Swordfish Technique: Advanced Sudoku Strategy

The Swordfish rule

A Swordfish needs three things:

1. Choose one digit, for example 2.
2. Find three rows where that digit is possible in only two or three cells each.
3. Check the columns of all those cells. Together they must use exactly three columns.

Each of the three rows must contain the digit once. In these rows the digit is possible only in the three columns, so three digits have to go into three columns. Two of them cannot share a column. Therefore each of the three columns gets exactly one of them.

That means the three columns already have their digit, inside the three rows. No other cell of those columns can hold it, so you remove the digit from all of them.

A row does not need the digit in all three columns. A row may have the digit in only two of them. What matters is the total: all cells of the three rows together must fit in three columns.

The rule in one sentence

Three rows, one digit, the same three columns: remove the digit from the rest of those three columns.

A worked example, step by step

In this position the easy moves are finished and there is no single left to place. Every empty cell shows its candidates, the digits that are still possible there.

Find the pattern

Look only at the digit 2 and count its cells row by row. Three of the rows fit together:

• Row 2: a 2 is possible only in R2C1 and R2C5.
• Row 3: a 2 is possible only in R3C5 and R3C8.
• Row 9: a 2 is possible only in R9C1 and R9C8.

These six cells use only three columns: column 1, column 5 and column 8. This is a Swordfish on 2. Each row has the digit in only two of the three columns, and each row leaves out a different column. The pattern is still complete, because the three rows together use exactly three columns.

Check both possibilities

• Option A: the 2 of row 2 is in R2C1. Then the 2 of row 9 cannot be in column 1, so it is in R9C8. Then the 2 of row 3 cannot be in column 8, so it is in R3C5.
• Option B: the 2 of row 2 is in R2C5. Then the 2 of row 3 is in R3C8, and the 2 of row 9 is in R9C1.

In both options, column 1, column 5 and column 8 each get one 2 from the pattern. So no other cell of these columns can be a 2. Remove 2 from R7C1 in column 1 and from R6C8 in column 8. Column 5 has no other cell with a candidate 2, so nothing changes there.

Use the result

A Swordfish places no digit by itself, but its eliminations often open the next move. R6C8 had two candidates, 2 and 8. The Swordfish removed the 2, so 8 is the only candidate left. That is a naked single: place 8 in R6C8.

The column-based Swordfish

The Swordfish also works when you exchange rows and columns. If a digit is possible in only two or three cells in each of three columns, and all those cells are in the same three rows, you remove the digit from every other cell of those three rows.

The logic is the same, turned by 90 degrees. The lines where you find the pattern keep their candidates, and the crossing lines lose theirs. Check both directions, because a puzzle may contain only one of them.

How to search for a Swordfish

A Swordfish is harder to find than an X-Wing, because three rows must match instead of two. You need complete pencil marks. Then search in this order:

1. Try the simpler techniques first. Singles, pairs, pointing pairs and X-Wings are faster to find and solve most cells.
2. Take one digit at a time. Start with a digit that has few candidates left on the grid. Fewer candidates mean fewer rows to check.
3. Note the useful rows. For each row, count the cells where the digit is possible. Keep the rows with two or three cells and remember their columns.
4. Combine three rows. If three of these rows together use exactly three columns, you have a Swordfish. If you find nothing, repeat the search with the columns.
5. Remove the candidates. Remove the digit from the other cells of the three crossing lines, then look for hidden singles and naked singles again.

Common mistakes

• Four columns instead of three. If the cells of the three rows use four columns, the pattern is not a Swordfish and the Swordfish rule removes nothing. Count the columns again before you remove a candidate.
• Removing in the wrong direction. A pattern found in rows removes candidates in columns. The three rows keep all their candidates.
• Incomplete pencil marks. One missing candidate can make a row look as if it fits the three columns when it really has a cell outside them.
• Expecting eliminations every time. A Swordfish is only useful when the three columns contain other cells with the digit. In the example, column 5 had none.

When to look for a Swordfish, and what comes next

A Swordfish is much rarer than an X-Wing, because three rows and three columns must match. Many hard puzzles never need one. Look for it only when you are stuck in a hard or expert puzzle and the simpler techniques give nothing more. When a puzzle does need one, the eliminations often give the next digit, as in the example.

After the Swordfish, learn the XY-Wing, which connects three cells that have two candidates each, and simple coloring, which follows one digit through a chain of cells. The finned X-Wing shows what you can still remove when a fish has one extra cell. For an overview, read the full list of sudoku techniques.