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Set E Puzzle 47

 
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daj95376



Joined: 23 Aug 2008
Posts: 3854
PostPosted: Thu Nov 27, 2008 9:50 pm    Post subject: Set E Puzzle 47 Reply with quote
This may take more than one step.

Code:
 +-----------------------+
 | . . . | . . . | 7 . 3 |
 | . . 6 | 7 5 . | . 8 . |
 | . 2 5 | . . . | . . . |
 |-------+-------+-------|
 | . 6 . | . . . | . 2 . |
 | . 9 . | . 4 . | . . 7 |
 | . . . | . . 7 | . . 4 |
 |-------+-------+-------|
 | 6 . . | . . . | 9 . . |
 | . 5 . | 8 . . | . 1 2 |
 | 2 . . | . 6 9 | . 7 5 |
 +-----------------------+

Play this puzzle online at the Daily Sudoku site
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Asellus



Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA
PostPosted: Fri Nov 28, 2008 3:51 am    Post subject: Reply with quote
daj95376 wrote:
This may take more than one step.

Indeed. It took me 4, and a couple of them are demanding, though interesting. I certainly won't claim that this is the most obvious route, and I expect that others will find less "extreme" paths. However, I believe that all paths are learning experiences.

The first 2 steps are fairly conventional. First, a Skyscraper on 4 in r28. Then, a 4-cell XY Chain:
(9)r2c1 - (9=1)r2c9 - (1=4)r2c6 - (4=3)r8c6 - (3=9)r8c1 - (9)r2c1; r2c1<>9

This gets us here, where I became obsessed with DPs:
Code:
+-----------------+-----------------+----------+
| 189  #148 #189  | 124   12   6    | 7    5 3 |
| 13    134  6    | 7     5    14   | 2    8 9 |
| 7     2    5    | 39    389  38   | 1    4 6 |
+-----------------+-----------------+----------+
| 4     6    7    | 359   389  358  | 38   2 1 |
| 358   9    38   | 12    4    12   | 358  6 7 |
| 1358  138  2    | 6     38   7    | 358  9 4 |
+-----------------+-----------------+----------+
| 6     7    14   | 1245  12   1245 | 9    3 8 |
| 39    5    349  | 8     7    34   | 6    1 2 |
| 2    #138 #138  | 13    6    9    | 4    7 5 |
+-----------------+-----------------+----------+

There is a 18UR in the cells marked #. One way to view the extra digits in this UR is as the <4> in r1c2 having a strong inference with the 39 group of <9> in r1c3 together with the grouped <3>s of r9c23, or:
18UR[((3)r9c23(9)r1c3)=(4)r1c2]
The parentheses around "(3)r9c23(9)r1c3" indicate that it is a group. A group is true if any of its members is true; and it is false if all of its members are false. A fairly simple branched AIC can render all of the members of this group false, leading to the next elimination:
Code:
       /(3)r8c6=(3)r9c4\
(3)r8c1                 18UR[((3)r9c23(9)r1c3)=(4)r1c2] -
       \(9)r8c1=(9)r8c3/

   (4)r1c4=(4)r7c4 - (4=3)r8c6 - (3)r8c1; r8c1<>3

That gets us here:
Code:
+----------------+------------------+----------+
| 18    148  9   |#124  #12    6    | 7    5 3 |
| 13    134  6   | 7     5     14   | 2    8 9 |
| 7     2    5   | 39    389   38   | 1    4 6 |
+----------------+------------------+----------+
| 4     6    7   | 359   389   358  | 38   2 1 |
| 358   9    38  |#12    4    #12   | 358  6 7 |
| 1358  138  2   | 6     38    7    | 358  9 4 |
+----------------+------------------+----------+
| 6     7    14  | 1245 #12   #1245 | 9    3 8 |
| 9     5    34  | 8     7     34   | 6    1 2 |
| 2     38   138 | 13    6     9    | 4    7 5 |
+----------------+------------------+----------+

There is a 6-cell 12DP marked #. The extra digits provide the folowing induced strong inference between the grouped <4>s in r1c4 and r7c6 and the <5> in r7c6:
6Cell12DP[(4)r1c4|r7c6=(5)r7c6]
This provides the following AIC:
(4)r7c4 - 6Cell12DP[(4)r1c4|r7c6=(5)r7c6] - ALS[(5)r4c6=(3)r34c6] - (3=4)r8c6 - (4)r7c4; r7c4<>4
This solves the puzzle.

Somebody please show me that I am overlooking something obvious!

[Edit to repair the "broken" display of the AIC in "code".]
[Edit again to correct the XY Chain address per Keith's edit, as shown in red above.]

Last edited by Asellus on Fri Nov 28, 2008 4:52 am; edited 1 time in total
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keith



Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
PostPosted: Fri Nov 28, 2008 4:09 am    Post subject: Reply with quote
Asellus wrote:

Code:
+-----------------+-----------------+----------+
| 189  #148 #189  | 124   12   6    | 7    5 3 |
| 13    134  6    | 7     5    14   | 2    8 9 |
| 7     2    5    | 39    389  38   | 1    4 6 |
+-----------------+-----------------+----------+
| 4     6    7    | 359   389  358  | 38   2 1 |
| 358   9    38   | 12    4    12   | 358  6 7 |
| 1358  138  2    | 6     38   7    | 358  9 4 |
+-----------------+-----------------+----------+
| 6     7    14   | 1245  12   1245 | 9    3 8 |
| 39    5    349  | 8     7    34   | 6    1 2 |
| 2    #138 #138  | 13    6    9    | 4    7 5 |
+-----------------+-----------------+----------+

Somebody please show me that I am overlooking something obvious!


The rectangular XY-wing 13 - 14 - 34 that takes out <3> in R8C1? Still needing another two XY-wings to put it to bed.

I got here after basics:
Code:
+----------------------+----------------------+----------------------+
| 189    148    1489   | 12469  129    1246   | 7      5      3      |
|13-9    134    6      | 7      5      14c    | 2      8      19d    |
| 7      2      5      | 139    1389   138    | 14     49     6      |
+----------------------+----------------------+----------------------+
| 4      6      7      | 1359   1389   1358   | 1358   2      19     |
| 1358   9      1238   | 12356  4      123568 | 1358   36     7      |
| 1358   138    1238   | 123569 12389  7      | 1358   369    4      |
+----------------------+----------------------+----------------------+
| 6      7      134    | 12345  123    12345  | 9      34     8      |
| 39a    5      349    | 8      7      34b    | 6      1      2      |
| 2      1348   1348   | 134    6      9      | 34     7      5      |
+----------------------+----------------------+----------------------+

The extended XY-wing abcd, cd <14> <19> = <49>, makes the elimination noted by Asellus. (Edit: This is the same chain as that noted by Asellus.)

Then, on to the same series of three XY-wings noted above.

Keith

Last edited by keith on Fri Nov 28, 2008 5:06 am; edited 3 times in total
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daj95376



Joined: 23 Aug 2008
Posts: 3854
PostPosted: Fri Nov 28, 2008 4:27 am    Post subject: Reply with quote
While composing my reply, I see that keith has already replied. I'm still going to post mine anyway.

Asellus wrote:
The first 2 steps are fairly conventional. First, a Skyscraper on 4 in r28. Then, a 4-cell XY Chain:
(9)r2c1 - (9=1)r2c9 - (1=4)r1c6 - (4=3)r8c6 - (3=9)r8c1 - (9)r2c1; r2c1<>9

This gets us here, where I became obsessed with DPs:

...

Somebody please show me that I am overlooking something obvious!

First, I'm interested in why you chose to make a loop out of the 4-cell XY-Chain. What would you have done if there had been more than one elimination?

Second, my head is still spinning from your DP explanation. Wow! All I could get is the (12) UR in [r17c45] => [r7c4]<>1.

Finally, you missed two obtuse XY-Wings:

Code:
         XY-Wing  [r2c6]/[r2c1]+[r8c6]   <> 3    [r8c1]          not necessary
         XY-Wing  [r8c6]/[r2c6]+[r9c4]   <> 1    [r1c4],[r7c6]

   c3b7  Locked Candidate 2              <> 1    [r9c2]

         XY-Wing  [r1c4]/[r2c6]+[r5c4]   <> 1    [r5c6]
         XY-Wing  [r2c6]/[r1c4]+[r5c6]   <> 2    [r5c4]          not necessary


Last edited by daj95376 on Fri Nov 28, 2008 4:41 am; edited 1 time in total
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Asellus



Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA
PostPosted: Fri Nov 28, 2008 4:38 am    Post subject: Reply with quote
I guess I should have obsessed on XY Wings! Confused

But, why take the easy route when the hard route is available?
daj95376 wrote:
First, I'm interested in why you chose to make a loop out of the 4-cell XY-Chain.

I don't really understand your question. ALL XY Chains that perform eliminations are (discontinuous) loops. If there happen to be multiple victims, they are included at the discontinuity. It is still a loop, regardless. In general:
Victim A|Victim B|Victim C|etc - AIC - Victim A|Victim B|Victim C|etc; A,B and C, etc., are false
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daj95376



Joined: 23 Aug 2008
Posts: 3854
PostPosted: Fri Nov 28, 2008 5:10 am    Post subject: Reply with quote
Asellus wrote:
I guess I should have obsessed on XY Wings! Confused

But, why take the easy route when the hard route is available?
daj95376 wrote:
First, I'm interested in why you chose to make a loop out of the 4-cell XY-Chain.

I don't really understand your question. ALL XY Chains that perform eliminations are (discontinuous) loops. If there happen to be multiple victims, they are included at the discontinuity. It is still a loop, regardless. In general:
Victim A|Victim B|Victim C|etc - AIC - Victim A|Victim B|Victim C|etc; A,B and C, etc., are false

Interesting perspective.

To me, many chains are simply a forcing net where the first inference stream is assumed. XY-Chains are a prime example. Take the heart of your AIC loop for example:

Code:
(9=1)r2c9 - (1=4)r2c6 - (4=3)r8c6 - (3=9)r8c1

Using a forcing net perspective:

Code:
[r2c9]=9                     => eliminations in peer cells of [r2c9] for (9)
[r2c9]<>9 => ... => [r8c1]=9 => eliminations in peer cells of [r8c1] for (9)

Thus, any cells that are peer cells of [r2c9] and [r8c1] will have (9) eliminated.

I thought the point of AICs being bidirectional was to allow a similar conclusion. If we assume that [r2c9]<>9, then reading the AIC chain from left-to-right guarantees that [r8c1]=9 is true. Similarly, if we assume that [r8c1]<>9, then reading the AIC from right-to-left guarantees that [r2c9]=9 is true. No matter what, the AIC guarantees that at least one of [r2c9],[r8c1] is true for (9).
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Asellus



Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA
PostPosted: Fri Nov 28, 2008 5:44 am    Post subject: Reply with quote
daj95376 wrote:
I thought the point of AICs being bidirectional was to allow an equivalent conclusion without resorting to forcing nets.

I can't imagine a more obvious AIC than an XY Chain. The strong inferences occur within the bivalue cells and the weak inferences occur between the bivalue cells. Alternation of inferences is assured. The pincer ends are each weakly linked to the victim(s), providing the weak link discontinuity for the closed loop. There is no need whatsoever to refer to forcing concepts. One only needs to note the sequence of inferences to perform the elimination(s). I can assure you that a forcing net never even entered my consciousness. (Why would one resort to forcing in the most obvious of all AICs unless one was, for some reason, reluctant to rely upon the inherent logic of AICs themselves?)

While the use of starting assumptions is, in an important sense, implicit in any AIC, the AIC structure obviates the need for any such explicit assumption (i.e. "forcing"). That is what makes it appealing. And, since almost any (if not every ... I await proof to the contrary) elimination or placement in sudoku can be expressed in terms of an AIC, forcing appears to be unnecessary.

[Edit: Somehow, between the time I quoted the previous message and the time I posted the response, the text I quoted had changed.]
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daj95376



Joined: 23 Aug 2008
Posts: 3854
PostPosted: Fri Nov 28, 2008 5:58 am    Post subject: Reply with quote
The reference to forcing nets was a tangent point that I (somehow) thought would provide a supporting perspective in attaining the same conclusion.

I knew that my original last paragraph wasn't right, but it took several iterations to make it better. (I'm a poor writer.) You managed to catch the first iteration.
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storm_norm



Joined: 18 Oct 2007
Posts: 1741
PostPosted: Fri Nov 28, 2008 7:46 am    Post subject: Reply with quote
ok, a little late to join the party. sorry!
I like the xy-chain first, and I really like the DP move on {1,2}. several UR eliminations also, but none help. put me down for the xy-wing ending.
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tlanglet



Joined: 17 Oct 2007
Posts: 2468
Location: Northern California Foothills
PostPosted: Fri Nov 28, 2008 7:14 pm    Post subject: Reply with quote
I had another variation on the theme without the UR endeavor. I also started with the skyscraper on <4> and then I found the xy-chain noted by Asellus but viewed in reverse order. Then I spotted a finned xy-wing on <134> that deleted a <1> from r1c4, but I do not think it was helpful in the long run. I finished with three xy-wings, two again on <134> and then <124> with a (probably unhelpful) coloring on <1> somewhere in that mix.

Ted
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