Ordinary Chain (Closed)
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An Ordinary Chain (Closed) is an AIC (alternating inference chain) that can combine strong links from conjugate pairs and bivalue cells. For now, strong links involving grouped nodes are left out. Its end nodes refer to different candidates, but their cells can see each other.
If the end nodes are [Ca:x] and [Cb:y], candidate y can be eliminated from Ca and candidate x from Cb, wherever present. It is called closed because eliminations are confined to the cells containing the end nodes, and ordinary because its shape has no distinctive features beyond alternating strong and weak links.
How to spot it
- Observe the strong links on the board, considering only conjugate pairs and bivalue cells.
- Connect these strong links using weak links to form an AIC. Its end nodes should refer to different candidates in two cells that see each other.
- Check each cell containing an end node for the candidate referred to by the other end node. Eliminate it if present. There may be an elimination in only one of the two cells, or in both.
How it works
As in any AIC, the end nodes [Ca:x] and [Cb:y] of an Ordinary Chain (Closed) form a logical strong link.
If candidate y is present in Ca, it sees [Ca:x] in the same cell and [Cb:y] because cells Ca and Cb see each other. Thus, [Ca:y] sees both end nodes, so candidate y can be eliminated from Ca.
Similarly, if candidate x is present in Cb, it also sees both end nodes, so candidate x can be eliminated from Cb.
Thus, when the end nodes are [Ca:x] and [Cb:y], candidate y can be eliminated from Ca and candidate x from Cb, wherever present.
Examples
The board above contains the AIC [R1C9:4]━[R3C9:4]┉[R3C9:7]━[R3C6:7]┉[R1C5:7]━[R1C3:7]. Its end nodes refer to candidates 4 and 7, but their cells can see each other.
By the AIC principle, at least one of the following must be true: R1C9 contains 4, or R1C3 contains 7.
If R1C3 contains 7, it cannot contain 4. If it does not contain 7, then R1C9 must contain 4, which also prevents R1C3 from containing 4. Candidate 4 can therefore be eliminated from R1C3.
Similarly, candidate 7 could be eliminated from R1C9 if present, but R1C9 has no candidate 7 in this example.
The board above contains the AIC [R1C4:3]━[R3C4:3]┉[R3C4:7]━[R3C7:7]┉[R3C7:8]━[R1C7:8]. Its end nodes refer to candidates 3 and 8, but their cells can see each other.
By the AIC principle, at least one of the following must be true: R1C4 contains 3, or R1C7 contains 8. In either case, R1C4 cannot contain 8 and R1C7 cannot contain 3. Both candidates are present, so both can be eliminated.