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2020-06-04T01:09:15Z

Created page with "The '''ALS-XY-Wing''' rule is a solving technique that uses three Almost Locked Sets. Make sure you understand the concept of Almost Locked Sets (ALSs) before you..."

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The '''ALS-XY-Wing''' rule is a [[solving technique]] that uses three [[Almost Locked Set]]s.

Make sure you understand the concept of [[Almost Locked Set]]s (ALSs) before you proceed.

== How it works ==
Say we have three [[Almost Locked Set]]s '''A''', '''B''' and '''C'''. Suppose
* '''A''' and '''C''' share a [[restricted common]] '''Y''',
* '''B''' and '''C''' share a [[restricted common]] '''Z''', and
* '''Y''' and '''Z''' are different [[digit]]s.

Then for any digit '''X''' that is distinct from '''Y''' and '''Z''' and is a common [[candidate]] for '''A''' and '''B''', we can [[eliminate]] '''X''' from any [[cell]] that sees all cells belonging to either '''A''' or '''B''' and having '''X''' as a candidate.

== Relations to other techniques ==
Observe that this [[technique]] generalizes [[XY-Wing]], and hence the name ALS-XY-Wing. Note that each of the three [[bivalue]] [[cell]]s in the XY-Wing is itself an ALS, hence we can set '''C''' as the [[pivot]] cell, and the other two [[pincer]] cells as '''A''' and '''B''' respectively.

Recall that the XY-Wing is a [[chain]] of length three, and can be generalized to [[XY-Chain]], which is a chain of arbitrarily length. Similarly, we can generalize ALS-XY-Wing to [[ALS-XY-Chain]], which is a chain whose [[node]]s are ALSs instead of single cells.

== Example ==
This example is taken from Ron Moore's [http://www.sudocue.net/forum/viewtopic.php?p=2717 walkthrough] for Ruud's [http://www.sudocue.net/olddaily.php?id=509&sol=0 2007 May 5 Nightmare].

[[Image:ALSXYWingExample.png]]

The ALS-XYZ-Wing is constructed as follows:
* The ALS '''A''' is the cells colored green in box 8,
* The ALS '''B''' is the cells colored light blue in box 3,
* The ALS '''C''' is the cell colored yellow in box 9,
* The restricted common '''Y''' between '''A''' and '''C''' is 2, and
* The restricted common '''Z''' between '''B''' and '''C''' is 1.

The cells '''r7c46''' in ALS '''A''' and the cell '''r2c9''' in ALS '''B''' contain the candidate 5. Since all of '''r7c46''' and '''r2c9''' have a common peer '''r7c9''', we can eliminate 5 from '''r7c9'''.

A possible Eureka notation for this elimination is:

(5=24791)ALS:r12c89,r3c8 - (1=2)r7c8 - (216=5)ALS:r7c46,r9c5 => r7c9 <> 5

Note, the reader must check the grid to ensure that r7c9 can see all the 5's in both ALS's and similarly for the links involving digits 1 and 2.

Using the ALS (245)r12c9 and a group node for (2)r12c8 we could also use ALS nodes in an [[AIC]]:

(5=42)ALS:r12c9 - (2)r12c8 = (2)r7c8 - (216=5)ALS:r7c46,r9c5 => r7c9 <> 5

which provides greater freedom for the locations of candidates in box 3.

In this example, since the pivot ALS '''C''' is a single cell, we can replicate the same elimination using [[Death Blossom]] instead.

== Notes on notation ==
Unlike [[ALS-XZ]] and [[XY-Wing]], which eliminates the [[digit]] '''Z''', the ALS-XY-Wing eliminates the [[digit]] '''X''' instead. Unfortunately, this inconsistent notation is widely used by the [[Sudoku]] community.

== External link ==
* [http://www.sudoku.com/forums/viewtopic.php?p=16334 Almost locked rules (for now)]

== See also ==
* [[XY-Wing]]
* [[Almost Locked Set]]
* [[ALS-XZ]]
* [[ALS-XY-Chain]]
* [[Death Blossom]]

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

ALS-XY-Wing - Revision history

Rooted: Created page with "The '''ALS-XY-Wing''' rule is a solving technique that uses three Almost Locked Sets. Make sure you understand the concept of Almost Locked Sets (ALSs) before you..."