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Cherylpc
Joined: 17 Jul 2006 Posts: 1

Posted: Mon Jul 17, 2006 5:00 pm Post subject: XWing & XY Wing 


Please explain what an XWing and an XYWing is? 

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

Posted: Mon Jul 17, 2006 9:58 pm Post subject: Lots of explanations online 


Hi, Cheryl! Welcome to the forum.
Please try reading this explanation of the "XWing" pattern. Then check out this explanation of the XYWing.
If you still have questions after reading that stuff, please feel free to ask again. dcb 

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jake3988
Joined: 22 Sep 2006 Posts: 5

Posted: Fri Sep 22, 2006 9:41 pm Post subject: 


Like Cheryl (but 2 months removed) am looking for an explaination. However, most explainations have horriblelooking diagrams associated (like on forums and such).
This has that advantage, but the explaination is utterly horrendous. First he says you can use it with rectangles. But, there's at least 3 other rectangles you can form with the 1's, never explaining why you can't use those.
Plus, there's no reasoning on why this method works, either. I won't just 'take his word for it', I have to have a reasoning as to why it works.
If anyone else can clearly explain it, or point me somewhere else, that would be immensely helpful. 

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

Posted: Fri Sep 22, 2006 10:32 pm Post subject: Which are you asking about? 


Hi, Jake!
I'm a little confused by your post. Are you asking abou the "XWing" pattern? Or about the "XYWing"? Since you mentioned a rectangle, I suppose you're asking about the "XWing".
The first thing you might try is to use the search button at the top of this page, and run a search on xwing in this forum. That should get you 15 or 20 examples, many of them clearly explained.
Here, I'll explain it one more time.
There are two possible kinds of XWings: an XWing on rows and an XWing on columns.
An XWing on rows occurs when three conditions are satisfied for a particular candidate digit x.
 In row a, there are only two spots where digit "x" can occur.
 In row b, there are only two spots where digit "x" can occur.
 The pairs of possible spots for "x" in rows a and b fall in the same two columns  let's call them m and n.
The "XWing rule" now says that any other possible "x"s in columns m and n can be eliminated. The logic is simple.
Either there's an "x" at Row a Column m and an "x" at Row b Column n, or else there's an "x" at Row a Column n and an "x" at Row b Column m; no other configurations are possible.
Another way to think of it is as follows. If an "x" occurred in column m and not in row a or row b, it would be impossible to complete rows a and b.
An XWing on columns is exactly analagous, but now you have two columns (in each of which "x" can only fit in two spots) and the "candidate cells" in these columns lie in just two rows. You can transform an XWing on rows into an XWing on columns by turning the puzzle sideways, so these are really the same pattern, just rotated by 90 degrees. dcb 

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jake3988
Joined: 22 Sep 2006 Posts: 5

Posted: Sat Sep 23, 2006 11:20 pm Post subject: 


Thanks for the reply.. I still don't understand the logic. Nor could I ever pick it out.
Let's say, we have something like:
(This is the exact [recreated] situation from the site refferal. Note, all the '1's are candidates for the one.)
1xxx1xxx1
xxxx1xxx1
1xxx111xx

xxxxxxxxx
xxxx111x1
xxxx1xxx1
Edit: Actually, according to your given definition, it would make the one specified on the website work. So I understand that.
However, on the website, it claims R3C6 is the upperleft corner of 'another' xwing which don't NOT apply to that definition. Which makes it really confusing.
Although, I'm not the type of guy who writes every single candidate for every single box. That's just sadistically overboard. So I guess this whole thing is moot anyway. 

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TKiel
Joined: 22 Feb 2006 Posts: 292 Location: Kalamazoo, MI

Posted: Sun Sep 24, 2006 2:33 am Post subject: 


The other Xwing is in columns. Columns 6 & 7 both have only two places for the 1 and those places are in rows 3 & 5. Eliminates all other 1 candidates in those two rows. 

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

Posted: Sun Sep 24, 2006 3:19 am Post subject: Play it again, Sam ... 


Hi, Jake! Thanks for writing back. And thanks for the diagram. Specific examples are easier to follow than abstract explanations.
Code:  1xxx1xxx1
xxxx1xxx1
1xxx111xx

xxxxxxxxx
xxxx111x1
xxxx1xxx1 
First, let's go through the logic for the XWing in rows 2 and 6 one more time.
 There has to be a "1" in row 2 somewhere. Just for the sake of argument, let's suppose that there's a "1" at r2c5. Then there cannot possibly be a "1" in these cells: r1c5, r3c5, r5c5, & r6c5.
 Still supposing that r2c5 = 1, we know that there has to be a "1" in row 6 somewhere. The only place it can fit is at r6c9. Therefore, if r2c5 = 1 we know that there cannot be a "1" at r1c9, r2c9, or r5c9.
 Summarizing, then, we see that if r2c5 = 1 there cannot be a "1" in any of these seven cells: r1c5, r3c5, r5c5, r6c5, r1c9, r2c9, r5c9.
 The "1" in row 2 does not have to appear at r2c5 ... it might be at r2c9 instead. So what happens if r2c9 = 1? Then there cannot be a "1" in any of these cells: r1c9, r5c9, & r6c9.
 Again, since there must be a "1" in row 6 somewhere, if r2c9 = 1 we must also have r6c5 = 1. Therefore, if r2c9 = 1 there cannot possibly ba a "1" in any of these cells: r1c5, r2c5, r3c5, & r5c5.
 Summarizing the case where r2c9 = 1 we see that "1" cannot appear in any of these seven cells: r1c5, r2c5, r3c5, r5c5, r1c9, r5c9, r6c9.
 Now, since one of these two cases must be true we can gain real insight into the situation by comparing the two lists derived in the third and sixth steps detailed above. We see that the following five cells appear in both lists: r1c5, r3c5, r5c5, r1c9, r5c9.
 The conclusion is inescapable  no matter where the "1" is placed in row 2 there cannot possibly be a "1" in any of these five cells: r1c5, r3c5, r5c5, r1c9, r5c9.
OK, that covers the "XWing on Rows" in the example you posted. As Tracy has pointed out, the second XWing in thie diagram is an "XWing on Columns". There are only two ways to fit a "1" in Column 6  at r3c6, or at r5c6. Similarly, there are only two ways to fit a "1" in Column 7  at r3c7, or at r5c7. This time the XWing pattern falls in columns 6 & 7, and the digits we can eliminate lie in rows 3 & 5. There cannot possibly be a "1" in any of these cells: r3c1, r3c5, r5c5, r5c9. The logic is entirely analagous to the argument I have explained in excruciating detail above.
Jake3988 wrote:  ... I'm not the type of guy who writes every single candidate for every single box. That's just sadistically overboard. 
I agree with you. About writing down the candidates, I mean. I generally work these puzzles without writing down any candidates in any of the boxes. I can usually spot the "XWing" pattern with ease. If I can do it, you can too. It's just a question of how hard you're willing to concentrate. dcb 

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jake3988
Joined: 22 Sep 2006 Posts: 5

Posted: Sun Sep 24, 2006 2:56 pm Post subject: 


thanks again. I still don't understand the logic, but I guess if I spot more examples I'll try and deduce why it actually works in my head. It'll come with practice.
That is, try and place ones in the other cells in that column and see why its impossible. I just don't see it from the example. 

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Marty R.
Joined: 12 Feb 2006 Posts: 5474 Location: Rochester, NY, USA

Posted: Sun Sep 24, 2006 3:50 pm Post subject: 


Quote:  Although, I'm not the type of guy who writes every single candidate for every single box. That's just sadistically overboard. 
I write 'em all in. But since this is a solo activity, wouldn't I be "masochistically overboard"? 

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TKiel
Joined: 22 Feb 2006 Posts: 292 Location: Kalamazoo, MI

Posted: Sun Sep 24, 2006 4:47 pm Post subject: 


jake3988,
Do you understand the concept of 'strong links' (also known as conjugate links)? It's pretty much essential to understanding Xwings.
Here is the link to an excellent post that explains the general concepts of 'strong links' and specifically explains the logic behind an Xwing:
http://www.sudoku.com/forums/viewtopic.php?t=3326 

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jake3988
Joined: 22 Sep 2006 Posts: 5

Posted: Sun Sep 24, 2006 10:37 pm Post subject: 


kinda. If he made his post about half as long and stayed with the quasireallife situation instead of making it confusing as heck with green letters and code, I would likely have it really understood.
He does a good job explaining the logic though in a very simple way.
But anyway, I found a real life situation in today's puzzle (the logic is completely different, and it wouldn't really qualify as a xwing, but basically the same logic applied.
That is, I had:
Code: 
5xx55xx5x
5xxx5x
xxxxxxxxx

(the dashes mean that row was completely filled already)
in my truncated example tje xwing box would be R1C1, R1C8, R2C1, an R2C8
It basically means, because that row is already filled in the middle, and ONE of the 5s HAS to be true in the top row of the side boxes so it eliminates 5 from the top row of the middle box.
I don't think its the same thing as an xwing at all, but I can sort of see how it works now. 

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TKiel
Joined: 22 Feb 2006 Posts: 292 Location: Kalamazoo, MI

Posted: Sun Sep 24, 2006 11:36 pm Post subject: 


Here is the position to which you refer:
Code: 
**
 7 5 28  4 289 28  6 1 3 
 248 3 1  25 7 6  25 9 48 
 2489 48 6  3 12589 258  25 7 48 
++
 6 7 9  8 4 3  1 2 5 
 48 1 48  7 25 25  9 3 6 
 3 2 5  9 6 1  4 8 7 
++
 258X 9 3  256 258 4  7 56X 1 
 245X 46 24  1 3 7  8 56X 9 
 1 68 7  56 58 9  3 4 2 
**

And it is an Xwing on columns, which means that the exclusions take place in the rows for the reasons you've stated, but your statement didn't go far enough. One of the top most cells in the outside boxes (in this example r7c1 or r7c8) must be 5 AND one of the middle cells in the outside boxes (r8c1 or r8c8) must also be 5. So, if there were unsolved cells in the middle row that contained candidate 5's, they could be excluded also. (I realize there are locked candidate exclusions that do the same thing but thought it best not to discuss them in this post). 

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jake3988
Joined: 22 Sep 2006 Posts: 5

Posted: Mon Sep 25, 2006 2:41 am Post subject: 


Right. Thanks. And yes, that's pretty much the exact point in the puzzle I saw it.
Solved it completely shortly thereafter too. (This hard was immensely easier than the previous day's very hard. It was indeed much harder than the medium and easy, but the very hard I only got about 2 numbers of the whole puzzle.) 

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