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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&nbsp;</td><td> | </td><td>&nbsp;&nbsp;&nbsp;4 </td><td>&nbsp;&nbsp;&nbsp;1 </td><td>&nbsp;&nbsp;278 </td><td>&nbsp;&nbsp;| </td><td>&nbsp;&nbsp;23 </td><td>&nbsp;&nbsp;37 </td><td>&nbsp;&nbsp;&nbsp;6 </td><td> | </td><td>&nbsp;&nbsp;&nbsp;9 </td><td>&nbsp;&nbsp;78 </td><td>&nbsp;&nbsp;&nbsp;5 </td><td>&nbsp;&nbsp;|</td> </tr><tr> <td>Row 7 &nbsp;</td><td> | </td><td>&nbsp;&nbsp;&nbsp;9 </td><td>&nbsp;&nbsp;37 </td><td>&nbsp;1347 </td><td>&nbsp;&nbsp;| </td><td>&nbsp;&nbsp;&nbsp;5 </td><td>&nbsp;&nbsp;34 </td><td> &nbsp;148 </td><td> | </td><td>&nbsp;&nbsp;&nbsp;6 </td><td>&nbsp;&nbsp;78 </td><td>&nbsp;&nbsp;&nbsp;2 </td><td>&nbsp;&nbsp;|</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]]

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