Ordinary Chain (Open)

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An Ordinary Chain (Open) 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. The digits may change along the chain, but both end nodes refer to the same candidate x, and their cells do not see each other.

It is called open because eliminations occur outside 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. Both end nodes should refer to the same candidate x, and their cells must not see each other. If they do, consider a continuous loop instead.
• Eliminate candidate x from any other cell that sees both end nodes. There may be several eliminations.

How it works

An Ordinary Chain (Open) is an AIC with no distinctive shape. Its end nodes form a logical strong link and must refer to the same candidate x, so at least one of the cells containing them must contain x. Candidate x can therefore be eliminated from any other cell that sees both end nodes.

Examples

The chain above combines two conjugate pairs and one bivalue-cell strong link:

[R7C3:1]━[R7C9:1]┉[R7C9:9]━[R9C9:9]┉[R9C4:9]━[R9C4:1]

• Candidate 1 appears only in R7C3 and R7C9 within row 7, forming a conjugate-pair strong link. If R7C3 does not contain 1, then R7C9 must contain 1.
• R7C9:1 and R7C9:9 share a cell, forming a weak link. If R7C9 contains 1, it cannot also contain 9.
• Candidate 9 appears only in R7C9 and R9C9 within column 9, forming another conjugate-pair strong link. If R7C9 cannot contain 9, then R9C9 must contain 9.
• R9C9:9 and R9C4:9 share a row, forming a weak link. If R9C9 contains 9, then R9C4 cannot contain 9.
• R9C4 contains only candidates 1 and 9, forming a bivalue-cell strong link. If it cannot contain 9, it must contain 1.
• Putting these inferences together, if R7C3 does not contain 1, then R9C4 must contain 1.

Conversely, starting from the assumption that R9C4 does not contain 1 also forces R7C3 to contain 1. Thus, R7C3:1 and R9C4:1 form a logical strong link: at least one of them must be true, so at least one of R7C3 and R9C4 must contain 1. Both R9C1 and R9C3 see these two cells, so candidate 1 can be eliminated from both.

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This longer example contains ten strong links:

[R4C6:2]━[R4C6:4]┉[R4C1:4]━[R6C1:4]┉[R6C1:9]━[R6C9:9]┉[R6C9:8]━[R6C8:8]┉[R8C8:8]━[R8C8:1]┉[R8C3:1]━[R8C3:4]┉[R8C5:4]━[R8C5:5]┉[R5C5:5]━[R5C4:5]┉[R5C4:7]━[R5C3:7]┉[R5C3:2]━[R6C3:2]

The digit changes several times along the chain, but both end nodes refer to candidate 2. At least one of R4C6 and R6C3 must therefore contain 2. Cell R6C6 sees both end nodes, so candidate 2 can be eliminated from it.