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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).

XYZ-Wing Exemplars

These puzzles require the XYZ-Wing strategy at some point but are otherwise trivial.
They also require one Naked Pair.
They make good practice puzzles.

Go back to Y-WingsContinue to WXYZ-Wing


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Saturday 8-May-2010

... by: Nassir M

Andrew Stuart writes:

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Article created on 11-April-2008. Views: 205102
This page was last modified on 27-December-2014.
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Copyright Andrew Stuart @ Syndicated Puzzles Inc, 2014