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# Franken Sword-Fish

I am testing the usefulness of this strategy which I have poached straight from Sudopedia.

Franken patterns extend the fish family which starts with X-Wings, Swordfish, Jellyfish and Squirmbags. Finned and Sashimi versions of these are covered under Finned X-Wings and Finned Swordfish. There is no Franken X-Wing.

With Franken Fish we look for a defining set and a secondary set. A defining set consists of three rows or columns. One of these will contain a conjugate pair and will belong to a different 'house' to the other two rows or columns. (A house is a set of rows or columns belonging to three boxes, such as columns 1, 2 and 3 or rows A, B and C).
A characteristic Franken Swordfish is set out in this first diagram. The defining set is the three columns 5, 7 and 8. One of these columns must contain a conjugate pair of a number X and this must much up exactly with other numbers X in the other columns. The empty yellow spaces are very important in this configuration. They must not contain X. The other two columns contain a bunch of X in box 3 (called the Franken Box) aligned on those two columns. All other cells (in white) may or may not contain X.

The secondary set consists of the two rows that the conjugate pair exist on - these are marked with light blue cells.

This Franken Swordfish contains 12 numbers of X which is maximal, but more often than not less will be available in that box. But at least one X must be present both columns and in the Franken Box.

The conjugate pair in E5 and H5 is the key. We don't know which will be the solution but the diagram shows the consequences of first one and then the other.
What happens when either is considered the solution is that the rest of the pattern makes an X-Wing style either/or set. Either the group marked A will be the solution or the group marked B.

If this is the case - and logically it is - then we have many places where X cannot go. You can remove any X in all the cells marked with a red X. We get to remove Xs in the rows of the secondary set and within the box defined by the defining set. The other consequence is that certain cells must have an X - which are marked in Blue. You may be certain which one, however, but I have added it to the diagram so show how the spaces configures when the logic is applied.
Fhe Franken box, box 3, does not have to contain all six candidates of X. In this version of the diagram I have left two remaining. The pattern and the eliminations are exactly the same. You can think of the Xs in this box and column as two Grouped Cells pointing down the columns. A Group Cell can contain 1, 2 or 3 cells as members.
Certain specific arrangements of X in this box give even more eliminations if X is aligned up correctly. To have two remaining X in the Franken Box that are aligned on the row causes them to be EITHER/OR as well. Which means the rest of that row can also have other candidates removed. The total extent of the elimination cells is shown on this diagram.
While I was testing this it occurred to me that the pattern seems to require that the conjugate pair we start will must exist in two separate boxes as well as aligned on the column (or row). Moving them up to the same box does work, but you need the same number of empty cells in the defining set's columns. I have written on the diagram to emphasis where the empty cells are - all the yellow ones.
In the final diagram I retain the conjugate pair in the centre box, but it is now only a pair within the box, not the column. This does not break the Franken pattern but it does severely limit the eliminations. These in fact they can only take place in Box 3 and Box 6. I have labelled the yellow cells 'empty' where these need to be empty.

(Is this a mutant Franken?)

The solver does not look for this alternative pattern as it is much less useful.
I have only noticed several Franken Swordfish per thousand very hard Sudoku so far. The position of the strategy in the testing list is important. It cannot go before the Finned and Sashimi Fish and I am testing it just after those strategies. I'll post the results here.

If it looks viable I will add some examples with diagrams.

 Go back to Finned Sword-Fish Continue to Pattern Overlay

## ... by: Sherman

Monday 12-May-2014
Andrew -

Franken Fish are more general than your description. Before I get into this, a correction to your comment "A house is a set of rows or columns belonging to three boxes...". In Sudopedia, a house is the same as a unit, i.e., ONE row or column or box.

I am disappointed in Sudopedia's description of Franken Swordfish, as it seems to imply that all Franken Swordfish have to contain a conjugate pair, and that is not the case. Also, it uses the conjugate pair to explain the eliminations, which while correct in this example, obscures the power of general fish patterns. I'd like to attempt an explanation based just on the general fish rules.

Sudopedia lists the general fish rules in http://sudopedia.enjoysudoku.com/Fish.html. I won't repeat them here. A couple of key points from the rules:

1) The number of units, N, in the defining set is equal to the number of units in the secondary set
2) The candidates in the defining set are a subset of the secondary set, i.e., there is no candidate in the defining set outside the secondary set

Basic Fish (X-Wing, Swordfish, Jellyfish) have N rows in their defining sets and N columns in their secondary sets or vice versa.

Franken Fish have one or more boxes in either the defining set or secondary set.

Mutant Fish have one or more boxes in both sets or a mix of rows and columns plus optionally boxes in one or both sets.

Overlapping units are allowed in a set, but the placement of candidates in the sets is restricted in that case to get a valid fish pattern. See the rules for details.

Now, in any general fish pattern, what happens if we assume one of the candidates in the secondary set that is not in the defining set is true? In this case, one of the N units in the secondary set is now solved, but none of the N units in the defining set has been solved. We are left with N-1 secondary set units to solve N defining set units, an impossibility. Thus, we can remove all secondary set candidates that are not part of the defining set.

In your Franken Swordfish example, the defining set contains columns 5, 7, 8. The secondary set contains rows 5, 8 and box 3 (you should color box 3 blue). The red X's are cells in the secondary set that are not in the defining set. What happens if one of these is assumed to be true, e.g., R1C9? If R1C9 were true, box 3 is solved, leaving two rows of the secondary set to fill three columns of the defining set. Impossible.

What Sudopedia doesn't show is a Franken Swordfish where the defining set contains a box and the secondary set does not. If you construct a maximal version of this, you will see that there are no conjugate pairs in it, e.g., look at Example 3B1 ccb\rrr in http://forum.enjoysudoku.com/the-ultimate-fish-guide-t4993.html. The rrr\ccb example next to this is essentially your Franken Swordfish with rows and columns exchanged.

BTW, N=2 Franken X-Wings do exist, but they are simply combinations of the basic strategies, e.g., pointing pairs and box/line reductions, so do not add anything new.

Franken Jellyfish is a simple extension of Franken Swordfish. You could add it to your solver and rename this strategy Franken Fish.