even named. Can anyone point me to such threads?

[eleven]

I remember, that i have solved a puzzle with such a 3-boxes loop, but i can't find it anymore. They are extremely rare. ([Added:]only found this related post)

The SK-loop does not need the 3 givens in the 4 boxes, but without them you hardly can reduce the candidates to that pattern.