Grouped Discontinuous Loop

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A Grouped Discontinuous Loop is a discontinuous loop that allows grouped nodes inside the chain. Its two end nodes must still be the same simple node. On returning to this node through a strong link, the next link is also strong, so we cannot continue into another circuit while maintaining the alternation.

Unlike a grouped continuous loop, a grouped discontinuous loop determines a digit directly: the candidate at the shared node must be true.

How to spot it

• Observe the strong links, including those involving grouped nodes. Look for a candidate x in a cell C that belongs to two strong links.
• Connect these strong links with weak links. If an AIC can be formed with both ends at the simple node [C:x], it forms a grouped discontinuous loop.
• An AIC requires at least one end node to be true, so cell C must contain digit x.

A grouped node is true when at least one of its candidates is true. A weak link to a grouped node must apply to every candidate in the group.

How it works

An AIC establishes a logical strong link between its two end nodes: at least one must be true. In a discontinuous loop, both ends are the same simple node [C:x]. Therefore, that node must be true, and cell C must contain digit x.

We can also prove this by assuming [C:x] is false. The first strong link forces the next node to be true; the following weak link forces the next node to be false. Continuing around the loop eventually forces [C:x] to be true, contradicting the assumption.

A strong link found directly on the board whose end nodes also see each other is also a weak link. Exactly one of those nodes must therefore be true; they cannot both be true. A logical strong link established by an AIC, however, guarantees only that at least one end node is true, without ruling out both being true. In a discontinuous loop, the two end nodes are the same node, so that node must be true.

A grouped discontinuous loop requires the shared end node to be a simple node. If it were a grouped node, proving it true would establish only that at least one candidate in the group is true, without identifying a single cell for the digit.

Example

The board above contains the discontinuous loop with two grouped nodes [R2C5:2]━[R2C5:8]┉[R4C5:8]━[R6C4:8, R6C5:8, R6C6:8]┉[R6C7:8]━[R4C8:8, R5C8:8]┉[R1C8:8]━[R1C8:2]┉[R1C4:2]━[R2C5:2]. The first and last nodes are both the simple node [R2C5:2], and both links incident to this node are strong.

Assume R2C5 is not 2. Then it must be 8, making R4C5:8 false. The first grouped node must therefore be true, which makes R6C7:8 false. The second grouped node must then be true, making R1C8:8 false. This forces R1C8 to be 2, so R1C4:2 is false. The final strong link now forces R2C5 to be 2, contradicting the assumption.

Therefore, R2C5 must contain 2.