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7/8/07 Nightmare...a nice multiple-methods example

 
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Sudtyro
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PostPosted: Sun Aug 05, 2007 1:50 pm    Post subject: 7/8/07 Nightmare...a nice multiple-methods example Reply with quote

After basics:
Code:
-------------------------------------------
135  14   345 | 6   7   2   | 349   8    49
9    26   7   | 13  8   134 | 5     34   26
38   26   348 | 5   9   34  | 267   27   1
--------------+-------------+--------------
367  8    2   | 137 4   136 | 139   39   5
36   5    1   | 9   36  8   | 24    24   7
4    79   39  | 137 2   5   | 13    6    8
--------------+-------------+--------------
1578 3    589 | 4   156 69  | 26789 279  26
178  1479 6   | 2   13  39  | 4789  5    49
2    49   459 | 8   56  7   | 469   1    3
-------------------------------------------

A check of the single-digit grids leads to...

Grouped Turbot chain (labels a-h in Grid 1 below):
(3): r5c1 = r5c5 – r8c5 = r8c6 – r3c6 = r2c46 – r2c8
= r1c7 => r1c1 <> 3, or
= r4c8 => r4c1 <> 3.

Code:
          Grid 1                         
3* . 3 | .  . .   | 3h .  .          
.  . . | 3f . 3f  | .  3g .
3  . 3 | .  . 3e  | .  .  .
-------+----------+------
3* . . | 3  .  3  | 3  3h .
3a . . | .  3b .  | .  .  .
.  . 3 | 3  .  .  | 3  .  .
-------+----------+------
.  . . | .  .  .  | .  .  .
.  . . | .  3c 3d | .  .  .
.  . . | .  .  .  | .  .  .


Finned X-Wing (labels x and f in Grid 2 below):
X-Wing = r24c48 with r6c4 = fin => r4c6 <> 3,
or grouped Turbot fish:
(3): r46c4 = r2c4 – r2c8 = r4c8 => r4c6 <> 3.

Code:
         Grid 2
. . 3 | .  . .  | 3 .  .
. . . | 3x . 3  | . 3x .
3 . 3 | .  . 3  | . .  .
------+---------+------
. . . | 3x . 3* | 3 3x .
3 . . | .  3 .  | . .  .
. . 3 | 3f . .  | 3 .  .
------+---------+------
. . . | .  . .  | . .  .
. . . | .  3 3  | . .  .
. . . | .  . .  | . . 
.

After follow-up:
Code:
--------------------------------------------
15   14   345 | 6    7   2   | 349   8    49
9    26   7   | 13   8   134 | 5     34   26
38   26   348 | 5    9   34  | 267   27   1
--------------+--------------+--------------
67   8    2   | 173* 4   16  | 139   39   5
36   5    1   | 9    36  8   | 24    24   7
4    79   39  | 137  2   5   | 13    6    8
--------------+--------------+--------------
1578 3    589 | 4    156 69  | 26789 279  26
178  1479 6   | 2    13  39  | 4789  5    49
2    49   459 | 8    56  7   | 469   1    3
--------------------------------------------

At this point, multi-digit methods abound for a single elimination (r4c4 <> 3):
1. WXYZ-Wing: W=6 and Z=3, using cells (36)r5c5 and (1369)r4c678 => r4c4 <> 3.
2. ALS-XZ rule:
ALS(A=[r5c5], B=[r4c678], X=6, Z=3) => r4c4 <> 3.
3. Grouped AIC:
(3=6)r5c5 – (6=193)r4c678 => r4c4 <> 3,
equivalent to both the WXYZ-Wing and the ALS-XZ rule.
4. Subset Counting: subset is (36)r5c5 and (1369)r4c678, with digit 3 having multiplicity of 2; all others have multiplicity of 1. Cell r4c4 = 3 would reduce total subset multiplicity from 5 to 3, which is one less than cell count => r4c4 <> 3.
5. APE (Aligned Pair Exclusion): cells (137)r4c4 and (16)r4c6 generate the two pair combinations, 3-1 and 3-6, and neither one is allowed => r4c4 <> 3.
6. Non-grouped AIC’s:
(3)r5c5 = (3-6)r5c1 = (6-7)r4c1 = (7)r4c4 => r4c4 <> 3.
(3=6)r5c5 - (6)r4c6 = (6-7)r4c1 = (7)r4c4 => r4c4 <> 3.
7. 3D Coloring: Coloring of digits in either of the above AICs quickly shows that (3)r5c5 and (7)r4c4 have opposite colors (parity) => r4c4 <> 3.
8. Discontinuous Nice loops:
[r4c4]-3-[r5c5]=3=[r5c1]=6=[r4c1]=7=[r4c4] => r4c4 <> 3.
[r4c4]-3-[r5c5]-6-[r4c6]=6=[r4c1]=7=[r4c4] => r4c4 <> 3.
These loops are equivalent to the above two non-grouped AIC’s.

Would anyone care to add to this “methods” list for the r4c4 <> 3 elimination? Contributions welcome!
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rep'nA
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Joined: 19 Jan 2007
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Location: Union City, California

PostPosted: Sun Aug 05, 2007 11:17 pm    Post subject: Re: 7/8/07 Nightmare...a nice multiple-methods example Reply with quote

Sudtyro wrote:

Would anyone care to add to this “methods” list for the r4c4 <> 3 elimination? Contributions welcome!

The cells (5,5), (4,6), (4,7), (4,8) could be viewed as an almost xy-chain, or one could also view the elimination as coming from one of Denis Berthier's xyzt-chains. I think his notation would look something like:

{3 6} - {6 1} - {1 9 (3#1)} - {9 3}.
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Sudtyro
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PostPosted: Mon Aug 06, 2007 8:28 pm    Post subject: Re: 7/8/07 Nightmare...a nice multiple-methods example Reply with quote

rep'nA wrote:

The cells (5,5), (4,6), (4,7), (4,8) could be viewed as an almost xy-chain ...


Thanks, rep'nA, for the feedback! I've learned just enough about "almost" structures from some of Ron Moore's postings to be almost dangerous Smile , but I'm not sure about an almost xy-chain. Can you show how that would work with the cells you listed?
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rep'nA
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Joined: 19 Jan 2007
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Location: Union City, California

PostPosted: Mon Aug 06, 2007 10:31 pm    Post subject: Re: 7/8/07 Nightmare...a nice multiple-methods example Reply with quote

Sudtyro wrote:
rep'nA wrote:

The cells (5,5), (4,6), (4,7), (4,8) could be viewed as an almost xy-chain ...


Thanks, rep'nA, for the feedback! I've learned just enough about "almost" structures from some of Ron Moore's postings to be almost dangerous Smile , but I'm not sure about an almost xy-chain. Can you show how that would work with the cells you listed?

Almost xy-chains are described here and a proof of their general applicability is given here.

In this case, there are two ways to think of the cells above as an almost xy-chain, one based on the approach given in the above links and one more in the spirit of Ron (and Carcul before him). I'll briefly describe them both. First, if it wasn't for the 3 in r4c7, then we would have an xy-chain in the cells I described. But since the 3 in r4c7 does not change the max multiplicity of 3 in the chain, it can be added without affecting the conclusions. Frankly, it is just subset counting on a particularly easy to spot pattern.

The second way to think about it is as follows: If the 3 wasn't in r4c7, we would have an xy-chain and conclude that r4c4<>3. If r4c7=3, then r4c4<>3. This is what I think is the more traditional meaning of an 'almost pattern'. However, since the pattern I use above is more common, I stole the name so that it might be used once in a while.
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Sudtyro
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PostPosted: Tue Aug 07, 2007 10:50 am    Post subject: Reply with quote

Thanks again...good explanation! I hadn't seen your threads in the other forum.

In Ron's jargon, (3)r4c7 must be the "spoiler" digit in the "almost" pattern, and in this case is more like the fin in a finned fish because the spoiler can see the victim cell. I'm guessing the AIC (in Ron's notation) would be something like:
[XY-Chain r5c5|r4c678] = (3)r4c7 => r4c4 <> 3.

Another interesting "method" to add to the list!
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