127.0.0.1: Created page with "A '''XY-Chain''' is a chain of cells which each contain only 2 candidates. Because the chain is entirely made up by bivalue cells, the link between the 2..."

2021-10-25T15:05:45Z

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A '''XY-Chain''' is a [[chain]] of [[cell]]s which each contain only 2 [[candidate]]s.

Because the chain is entirely made up by [[bivalue]] cells, the [[link]] between the 2 candidates in each cell cause strong [[inference]], which allows us to use weak inference between the cells.

The shortest XY-Chain is an [[XY-Wing]] with only 3 cells.

A XY-Chain is both a [[Double Implication Chain]] and a [[Alternating Inference Chain]].

== Example ==
The following example shows a XY-chain resulting in an elimination.

[[Image:XYChain.png]]

This chain involves four [[cell]]s. In [[Eureka]] notation:
(5=1)r3c3-(1=6)r7c3-(6=1)r8c1-(1=5)r8c5

What does this XY-chain mean? Note that both ends of the chain involves the [[digit]] 5 as a [[candidate]]. Now, if '''r3c3<>5''', then we have '''r3c3=1''', which implies '''r7c3=6''' and '''r8c1=1''', so '''r8c5=5'''. Similarly, by reversing the direction, if '''r8c5<>5''', then '''r3c3=5'''. Thus, this is a [[Double Implication Chain]] and we showed that either '''r3c3=5''' or '''r8c5=5'''. This means that any cell that is seen by both '''r3c3''' and '''r8c5''' cannot contain the [[digit]] 5, so we [[eliminate]] 5 from '''r3c5'''.

A single inference xy-chain is also useful for finding a wrap rather than a trap as in the above example. To illustrate this consider rows 6 and 7 of a puzzle shown below.
<table>
<tr>
<td>Row 6 </td><td> | </td><td>   4 </td><td>   1 </td><td>  278 </td><td>  | </td><td>  23 </td><td>  37 </td><td>   6 </td><td> | </td><td>   9 </td><td>  78 </td><td>   5 </td><td>  |</td>
</tr><tr>
<td>Row 7  </td><td> | </td><td>   9 </td><td>  37 </td><td> 1347 </td><td>  | </td><td>   5 </td><td>  34 </td><td>  148 </td><td> | </td><td>   6 </td><td>  78 </td><td>   2 </td><td>  |</td>
</tr></table>
The 3 digits in column 5 are a conjugate pair. Now r7c5 = 3 implies r7c2 = 7 implies r7c8 = 8
implies r6c8 = 7 implies r6c5 = 3, which is a wrap. Therfore r7c5 = 4 and r6c5 = 3.

== See also ==
* [[Chain]] and [[Loop]]
* [[XY-Wing]]
* [[X-Chain]]
* [[Remote Pairs]]
* [[Nice Loop]]
* [[Double Implication Chain]]
* [[Alternating Inference Chain]]

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

XY-Chain - Revision history

127.0.0.1: Created page with "A '''XY-Chain''' is a chain of cells which each contain only 2 candidates. Because the chain is entirely made up by bivalue cells, the link between the 2..."