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.
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* 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:
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. 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.
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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:
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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.
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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@ |
+-------------------+------------------+--------------------+
