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2020-06-04T02:58:00Z

Created page with "A '''X-Chain''' is a single-digit solving technique which uses a chain consisting of links that alternate between strong links and weak links, with the..."

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A '''X-Chain''' is a [[single-digit]] [[solving technique]] which uses a [[chain]] consisting of [[link]]s that alternate between [[strong link]]s and [[weak link]]s, with the starting and ending link being strong. In other words, the chain involves an even number of [[cell]]s having the target [[digit]] as a [[candidate]]. The target digit can be [[eliminate]]d from any cell that is seen by both ends of the X-Chain.

The X-Chain technique and the [[Multi-Colors]] technique are equally powerful. Also, a X-Chain is also a [[Double Implication Chain]] and an [[Alternating Inference Chain]].

== Example ==
This is an example of a X-Chain of length six for the digit 3.

[[Image:XChain.png]]

In this example, the X-Chain is marked using blue lines, with solid lines indicating strong links and dotted lines indicating weak links. In [[Nice Loop]] notation, the X-Chain can be expressed as follows:
[r4c4]=3=[r5c5]-3-[r5c9]=3=[r6c8]-3-[r9c8]=3=[r8c7]

The two ends of the chain are '''r4c4''' and '''r8c7''', and either end not being 3 will result in the other end being 3. This means that '''r8c4''', which is [[see]]n by both '''r4c4''' and '''r8c7''', cannot contain 3.

One proof for this is that we can formulate these two [[Forcing Chain]]s, with the second due to the X-Chain:
r4c4=3 => r8c4<>3
r4c4<>3 => r5c5=3 => r5c9<>3 => r6c8=3 => r9c8<>3 => r8c7=3 => r8c4<>3

== X-Chain and Multi-Colors ==
X-Chain can be replicated by [[Multi-Colors]] by treating each pair of [[cell]]s connected by a [[strong link]] as a color [[cluster]] and [[weak link]]s as [[bridge]]s between clusters. Multi-Colors can be replicated by X-Chain by the reverse of the above, and treating some of the strong links induced by the color clusters as weak links.

== Minimum Length of X-Chain ==
In an ordinary Sudoku, the shortest useful X-Chain that allows eliminations to be made must involve at least four cells.

However, in [[Sudoku Variants]] such as [[Sudoku-X]] and [[Killer Sudoku]], the shortest useful X-Chain can involve only two cells. An example is illustrated in the [[Crossover]] technique for [[Sudoku-X]], made possible due to the [[diagonal]] [[constraint]]s. When we eliminate candidates using a X-Chain of two cells, the same elimination can be performed using [[Common Peer Elimination]].

== See Also ==
* [[Chain]] and [[Loop]]
* [[Multi-Colors]]
* [[XY-Chain]]
* [[Nice Loop]]
* [[Double Implication Chain]]
* [[Alternating Inference Chain]]

[[Category:Chains and Loops]]
[[Category:Solving Techniques]]

X-Chain - Revision history

Rooted: Created page with "A '''X-Chain''' is a single-digit solving technique which uses a chain consisting of links that alternate between strong links and weak links, with the..."