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XYZ-Wing

This is an extension of Y-Wing or (XY-Wing). John MacLeod defines one as "three cells that contain only 3 different numbers between them, but which fall outside the confines of one row/column/box, with one of the cells (the 'apex' or 'hinge') being able to see the other two; those other two having only one number in common; and the apex having all three numbers as candidates."

It follows therefore that one or other of the three cells must contain the common number; and hence any extraneous cell (there can only be two of them!) that "sees" all three cells of the Extended Trio cannot have that number as its true value.

XYZ-Wing theory
XYZ-Wing theory

It gets its name from the three numbers X, Y and Z that are required in the hinge. The outer cells in the formation will be XZ and YZ, Z being the common number.
Y-Wing Equivalent
Y-Wing Equivalent

It is worth comparing the XYZ-Wing to the Y-Wing for a moment. Lets drop the Z candidate from the hinge. The diagram on the right is the result. With just two Zs in the pincer cells we get more cells elsewhere which could potentially contain a Z to eliminate. The overlap is greater with less cells to line up.
XYZ-Wing example 1
XYZ-Wing example 1 : Load Example or : From the Start

In this example the candidate number is 1 and F9 is the Hinge. It can see a 1/2 in D9 and a 1/4 in F1. We can reason this way: If D9 contains a 2 then F1 and F9 become a naked pair of 1/4 - and the naked pair rule applies. Same with F1. If that's a 4 then D9 and F9 become a naked pair of 1/2 each. If any of the three are 1 then 1 is still part of the formation. Any 1 visible to all three cells must be removed, in this case in F7.
XYZ-Wing example 2
XYZ-Wing example 2 : Load Example or : From the Start
The second example shows an XYZ-Wings based on a row and a box.

The hinge cell is on E8 and the common candidate is 6. The 6 in E9 can see all the 6s in the whole XYZ formation.

Aligned Pair Exclusion



The logic on an XYZ-Wing is completely different and lot simpler than the Aligned Pair Exclusion described below but the funny thing is that XYZ-Wing is a total sub-set of APE. Every XYZ-Wing can be solved by APE (but not vis versa).
Go back to Y-WingsContinue to WXYZ-Wing


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Comments Talk

Sunday 1-Dec-2013

... by: Pieter, Newtown, Australia

Hi Andrew
Beside Example 1 you say "the candidate number is 1 and F9 is the Hinge. It can see a 1/2 in D9 and a 1/4 in F1. We can reason this way: ... "

I find it interesting your reasoning or explanation here starts from the wings and then uses Naked Pairs, and doesn't work outwards from the Hinge (as you do in Y-Wings). I always seem to reason from the Hinge outwards. If F9 is a 2 then D9 is a 1, or if F9 is a 4 then F1 is a 1, or F9 could be a 1, therefore any 1 visible to all three cells must be removed, in this case in F7.

In WXYZ-Wing you explain it quite succinctly by saying "The easy principle is that each possible value of the hinge cell results in a Z value in one of the cells in the [WXYZ]-Wing pattern, thus leaving no room for a Z on any cell all [four] can 'see'."

I like the idea of "Bent Quads" for WXYZ-Wing, so I suppose XYZ-Wing could be considered a "Bent Triple"?!

Sunday 7-Apr-2013

... by: Richard Goodrich

Which John MacLeod are you referring to?
Article created on 11-April-2008. Views: 96045
This page was last modified on 20-July-2013.
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Copyright Andrew Stuart @ Syndicated Puzzles Inc, 2013