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Continuous Loop ​

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A Continuous Loop is an AIC (alternating inference chain) whose ends are joined. 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.
  • Connect these strong links with weak links to form an AIC. Its end nodes must see each other; otherwise, it isn't a continuous loop, consider an ordinary chain.
  • For each weak link, eliminate candidates that see both of its end nodes. This can produce many eliminations.

How it works ​

Continuous Loop

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 weak link has been “converted into a strong link.”

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.

Examples ​

67681617846468684671462612617241812848127271214714727168161248147146746167246162461681482472667465294331275989536395758439693535892937585193

The board above contains the continuous loop [R1C5:1]━[R1C9:1]┉[R1C9:8]━[R2C8:8]┉[R6C8:8]━[R6C5:8]┉[R6C5:6]━[R1C5:6]┉[R1C5:1].

Candidates 1 and 8 in R1C9 originally have only a weak link within the cell, because the cell has three candidates. The rest of the loop establishes a logical strong link between them, requiring R1C9 to contain either 1 or 8. Candidate 7 can therefore be eliminated from that cell.

Similarly, R6C5 must contain either 8 or 6, so candidate 1 can be eliminated.

1683513589135938968918351358935823891238168134813438136825258258235935235925258252818381233589289358181348134683684277642975196473352618479874619376146475297

The board above contains this continuous loop:

[R2C9:1]━[R3C9:1]┉[R3C9:6]━[R9C9:6]┉[R9C8:6]━[R9C8:8]┉[R7C8:8]━[R7C4:8]┉[R8C5:8]━[R2C5:8]┉[R2C2:8]━[R2C2:1]┉[R2C9:1].

Checking the original weak links gives these eliminations:

  • Candidates 1 and 6 in R3C9 form a strong link. Eliminate 3 and 8 from that cell.
  • Candidate 8 in R9C8 and R7C8 forms a strong link. Eliminate 8 from R1C8, R8C7, R8C8, R8C9, and R9C9, which see both nodes.
  • Candidate 8 in R7C4 and R8C5 forms a strong link. Eliminate 8 from R9C4.
  • Candidate 8 in R2C5 and R2C2 forms a strong link. Eliminate 8 from R2C4, R2C7, and R2C9.
  • Candidate 1 in R2C2 and R2C9 also forms a strong link. Eliminate 1 from R2C4.

This loop eliminates 12 candidates, a total that is difficult to achieve with other techniques.