## Advanced Application of xyz-wing - xyz-chain

Advanced methods and approaches for solving Sudoku puzzles

### Advanced Application of xyz-wing - xyz-chain

The specific requirements of an xyz-wing render the technique rather limited in its application. However, after the underlying logic is understood, it can be used as a building block in conjunction with other simple logic to solve puzzles. We can use xyz-wing to extend our thinking and help us to logically identify cells for candidate elimination; which normally would not be possible.

Consider this example

r3c9-r3c1-r1c2 is an xyz-wing of 9. But, there is no elimination available at the intersection r3c2 or r3c3 since the cells are already occupied by 5 and 2 respectively. However, simple logic tells me that if r3c9=9 then r2c4=9. With this piece of additional information, the intersections of the xyz-wing can now be extended to cover cells r2c1, r2c2 and r2c3 also. This allows a 9 to be eliminated from r2c1 and r2c3; and the puzzle can be easily solved from here.

With simple xyz-wing:
When r3c9=9 => r3c2,3 not=9.
When r3c9=7 => r3c1 & r1c2=89 => r3c2,3 not=9.

With this extended xyz-wing (xyz-chain):
When r3c9=9 => r2c4=9 => r2c1,2,3 not=9.
When r3c9=7 => r3c1 & r1c2=89 => r2c1,2,3 not=9.
Last edited by Jeff on Sat Sep 10, 2005 3:57 pm, edited 2 times in total.
Jeff

Posts: 708
Joined: 01 August 2005

I would call this a grade-2 (using a pair) forcing chain over 5 cells:
r3c9=7, r3c1={8.9}
with r1c2={8.9}
-> r2c1<>9
r3c9=9->r3c4=3->r2c1<>9
still enjoyable for me (my current enjoyable cell limit is 6)
Wolfgang

Posts: 208
Joined: 22 June 2005

### Re: Advanced Application of xyz-wing

Jeff wrote:However, simple logic tells me that if r3c9=9 then r2c4=9. With this piece of additional information, the xyz-wing can now be referred to as r2c4-r3c1-r1c2

I fail to see the logic in that.

If r2c4=9 implied r3c9=9, then I would see it. But the other way around, I don't.
Nick70

Posts: 156
Joined: 16 June 2005

### xyz-wing: application

Nick

Look closer.

With simple xyz-wing:
When r3c9=9 => r3c2,3 not=9.
When r3c9=7 => r3c1 & r1c2=89 => r3c2,3 not=9.

With this transformed xyz-wing:
When r3c9=9 => r2c4=9 => r2c1,2,3 not=9.
When r3c9=7 => r3c1 & r1c2=89 => r2c1,2,3 not=9.

The condition requires r2c4=9 when r3c9=9; not the other way around.
Jeff

Posts: 708
Joined: 01 August 2005

### Re: xyz-wing: application

Right, I see it now. What mislead me is that you said

Jeff wrote:With this piece of additional information, the xyz-wing can now be referred to as r2c4-r3c1-r1c2.

Which doesn't seem to be the case since r3c9 is still an integral part of the reasoning.
Nick70

Posts: 156
Joined: 16 June 2005

### Re: xyz-wing: application

Post edited. Thanks Nick.
Jeff

Posts: 708
Joined: 01 August 2005