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Strategies for Popular Number Puzzles

# Jellyfish Strategy

Jellyfish extends Swordfish one further row and column. We are looking for either

• four rows such that, in total, four cells are occupied in the row by a candidate number; or
• four columns such that, in total, four cells are occupied in the column by a candidate number

If this configuration is found then we can look in the opposite direction (if by row then down the column, if by column then across the row. If any candidates are found they can be eliminated. After the elimination both conditions above will hold.

This diagram shows a full Jellyfish with four X found in four columns and aligned on four rows. It means we can remove any X found in the columns and it leaves behind a 4 by 4 grid.

How does it work? Pick any yellow cell in the example above that contains an X. Keeping an eye on it. Pretend the solution actually is X. All others Xs in the row and columns are suppressed. What we're left with is a Swordfish. The Swordfish logic then applies. Pick any X in the Swordfish and it reduces to an X-Wing. Since any combination of Xs on the grid are possible there is no room for Xs outside the grid - that align on the grid rows and columns.

Finding exactly candidate X in all sixteen cells (a 4-4-4-4 formation) is extremely unlikely. But the logic of Jellyfish (and Swordfish) is that we only need four X in total over the four rows and columns. Here is a minimal spread of a particular candidate X - a 2-2-2-2 formation. Most of the time you will have a mixture of two, three or four candidates lined up.

One way to double check the logic is to pretend any of the crossed out Xs is a solution. When you do that and trace the consequences you will find at least one row (or column) with no X left - clearly a bad consequence.

This is a real Jellyfish example, one of the very few I have that does actually show up in the solver - a crazy hard extreme. It is based on the candidate 7 as show by the green highlighted candidates. The cells in the pattern are orange. The pattern establishes that we have 7 common to four rows (B,D,E and H) which are aligned on four columns (1,5,7 and 9). Therefore four 7s must go in those yellow cells.

This is a 2-3-3-2 formation Jelly Fish since there are 7s absent in B1 and H2.

I am pleased to report a massive catch by Klaus Brenner from Germany. He has created a 31 clue Sudoku with a required JellyFish containing an amazing 18 eliminations.

Well done Klaus!

8th October 2012

(Turn off Rectangle Elimination)

To my knowledge this is the first Perfect 4-4-4-4 Jellyfish. Perfect in the sense that every 2 in the sixteen cells that form the pattern - the 2 is still a candidate. In the examples above, each contain some cells that are clues or solutions. This type of formation is pretty rare in Swordfishes, let alone Jellyfish. Klaus Brenner also found this puzzle.

13th July 2014

I have a new crop of Jelly-Fish discovered by Klaus. This one is a maximally eliminating pattern taking off all twenty 9s.

## Jellyfish Exemplars

These puzzles require the Jellyfish strategy at early on (but are diabolical puzzles overall).
They make good practice puzzles.
All discovered by Klaus Brenner

Exotic variations of the Swordfish continue with the Finned Swordfish and Franken Swordfish.

 Go back to Swordfish Strategy Continue to Singles Chains

## ... by: Jonesvan

Thursday 2-Nov-2017
Regarding the 3-2-3-3 formation above why did you select rows 1,4,6 & 9 and columns 1,3,7,9. Could you not have selected row 3 &7 and columns 3&8? Or rows 2&8 and columns 4&6? Iâ€™m confused which rows and columns to select to make the patter? Iâ€™m having a mentally block with this pattern as it seems there are multiple options?
Andrew Stuart writes:
(Example has been replaced) - [Del]

## ... by: speter

Monday 22-May-2017
In your first example (top row = ..17538..); I can't work out how to discard candidate 2's at B1, G1, H1 & H9; and unless they are eliminated the jellyfish isn't there. ;-)

Can you please explain how the candidate 2 at B1 (for example) is eliminated!?
 REPLY TO THIS POST

## ... by: Pieter, Newtown, Oz

Monday 29-Oct-2012
Hi Andrew
I found an interesting Jelly-Fish (my first ever, actually) in the local paper [Load Puzzle].

It's interesting cuz there are 2 simultaneous Jelly-Fish on 4's! The Solver finds CDFG2357 (a 3-4-2-3 J/F), but there is also one at ABEJ1469 (a 3-4-4-3), and also interestingly, the same 7 eliminations are made by both.

BTW - Since I use Letters & Numbers to define cells, I found the rYcX notation totally confused me at first! "JF=r3467c2357"???
Andrew Stuart writes:
This is an interesting example. To see the second Jelly Fish, use the [<<] back button and then rotate once 90 degress (either way). Hit take step a few times. - [Del]

## ... by: John Francis

Saturday 23-Jan-2010
John White observes:

You can extend all of these in the following ways:

A. You are not restricted to rows and colums.
B. As with X-Wing and Swordfish, you can 'Fin' Jellyfish.
C. Why stop at 4......

The reason for stopping at 4 is because of the symmetry between possible candidates in a cell and potential sites for a given digit.

It's best explained by considering Naked and Hidden n-tuples.
Consider a Naked heptuple. This is a set of seven cells in one set (row, col or box) which between them only have seven different candidates. This means that these seven cells will contain those seven values in some order.

Now describe that situation it another way. The cells *not* in the n-tuple are the only ones which can include numbers not in those n candidates.
This is the description of a hidden (9-n)-tuple (hidden pair, in this case).

This means there's no need to go beyond 4 in describing a strategy; Naked quints are the same as hidden quads, and so on. For a strategy described using numbers greater than four there is an equivalent description using smaller numbers (and interchanging the roles of candidates in a cell and positions for a digit).

 REPLY TO THIS POST

## ... by: John White

Friday 24-Jul-2009
Jelly fish is a sub category of a more general system:

Take any group of N disjoint (not overlapping) sets(row,column or box) and compare them to any other group of N disjoint sets. If the overlap of these two groups contains all the possible candidates for any given number then you can remove all the candidates in the other group that are not in the overlapping area.

If N=2 you get X-Wing
If N=3 you get Swordfish
If N=4 you get Jellyfish

What this means is that you can extend all of these in the following ways:

A. You are not restricted to rows and colums.
B. As with X-Wing and Swordfish, you can 'Fin' Jellyfish.
C. Why stop at 4......

(Finning: if all candidates from one of the groups that are not contained in the overlap can be contained by one more set (row, colum or box) (the set must not be in the original 2 groups!), then the candidates contained in overlap of this extra set and the second group can be removed.
 Philipp Huebner replies: Saturday 22-Aug-2009 Shouldn't John White's main sentence be: "If the overlap of these two groups contains all the possible candidates for any given number in one of the groups then you can remove all the candidates in the other group that are not in the overlapping area." (I added: "... in one of the groups ...")