The Ultimate FISH Guide

Advanced methods and approaches for solving Sudoku puzzles

Postby Ruud » Thu Nov 23, 2006 2:43 am

Studied this interesting mutant Swordfish:

Code: Select all
 *  *  X  | .  .  .  | .  *  .        /  /  X  | .  .  .  | .  /  .
 *  *  X  | .  .  .  | .  *  .        /  /  X  | .  .  .  | .  /  .
 X  X  /  | /  /  /  | /  X  /        X  X *X  | *  *  *  | *  X  *
----------+----------+---------      ----------+----------+---------
 .  .  /  | .  .  .  | .  *  .        .  .  *  | .  .  .  | .  /  .
 .  .  /  | .  .  .  | .  *  .        .  .  *  | .  .  .  | .  /  .
 .  .  /  | .  .  .  | .  *  .        .  .  *  | .  .  .  | .  /  .
----------+----------+---------      ----------+----------+---------
 X  X  /  | /  /  /  | /  X  /        X  X *X  | *  *  *  | *  X  *
 *  *  X  | .  .  .  | .  *  .        /  /  X  | .  .  .  | .  /  .
 *  *  X  | .  .  .  | .  *  .        /  /  X  | .  .  .  | .  /  .
 Fig 3C: rrc\cbb                      Fig 3C inverse: cbb\rrc
 rcc\rbb transpose                    rbb\rcc transpose
 Mutant swordfish


These are finned varieties:
Code: Select all
 *  *  X  | .  .  .  | .  .  .        #  #  X  | .  .  .  | .  /  .
 *  *  X  | .  .  .  | .  .  .        #  #  X  | .  .  .  | .  /  .
 X  X  #  | /  /  /  | /  X  /        X  X  *  | .  .  .  | .  X  .
----------+----------+---------      ----------+----------+---------
 .  .  /  | .  .  .  | .  .  .        .  .  .  | .  .  .  | .  /  .
 .  .  /  | .  .  .  | .  .  .        .  .  .  | .  .  .  | .  /  .
 .  .  /  | .  .  .  | .  .  .        .  .  .  | .  .  .  | .  /  .
----------+----------+---------      ----------+----------+---------
 X  X  /  | /  /  /  | /  X  /        X  X  X  | .  .  .  | .  X  .
 .  .  X  | .  .  .  | .  .  .        /  /  X  | .  .  .  | .  /  .
 .  .  X  | .  .  .  | .  .  .        /  /  X  | .  .  .  | .  /  .

The pattern on the left is an Empty Rectangle. The pattern on the right has no name, but shouldn't we call this an Inversed Empty Rectangle?

I'm sure Havard will soon point us to an ancient thread that already describes this pattern...

Ruud
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Mutant Fish

Postby Mike Barker » Sat Nov 25, 2006 6:20 am

Here's a shot at a description for mutant fish. Others probably can and should try to improve it, but hopefully it hits the salient points.

All fish are forms of Constraint Subsets. Basic fish consist of cells which contain the same digit located in "n" rows or in "n" columns. All of these cells can also be contained in "n" columns or "n" rows, respectively. Franken Fish consist of cells which contain the same digit located in "n" rows and boxes or in "n" columns and boxes. All of these cells can also be contained in "n" columns and boxes or in "n" rows and boxes, respectively.

Mutations are possible where the fish digit is contained in a combination of "n" rows, columns, and possibly boxes as well. These units form the base sets. To be a valid fish no cell which contains the fish digit can belong to more than one unit of the base set. For example, the following fish consists of r2, c2, and b9. r2c2 cannot contain the fish digit since this cell is common to both r2 and c2.
Code: Select all
 *  X  *  | .  .  .  | .  *  .
 X  /  X  | /  /  /  | /  X  /
 *  X  *  | .  .  .  | .  *  .
----------+----------+---------
 .  /  .  | .  .  .  | .  *  .
 .  /  .  | .  .  .  | .  *  .
 .  /  .  | .  .  .  | .  *  .
----------+----------+---------
 .  /  .  | .  .  .  | /  X  /
 *  X  *  | *  *  *  | X *X  X
 .  /  .  | .  .  .  | /  X  /

To be an unfinned mutant fish all of the cells which contain the fish digit must also be contained in "n" different rows, columns, and/or boxes. These units form the cover set. Note that to be a true mutant either the base set, the cover set, or both should include both rows and columns. In the above example the cover set consists of r8, c8, and b1. Just as in the case of a basic fish, the fish digit can be removed from any cell which is part of the cover set, but is not part of the base set. The possible eliminations in the above example are shown with "*"s. The reason is as follows. If one of these cells contain the fish digit, then none of the other cells in the unit can, including any cells which make up the fish. There are only "n-1" units remaining in which to place "n" fish digits creating a contradiction. For example, if r1c1 contains the digit then r2c13 and r13c2 cannot which leaves only r8 and c8 in which to place the three digits required by r2, c2, and b9.

In the case of mutant fish it is also possible for a candidate to exist in a cell common to two units of the cover set. This candidate can also be eliminated since placing a candidate in this cell will reduce the number of units available in the covering set by two leaving too few units in which to place digits. For example, placing a digit in r8c8 leaves only b1 in which to place two candidates for r2 and c2.

Finned mutant fish are identical to finless except that not all of the cells which make up the base set can be included in the "n" units of the cover set. The remaining cells are the fins. There may be more than one fin. An elimination can occur if a cell which is not part of the fish sees all of the cells in one unit of the cover set and all of the fins. For example, r8c13 see r8 and the fin in r9c2 in the following fish:
Code: Select all
 .  X  .  | .  .  .  | .  .  .
 X  /  X  | /  /  /  | /  X  /
 .  X  .  | .  .  .  | .  .  .
----------+----------+---------
 .  /  .  | .  .  .  | .  .  .
 .  /  .  | .  .  .  | .  .  .
 .  /  .  | .  .  .  | .  .  .
----------+----------+---------
 .  /  .  | .  .  .  | /  X  /
 *  X  *  | .  .  .  | X  X  X
 .  #  .  | .  .  .  | /  X  /

Here is an example of a finned mutant swordfish with the base set equal to {r6,c9,b7}. The cover set is {r8,c3,b6} with a fin in r3c9. The cell, r3c3, sees c3 and the fin allowing the elimination.
Code: Select all
Finned Mutant Swordfish: r6c9b7/r8c3b6,r3c9 => r3c3<>9
+-------------------+---------------+--------------------+
|   1    6     478  | 25   28    9  | 3457     45   235  |
|  29   39     379  |  4   12  157  |    8    159     6  |
| 249    5   478-9  |  3    6  178  | 2479    149    29# |
+-------------------+---------------+--------------------+
| 459  349       1  | 69    7    2  |  345  45689  3589* |
| 459    7    3469  | 69  348  348  |    1      2   359* |
|   8  234   23469* |  1    5   34  |  349*   469*    7  |
+-------------------+---------------+--------------------+
|   3    8      29* | 25  249   45  |    6      7     1  |
|   6  249*    249* |  7   13   13  |  259    589  2589* |
|   7    1       5  |  8   29    6  |   29      3     4  |
+-------------------+---------------+--------------------+

The following example is similar with the base set equal to {r39,c4}, the cover set={c18,b8} and a fin in r1c4:
Code: Select all
Finned Mutant Swordfish: r39c4/c18b8,r1c4 => r1c18<>9
+--------------------+-------------------+---------------------+
| 12578-9  178  249  |   49#   47   579  | 2358  2358-9     6  |
|   25789   78  249  |    3   467  5679  |    1    2589    89  |
|      59*   6    3  |    1     2     8  |    4      59*    7  |
+--------------------+-------------------+---------------------+
|       4    9    8  |    5     3     1  |    6       7     2  |
|      36    2   57  |  468     9    67  |  358       1   348  |
|     136   13   57  | 2468  4678   267  |    9    3458   348  |
+--------------------+-------------------+---------------------+
|   23789  378   29  | 2689*    5     4  | 2378    2368     1  |
|    2378    5    6  |  289*    1   239  | 2378   23489  3489  |
|    2389*   4    1  |    7    68  2369* |  238   23689*    5  |
+--------------------+-------------------+---------------------+

The largest known fish is a Finned Mutant Squirmbag, but larger fish could possibly (although unlikely) exist. Examples exist here.

Just as there can be Kraken Fish, there can be Kraken Mutant Fish. Here is an example of a Kraken Mutant Swordfish with r34,c7 forming the base set, c6,b34 forming the cover set, and strong links r9c9=3=r9c3 and r7c1=3=r9c3 and a direct link linking the fins, r4c9,r57c7, to the candidate elimination cell. This allows the CEC to see the fins and b4 of the cover set allowing the elimination.
Code: Select all
1-link Mutant Swordfish: r34c7/c6b34,r4c9,r57c7 (r4c9-3-r9c9=3=r9c3-3-, r5c7-3-, r7c7-3-r7c1=3=r9c3-3-)  => r5c3<>3
+-------------------+------------------+--------------------+
|    5    3    468  |  468    1     9  |    7     2    468  |
| 1468    7      9  | 3468   48     2  | 3458* 1345  34568  |
|  468   14      2  |    5  478   367* |    9   134*  3468* |
+-------------------+------------------+--------------------+
| 1349*  19    345* |    2  457   137* |    6     8   3457# |
|    7    2  456-3  | 3468  458    36  |  345#    9      1  |
| 1346    8   3456  |  346    9  1367  |    2   345   3457  |
+-------------------+------------------+--------------------+
|   38$   6      1  |    7    2    45  |  348#  345      9  |
|   49  459      7  |    1    3     8  |   45     6      2  |
|    2   45    38@$ |    9    6    45  |    1     7     38@ |
+-------------------+------------------+--------------------+
Last edited by Mike Barker on Sat Nov 25, 2006 2:13 pm, edited 4 times in total.
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Re: Mutant Fish

Postby ronk » Sat Nov 25, 2006 4:43 pm

Mike Barker wrote:The following example is similar with the base set equal to {r39,c4}, the cover set={c18,b8} and a fin in r1c4:
Code: Select all
Finned Mutant Swordfish: r39c4/c18b8,r1c4 => r1c18<>9
+--------------------+-------------------+---------------------+
| 12578-9  178  249  |   49#   47   579  | 2358  2358-9     6  |
|   25789   78  249  |    3   467  5679  |    1    2589    89  |
|      59*   6    3  |    1     2     8  |    4      59*    7  |
+--------------------+-------------------+---------------------+
|       4    9    8  |    5     3     1  |    6       7     2  |
|      36    2   57  |  468     9    67  |  358       1   348  |
|     136   13   57  | 2468  4678   267  |    9    3458   348  |
+--------------------+-------------------+---------------------+
|   23789  378   29  | 2689*    5     4  | 2378    2368     1  |
|    2378    5    6  |  289*    1   239  | 2378   23489  3489  |
|    2389*   4    1  |    7    68  2369* |  238   23689*    5  |
+--------------------+-------------------+---------------------+

Great find. As that one is not currently in the exemplars, I will add it as ...
Code: Select all
 .  .  . |  .  /  . |  .  /  .        .  .  . |  .  *  . |  .  *  .
 *  *  * |  *  X  * |  *  X  *        /  /  / |  /  X  / |  /  X  /
 .  .  . |  .  /  . |  .  /  .        .  .  . |  .  *  . |  .  *  .
---------+----------+----------      ---------+----------+----------
 .  .  . |  *  X  * |  .  /  .        .  .  . |  /  X  / |  .  *  .
 /  /  / |  X  /  X |  /  /  /        *  *  * |  X *X  X |  *  *  *
 .  .  . |  *  X  * |  .  /  .        .  .  . |  /  X  / |  .  *  .
---------+----------+----------      ---------+----------+----------
 .  .  . |  .  /  . |  .  /  .        .  .  . |  .  *  . |  .  *  .
 *  *  * |  *  X  * |  *  X  *        /  /  / |  /  X  / |  /  X  /
 .  .  . |  .  /  . |  .  /  .        .  .  . |  .  *  . |  .  *  .
 Fig 3x: rcc\rrb                      Fig 3x inverse: rrb\rcc
 rrc\ccb transpose                    ccb\rrc transpose
 mutant swordfish

The actual 'x' of 'Fig 3x' to be determined.

Excellent description of the mutant fish as well. However, the distinction between franken fish and mutant fish seems lost. Maybe you should empasize that mutant fish must have both rows and cols in either the base set, the cover set, or both.
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Postby Ruud » Sat Nov 25, 2006 9:17 pm

I am a little confused about Sashimi fish.

When Myth Jellies introduced this type, only an X-Wing example was given. For a Sashimi X-Wing, the only vertex that can safely be dropped is the one in the box containing the fin. The 3 other vertices must remain in place, or singles emerge.

In the introduction, Myth stated that a Sashimi Fish is an incomplete fish, stabilized by the fin. Without the fin, the underlying fish would degenerate.

Somehow, this definition has shifted towards a Sashimi fish being a finned fish that does not have a candidate in the fin-box that belongs to the covered set. For an X-Wing, this shift does not make any difference, but for bigger Sashimi fish, there are some implications.

The following diagrams may illustrate my point:

Code: Select all
 .  /  .  | .  /  .  | .  /  .
 .  /  .  | .  X  .  | .  X  .
 .  /  .  | .  /  .  | .  /  .
----------+----------+---------
 .  /  .  | .  #  .  | .  /  .
 .  X  .  | *  /  *  | .  /  .
 .  /  .  | .  #  .  | .  /  .
----------+----------+---------
 .  /  .  | .  /  .  | .  /  .
 .  X  .  | .  /  .  | .  X  .
 .  /  .  | .  /  .  | .  /  .
True Sashimi Swordfish

 .  /  .  | .  /  .  | .  /  .
 .  X  .  | .  X  .  | .  /  .
 .  /  .  | .  /  .  | .  /  .
----------+----------+---------
 .  /  .  | .  #  .  | .  /  .
 .  X  .  | *  /  *  | .  X  .
 .  /  .  | .  #  .  | .  /  .
----------+----------+---------
 .  /  .  | .  /  .  | .  /  .
 .  /  .  | .  X  .  | .  X  .
 .  /  .  | .  /  .  | .  /  .
Fake Sashimi Swordfish

There is no doubt about the first diagram. When the fin is cleared, the pattern degenerates. The second diagram is the one I'm not sure about. There is no candidate for the underlying Swordfish in box 5. However, removing the fin leaves a perfectly valid Swordfish. Should we call this a Finned Swordfish or a Sashimi Swordfish?

Ruud.
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Postby tarek » Sat Nov 25, 2006 9:40 pm

I think that the wise thing to do is actually to leave "Fin, vertix & elimination cell in the same box" as an observation....not a rule.....

Therefore -for learning curve purposes- I would agree with Ruud on his diagrams .......

take the fin(s) out:
1. if you have a viable basic fish -----------> Finned Fish
2. if you have a non-viable basic fish -----> Sashimi Fish


IMO the importance would be on always explaining the Finned fish in relation to basic fish & on always explaining Sashimi fish in relation to finned fish.

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Postby Myth Jellies » Sun Nov 26, 2006 1:22 pm

My original intent for "sashimi" matches what Tarek wrote. The idea was to show that otherwise impossible fish structures could now be possible with a stabilizing fin. That point has been rendered somewhat moot by your work here showing that it is the cells where the candidates must be missing that really define the fish and the fin.
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Postby tarek » Sun Nov 26, 2006 4:55 pm

1st post updated........

Attempts at addressing fish name as a combination of Shape + Size:idea:

the 1-fish (equ. to Sector*Sector interaction) was added as a fish size

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Postby ronk » Mon Nov 27, 2006 1:28 pm

Ruud, would you please use your program to confirm the exclusions in the hidden pattern below? Are there more:?:
Code: Select all
 .  .  * |  .  .  . |  .  .  *
 .  .  * |  .  .  . |  .  .  *
 /  /  / |  /  /  . |  /  .  .
---------+----------+----------
 /  /  . |  /  /  . |  /  /  .
 .  .  * |  .  .  * |  .  .  *
 /  /  . |  .  .  . |  /  .  .
---------+----------+----------
 .  .  * |  .  .  . |  .  .  *
 .  .  * |  .  .  . |  .  .  *
 /  /  . |  /  /  / |  /  /  .

Without success I've tried to find a single fish to explain all the exclusions. Does anyone see one?

[edit: Original post deleted and replaced with this.]
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Postby Ruud » Mon Nov 27, 2006 3:32 pm

Your eliminations are correct, but it does not seem to be a single fish.

Here are some elements:

Finned Swordfish:
Code: Select all
 .  .  . |  .  .  . |  .  .  *
 .  .  . |  .  .  . |  .  .  *
 /  /  / |  /  /  X |  /  #  X
---------+----------+----------
 /  /  X |  /  /  X |  /  /  X
 .  .  . |  .  .  . |  .  .  .
 .  .  . |  .  .  . |  .  .  .
---------+----------+----------
 .  .  . |  .  .  . |  .  .  .
 .  .  . |  .  .  . |  .  .  .
 /  /  X |  /  /  / |  /  /  X


Finned Jellyfish:
Code: Select all
 .  .  . |  .  .  . |  .  .  .
 .  .  . |  .  .  . |  .  .  .
 /  /  / |  /  /  X |  /  X  X
---------+----------+----------
 /  /  X |  /  /  X |  /  /  X
 .  .  . |  .  .  * |  .  .  .
 /  /  X |  #  #  X |  /  X  X
---------+----------+----------
 .  .  . |  .  .  . |  .  .  .
 .  .  . |  .  .  . |  .  .  .
 /  /  X |  /  /  / |  /  /  X


Empty Rectangle:
Code: Select all
 .  .  . |  .  .  . |  .  .  .
 .  .  . |  .  .  . |  .  .  .
 /  /  / |  /  /  . |  /  .  .
---------+----------+----------
 /  /  A |  /  /  . |  /  /  .
 B  B AB |  .  .  . |  .  .  *
 /  /  A |  .  .  . |  /  .  .
---------+----------+----------
 .  .  . |  .  .  . |  .  .  .
 .  .  . |  .  .  . |  .  .  .
 /  /  B |  /  /  / |  /  /  A


The remaining eliminations can be explained by testing the implications for both candidates in row 9, but I do not see any suitable fish.

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Postby ronk » Tue Nov 28, 2006 5:43 pm

Ruud wrote:Your eliminations are correct, but it does not seem to be a single fish.

Here are some elements: Finned Swordfish: ... Finned Jellyfish: ... Empty Rectangle:
...
The remaining eliminations can be explained by testing the implications for both candidates in row 9, but I do not see any suitable fish.

Thanks for checking with your program.

One can use two "remora" -- based upon r6c8<>X and r6c8=X -- to obtain two elimination sets ...
Code: Select all
 .  .  * |  .  .  . |  .  .  *
 .  .  * |  .  .  . |  .  .  *
 .  .  * |  .  .  . |  .  .  *
---------+----------+----------
 /  /  X |  .  .  . |  /  /  X
 X  X *X |  *  *  * |  X  X *X
 /  /  X |  .  .  . |  /  /  X
---------+----------+----------
 .  .  * |  .  .  . |  .  .  *
 .  .  * |  .  .  . |  .  .  *
 /  /  X |  /  /  / |  /  /  X
 if r6c8<>X, franken swordfish r9b46\r5c39

 .  .  * |  .  .  * |  .  .  *
 .  .  * |  .  .  * |  .  .  *
 /  /  / |  /  /  X |  /  /  X
---------+----------+----------
 /  /  X |  /  /  X |  .  .  *
 X  X *X |  X  X *X |  *  *  *
 /  /  X |  /  /  X |  .  .  *
---------+----------+----------
 .  .  * |  .  .  * |  .  .  *
 .  .  * |  .  .  * |  .  .  *
 /  /  X |  /  /  / |  /  /  X
 if r6c8=X, franken jellyfish r39b45\r5c369

... and then take the intersection of those elimination sets to obtain the combined effect.
Code: Select all
 .  .  * |  .  .  . |  .  .  *
 .  .  * |  .  .  . |  .  .  *
 /  /  / |  /  /  X |  /  X  X
---------+----------+----------
 /  /  X |  /  /  X |  /  /  X
 X  X *X |  X  X *X |  X  X *X
 /  /  X |  X  X  X |  /  X  X
---------+----------+----------
 .  .  * |  .  .  . |  .  .  *
 .  .  * |  .  .  . |  .  .  *
 /  /  X |  /  /  / |  /  /  X

A candidate must be eaten by both remora to be truly eliminated ... so it's not a pretty analogy.:D
Last edited by ronk on Tue Nov 28, 2006 9:13 pm, edited 1 time in total.
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X-wing etc - type b (boxes-to-lines) and c (lines-to-boxes)

Postby Pat » Tue Nov 28, 2006 6:11 pm

while this Topic is mostly concerned with the "finned" types etc, i'd like to ensure the completeness of the list of standard (basic) X-wings etc —


    X-wing, Swordfish, Jellyfish, etc, type bboxes-to-rows (or boxes-to-columns)
    X-wing, Swordfish, Jellyfish, etc, type crows-to-boxes (or columns-to-boxes)

example —
X-wing type b: box1 , box3 can have the digit only in c1 , c2; exclude the digit elsewhere in those Columns

Code: Select all
 X X / / | . . . .
 X X / / | . . . .
---------+---------
 X X / / | . . . .
 X X / / | . . . .
---------+---------
 * * . . | . . . .
 * * . . | . . . .
---------+---------
 * * . . | . . . .
 * * . . | . . . .


equivalent example —
X-wing type c: c3 , c4 can have the digit only in box5 , box7; exclude the digit elsewhere in those Boxes

Code: Select all
 . . / / | . . . .
 . . / / | . . . .
---------+---------
 . . / / | . . . .
 . . / / | . . . .
---------+---------
 * * X X | . . . .
 * * X X | . . . .
---------+---------
 * * X X | . . . .
 * * X X | . . . .




— these may be needed where max ( rows_per_box , columns_per_box ) > 3

~ Pat
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Postby tarek » Tue Nov 28, 2006 7:28 pm

I think that this thread would address hopefully all sudoku variants in one way or another.....

The example that you mentioned does exist also in the vanilla sudoku word.....

Name of fish = Shape + Size

Size=2 ----> X-wing
Shape= N(Rows+Boxes) * N(Columns+Boxes) or vv. (no fins) = 2 Columns * 2 boxes ----->franken

Therefore it is a Franken x-wing regrdless of box size or dimensions of grid........

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re: X-wing - type b (boxes-to-lines) and c (lines-to-boxes)

Postby Pat » Thu Nov 30, 2006 1:50 pm

tarek wrote:it is a Franken X-wing


thanks, tarek — i do look rather silly jumping into the discussion after 115 posts (plus many more in earlier Topics...)


i did look in the first post —
3. Franken (no fins): N (rows+boxes) * N (columns+boxes)
.

— i certainly admit that "N (rows+boxes) * N (columns+boxes)" does include the simpler cases of
  • N (boxes) * N (columns)
  • N (rows) * N (boxes)
what i called type b and type c are (to me) a natural extension of the standard (basic) type (type a)
    type aRows-to-Columns (or Columns-to-Rows)
    type bBoxes-to-Rows (or Boxes-to-Columns)
    type cRows-to-Boxes (or Columns-to-Boxes)
i haven't tried mixing rows+boxes (or columns+boxes) — when i get that figured out, my classification may need a type d


by the way, your first diagram "1-FISH" doesn't identify it as a Franken fish.

~ Pat
Last edited by Pat on Thu Nov 30, 2006 11:16 am, edited 1 time in total.
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Postby Ruud » Thu Nov 30, 2006 3:03 pm

Here is an interesting observation for a Sashimi Swordfish with the fin in the same box as 2 lines from the covered set.
Code: Select all
 *  /  *  | .  #  .  | *  /  *
 .  X  .  | *  /  *  | .  X  .
 .  X  .  | *  /  *  | .  X  .
----------+----------+---------
 .  /  .  | .  /  .  | .  /  .
 .  X  .  | .  X  .  | .  X  .
 .  /  .  | .  /  .  | .  /  .
----------+----------+---------
 .  /  .  | .  /  .  | .  /  .
 .  /  .  | .  /  .  | .  /  .
 .  /  .  | .  /  .  | .  /  .

Because column 5 only has a single candidate in the basic swordfish, it is a true sashimi fish. The 2 empty cells in r23c5 combined with the finned swordfish eliminations in r23c46 will trigger Locked Candidates in box2\row 1, causing the remaining eliminations in row 1. This only happens in the Sashimi version and not in the regular Finned version.

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Re: Franken Fish

Postby tarek » Thu Nov 30, 2006 3:28 pm

Hi Pat, The subject & some of the terms addressed in this subject are fairly new & are still evolving with plenty of room for modification

Pat wrote:and while i have no idea what "vv" means
That is my short hand for vice versa:D

Pat wrote:by the way, your first diagram "1-FISH" doesn't identify it as a Franken fish.
For the sake of completeness, I think you are right........ by default,you can't mix columns & rows together in a 1-fish because you need N>=2 for that.... & the 1-fish is N=1 so there is no mutant variety of this fish under the current definitions

It has plenty of potential applications......Diagonal box interaction in sudoku x, DG line interaction in DG variant & DG diagonal interaction in DGX variant ......... it is still the same, a Franken 1-Fish or a term I like is the Franken Cyclopsfish.

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