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David Bryant
Joined: 29 Jul 2005 Posts: 559 Location: Denver, Colorado

Posted: Mon Oct 10, 2005 3:42 pm Post subject: XYWing Example #2 


This is a continuation from the topic "A Very Tough Minimal Sudoku."
Someone_Somewhere wrote: 
Hi,
one more:
014050000
000006008
002000000
000800002
005010900
600007000
000000100
700300000
000090540 
The early going on this puzzle is quite similar to the first few moves on your previous example . The pairs one must look at are {1, 5} in column 9, {5, 9} in the middle center 3x3 box, and {4, 5} in row 8  that's just to get the first few numbers placed, so one can make some more progress.
Anyway, after 35 moves I arrived at the following position:
Code: 
*8/9* 1 4 *7/9* 5 3/8 . . .
5 7 3 1/9 2 6 4 1/9 8
*8/9* 6 2 4 *7/8* . 3/7 . 5
1 4 7 8 3 9 6 5 2
3 8 5 6 1 2 9 7 4
6 2 9 5 4 7 3/8 3/8 1
4 9 6/8 2 . 5 1 . .
7 5 1 3 6/8 4 2/8 . 6/9
2 3 6/8 1/7 9 1/8 5 4 6/7 
where the "XYWing" formation is marked with asterisks.
If we concentrate on the {7, 8} pair in r3c5 we can see that
r3c5 = 7 ==> r1c4 = 9 ==> r1c1 = 8
and also
r3c5 = 8 ==> r3c1 = 9 ==> r1c1 = 8
So whichever number goes in r3c5, we _must_ put the "8" in r1c1. This move uncovers a "3" at r1c6 and a "9" at r3c1, and the rest of the puzzle is fairly simple from there. dcb
PS One can also look at the "XYWing" in reverse, which is actually easier for me to see. If we assume r1c1 = 9 we obtain a contradiction:
r1c1 = 9 ==> r3c1 = 8 ==> r3c5 = 7 ==> r1c4 = 9 tilt!  two "9"s in row 1. 

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alanr555
Joined: 01 Aug 2005 Posts: 194 Location: Bideford Devon EX39

Posted: Wed Oct 12, 2005 9:14 pm Post subject: Re: XYWing Example #2 


> [Anyway, after 35 moves I arrived at the following position:
> *8/9* 1 4 *7/9* 5 3/8 . . .
> 5 7 3 1/9 2 6 4 1/9 8
> *8/9* 6 2 4 *7/8* . 3/7 . 5
> where the "XYWing" formation is marked with asterisks.
This XYWing can be viewed in TWO ways.
The skill is in spotting where one occurs.
One way is to find a quadrilateral of which three corners are composed
each of any two digits from three. Then the fourth corner CANNOT be
the digit that is NOT in its opposite corner but IS in the other two.
Here the three corners could be r1c1,r1c4,r3c5 This leads to digit 8
not be possible for r3c1 (ie must be 9)
Alternatively the three corners could be r1c4,r3c5,r3c1. This leads to
digit 9 being impossible for r1c1 (ie must be 8)
Note: This example can give information only about r1c1 and r3c1 as
these two cells have identical candidates and so CANNOT form the
"second" corner of a quadrilateral which meets the rule above.
(eg 8/9,8/9,7/8 does NOT have any digit which in the first and third
positions but is not in the second. Similarly 8/9,8/9,7/9 is not one).
> PS One can also look at the "XYWing" in reverse, which is actually
> easier for me to see. If we assume r1c1 = 9 we obtain a contradiction:
> r1c1 = 9 ==> r3c1 = 8 ==> r3c5 = 7 ==> r1c4 = 9 tilt!  two "9"s in > row 1.
This is true  and useful in formulating a logical rule  but it invokes the
big confront of "IFTHEN". My impression is that postulation 0f a value
for a cell is, BY AGREEMENT, not regarded as within the ranges of
solution methods being promoted on THIS site (or needed for any of
the Daily sudoku puzzles)  however much it may be tolerated or even
promoted on OTHER sites.
ie  the XYwing method is a means of resolving a potential "IFTHEN"
without actually invoking the possibility of backtrack if an incorrect
postulation is made!!
Alan Rayner BS23 2QT 

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