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Sudopedia: Kraken Fish

 
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keith



Joined: 19 Sep 2005
Posts: 3178
Location: near Detroit, Michigan, USA

PostPosted: Wed Nov 21, 2012 7:30 am    Post subject: Sudopedia: Kraken Fish Reply with quote

Kraken Fish

A Kraken Fish is a fish pattern which is connected to a candidate elimination cell (CEC) in such a way that placing this candidate would reduce the secondary set to a size smaller than the defining set.
Here is a diagram depicting a Kraken X-Wing. It has 2 CECs.
Code:
.-------.-------.-------.
| . / . | . / . | . . . |
| . X . | . X . | . * . |
| . / . | . / . | . . . |
:-------+-------+-------:
| . / . | . / . | . . . |
| . X . | . X . | . * . |
| . / . | . / . | . . . |
:-------+-------+-------:
| . / . | . / . | . . . |
| . X . | . / . | . . . |
| / / X | / / / | / X / |
'-------'-------'-------'


There is a potential X-Wing in columns 2 & 5 and rows 2 & 5. The candidate in r8c2 acts as a fin, but because it does not share a box with one of the candidates forming the X-Wing, it does not cause any eliminations that would normally occur in a Finned X-Wing. The pattern has 2 columns covering 3 rows.

Now consider what would happen if either r2c8 or r5c8 would contain digit X. Both would eliminate all candidates in their own row and, through the strong link in row 9, the candidate in r8c2, leaving a pattern with 2 columns covering 1 row, an impossible situation.

A Kraken-related Extended 2-String Kite Example

Next consider the example below. This is identical to the Kraken X-wing example except the candidate has benn removed from cell r2c2. The resultant pattern is a 2-string kite consisting of conjugate pairs AB and CD and an additional conjugate pair EF. This is an example of a more general technique called the extended 2-string kite technique.
Code:
.-------.-------.-------.
| . - . | . - . | . . . |
| . - . | . F . | . * . |
| . - . | . - . | . . . |
:-------+-------+-------:
| . - . | . - . | . . . |
| . D . | . E . | . * . |
| . - . | . - . | . . . |
:-------+-------+-------:
| . - . | . - . | . . . |
| . C . | . - . | . . . |
| - - A | - - - | - B - |
'-------'-------'-------'

How it works. Since A and C are peers, at most only one of them can be X. This means that either or both of their conjugates B and D must be X. If B is X then r25c8 cannot be X. If B is not X, then D must be X and its peer E must be not X. Therefore E's conjugate F must be X. Therefore r25c8 cannot be X. So you get the same eliminations as the Kraken X-wing. A similar argument holds if r5c2 had been eliminated instead of r2c2. This means that either r2c2 or r5c2 but not both can be eliminated from the Kraken X-wing pattern without changing the results. The extended 2-string kite pattern occurs more often then the Kraken X=wing in puzzles.

Connection Types

In the example, the CECs are connected to the pattern with a direct weak link and an indirect connection via a strong link. However, there are several other methods to connect the CEC to the fish pattern:

Strong links
Grouped strong links
Almost Locked Sets

As a result, the CEC does not even need to be a candidate for the same digit as the pattern. The following diagram shows how an X-Wing pattern for digit X can eliminate a candidate for digit Y.
Code:
.  -  .  | .  -  .  |  .  .  .
.  X  .  | .  X  .  |  .  XY .
.  -  .  | .  -  .  |  .  .  .
---------+----------+---------
.  -  .  | .  -  .  |  .  .  .
.  X  .  | .  X  .  |  .  .  .
.  -  .  | .  -  .  |  .  .  .
---------+----------+---------
.  -  .  | .  -  .  |  .  .  .
.  X  .  | .  -  .  |  .  .  .
.  -  XY | .  -  .  |  . CEC .

In this example, a similar Finned X-Wing is present, but r2c8 and r9c3 are both bivalue cells with candidates XY. Placing Y in the CEC at r9c8 would force both these cells to X and reduce the fish pattern to a single row and 2 columns.

External Link

Kraken Fish (New Sudoku Players' Forums)
http://forum.enjoysudoku.com/viewtopic.php?t=5143


Last edited by keith on Wed Nov 21, 2012 7:56 am; edited 4 times in total
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keith



Joined: 19 Sep 2005
Posts: 3178
Location: near Detroit, Michigan, USA

PostPosted: Wed Nov 21, 2012 7:32 am    Post subject: Nienhaus Comments Reply with quote

Talk: Kraken Fish

If you eliminate either r2c2 or r2c4 from the Kraken X-wing example, you still get the same eliminations. The resultant pattern is a 2 string kite and a conjugate pair, where one cell of the conjugate pair is a peer of one cell of the 2 string kite. This occurs more often than the Kraken X-wing. The Kraken X-wing pattern has one cell of an x-wing merged with a one cell of a 2 string kite. This concept of adding a conjugate pair to an existing pattern also works for the empty rectangle technique. It gives an extra cell elimination.

Nienhaus
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arkietech



Joined: 31 Jul 2008
Posts: 1727
Location: Northwest Arkansas USA

PostPosted: Wed Nov 21, 2012 2:02 pm    Post subject: Reply with quote

Kraken x-wing = x-wing with extended fin

I can see the latter easier. Very Happy
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keith



Joined: 19 Sep 2005
Posts: 3178
Location: near Detroit, Michigan, USA

PostPosted: Wed Nov 21, 2012 2:49 pm    Post subject: Reply with quote

arkietech wrote:
Kraken x-wing = x-wing with extended fin

I can see the latter easier. Very Happy

Except, in the example, there is no fin:
Code:
.-------.-------.-------.
| . / . | . / . | . . . |
| . X . | . X . | . * . |
| . / . | . / . | . . . |
:-------+-------+-------:
| . / . | . / . | . . . |
| . X . | . X . | . * . |
| . / . | . / . | . . . |
:-------+-------+-------:
| . / . | . / . | . . . |
| . X . | . / . | . . . |
| / / X | / / / | / X / |
'-------'-------'-------'

If you take away the "extension", the strong link in R9, there are no possible eliminations from the X-wing is true or R8C2 is true.

Keith
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ronk



Joined: 07 May 2006
Posts: 397

PostPosted: Wed Nov 21, 2012 5:04 pm    Post subject: Reply with quote

Who authored the main article?
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arkietech



Joined: 31 Jul 2008
Posts: 1727
Location: Northwest Arkansas USA

PostPosted: Wed Nov 21, 2012 7:24 pm    Post subject: Reply with quote

keith wrote:
If you take away the "extension", the strong link in R9, there are no possible eliminations from the X-wing is true or R8C2 is true.


The fin is r8c2 without it you have an x-wing

with it you have the same deletes in column 8 with the extension

What am I missing Confused
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keith



Joined: 19 Sep 2005
Posts: 3178
Location: near Detroit, Michigan, USA

PostPosted: Wed Nov 21, 2012 8:44 pm    Post subject: Reply with quote

arkietech wrote:
keith wrote:
If you take away the "extension", the strong link in R9, there are no possible eliminations from the X-wing is true or R8C2 is true.


The fin is r8c2 without it you have an x-wing

with it you have the same deletes in column 8 with the extension

What am I missing Confused

A finned X-wing makes eliminations without an extension.

Keith
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keith



Joined: 19 Sep 2005
Posts: 3178
Location: near Detroit, Michigan, USA

PostPosted: Wed Nov 21, 2012 9:00 pm    Post subject: Reply with quote

ronk wrote:
Who authored the main article?

Who knows? Nienhaus is the only discernible credit.

Quote:
From Sudopedia, the free Sudoku reference guide

Talk:Kraken Fish

From Sudopedia

Revision history
Jump to: navigation, search
(Latest | Earliest) View (previous 50) (next 50) (20 | 50 | 100 | 250 | 500).
Diff selection: mark the radio boxes of the versions to compare and hit enter or the button at the bottom.
Legend: (cur) = difference with current version, (last) = difference with preceding version, M = minor edit.


(cur) (last) 23:51, 18 July 2008 Nienhaus

(Latest | Earliest) View (previous 50) (next 50) (20 | 50 | 100 | 250 | 500).
Retrieved from "http://www.sudopedia.org/wiki/Talk:Kraken_Fish"


Please feel free to rewrite history!

Keith
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ronk



Joined: 07 May 2006
Posts: 397

PostPosted: Sun Nov 25, 2012 4:08 pm    Post subject: Reply with quote

elsewhere tlanglet wrote:

Another x-wing, which I believe is termed a Kraken x-wing, makes the same deletion.

Kraken x-wing(9)r45c16 with fin 9r3c1
9r3c1-r2c2=9r2c9 => r5c9<>9
x-wing(9)r45c16 => r5c9<>9

Code:

au tough November 19, 2012
.5.9..7..3.....48....6........7....1.78.2.53.5....3........2....49.....5..1..4.6.

 12468  5      246     | 9      34     18      | 7      12     236
 3      169    67      | 2      157    157     | 4      8      69
 12489  1289   247     | 6      34     178     | 1239   5      239
-----------------------+-----------------------+---------------------
 2469   269    3       | 7      568    569     | 2689   249    1
 1469   7      8       | 14     2      169     | 5      3      4-9
 5      1269   246     | 148    168    3       | 2689   2479   24789
-----------------------+-----------------------+---------------------
 678    368    5       | 138    16789  2       | 1389   1479   34789
 2678   4      9       | 138    1678   167     | 1238   127    5
 278    238    1       | 5      789    4       | 2389   6      23789

Code:

*.  .  . |  .  .  / |  .  .  .
 / *9 *. |  /  /  / |  /  / f9
*9  .  . |  .  .  / |  .  .  .
---------+----------+----------
*9  .  . |  .  . *9 |  .  . -.
*9  .  . |  .  . *9 |  .  . -9
 /  .  . |  .  .  / |  .  .  .
---------+----------+----------
 /  .  . |  .  .  / |  .  .  .
 /  .  . |  .  .  / |  .  .  .
 /  .  . |  .  .  / |  .  .  .

 sashimi mutant swordfish body: 9r2c16\r45b1 + fin: 9r2c9 ==> r45c9<>9

According to The Ultimate Fish Guide, the pattern is formally a sashimi mutant swordfish (Fig 3V). The pattern is ...
  • swordfish because the defining set (base set) is comprised of three strong inference sets, 9r2c16
  • mutant because the base set is composed of both rows and columns
  • sashimi because the pattern would collapse (to that of an x-wing) if fin r2c9 did not exist
Note the pattern is still valid with candidates <9> at r1c1 and r2c3 and can make an additional exclustion r4c9<>9.

FWIW Nienhaus is Bud on this forum and The New Players' Forums.
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