pure-sudoku.app

English

简体中文

English

简体中文

Appearance

X-Chain ​

Loading demo...

An X-Chain is an AIC (alternating inference chain) involving a single candidate digit. Every node in the chain refers to the same candidate digit, and its strong links come from conjugate pairs for that digit.

How to spot it ​

  • For each candidate x from 1 to 9, find its conjugate-pair strong links: houses in which only two cells can contain x.
  • Connect several strong links using weak links between nodes that see each other. The resulting chain must alternate between strong and weak links, and start and end with strong links, for example [Ca:x]━[Cb:x]┉[Cc:x]━[Cd:x]┉…┉[Cy:x]━[Cz:x].
  • Check the other cells that see both end nodes, [Ca:x] and [Cz:x], and eliminate candidate x from them. There may be more than one elimination.

How it works ​

X-Chain
  • An X-Chain is an AIC for a single candidate x, with strong links supplied only by conjugate pairs.
  • In an X-Chain of the form [Ca:x]━[Cb:x]┉[Cc:x]━[Cd:x]┉…┉[Cy:x]━[Cz:x], if Ca does not contain x, then Cb must contain x (strong link). This prevents Cc from containing x (weak link), which forces Cd 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.

Examples ​

123123812381381238238281381238138123581235812242413813483483823238238123123535121414528125818123138138284965775649974667193855796296514789674976367459

On the board above, candidate 3 forms the X-Chain [R6C1:3]━[R1C1:3]┉[R1C9:3]━[R2C9:3]┉[R2C5:3]━[R5C5:3]. By the rules of AICs, at least one of R6C1 and R5C5, the cells containing the end nodes, must contain 3. Candidate 3 can therefore be eliminated from any other cell that sees both end nodes. Note that there may be several such cells.

694934646836827467246868468274634672334668368368494669492424889369344669696896893682362668517231598915512974875321361751785221754345917

On the board above, candidate 4 forms the X-Chain [R2C9:4]━[R3C9:4]┉[R3C3:4]━[R6C3:4]┉[R5C2:4]━[R5C5:4]┉[R7C5:4]━[R7C6:4]. At least one of R2C9 and R7C6, the cells containing the end nodes, must contain 4, so candidate 4 can be eliminated from any other cell that sees both end nodes.

24242467462478989242646289268912657278128564646248238123457278128454534634634545893893756746493592459247574745969185733511783954375218399682917121368

The board above contains an X-Chain for candidate 4 whose end nodes happen to lie in the same column.