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Definition of XY-Wing - Clarification

 
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alanr555



Joined: 01 Aug 2005
Posts: 194
Location: Bideford Devon EX39

PostPosted: Mon Oct 17, 2005 9:53 pm    Post subject: Definition of XY-Wing - Clarification Reply with quote

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In discussion of a specific puzzle the following general theme emerged
concerning identifying XY-wing. This topic is designed to take the point
further to assist us all in being able to use the technique successfully.

> The rule is that you need a quadruplet that lies in the logical equivalent
> of a straight line and can be separated into a "naked triplet" and the
> fourth value, which must stand alone.

Thank you for this. It makes sense - although I had never considered
the "naked triplet" aspect. They were just three corners of a rectangle!

> Here's an example of an "XY-Wing":
  3/5      3/7
  5/7      7/8
> Now, what would you call this pattern?
  3/5  3/7  5/7  7/8
> I guess most of us would call the second pattern a "naked triplet" and
> immediately resolve the fourth cell to an "8". But the logic involved is
> really the same as the logic in the "XY-Wing" -- in either case the
> values {3, 5, 7} must all appear in three of the cells, leaving the "8" as > the only possibility for the fourth cell.

Admirably (!) clear!

> As to what the "logical equivalent of a straight line" is, I think I'll leave > the verbose definition up to someone else. Oh, here, I'll take a stab at > it -- the four cells must form a closed loop of some sort, so that they're > the ONLY candidates for the quadruplet of values in question.   dcb  :)

This is the key point.
Knowing what to do with the pattern is one thing,
Spotting its occurrence is another!

The definition would seem to allow the cells to appear in any order but
if a rectangle is the "logical equivalent of a straight line " (logequiv)
then what distinguishes it and why cannot ANY line joining four cells
be regarded as a log-equiv?

In the example above.
If the {3/5  3/7  5/7} triplet is in a row or column say, what are the
constraints on the placement of the 7/8? Clearly it is OK if the 7/8 is
in the same row/column but what if it is a chess-knight's move away?

Using the rectangle version, it would appear that the fourth cell is
connected to TWO of the other three cells by means of being in the
same row,column or region. Is this clue? Is it required that of the four
cells EVERY one of them MUST be connected to at least TWO of the others
by being in the same row, column or region as such other cell? 

> PS You might think of the case where the four cells all lie within a single
> 3x3 box. These might even form a rectangle, but most of us would still > see it as a triplet plus a fourth value.

I always saw it as a "congruent group" and "stragglers" or "outsiders"
such that those outsiders could be eliminated. In fact, I never gave them
much thought - I just eliminated them!

> The "XY-Wing" terminology came into use, I think, because the pattern
> seems different, somehow, when the four cells don't all lie in the same
> row or column or 3x3 box.

> It's fundamentally the same pattern no matter how twisted or grotesque > it appears to be when mapped onto the two-dimensional grid.

Yes - that is why we need to define the constraints. A set of four cells
in r1c2, r5c3, r6c8, r3c7 form a set of four cells. It is possible to join a
closed line between them and create a "grotesque" shape but I doubt
that XY-wing would apply to them - or would it????

A little earlier, I suggested that any one cell of the four might need to be
"connected" to at least two of the others. Is this both a necessary and a
sufficient condition? If not what is the condition that is?

Alan Rayner BS23 2QT
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