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March 19 from sudoku.org.uk

 
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PostPosted: Sun Mar 19, 2006 11:21 pm    Post subject: March 19 from sudoku.org.uk Reply with quote

Is there any logical step from here?
Code:

251|748|963
394|652|7--
678|139|452
---+---+---
5--|983|-4-
183|427|596
94-|516|-3-
---+---+---
83-|2-4|--5
4-5|8-1|3-9
71-|3-5|--4

With candidates:
2 5  1  |7 4   8|9  6  3
3 9  4  |6 5   2|7  18 18
6 7  8  |1 3   9|4  5  2
--------+-------+--------
5 26 267|9 8   3|12 4  17
1 8  3  |4 2   7|5  9  6
9 4  27 |5 1   6|28 3  78
--------+-------+--------
8 3  69 |2 679 4|16 17 5
4 26 5  |8 67  1|3  27 9
7 1  269|3 69  5|68 28 4



It can be solved by guessing because many cells have only 2 candidates but is there any logical method to this?
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David Bryant



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

PostPosted: Mon Mar 20, 2006 12:30 pm    Post subject: Focus on rows 7 & 8 Reply with quote

guest wrote:
Code:
2 5  1  |7 4   8|9  6  3
3 9  4  |6 5   2|7  18 18
6 7  8  |1 3   9|4  5  2
--------+-------+--------
5 26 267|9 8   3|12 4  17
1 8  3  |4 2   7|5  9  6
9 4  27 |5 1   6|28 3  78
--------+-------+--------
8 3  69 |2 679 4|16 17 5
4 26 5  |8 67  1|3  27 9
7 1  269|3 69  5|68 28 4

I see several ways to proceed, all of them depending on some sort of "forcing chain." Here's the shortest chain I noticed.

A. r7c3 = 6 ==> r7c7 = 1 ==> r7c8 = 7 ==> r8c8 = 2
B. r7c3 = 6 ==> r8c2 = 2

But we can't have two "2"s in row 8, so r7c3 must be "9".

You might think of this as a form of "guessing" -- I call it a "5-star constellation" that involves a contradiction. dcb
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george woods1
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PostPosted: Tue Mar 28, 2006 1:22 pm    Post subject: Reply with quote

another way of looking at this 5 star constellation is to note that whichever of the 2's is chosen in block 7, it leads to a 9 in the "correct" position
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keith



Joined: 19 Sep 2005
Posts: 3162
Location: near Detroit, Michigan, USA

PostPosted: Tue Mar 28, 2006 10:47 pm    Post subject: Unchained! Reply with quote

There is a method to solve this puzzle without chains:

Code:


2 5  1  |7 4   8|9  6  3
3 9  4  |6 5   2|7  18 18
6 7  8  |1 3   9|4  5  2
--------+-------+--------
5 26 267|9 8   3|12 4  17
1 8  3  |4 2   7|5  9  6
9 4  27 |5 1   6|28 3  78
--------+-------+--------
8 3  69 |2 679 4|16 17 5
4 26 5  |8 67  1|3  27 9
7 1  269|3 69  5|68 28 4 



First, there is an X-wing on <9>. The corners are R7C3, R7C5, R9C3, and R9C5. This does not lead to any eliminations, but note that either R7C3 and R9C5 are <9>, or R7C5 and R9C3 are <9>.

Second, there is a possible non-unique rectangle on <69> lurking in the same squares as the X-wing. So, R7C5 and R9C3 CANNOT be <9>, for then R7C3 and R9C5 must be <6>, and there will be a non-unique rectangle.

So, R7C3 and R9C5 must be <9>.

Finally, to solve the puzzle, there is a BUG pattern which forces R4C3 = <2>.

Pretty cool, I think! I have not seen this variation of a unique rectangle described before.

Best wishes,

Keith
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TKiel



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

PostPosted: Wed Mar 29, 2006 1:28 pm    Post subject: Reply with quote

Keith,

Good spot on that combination. I think maybe you can claim naming rights on that. Smile
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Steve R



Joined: 24 Oct 2005
Posts: 289
Location: Birmingham, England

PostPosted: Wed Mar 29, 2006 6:57 pm    Post subject: Unchained! Reply with quote

A very fine observation, Keith.

I’m not too sure about priorities, though. In this particular example the pattern seems to match an AUR:

“Almost Unique Rectangle (AUR): a set of four cells populated in such a way that if one or two of them does not have a specific candidate 'x', then those four cells form a unique rectangle.”

Perhaps your thoughts will develop in a different way.

Steve
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keith



Joined: 19 Sep 2005
Posts: 3162
Location: near Detroit, Michigan, USA

PostPosted: Wed Mar 29, 2006 11:33 pm    Post subject: Reply with quote

Tracy and Steve,

Thank you very much for the kind words. Embarassed

Tracy,

If I had naming rights, I would call it something like a "Diagonal Pair in an X-wing". The real problem with naming is that the existing names are not good: An X-wing is really a fish, not a wing (etc.).

Steve,

The AUR sounds a little open-ended to me. I will do some research in the forums for solution methods and algorithms before I stake my claim! (And, to be pedantic, the pattern is a non-unique rectangle. Unique rectangles are quite welcome!)


Anyway, it was a thrill to find this. I have no idea how common this pattern may be, because the software I use to learn does not spot it! In fact, it will not even show the X-wing, because it is not "useful" (in that it does not lead to eliminations). It also does not identify the Rectangle.

Best wishes,

Keith
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keith



Joined: 19 Sep 2005
Posts: 3162
Location: near Detroit, Michigan, USA

PostPosted: Fri Mar 31, 2006 10:25 pm    Post subject: Some general comments Reply with quote

I have spent some time pondering this. I think there are two observations:

1. There are useful variations of Unique Rectangles in which there is a "Diagonal Pair". For example, consider the following two blocks:

Code:


+--------------+
|  x   -    -  |
| 46   -   246 |
|  x   -    -  |
+--------------+
| 246  -   46  |
|  -   -    x  |
|  -   -    x  |
+--------------+



To avoid a non-unique rectangle, one of the squares with possibilities <246> must be <2>. Therefore, the squares labelled "x" cannot be <2>. This is comparable to a "Type-2 Unique Rectangle" (at least as I understand the explanation in the Susser manual of 12/31/2005).

2. If the "Diagonal Pair" defines a possible non-unique rectangle, and if that rectangle is also an X-wing on one of the components of the pair, then the diagonal pair must have the value of that component. This is the point of my original posts in this thread. In the example of this post, if the rectangle is also an X-wing on <4>, the top left and bottom right squares must be <4>, i.e.,

Code:


+------------+
| -   -   -  |
| 4   -   26 |
| -   -   -  |
+------------+
| 26  -   4  |
| -   -   -  |
| -   -   -  |
+------------+



Best wishes,

Keith
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Steve R



Joined: 24 Oct 2005
Posts: 289
Location: Birmingham, England

PostPosted: Sun Apr 02, 2006 12:50 pm    Post subject: Reply with quote

Good stuff, Keith.

Have you considered the effect of weakening the X-wing assumption in your second example? It seems to me that, if there are just two places for 4 in the top row, the bottom left <246> reduces to <26>. Also, the top right <246> can contain any number of candidates as long as a 4 is amongst them.

If there is anything in this, the result is much less elegant than the X-wing but possibly of wider application.

Regards

Steve
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keith



Joined: 19 Sep 2005
Posts: 3162
Location: near Detroit, Michigan, USA

PostPosted: Sun Apr 02, 2006 10:56 pm    Post subject: Type 5 and Type 6 Unique Rectangles Reply with quote

Steve,

I think you are definitely on the right track. This "Unique X-wing" is certainly cute, but I think it is more important to recognize that diagonal pairs might be useful to identify possible Unique Rectangles and, perhaps, X-wings. Once identified, there are a number of logical steps to be considered.

I believe the masters have declared Unique Rectangles with a diagonal pair to be "Type 5", and Ruud has proposed that the "Unique X-wing" be "Type 6". Take a look at (thank you, Tracy!)

http://www.sudoku.com/forums/viewtopic.php?t=3709

Keith's view:

This "Unique X-wing" is certainly a special case of something, for it enables you to immediately place values in squares.

Unique Rectangles are a way to eliminate possibilities, and these eliminations often result in values forced in some squares.

The next abstraction is "Almost Unique Rectangles", which seem to be mostly useful to define interesting cycles which may lead to possibility eliminations. If you follow the URL above, you will see discussions on "strong" and "weak" links in patterns and chains, which I think is exactly your point about "weakening the X-wing assumption".

Beat wishes,

Keith
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keith



Joined: 19 Sep 2005
Posts: 3162
Location: near Detroit, Michigan, USA

PostPosted: Mon Apr 03, 2006 9:10 pm    Post subject: Another example Reply with quote

Today's SudoCue Nightmare puzzle has another example of this "Unique X-wing". You can see my message on the Nightmare discussion forum:

http://www.sudocue.net/forum/viewtopic.php?t=109

Best wishes,

Keith
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