W-Wing ?

Everything about Sudoku that doesn't fit in one of the other sections

Postby Steve K » Fri Jul 04, 2008 11:08 pm

Code: Select all

     8 |     1 |     4 ||     9 |     2 |   #36 ||  3**6 |     5 |     7
-------+-------+-------||-------+-------+-------||-------+-------+-------
    27 |   236 |  3567 ||    58 |     4 | 35678 ||     9 |   236 |     1
-------+-------+-------||-------+-------+-------||-------+-------+-------
     9 |   236 |  3567 ||     1 |   367 |  3567 ||     8 |     4 |   236
=======================||=======================||=======================
     3 |     4 |     1 ||    58 |    67 |    58 ||   267 |  2679 |   269
-------+-------+-------||-------+-------+-------||-------+-------+-------
     6 |     5 |    79 ||     2 |   379 |     4 ||  3**7 |     1 |     8
-------+-------+-------||-------+-------+-------||-------+-------+-------
    27 |    29 |     8 ||  3*67 |     1 | 3*679 ||     5 |  3*67 |     4
=======================||=======================||=======================
     4 |     8 |   369 ||   367 |  3679 |     1 ||23**67 | 23679 |     5
-------+-------+-------||-------+-------+-------||-------+-------+-------
     5 |   369 |     2 ||     4 |     8 | 3-679 ||     1 |   367 |   #36
-------+-------+-------||-------+-------+-------||-------+-------+-------
     1 |     7 |   369 ||   #36 |     5 |     2 ||     4 |     8 |   369





It seems the latter elimination, regardless of name, is valid in the first grid, perhaps seen as at least one of two w wings, much like 2 ALS with a restricted common:
[(6=3)r1c6-(3)r6c6=*(3)r4c6-(3=6)r9c4]=(3*)r6c8-(3**)r5c7=[(6=3)r1c6-(3)r1c7=**(3)r7c7-(3=6)r8c9] => r8c6<>6
Naming patterns is a convenience so that we can communicate with each other. If one makes the pattern names too general, then the communication gets fuzzy. If one makes the pattern names too specific, there are too many names. I think this problem (nomenclature) is common in language, and generally not resolvable to the satisfaction of everyone.

More interesting to me is that this specific pattern, if one were to label it a pattern, still exists regardless of the elimination of any arbitrary (3)'s in r6 or c7, much like an ALS becoming a LS after eliminating a restricted common. The general pattern: linking known patterns with restricted commons is a common theme. Finding them, much like finding naked locked sets, is often a function of understanding what is not required versus what is required to the pattern validity.
Naturally, extending patterns with any valid strong inference extension pattern is always a manner to further generlize a smaller pattern, thus a turbot extension of the three sis w wing is a natural extension of the base pattern. Naming the extensions - that is perhaps more difficult than finding and understanding them!
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Postby daj95376 » Sat Jul 05, 2008 5:26 am

I'm sure those who understand ALS and read Eureka notation can follow Steve K's reply. I was muddling along until I encountered *(3)r4c6 ... and that stopped me cold. So, I decided to see what I could find for [r8c6]<>6.

Code: Select all
 Original PM
 +-----------------------------------------------------------------------+
 |  8      1      4      |  9      2      36     |  36     5      7      |
 |  27     236    3567   |  58     4      35678  |  9      236    1      |
 |  9      236    3567   |  1      367    3567   |  8      4      236    |
 |-----------------------+-----------------------+-----------------------|
 |  3      4      1      |  58     67     58     |  267    2679   269    |
 |  6      5      79     |  2      379    4      |  37     1      8      |
 |  27     29     8      |  367    1      3679   |  5      367    4      |
 |-----------------------+-----------------------+-----------------------|
 |  4      8      369    |  367    3679   1      |  2367   23679  5      |
 |  5      369    2      |  4      8      3679   |  1      367    36     |
 |  1      7      369    |  36     5      2      |  4      8      369    |
 +-----------------------------------------------------------------------+

Code: Select all
                 -3- r6c6       r5c7 -3- r6c8 =3= r6c4 -3- r9c4 -6 => [r8c6]<>6
                  +           /
Network: 6- r1c6 -3- r1c7 =3=
                              \
                                r7c7 -3-                   r8c9 -6 => [r8c6]<>6
_______________________________________________________________________________

Not something I'd normally encounter. I'd more likely find:

Code: Select all
[r8c6]=3 => ([r9c4],[r1c6],[r8c9]=6 => [r4c7]=6) => [b5]~6 ==> [r8c6]<>3
[r8c6]=6 => ([r9c4],[r1c6],[r8c9]=3 => [r2c8]=3) => [r6]~3 ==> [r8c6]<>6
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Postby Glyn » Sat Jul 05, 2008 8:44 am

Here is Steve's elimination as I see it, again in Eureka notation.
Code: Select all
          (3)r6c4-(3=6)r9c4
          ||
(6=3)r1c6-(3)r6c6     
          ||     
          (3)r6c8=(3)r5c7
                  ||
                  (3)r1c7-(3=6)r1c6
                  ||
                  (3)r7c7-(3=6)r8c9


The bilocation 3's in Box 6 force the Kraken in either r6 or c7 to degenerate into an AIC.

Code: Select all
Either (i) or (ii) must be true
(i) (6=3)r1c6-(3)r6c6=(3)r6c4-(3=6)r9c4
(ii) (6=3)r1c6-(3)r1c7=(3=6)r8c9

The overall conclusion is r8c6<>6.

EDIT:Here is an effort to do this in NL. It's obviously wrong as it lacks a mechanism for promoting - to = as required. Must it be expressed as a net?

Code: Select all
r8c6-6-r1c6-3-(r6c6-3-r6c4-3-r9c4,r1c7-3-r7c7-3-r8c9)-6-r8c6
                 |                   |
                 3-r6c8 = 3 = r5c7 - 3
Last edited by Glyn on Sat Jul 05, 2008 8:12 am, edited 2 times in total.
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Postby Steve K » Sat Jul 05, 2008 9:04 am

Conceptually, what I was driving at was: wwing =3-3=wwing, meaning:
wwing using (36)r1c6,(36)r9c4, (3)r6c46 OR[ (3)r6c8 which forbids (3)r5c7] OR wwing using (36)r1c6, (3)r17c7,(36)r8c9. In other words, we have two almost wwings, each with one degree of freedom. They share a restricted common, thus we must have at least one wwing. Since the wwings have a common target, it is eliminated.

Of course, there is more than one way to dissect the pattern.

More generally, Pattern A with one extra Boolean D plus Pattern B with extra Boolean C plus C-D => if there exists a common target or targets, it/they can be eliminated. C,D are similar to restricted commons in ALSxALS.
Last edited by Steve K on Sat Jul 05, 2008 5:28 am, edited 2 times in total.
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Postby ronk » Sat Jul 05, 2008 9:22 am

Would someone please remind me why I bothered to learn AIC notation in order to post on the Eureka! forum.
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Postby Steve K » Sat Jul 05, 2008 9:30 am

Ronk, perhaps a dig at me? I should learn to use NL notation here? It may take me a while.....
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Postby Glyn » Sat Jul 05, 2008 9:47 am

Sorry Ronk I didn't know how to do the Kraken bit in NL notation (I'm sure it's mentioned somewhere) so I didn't want to mix two modes in one post.

As the original sketch shows simply covering up the appropriate three lines leaves the two chains toggled by 3's in Box 6.

I've attempted add an NL version in the post above which is wrong. I don't see how to toggle between the two alternative loops in an appropriate way i.e. selecting the desired route and switching to strong inferences when required. Hopefully someone will help us out here.
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Postby daj95376 » Sat Jul 05, 2008 1:30 pm

Glyn wrote:EDIT:Here is an effort to do this in NL. It's obviously wrong as it lacks a mechanism for promoting - to = as required. Must it be expressed as a net?

From what little I can tell, you and Steve K are using Eureka! notation to express networks. That's why I included one (as best I could) in NL notation. However, there's always the non-net approach:

Code: Select all
Skyscraper:    r15\c7+56                                             => [r3c5],[r6c6]<>3
Chain:         r7c7 -3- r89c9 =3= r3c9 =2= r2c8 -2- r7c8 =2= r7c7    => [r7c7]       <>3

W-Chain:    6- r1c6 -3- r1c7  =3= r5c7 -3- r6c8 =3= r6c4 -3- r9c4 -6 => [r8c6]       <>6

________________________________________________________________________________________
Last edited by daj95376 on Sat Jul 05, 2008 9:54 am, edited 1 time in total.
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Postby Mike Barker » Sat Jul 05, 2008 1:40 pm

Code: Select all
+----------------+-------------------+-------------------+
|  8    1     4  |   9     2     36* |   36b     5    7  |
| 27  236  3567  |  58     4  35678  |    9    236    1  |
|  9  236  3567  |   1   367   3567  |    8      4  236  |
+----------------+-------------------+-------------------+
|  3    4     1  |  58    67     58  |  267   2679  269  |
|  6    5    79  |   2   379      4  |   37b     1    8  |
| 27   29     8  | 367a    1   3679a |    5    367a   4  |
+----------------+-------------------+-------------------+
|  4    8   369  | 367  3679      1  | 2367b 23679    5  |
|  5  369     2  |   4     8  379-6  |    1    367   36* |
|  1    7   369  |  36*    5      2  |    4      8  369  |
+----------------+-------------------+-------------------+
As far as a name, the elimination is a multi-inference nice loop. Given a single multi-inference node Jeff would have refered to it as a triple chain (so you could refer to it as a triple w-chain) and I would have included it in the Kraken family as a Kraken row. With two multi-inference nodes, I'd probably just stick with multi-inference nice loop.

Using multi-line nice loop notation, I'd probably show the eliminations as one of the following. The "||" doesn't translate over well so I tend to use a "|" and "+", but this is not standard, of course, so is all multi-line notation.
Code: Select all
r6c4 -3- r9c4 -6-
r6c6 -3- r1c6 -6-
r6c8 -3- r5c7=3=r1c7 -3- r1c6 -6- => r8c6<>6
            |
            +=3=r7c7 -3- r8c9 -6-

or

r1c6 -3- r6c6=3=r6c4 -3- r9c4 -6-
            |
            +=3=r6c8 -3- r5c7=3=r1c7 -3- r1c6 -6- => r8c6<>6
                            |
                            +=3=r7c7 -3- r8c9 -6-
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Postby daj95376 » Sat Jul 05, 2008 2:01 pm

Mike Barker uses all of the 3s in [r6] to get [r8c6]<>6, and I used [r8c6]=6 to eliminate all of the 3s in [r6] for a contradiction. Just goes to show that I'm always putting the cart-before-the-horse!:D
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Postby Glyn » Sat Jul 05, 2008 3:56 pm

Thanks Mike for the definitive version of that in NL notation.
Thanks as well Daj I guess you had implicitly used the 3's of r6 in your network to give r6c68=3=r6c4. I tried a minor change to show that.
Code: Select all
                          --------3-----------
                         /                    \
                  3- r6c6       r5c7 -3- r6c8- +=3= r6c4 -3- r9c4 -6
                  +           /
Network: 6- r1c6 -3- r1c7 =3=
                              \
                                r7c7 -3- r8c9 -6
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Postby eleven » Sat Jul 05, 2008 10:16 pm

Nice catch, Steve K.
For me the discussions about notation and namings often have an imbreeding character.
The point is, how to find such things. One way is to see the almost w-wing 36 in r1c6 and r8c9 with the 3's in column 7. Without the 3 in r5c7 it says r1c6=6 or r8c9=6. Now what does r5c7=3 imply ? In box 5 either r6c4=3 (and r9c4=6) or r6c6=3 (and r1c6=6 again).
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Postby Mike Barker » Sat Jul 05, 2008 11:21 pm

I am not a good source for the definitive nomenclature, as Ron can attest to. What I posted was reasonable extension of what exists. Even with this I was thinking that starting each line with some symbol to show the connectedness along the lines of what Champagne does might be a nice addition:
Code: Select all
[] r6c4 -3- r9c4 -6-
[] r6c6 -3- r1c6 -6-
[] r6c8 -3- r5c7=3=r1c7 -3- r1c6 -6- => r8c6<>6
               |
               +=3=r7c7 -3- r8c9 -6-

The naming and nomenclature only serves to help communication. I agree with eleven - the trick is finding the beasties.
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Postby eleven » Sun Jul 06, 2008 9:46 am

I have to apologize, if someone feels offended by my rude statement above.

Of course i see, that notations make sense. It just happened, that there are at least two major notations, both with their own advantages and disadvantages, which need some study to understand them and more time to be able to write them. Moreover also the experts seem to have difficulties to express more complex things correctly (one reason for that is the lack of precise definitions).

So i lost my temper yesterday, because it was much harder to follow the discussion than to see the idea of the 2 almost w-wings directly in the grid (connected by the strong link for 3 in box 6).
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Postby ronk » Sun Jul 06, 2008 12:47 pm

eleven wrote:it was much harder to follow the discussion than to see the idea of the 2 almost w-wings directly in the grid (connected by the strong link for 3 in box 6).

Agreed, and in NL notation it might look like ...

w-wing if r5c7<>3: r8c6 -6- r1c6 -3- r1c7 =3= r7c7 -3- r8c9 -6- r8c6

w-wing if r6c8<>3: r8c6 -6- r1c6 -3- r6c6 =3= r6c4 -3- r9c4 -6- r8c6

At most one of r5c7=3 and r6c8=3 can be true (a weak inference BTW). Therefore r8c6<>6.

[edit: thanks to Pat, typos corrected]
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