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[ZeroAssoluto]

Hi everyone,

[Mogulmeister]

r2c5 can't be 6 because if you do then r2c3 is 4 and then r3c1 is 7 and r3c4 is 6.

This gives two 6's in block 2 so r2c5 <>6.

[TomC]

[Mogulmeister]

No Tom, although as you have noted, it used the same cells as the XY Wing (sometimes just called a Y wing).

However the execution is different. This is a “Forcing Chain” that creates a contradiction showing that the initial assumption is wrong. It involves plugging in an actual value.

The XY wing is a logical structure that doesn’t care about the values that the cells might eventually take. The internal logic is saying that one end (pincer) or the other (r2c3 or r3c4) must be 6 which means there can be no 6 in r2c5.

The XY wing is of course just the shortest XY chain. Some people refer to them as “bent triples”

[Mogulmeister]

Lots of discussion about this in the past - some purists would argue against forcing chains mostly on aesthetic grounds.

Personally, I use them all the time and they’re quite valuable - even essential. I don’t think solvers care too much about these distinctions these days. The contradiction is an essential part of the armoury imho.

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[TomC]

I've been under the impression for a very long time that this was the way to solve XY wings. When I find the 3 candidate cells the contradiction is what is most apparent to me.

Is a W wing a form of forcing chain too?

[Mogulmeister]

The classical use of "wings" is to produce a situation where one or other of the ends (pincers) contains the number which will cause the elimination. The situation works because one or other of the ends will always be true for that number.

[Mogulmeister]

On a general point everyone sees a puzzle differently which is why I put in the contradiction in after Gianni's XY wing. All wings are also chains. Not only is the XY Wing a chain but it is also an ALS xz. Using exactly the same squares.

ALS

Set A {4,6,7) r2c3 and r3c1
Set B {6,7} r3c4

Restricted common is 7 in r3c14

Common candidate to be removed is the 6 in r2c5 because it can see all other 6's in Set A&B.

[Mogulmeister]

The classic XY wing above {1,2,7} has three candidates and most importantly 1 in both pincers (pale blue). The green square is the pivot that links them.

We don't know what values the blue squares will be but we can see that that one of them will have to be 1. Any 1(pink) that can be seen by both pincers is eliminated. You don't have to see a contradiction for the wing to work.

Diagram below is an XY chain at work that just happens to be a W Wing.

[Mogulmeister]

Finally, on a philosophical point regarding Sudoku solving:





There are some who see strange "yellow" teeth in a black mouth rather than the famous symbol!