Grouped Ordinary Chain (Closed)
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A Grouped Ordinary Chain (Closed) is an Ordinary Chain (Closed) that allows grouped nodes. The chain can combine strong links from conjugate pairs, bivalue cells, and grouped nodes. Grouped nodes may occur in the middle of the chain, but both end nodes must be simple nodes. The 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. As in an Ordinary Chain (Closed), eliminations are confined to the cells containing the end nodes.
How to spot it
- Observe the strong links on the board, including conjugate pairs, bivalue cells, and strong links involving grouped nodes.
- Connect these strong links using weak links to form an AIC (alternating inference chain). Both end nodes should be simple nodes that refer to different candidates, but their cells must see each other. Grouped nodes may occur only in the middle of the chain.
- 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. Sometimes a candidate within a grouped node in the chain also lies in a cell containing an end node and can be eliminated; see Cannibalistic Chains.
How it works
A Grouped Ordinary Chain (Closed) allows grouped nodes in the middle, but both end nodes must be simple nodes. They refer to different candidates, and their cells see each other. As in any AIC, the end nodes [Ca:x] and [Cb:y] 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 [R9C2:2]━[R5C2:2]┉[R4C3:2]━[R4C3:1]┉[R4C1:1, R6C1:1]━[R8C1:1]. Its end nodes refer to candidates 2 and 1, but their cells can see each other.
By the AIC principle, at least one of the following must be true: R9C2 contains 2, or R8C1 contains 1.
If R9C2 contains 2, it cannot contain 1. If it does not contain 2, then R8C1 must contain 1, which also prevents R9C2 from containing 1. Candidate 1 can therefore be eliminated from R9C2.
Candidate 2 could also be eliminated from R8C1 if present, but it is absent in this example.
The board above contains the AIC [R2C2:6]━[R2C5:6]┉[R8C5:6]━[R8C5:8]┉[R8C1:8, R8C2:8]━[R9C2:8]. Its end nodes refer to candidates 6 and 8, but their cells can see each other.
By the AIC principle, at least one of the following must be true: R2C2 contains 6, or R9C2 contains 8. In either case, R2C2 cannot contain 8 and R9C2 cannot contain 6. Both candidates are present, so both can be eliminated.
Cannibalistic chains
Sometimes, part of a grouped node lies in a cell containing an end node and includes the candidate to be eliminated. That candidate can still be safely eliminated from the grouped node. A chain that eliminates a candidate used within the chain itself is called a cannibalistic chain.
For example, the board above contains the AIC [R2C6:9]━[R4C6:9, R5C6:9]┉[R5C4:9, R6C4:9]━[R8C4:9]┉[R8C4:2]━[R7C5:2, R8C5:2]┉[R4C5:2]━[R4C6:2]. The grouped node [R4C6:9, R5C6:9] includes R4C6, the cell containing the other end node, and candidate 9 in that cell is the elimination target.
Candidate 9 can still be eliminated from R4C6. If R2C6 does not contain 9, the strong link makes the grouped node [R4C6:9, R5C6:9] true. Following the chain then forces R4C6 to contain 2, so it cannot contain 9. In this case, [R5C6:9] being true is enough to make the grouped node true, so eliminating [R4C6:9] causes no contradiction. If R2C6 does contain 9, Sudoku rules also prevent R4C6 from containing 9 because they share a column. In either case, candidate 9 can be eliminated from R4C6.