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strmckr
Joined: 18 Aug 2009 Posts: 64
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Posted: Tue Jan 26, 2010 7:00 am Post subject: M-wings and m-rings: exemplars and examples |
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the original thread by Ronk has been lost and now is partially restored here
this the quoted from Ronk in the above link.
Quote: | Here is what I believe to be a minimal set of fourteen m-wing and four m-ring exemplars. It is minimal in the sense that any valid m-wing (or m-ring) will match only one of the exemplars in this set. The lone exeption is shown in the Extensions section. For additional extensions, see StrmCkr's Addendum -- Useful Extensions to the Minimal Set{not restored} This thread was inspired by 999_Spring's and StrmCkr's postings {not restored} Thanks to both for the kickoff.
What I'm calling m-wing is generally known as "generalized" m-wing elsewhere. The original m-wing introduced by keith here is defined with five strong inferences, rather than the minimum three strong inferences required, as for an xy-wing and a w-wing. As the two extra strong inferences produce no extra eliminations AFAIK, it seems fair to say the original m-wing is "over-specified." Hence, the adjective "generalized" is dropped.
It is neither my expectation nor my intent that solvers use the "Type" numbers below. They are included merely to facilitate unambiguous discussion in this thread.
A general note about the exemplars: All cells required to be void (empty) of candidates 'a' and 'b' are not explicitly marked with '/'. However, there are only two grouped conjugate links and the unit (row, column, box) containing each should be clear. If not, I'm willing to consider changing the presentation. |
MINIMAL EXEMPLAR SET:
M-WINGS:
Code: |
Type 1A: Type 1B:
. . . | . . . | . . . . . . | . / . | . . .
. ab . | . . . | . -b . . ab . | . a . | . -b .
. . . | . . . | . . . . . . | . / . | . . .
----------+----------+--------- ----------+----------+---------
. . . | . . . | . . . . . . | . / . | . . .
/ a / | / ab+ / | / b / / / / | / ab+ / | / b /
. . . | . . . | . . . . . . | . / . | . . .
----------+----------+--------- ----------+----------+---------
. . . | . . . | . . . . . . | . / . | . . .
. . . | . . . | . . . . . . | . / . | . . .
. . . | . . . | . . . . . . | . / . | . . .
1A: r2c8 -b- r2c2 -a- r5c2 =a= r5c5 =b= r5c8 -b- r2c8 --> r2c8<>b
1B: r2c8 -b- r2c2 -a- r2c5 =a= r5c5 =b= r5c8 -b- r2c8 --> r2c8<>b
Type 2A: Type 2B:
. . . | . . . | . . . . . . | . / . | . . .
. ab . | . . -b | . . . . ab . | . a -b | . . .
. . . | . . . | . . . . . . | . / . | . . .
----------+----------+--------- ----------+----------+---------
. . . | / / b | . . . . . . | / / b | . . .
/ a / | / ab+ b | / / / . . . | / ab+ b | . . .
. . . | / / b | . . . . . . | / / b | . . .
----------+----------+--------- ----------+----------+---------
. . . | . . . | . . . . . . | . / . | . . .
. . . | . . . | . . . . . . | . / . | . . .
. . . | . . . | . . . . . . | . / . | . . .
2A: r2c6 -b- r2c2 -a- r5c2 =a= r5c5 =b= r456c6 -b- r2c6 --> r2c6<>b
2B: r2c6 -b- r2c2 -a- r2c5 =a= r5c5 =b= r456c6 -b- r2c6 --> r2c6<>b
Type 3A: Type 3B:
. . . | . b . | . . . . . . | . b . | . . .
. ab . |-b b -b | . . . . ab . |-b ab -b | . . .
. . . | . b . | . . . . . . | . b . | . . .
----------+----------+--------- ----------+----------+---------
. . . | . / . | . . . . . . | . / . | . . .
/ a / | / ab+ / | / / / . . . | . ab+ . | . . .
. . . | . / . | . . . . . . | . / . | . . .
----------+----------+--------- ----------+----------+---------
. . . | . / . | . . . . . . | . / . | . . .
. . . | . / . | . . . . . . | . / . | . . .
. . . | . / . | . . . . . . | . / . | . . .
3A: r2c46 -b- r2c2 -a- r5c2 =a= r5c5 =b= r123c5 -b- r2c46 --> r2c46<>b
3B: r2c46 -b- r2c2 -a- r2c5 =a= r5c5 =b= r123c5 -b- r2c46 --> r2c46<>b
Type 4A: Type 4B:
. . . | . / . | . . . . . . | . / . | . . .
. ab . |-b / -b | . . . . ab . |-b a -b | . . .
-b -b -b | . b . | . . . -b -b -b | . b . | . . .
----------+----------+--------- ----------+----------+---------
. . . | . / . | . . . . . . | . / . | . . .
/ a / | / ab+ / | / / / . . . | . ab+ . | . . .
. . . | . / . | . . . . . . | . / . | . . .
----------+----------+--------- ----------+----------+---------
. . . | . / . | . . . . . . | . / . | . . .
. . . | . / . | . . . . . . | . / . | . . .
. . . | . / . | . . . . . . | . / . | . . .
4A: r2c46 -b- r2c2 -a- r5c2 =a= r5c5 =b= r3c5 -b- r2c46 --> r2c46<>b,r3c123<>b
4B: r2c46 -b- r2c2 -a- r2c5 =a= r5c5 =b= r3c5 -b- r2c46 --> r2c46<>b,r3c123<>b
Type 5A: Type 5B:
. . . | . / . | . . . . . . | / / / | . . .
. ab . | . / . | . . . . ab . | a a a | . . .
a a a | / ab+ / | / / / . . . | / ab+ / | . . .
----------+----------+--------- ----------+----------+---------
. . . | . / . | . . . . . . | . / . | . . .
. -b . | . b . | . . . . -b . | . b . | . . .
. . . | . / . | . . . . . . | . / . | . . .
----------+----------+--------- ----------+----------+---------
. . . | . / . | . . . . . . | . / . | . . .
. . . | . / . | . . . . . . | . / . | . . .
. . . | . / . | . . . . . . | . / . | . . .
5A: r5c2 -b- r2c2 -a- r3c123 =a= r3c5 =b= r5c5 -b- r5c2 --> r5c2<>b
5B: r5c2 -b- r2c2 -a- r2c456 =a= r3c5 =b= r5c5 -b- r5c2 --> r5c2<>b
Type 6A: Type 6B:
-b -b -b | b b b | . . . -b -b -b | b b b | . . .
. ab . | / / / | . . . . ab . | a a a | . . .
a a a | / ab+ / | / / / . . . | / ab+ / | . . .
----------+----------+--------- ----------+----------+---------
. . . | . . . | . . . . . . | . . . | . . .
6A: r1c123 -b- r2c2 -a- r3c123 =a= r3c5 =b= r1c456 -b- r1c123 --> r1c123<>b
6B: r1c123 -b- r2c2 -a- r2c456 =a= r3c5 =b= r1c456 -b- r1c123 --> r1c123<>b
Type 7A: Type 7B:
. . . | . . . | . . . . . . | / / / | . . .
. ab . | . . . |-b -b -b . ab . | a a a |-b -b -b
a a a | / ab+ / | b b b / / / | / ab+ / | b b b
----------+----------+--------- ----------+----------+---------
. . . | . . . | . . . . . . | . . . | . . .
7A: r2c789 -b- r2c2 -a- r3c123 =a= r3c5 =b= r3c789 -b- r2c789 --> r2c789<>b
7B: r2c789 -b- r2c2 -a- r2c456 =a= r3c5 =b= r3c789 -b- r2c789 --> r2c789<>b |
M-RINGS:
Code: |
Type A:
. -a . | . / . | . . .
-b ab -b |-b b -b |-b -b -b
. -a . | . / . | . . .
------------+----------+---------
. -a . | . / . | . . .
/ a / | / ab+ / | / / /
. -a . | . / . | . . .
------------+----------+---------
. -a . | . / . | . . .
. -a . | . / . | . . .
. -a . | . / . | . . .
In addition, r5c5=ab
r2c2 -a- r5c2 =a= r5c5 =b= r2c5 -b- r2c2 - continuous loop
Type B:
-a -a -a | / / / | . . .
-ab ab -ab| b b b |-b -b -b
a a a | / ab+ / | / / /
----------+-----------+---------
. . . | . . . | . . .
In addition, r3c5=ab
r2c2 -a- r3c123 =a= r3c5 =b= r2c456 -b- r2c2 - continuous loop
Type C:
-ab -ab -ab | . . . | . . .
-ab ab -ab | . . . | . . .
ab* ab* ab* | / ab+ / | / / /
------------+----------+---------
. . . | . . . | . . .
In addition, r3c5=ab
r2c2 -a- r3c123 =a= r3c5 =b= r3c123 -b- r2c2 - continuous loop
Type D:
. . . | / / / | . . .
-ab ab -ab |ab* ab* ab*|-ab -ab -ab
. . . | / ab+ / | . . .
------------+-----------+-----------
. . . | . . . | . . .
In addition, r3c5=ab
r2c2 -a- r2c456 =a= r3c5 =b= r2c456 -b- r2c2 - continuous loop |
'ab' means a bivalued cell with candidates 'a' and 'b'
'ab+' means the cell must contain both 'a' and 'b' candidates, and possibly others
'ab*' means the cell may contain 'a' or 'b' or both, and very likely others
'-ab' means both 'a' and 'b' are eliminated
If the above is correct, any valid m-wing or m-ring will match only one of the above exemplars, with Type 2C (below) being the only known exception.
USEFUL EXTENSIONS to MINIMAL EXEMPLAR SET
Code: |
Type 2C: (Simultaneously a Type 2A and a transposed Type 2B)
. . . | . . . | . . .
. ab . | . . -b | . . .
. . . | . . . | . . .
--------+---------+-------
. . . | / / / | . . .
/ a / | / ab+ / | / / /
. -b . | / / b | . . .
--------+---------+-------
. . . | . . . | . . .
2: r2c6 -b- r2c2 -a- r5c2 =a= r5c5 =b= r6c6 -b- r2c6 --> r2c6<>b,r6c2<>b |
Last edited by strmckr on Thu May 27, 2010 6:49 pm; edited 1 time in total |
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Bud
Joined: 06 May 2010 Posts: 47 Location: Tampa, Florida
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Posted: Tue May 25, 2010 12:17 am Post subject: |
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Thanks Strmckr. This has been very helpful. |
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strmckr
Joined: 18 Aug 2009 Posts: 64
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Posted: Tue May 25, 2010 1:08 am Post subject: |
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your welcome I am sorry that the links no longer work since the crash of sudoku.com
the thread might get rebuild on the new site but lacking some of the information as it was lost permanently
how ever the addendum is merely traversals of finding the patterns {or are subsumed by easier techniques} thus they are still covered within the main types listed above |
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kcplayer
Joined: 30 Jun 2010 Posts: 6
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Posted: Fri Jul 16, 2010 4:55 pm Post subject: |
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strmCkr, I really want to understand these exemplars. you explain what 'ab' 'ab+' 'ab*' and '-ab' , would you also explain what is 'a' 'b' in the code, is that mean the box contain the value 'a' , 'b' plus other. Also what is the meaning for =a= and -a- |
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strmckr
Joined: 18 Aug 2009 Posts: 64
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Posted: Mon Jul 26, 2010 3:24 am Post subject: |
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Quote: | meaning for =a= and -a- | a strong and weak inference in respect to the chain displayed { they are written in nice loop format }
the symbols A and b respectively represent a singular different digit found on the grid. digits 1-9
ab represents a cell that has only the candidates A and B
ab+ represents a cell that has the digits a&b plus extras candidates.
here is a sample pattern converted to numbers hope it helps a bit.
Code: |
Type 1A: Type 1B:
. . . | . . . | . . . . . . | . / . | . . .
. 12 . | . . . | . -2 . . 12 . | . 1 . | . -2 .
. . . | . . . | . . . . . . | . / . | . . .
----------+----------+--------- ----------+----------+---------
. . . | . . . | . . . . . . | . / . | . . .
/ 1 / | / 12+ / | / 2 / / / / | / 12+ / | / 2 /
. . . | . . . | . . . . . . | . / . | . . .
----------+----------+--------- ----------+----------+---------
. . . | . . . | . . . . . . | . / . | . . .
. . . | . . . | . . . . . . | . / . | . . .
. . . | . . . | . . . . . . | . / . | . . .
11: r2c8 -2- r2c2 -1- r5c2 =1= r5c5 =2= r5c8 -2- r2c8 --> r2c8<>2
12: r2c8 -2- r2c2 -1- r2c5 =1= r5c5 =2= r5c8 -2- r2c8 --> r2c8<>2 |
my best explanation is simply look at each cell and place the digits "1" or "2" in that cell and based on limitations of placement you should find the exact same elimination from all points plotted above. |
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