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Grouped Ordinary Chain (Open) ​

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A Grouped Ordinary Chain (Open) is an Ordinary Chain (Open) that allows grouped nodes. The digits may change along the chain, but both end nodes must refer to the same candidate. Grouped nodes may occur in the middle of the chain or as its 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 refer to the same candidate x, and they must not see each other. If they do, consider a grouped continuous loop instead.
  • Eliminate candidate x from any other cell that sees both end nodes. There may be several eliminations.

How it works ​

Ordinary Chain (Open)

A Grouped Ordinary Chain (Open) is also an AIC with no distinctive shape, but grouped nodes may occur in the middle or at either end. Both end nodes refer to the same candidate x, and the AIC establishes a logical strong link between them. At least one end node must therefore be true. Candidate x can be eliminated from any other cell that sees both end nodes.

Examples ​

124912492334292425257478278292373636792679189494681468168179156795694935134715747845745578237925679569372556868947956794759692349459342591915978656391581432572863192141823867

The chain above has one grouped end node and one simple end node:

[R2C8:7, R3C8:7]━[R5C8:7]┉[R5C8:3]━[R5C7:3]┉[R5C7:5]━[R7C7:5]┉[R7C6:5]━[R2C6:5]┉[R2C6:7]━[R3C6:7]

  • Candidate 7 appears only in R2C8, R3C8, and R5C8 within column 8. The grouped node [R2C8:7, R3C8:7] and simple node [R5C8:7] therefore form a strong link: if neither R2C8 nor R3C8 contains 7, then R5C8 must contain 7.
  • Candidates 7 and 3 in cell R5C8 form a weak link: if R5C8 contains 7, it cannot contain 3.
  • Continuing along the alternating strong and weak links eventually forces R3C6 to contain 7.
  • Conversely, if R3C6 does not contain 7, the reverse reasoning forces at least one of R2C8 and R3C8 to contain 7.

Cell R3C9 sees both cells in the grouped end node and cell R3C6 in the simple end node; Candidate 7 can therefore be eliminated from R3C9.

134194515391383484681584563454818919494545345346456181836383683613516355634564693594563569345346469355962789277523697136287922549719278428177128

Both end nodes of this chain are grouped nodes:

[R9C1:4, R9C2:4]━[R9C4:4]┉[R9C4:9]━[R9C8:9]┉[R9C8:5]━[R1C8:5]┉[R1C6:5]━[R1C6:4]┉[R1C2:4]━[R2C3:4, R3C3:4]

At least one of the end nodes [R9C1:4, R9C2:4] and [R2C3:4, R3C3:4] must be true. This means that 4 must appear in at least one of the two groups: in R9C1 or R9C2, or in R2C3 or R3C3. Cell R8C3 sees all four cells in these two groups, so candidate 4 can be eliminated from it.