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Grouped X-Chain ​

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A Grouped X-Chain is an X-Chain that allows grouped nodes, while still involving only one candidate digit. Grouped nodes may occur in the middle of the chain or as its end nodes.

How to spot it ​

  • For each candidate x from 1 to 9, find conjugate pairs or strong links involving grouped nodes.
  • Use weak links to connect those strong links, forming an AIC (alternating inference chain).
  • Eliminate candidate x from other cells that see both end nodes of the chain.

How it works ​

X-Chain
  • A Grouped X-Chain is also an AIC for a single candidate x, but it allows grouped nodes.
  • In a chain of the form [Ca:x]━[Cb:x, Cc:x]┉[Cd:x, Ce:x]━[Cf:x]┉…┉[Cy:x]━[Cz:x], if Ca does not contain x, then at least one of Cb and Cc must contain x (grouped strong link). This prevents both Cd and Ce from containing x (grouped weak link), which forces Cf to contain x (strong link). Continuing along the chain eventually forces Cz to contain x. The same reasoning works in reverse: if Cz does not contain x, then Ca must contain x.
  • Therefore, the end nodes [Ca:x] and [Cz:x] form a logical strong link: at least one of Ca and Cz must contain x.
  • Any other cell that sees both cells cannot contain x, so candidate x can be eliminated from it.

The same conclusion applies when one or both end nodes are grouped: candidate x can be eliminated from any other cell that sees every cell in both end nodes.

Examples ​

7927912791239123946914679167926957912591259169169591591689789791716689679679257892724572456172572571254679679567956794791267916796792723792394792479792498546382384752433134698321854881356

On the board above, consider an AIC for candidate 7, with end nodes [R1C1:7] and [R6C1:7]. Reasoning from each end:

  • If R1C1 is not 7, at least one cell in the group [R1C4, R1C5] must contain 7 (strong link). Although we do not know which cell it is, either choice prevents R2C6 from containing 7 (weak link). This forces R4C6 to contain 7 (strong link), ruling out 7 from both cells in the group [R6C4, R6C5] (weak link). Therefore, R6C1 must contain 7 (strong link).
  • If R6C1 is not 7, the same reasoning in reverse forces R1C1 to contain 7.

Thus, at least one of R1C1 and R6C1 must contain 7. Either cell rules out 7 from R9C1, so candidate 7 can be eliminated from R9C1.

568958916812856812585681685681586826828579569567357927923537957835879583791571569356867957895789791915579127814815727357845781358128914891626778947891843723749179543841421624348266953

On the board above, an AIC for candidate 7 has the grouped node [R5C4:7, R5C6:7] at one end and the simple node [R9C6:7] at the other. A cell eligible for elimination must see both R5C4 and R5C6, as well as R9C6.

347934745894589574571491416846848914745945913573457347914714748348347847564593459367347561581458468483673671591595937457457485781345134548358358612537228626958213972842236919726

Both end nodes of the chain above are grouped nodes. A cell eligible for elimination must see every cell in both groups.

68989393456857845745671616767168133456858564569456935891581589135459491515615693578979895939141241258456815614568147145145745147912479146812781467846146821245983726472273812643976839253

In the chain above, two grouped nodes lie in the same box, one along a row and the other along a column. Do not confuse this with an Empty Rectangle: in an Empty Rectangle, the two perpendicular grouped nodes in the same box form a strong link; here, they form a weak link.