Grouped Continuous Loop

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A Grouped Continuous Loop is a Continuous Loop that allows grouped nodes. Strong and weak links alternate around the entire loop, including where the ends meet. Once the loop is formed, there is no uniquely defined starting node.

How to spot it

• Observe the strong links on the board, including those involving grouped nodes.
• Connect these strong links with weak links to form an AIC. Its end nodes must see each other; otherwise, consider an ordinary chain.
• For each weak link, eliminate candidates that see both of its end nodes. A candidate can still be eliminated even if its node belongs to a grouped node within the loop; see Cannibalistic Loops.

How it works

For any weak link A┉B in the loop, following the loop in the other direction gives an AIC from A to B. This establishes a logical strong link between A and B. The presence of grouped nodes does not change this reasoning.

Each weak link can therefore also be treated as a strong link. Applying the elimination logic for strong and weak links, we can eliminate the candidates in nodes that see both ends of that weak link.

If one end is a grouped node, a candidate to be eliminated must see every simple node in that group. If both ends are grouped nodes, it must see every simple node in both groups.

Examples

The board above contains the continuous loop [R3C7:2]━[R3C9:2]┉[R3C9:5]━[R7C9:5, R8C9:5]┉[R9C8:5]━[R9C8:8]┉[R9C7:8]━[R9C7:2]┉[R3C7:2].

Checking the original weak links gives these eliminations:

• Candidates 2 and 5 in R3C9 form a logical strong link. Eliminate 6 and 8 from that cell.
• The grouped node [R7C9:5, R8C9:5] and [R9C8:5] form a logical strong link. Candidate 5 in R8C8 sees all three simple nodes, so it can be eliminated.
• Candidate 8 in R9C8 and R9C7 forms a logical strong link. Eliminate 8 from R8C8, R8C9, R9C2, and R9C3.

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Cannibalistic Loops

Sometimes a candidate to be eliminated belongs to a grouped node within the loop itself. Such an elimination from part of the loop is called a cannibalistic loop.

The board above contains the continuous loop [R1C1:5]━[R5C1:5]┉[R5C7:5, R5C9:5]━[R4C7:5]┉[R2C7:5]━[R2C7:9]┉[R1C7:9, R1C8:9]━[R1C1:9]┉[R1C1:5].

Notice the grouped node [R5C7:5, R5C9:5]. Although it is a node within the loop, its simple node [R5C7:5] sees both ends of the logical strong link between [R4C7:5] and [R2C7:5]. Candidate 5 in R5C7 can therefore be eliminated.

When the grouped node [R5C7:5, R5C9:5] is assumed to be true, [R5C9:5] alone being true is sufficient to satisfy it, so there is no contradiction.