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Discontinuous Loop ​

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A Discontinuous Loop is an AIC (alternating inference chain) whose two end nodes are the same simple node. Strong and weak links alternate along the chain, but on returning to the starting node through a strong link, the next link is also strong. We therefore cannot continue into another circuit while maintaining the alternation, which gives the loop its name.

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

How to spot it ​

  • Observe the strong links on the board. 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 [C:x], it forms a discontinuous loop.
  • An AIC requires at least one end node to be true, so cell C must contain digit x.

How it works ​

Discontinuous Loop

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 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.

Example ​

234823484823283813481234848938923838237378137913978379378189178978989897893489347837912892781289187938938937893782713913491234912347979561675145964526352646215653446512586

The board above contains the discontinuous loop [R3C1:2]━[R3C8:2]┉[R3C8:7]━[R3C9:7]┉[R7C9:7]━[R9C7:7]┉[R9C2:7]━[R9C2:2]┉[R7C1:2]━[R3C1:2]. The first and last nodes are both [R3C1:2], and both links incident to this node are strong.

If R3C1 is not 2, the first strong link forces R3C8 to be 2. Following the alternating links then forces R3C9 to be 7, R9C7 to be 7, and R9C2 to be 2. Consequently, R7C1 cannot be 2, and the final strong link forces R3C1 to be 2, contradicting the assumption.

Therefore, R3C1 must contain 2.