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The definite method for identification of XYZ wing

 
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IUSUT
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Joined: 11 Feb 2007
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Location: ROMANIA

PostPosted: Tue Apr 17, 2007 10:22 pm    Post subject: The definite method for identification of XYZ wing Reply with quote

I invite you to talk about
[b]Identification methods of the XYZ wing patterns in the Sudoku grids,[/b]
my essay from page
http://www.sudoku.org.uk/discus/messages/2/3982.html?1176761976

MIHAIL IUSUT
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Joined: 09 May 2007
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Location: Belgium - Ghent

PostPosted: Wed May 09, 2007 9:25 pm    Post subject: Reply with quote

Hi IUSUT,

Like (X)Y-Wings, XYZ-Wings are members of what I call the Y-Wing family, because they can be expanded to WXYZ-Wings, VWXYZ-WINGS and so on...

Historical, an Y-Wing was given that name because of the Y-pattern the 3 involved cells form if you represent them as a tree with the pivot as stem and both pincers as branches. The XYZ-Wing can be represented the same way, so, to avoid misunderstandings they called that pattern an XYZ-Wing based upon the number of candidates of the stem cell {X,Y,Z}. This, of course had implications on naming the Y-Wing, who now, based upon the number of candidates of the stem cell {X,Y}, could be named XY-Wing.

As a result of my investigations about (Extended) Aligned Pair Exclusions, I came up with a new, more logical, naming. But I think most of the sudoku programmers will stick to the old namings, and so will I.

Nevertheless, there is a logic ground on naming Y-Wing members like I see it:
It's maybe a little bit confusing, because both (X)Y-Wing and XYZ-Wing could have been named accordingly to the total number of candidates that are included in all cells belonging to the pattern {X,Y,Z}. This means that both (X)Y-Wings and XYZ-Wings should be named XYZ-Wings. The only difference is, that a regular Y-Wing's stem cell does not have the Z-candidate (also called the attacking candidate). This has some implications, because 2 attacking cells (in a regular Y-Wing) can attack more cells (outside the pivot's box) then 3 attaking cells (in a regular XYZ-Wing). Therefore I call a regular Y-Wing, who is more powerfull than a regular XYZ-Wing, an Extended XYZ-Wing. This approach, allows me to make simple distinguises between other (larger) members from the Y-Wing family with their stem cell holding or not holding the attacking candidates as follows:

Stem cell with attacking candidate / Stem cell without attacking candidate
XYZ-Wing / Extended XYZ-Wing
WXYZ-Wing / Extended WXYZ-Wing
VWXYZ-Wing / Extended VWXYZ-Wing
UVWXYZ-Wing / Extended UVWXYZ-Wing
TUVWXYZ-Wing / Extended TUVWXYZ-Wing

Note that large Y-Wing members like UVWXYZ-Wings and so on... probably don't exist in practise.

But there is more to say about the Y-Wing family...
The pincers are in fact bi-value cells, and as we all know, bi-value cells are, in fact, the smallest possible Almost Locked Sets, also known as ALS. Now, those pincers are not limited to a single cell ALS, they can be multiple cell ALS's as well and will, in fact, form a pattern that is allready known as Death Blossom.

But that is only the beginning...

Mike Barker quoted:
Quote:
Strong links and grouped strong links can be used for pincers. Likewise chains can be used as petals. With these additions, you have what is called Kraken Blossom.

And if that doesn't work you can replace the pivot cell with any restricted set you wish: a row, column, or box; a basic, franken, or mutant fish; a UR, BUG-lite, ALS, multi-celled AALS, etc. So there are lots of options to use the knowledge you've gained looking at Y-Wing members this way - not to mention all of the more traditional ALS techniques!


Lunatic.
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