This revised post explains what mutant fish are, and the rule used to eliminate candidates. It offers a hard-to- find proof that this rule works, and some spotting tips.

Regular fish, finned fish, and sashimi fish are candidate elimination methods where a set of *n* parallel lines limit the placement of candidates on crossing lines. If these *n* lines offer only *n* positions for candidates of a number to be placed, then each of the n lines gets one of the positions for its true candidate of that number, and crossing lines at those positions can have no additional candidates.

A mutant fish, also called a Franken fish, is a set of n units, including lines and boxes, that similarly offer n positions for true candidates to be placed, and which restrict placements of candidates of the same number in an intersecting set of units.

To accommodate boxes as well as lines, the restrictions of one set of units on another is formulated in a more general way. The set of n parallel lines (rows or columns) becomes a set of *base* units(rows, columns and boxes) which share no candidates of the fish number. The crossing lines are generalized to a set *cover* units which collectively cover all of the base set candidates, again without sharing any base candidates among the cover units. The cover units are not restricted n candidates collectively, as the base units are.

The remarkable mutant fish elimination rule is that any candidate in a cover unit but not in a base unit can be removed!

Here is an X-panel from a mutant swordfish from www.sudokusnake.com . The base units r4, c1 and S are marked solid, and the cover units r8,c5 and W are dashed. The victims are v’s and the other candidates, x’s.

This looks nothing like the fish we are accustomed to, and you might wonder how you could ever find such a thing. Actually, we can account for these removals in another, surprising way.

Take the unfortunates in cover box W. If either is true, then groups in base lines r4 and c1 are false, forcing both r4c5 and r8c1 true and leaving base box S without candidates.

We find the same cause and effect for the other removals. Each v candidate sees candidates or groups of two base units, forcing these two units to rely on another of their candidates or groups. These candidates or groups see all the potential candidates of a third base unit.

This suggests a possible relationship between grouped chains and mutant fish. To apply that here, we expand our repertoire of grouped winks. To complete the grouped nice loop at the left, which is channeled by the base and cover sets, we need a wink between base groups in W. No problem. In fact divide the X-candidates in a unit any way you like, and there is a wink between any two subsets you create.

So in this example, the mutant swordfish construction rules produce a grouped nice loop, and every victim sees two consecutive nodes of this loop. Hmmm.

The mutant swordfish limit of three candidates per this limit keeps the slinks of the loop strong, but grouping could allow more candidates while maintaining the base slinks, so this explanation for the mutant swordfish removals suggests that the limit of three candidates per primary is not always necessary. In fact, we’ll see shortly that it applies to a mutant fish only in a special way.

But is the grouped nice loop equivalence to Snake’s mutant swordfish a freak of nature? Let’s look at another example. This one is from www.sudoku9981.com . But just for fun, you get only the candidates and victims in this post. I’ll bring along the mutant swordfish and the equivalent grouped nice loop in the next post.

Just when I was beginning to think that grouped nice loops were the rationale for mutant fish, I was lucky enough to come across HoDoKo’s mutant jellyfish at http://hodoku.sourceforge.net/en/tech_fishc.php . Hodoku calls fish with box units, Franken Fish.

Base units are three rows and a box. Covers are four columns. Victims are identified by the mutant fish rule. Slinks are not found within all base units. I was soon convinced that a grouped nice loop erasing all of these victims at once was not defined by this mutant’s structure.

As to spotting mutant fish, generally we want fewer base candidates and the more cover candidates. We are trying to add base units with minimum expansions of covers. The three lines of Hodoku’s Franken fish require only four columns for a cover. Then it’s easy to see that the SW box adds a fourth base unit without introducing more cover columns.

In harder- to-see cases, we can use suset enumeration. When looking for a fourth row, we evaluate the box by combining its rows. In this case, starting with a suset list of

1/125, 2/1257, 5/1257, . . .

The SW box suset is SW/12, and the row suset of the Franken fish is 125W/1257.

So why does the mutant fish rule work? To state it succinctly:

*Disjointly covered* means *with no shared X-candidates*.

Here’s why it works:

In the solution, one of the X-candidates in a base unit will be the unit’s true X. This true candidate must also occupy one of the cover units, since the covers cover all base candidates. All other X-candidates of that cover unit see that true candidate, and must be false. Since every base unit has a true X, it also has a cover partner that contains that true X. Since covers don’t share candidates, the base unit’s partner is paired with it alone. With n base units and n covers, there is no cover that does not contain a base true X. Therefore any cover candidate outside of any base does nevertheless see a base true X.

If that paragraph degenerated into a mass of meaningless words, use your knowledge of regular fish to tie it down. Substitute “row” for “base unit”, “column” for “cover unit”. It then expresses the fish economics you are familiar with.

I’ve added a ©PowerPoint slide for mutant fish to my toolbox. It has a single panel into which I can copy a panel, and pre-fitted blue and red rectangles I can just drag in and out. The red and blue rectangles can even interchange roles as base and cover as I experiment.

Now you can find alternative ways of solving the Hodoku Franken Fish puzzle, and the additional Franken fish offered on Hodoku, at my post Hodoku Franken Fish of 12/22/15.

Next time, we look at the natural follow up to mutant fish, which is finned mutant fish.

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