SK Loops are in the ‘diabolical’ techniques rather than the ‘extreme’ since they have unique patterns.
However, it was found (invented?) while attempting to solve the Easter Monster, and it necessitates a large number of candidates, as is typical of the most difficult riddles.
First and foremost, credit goes to I owe a debt of gratitude to pjb or Phil from Sydney, Australia, for his article and the numerous samples he’s gathered on his website.
The solver will locate SK Loops for all of Phil’s types, from vanilla through Type H, but not “Related Loops” yet.
I don’t distinguish ‘types’ in my implementation because I’ve generalized the detection. SK Loops appear to be a subset of something broader, but I haven’t looked into it yet.
Easter Monster, SK Loop:
I will start with the Easter Monster, as Phil does because the design is neatly spaced out and symmetrical. SK Loops are bi-directional continuous loops, which means you may start the loop at any point and trace the links in any direction.
First, distribute four solved or provided cells over four boxes to form a rectangle. Phil distinguishes between givens (clues) and solved cells, but I don’t, and as a result, I’ve never had a false positive.
Four such numbers are shown in the diagram, surrounded by eight 2-cell pairs containing Hidden Pairs.
The pattern is then defined.
- The loop requires four boxes in two bands and two stacks, as well as eight links, one in each box, one in each row, and one in a column.
- In a box, the cell at the row and column cells intersection is given.
Loops can have one, two, or three connections as long as the total number of links is less than or equal to 16.
- •One of the two cells in a row or column within a box can be solved, but it’s not a guarantee.
Because there are eight pairs of cells, we’ll need sixteen integers to fill them all.
Candidates are highlighted in green and blue by the solver. These are some of the other options. Green or blue numbers will be present in all sixteen cells, or one green and one blue in each pair. (It may appear hazy, but the first case above is an example of the latter, as Robert pointed out in the comments.)
The sum of the connections must now be less than or equal to 16 for the pattern to be defined.
Here’s the formal specification of the loop to show what we’re counting.
(27=38) (38=16) B13- B79- (16=39) B79- (16=39) B79- (16=39) (39=27) AC8- GJ8- (45=27) H79- (45=16) H79- (45=16) H79- (45=16 H13- (16=48) H13- (16=48) H13- (16=48) AC2- (48=27) GJ2- (48=27)
Pair B13 begins when the solver starts from the top-left most provided cell in the row and loops clockwise. (Phil’s documentation begins at a different point.) On the Hidden Pair 3,8, B13 is strongly linked to B79. SK Loop Type 2-2-2-2-2-2-2-2-2
We can delete along with the unit of alignment now that the content of these sixteen cells has been designated as a Locked Set.
For example, we can rule out 3 and 8 from B5 and B6 since B13 has a blue 3,8 and B79 has a green 3,8, and the solution will be one of the two.
3-1-3-1-3-1-3-1-SK Loop: From the Beginning
Instead of the traditional doubles, this SK Loop alternates Triple and Single links all around. So the type is now 3-1-3-1-3-1-3-1, yet the numbers add up to 16.
Consider tracing some of the relationships in the graphic and comparing them to the loop’s definition:
(579=8) 8=469) A23- BC9- (2=457)GH9- (457=1) A78- (469=2) A78- (469=2) A78- (469=2) A78- (469=2) A78- (469=2) J78- (1=469)J23- (469=3)J78- (1=469)J23- (469=3)J78- (1=469) BC1- GH1- (3=579)
From the Beginning: SK Loop with Solved Cells
Finally, the fourth aspect of the SK Loop definition may be considered: When one of the cells in the pair is a solved cell, the pair is said to be solved. Phil asserts that this cannot be assumed, but I’m still hunting for a counterexample, and the solver can’t distinguish between the two.
Because two of the cells (colored in beige) have been used up, the type for this loop, 2-2-1-2-2-2-1-2-, now adds up to just 14, which is proper.
B13- (69=1)(34=69)(34=69)(34=69)(34=69)(34=69 B79- (1=68) B79- (1=68) B79- (1=68) AC8- (68=34) AC8- (68=34) AC8- (68=34 (34=68) GJ8 (68=2) H79- GJ2- (56=34)AC2- H13- (2=56)H13- (2=56)H13- (2=56)H13- (2=56)H13-
SK Loop (classical)
The first step is to recognize the Givens signature:
• Givens in four Cells in various Squares defining the rectangle’s edges (these Cells are called pivots)
If the Givens signature is confirmed, the Squares holding the pivots must meet the following requirements:
1. Squares share a pair of Candidates in the same Band (we call it an external link).
2. The two Cells in the mini-Row holding the pivot jointly contain just four Candidates inside these Squares (these two Cells form what we call a Node).
3. There are only four candidates in each of the two cells in the mini-column holding the pivot (these two Cells also form a Node).
4. There are two Candidates in common between this mini-Row and this mini-Column that are not part of the external links; these two Candidates form an internal link.
When most of these criteria are met, the following can be ruled out:
1. All Candidates of the external link can be removed outside the Squares containing the pivots in a Row or Column containing two pivots.
2. All Candidates of the inner connection can be removed outside the mini-Row and mini-Column in a Square with a pivot.
1. Suppose two Candidates of an internal link are true in a Node in a particular Square. In that case, the two Candidates of the external connection in this Node must be in the other Node in the same Square, which excludes them from the connected Square and compels the two Candidates of this Node to be in the other Node in the same Square. Square’s internal link is true in its other Node, and this process repeats until the loop is closed.
If one Candidate of an internal link is true in one Node of a Square, then the other Candidate is true in the other Node of the Square. Thus, 1 Candidate of the linked external link is deleted from the Node in the connected Square in the other Nodes cell. In this second Square, this Node must now include the other Candidate of this external connection and 1 Candidate of the inner link. And the cycle continues till the loop is closed.
This removes them from the relevant Node in the linked Square, requiring the two inner link Candidates to be true, putting us back in the same predicament as before.
Variant SK Loop
In addition, the variants of the standard SK Loop take into account:
- Four pivots represent the Givens signature.
- The Pivots are related to the 16 Cells of the mini-Rows and mini-Columns.
The following are the new requirements for the Variants to the Classic SK Loop:
- The inner and outer linkages can each employ one, two, or three Candidates, as long as all of the Candidates are present in the 16 Cells of the Pivots’ mini-Rows and mini-Columns.
- The Pivots’ mini-Rows and mini-Columns can each have one solution, or one supplied, plainly excluded from the linkages.
When these variant requirements are met, the same eliminations as in the classical SK Loop can be performed, i.e., in a Row or Column containing two pivots, All external link candidates outside of the Squares containing the pivot can be eliminated, and all inner link candidates outside of the mini-Row and mini-Column can be deleted in a Square containing a pivot.
This first example uses single, double, and triple connections to create a Variant SK Loop:
The second example illustrates a Variant SK Loop with Pivots in the mini-Rows and mini-Columns and solutions in the mini-Rows and mini-Columns.
There are 8 x 2 = 16 links in the typical SK loop.
However, there can be triple or single linkages, and the logic still applies if the total number of links is 16. One or more of the sixteen cells that make up the loop can be solved, but they are not disclosed.
The following rules can be summarized as follows:
1) The loop calls for 16 cells in four boxes, two bands, and two stacks.
2) Loops can have one, two, or three links as long as the total number of connections is 16.
3) The cell that lies at the intersection of the row and column cells is a given in each box.
4) Only one of the two cells in a row or column within a box can be solved or provided.
5) The link count should be increased by one for each solved or provided cell.