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XY-Chain ​

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An XY-Chain is an AIC (alternating inference chain) whose strong links all come from bivalue cells. The candidates may change along the chain, but its two end nodes refer to the same candidate digit x.

How to spot it ​

  • Observe the bivalue cells on the board. Each contains exactly two candidates, which naturally form a strong link within the cell.
  • Try connecting some of these cells using weak links between matching candidates in cells that see each other. The chain must alternate between strong and weak links, start and end with strong links, and have the same candidate x at both end nodes.
  • Eliminate candidate x from any other cell that sees both end nodes. There may be several eliminations.

How it works ​

XY-Chain
  • All strong links in an XY-Chain come from bivalue cells, so all its nodes are simple nodes. Under this definition, an XY-Chain has no grouped variant.
  • In an XY-Chain of the form [Ca:x]━[Ca:a]┉[Cb:a]━[Cb:b]┉[Cc:b]━[Cc:c]┉[Cd:c]━[Cd:x], the two nodes of each strong link belong to the same cell. If Ca does not contain x, it must contain a (strong link). This prevents Cb from containing a (weak link), so Cb must contain b (strong link). Continuing along the chain eventually forces Cd to contain x. The same reasoning works in reverse: if Cd does not contain x, then Ca must contain x.
  • Therefore, the end nodes [Ca:x] and [Cd:x] form a logical strong link: at least one of Ca and Cd must contain x.
  • Any other cell that sees both cells cannot contain x, so candidate x can be eliminated from it.

Examples ​

68585639463659456131436868461946146889293741416358972785926431827157922537879514263534692718621873549

On the board above, the four bivalue cells form the XY-Chain [R4C1:9]━[R4C1:3]┉[R4C5:3]━[R4C5:6]┉[R6C5:6]━[R6C5:8]┉[R6C8:8]━[R6C8:9].

  • If R4C1 does not contain 9, it must contain 3 (strong link). This prevents R4C5 from containing 3 (weak link), forcing it to contain 6. Continuing along the chain eventually forces R6C8 to contain 9.
  • The same reasoning works in reverse: if R6C8 does not contain 9, then R4C1 must contain 9.

Candidate 9 can therefore be eliminated from any other cell that sees both R4C1 and R6C8.

684824682416788946784789146141674794679246749124656567827828787867868471478147816895791357179158189361689689491341483619145145149315972355389314542936123594252737862

The board above shows another XY-Chain, this time formed by five bivalue cells.