Fish & Wings
Jellyfish
A Jellyfish removes one digit from four columns at once. It appears when, in four rows, the digit has only two, three or four possible cells each and all of those cells lie in the same four columns: each row must put the digit in a different one of those columns, so the four columns are used up and the digit can be removed from every other cell in them. It is the Swordfish argument stretched to four rows, and it is rare, because it can only be the simplest move while the digit is on the grid at most once; it works the same way with rows and columns swapped.
What you need to see on the grid
A Jellyfish is the last fish to try, not the first. Hunt for one only when singles, subsets, X-Wing and Swordfish have all stopped producing moves, and only on a digit that is on the grid at most once: for any other digit, a smaller pattern makes the same eliminations (see the last mistake below). Four terms carry the whole method:
- Base rows
- The four rows you build on. In each one the digit must have two, three or four candidate cells. Four is allowed now, because there are four columns to hold them. One place is a Hidden Single you should simply play, and five places can never fit into four columns. The rows need not be neighbours or share a band of boxes.
- Cover columns
- The four columns that, between them, hold every candidate of the digit from the base rows. They cross the base rows in sixteen cells, and a real Jellyfish uses only some of them. Eliminations happen only in these columns, only outside the base rows, and a cover column may well have nothing to lose.
- One argument, three sizes
- X-Wing is two rows on two columns, Swordfish three on three, Jellyfish four on four. The reasoning never changes: N rows whose places fit into N columns use those columns up. Only the work grows. There are more groups of four rows to test than pairs or trios, each with up to sixteen cells to read, which is why a Jellyfish is so easy to miss.
- The column version, and its twin
- Swap the roles: four base columns whose candidates fall into four cover rows clear the rest of those rows. When the digit is on the grid exactly once, every row Jellyfish has a column twin that removes exactly the same candidates. In the example below, columns 1, 2, 5 and 6 hold 5 only in rows 1, 3, 4 and 7. So one direction is enough to search.
How to apply it, step by step
One example runs through all five steps: the digit 5, part-way through a hard puzzle. Every diagram shows the whole grid. The cells a step needs are filled in; the others are left blank and hatched, so you know you can ignore them.
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Step 1Pick one digit and list its places, row by row
Choose a digit and read the grid one row at a time, noting every cell where that digit is still a candidate. Here row 5 leaves 5 four places: R5C3, R5C4, R5C7 and R5C9. Write it down as "R5: C3, C4, C7, C9".
If you are used to Swordfish, you would cross this row off at once. Keep it: in a Jellyfish a row may have up to four places. Do the same for every row where the digit is not placed yet, and only with pencil marks you trust. One candidate you forgot to erase turns a four-place row into a five-place row, and the pattern is gone.
Only row 5 matters here. 5 is a candidate in four of its cells: R5C3, R5C4, R5C7 and R5C9. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 6 R6 R7 R8 R9 - Digit already on the grid
- Row being read
- A place for 5
- Candidate 5, the digit studied
- Not needed for this step
Filled in: row 5. The other 72 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 2Keep the rows where it has two to four places
Sort the rows by how many places the digit has left. A row with two, three or four places can serve as a base row. A row with five or more cannot: five cells of one row always sit in five different columns.
Row 8 already has its 5, at R8C8, so it is out of the count. Rows 2 and 9 have two places, row 6 has three, and rows 3, 5 and 7 have four: these six rows stay on the list. Rows 1 and 4 give 5 six places each and drop out.
Rows 1 to 7 and row 9 in full, and the 5 already placed in row 8. Kept rows 2, 3, 5, 6, 7 and 9 have two, four, four, three, four and two places for 5; rows 1 and 4 have six each. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 4 7 1 R2 7 1 4 2 R3 4 2 1 7 3 R4 8 6 7 R5 6 R6 6 1 7 4 8 R7 4 8 9 2 R8 5 R9 8 9 2 6 4 - Digit already on the grid
- Kept: two to four places for 5
- A place for 5 in a kept row
- Candidate 5, the digit studied
- Not needed for this step
Filled in: rows 1, 2, 3, 4, 5, 6, 7 and 9; cell R8C8. The other 8 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 3Find four rows whose places use only four columns
Among the kept rows, look for four whose places for the digit, taken together, fall in exactly four columns. Rows 2, 5, 6 and 9 do: their 5s sit only in columns 3, 4, 7 and 9. These are the base rows; columns 3, 4, 7 and 9 are the cover columns.
The shape is 2-4-3-2. Only row 5 reaches all four columns, and only 11 of the 16 crossings hold a 5. Rows 3 and 7 have four places too, but neither can join: row 3 reaches columns 2 and 6, and row 7 misses the cover columns entirely. Six kept rows give fifteen groups of four to test, and that is why a Jellyfish hides so well. Work through the groups in order instead of trusting your eye.
Technique: Jellyfish
Base rows 2, 5, 6 and 9 keep 5 inside columns 3, 4, 7 and 9. Those four columns are shown top to bottom, because the next step clears them; rows 3 and 7 are shown because they reach other columns. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 4 R2 7 1 4 2 R3 4 2 1 7 3 R4 8 6 R5 6 R6 6 1 7 4 8 R7 4 8 9 2 R8 6 9 8 7 R9 8 9 2 6 4 - Digit already on the grid
- Base rows
- Cover columns
- A place for 5 in a base row
- Candidate 5, the digit studied
- Not needed for this step
Filled in: rows 2, 3, 5, 6, 7 and 9; columns 3, 4, 7 and 9. The other 15 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 4Remove the digit from the rest of those four columns
Each base row must contain a 5, and it can only go into one of the cover columns. Two base rows cannot both use the same column, because a column holds 5 only once. So rows 2, 5, 6 and 9 use up columns 3, 4, 7 and 9 between them, one column each, even though we do not yet know which row takes which.
Every other cell of those four columns therefore loses 5: R1C3 and R3C3 in column 3, R1C7 and R4C7 in column 7, R1C9, R3C9 and R4C9 in column 9. Column 4 loses nothing, because every 5 it can still take is in a base row. That is normal, since a cover column does not have to give anything. Check all nine cells of each cover column. The base rows keep all their candidates.
Technique: Jellyfish
The base rows and the cover columns, every cell of them. 5 leaves the seven cover-column cells outside the base rows that still list it. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 4 R2 7 1 4 2 R3 2 7 R4 8 6 R5 6 R6 6 1 7 4 8 R7 4 8 2 R8 6 9 8 7 R9 8 9 2 6 4 - Digit already on the grid
- Base rows
- Cover columns
- A place for 5 in a base row
- Candidate 5, the digit studied
- Candidate removed
- Not needed for this step
Filled in: rows 2, 5, 6 and 9; columns 3, 4, 7 and 9. The other 25 cells are left blank: they are not needed for this step. Removed: 5 from R1C3, R1C7, R1C9, R3C3, R3C9, R4C7 and R4C9.
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Step 5Collect what the elimination unlocks
A Jellyfish never places a digit by itself, so look at the cells it just cleared. Three of them are in box 3 (top right), which had four places for 5: R1C7, R1C9, R2C9 and R3C9. Only R2C9 is left, so it must be 5. That is a Hidden Single in the box, and base row 2 now knows which of its two places is the real one.
There is more: R1C7 and R3C3 are down to one candidate, 9, so each is a Naked Single too. Check every row, column and box that lost a candidate before you move on. After a search this long, that is where the Jellyfish pays you back.
Technique: Hidden Single
Box 3 after the Jellyfish. 5 now fits only R2C9. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 1 R2 4 2 5 R3 7 3 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- Box 3: one place left for 5
- The cell that must be 5
- Candidate 5, the digit studied
- Digit placed
- Not needed for this step
Filled in: box 3. The other 72 cells are left blank: they are not needed for this step. Placed: R2C9 = 5.
Common mistakes
Dropping rows with four places
The Swordfish limit of two or three places is a habit to break here. A Jellyfish has four cover columns, so a base row may have four places, like row 5 in the example. The limit moves up to four and stops there: five places in one row means five different columns, so that row can always put the digit outside your four.
Waiting for a full 4×4 block
Sixteen candidates in a neat square almost never happen. Real Jellyfish are sparse: the example uses 11 of the 16 crossings, in a 2-4-3-2 shape. The only test is that the four rows together use exactly four columns. Looking for a shape instead of counting columns is how most Jellyfish are missed.
Hunting on a digit placed twice or more
Count how often the digit is already on the grid. If it is there two or more times, the rows and columns your Jellyfish leaves out always form a smaller pattern in the other direction that removes exactly the same candidates: a Swordfish, an X-Wing or even a Hidden Single, depending on how many copies are placed. Look for that one instead. A Jellyfish only earns its search while the digit is placed at most once.
Learn the move, not just the digit
In a real game, when a Jellyfish is what opens the next cell, the hint in the app names it and shows the cell it unlocks — the same move as step 5 above.