X-Wing
An X-Wing is four cells that form a rectangle: in two rows or columns a digit can only go in the same two places. Then the digit can be removed from the rest of the crossing lines.
Learn the techniqueSwordfish
Swordfish is X-Wing extended to three lines: a digit that in three rows can only go in the same three columns can be removed from the rest of the columns.
See the technique in practice
Work through the examples step by step. Each step explains what you see on the puzzle and why the conclusion holds.
Example:
- We follow digit 3 through three columns, namely column 1, column 7 and column 8. The notes show where 3 can still go.
- Here is the pattern: all the highlighted cells lie in the same three rows, namely row 2, row 8 and row 9.
- Each of the three columns must get digit 3 in one of its highlighted cells. Two cannot use the same row, because then it would get 3 twice. So each takes its own.
- Thus the spot for 3 in the three rows is used up by the pattern. We remove 3 from the notes in row 8, column 3 and row 9, column 3.
How to recognize the pattern
Swordfish is X-Wing extended from two to three lines. You look for one digit that in three different rows can only go in the same three columns. The rows do not need to use all three columns, because two possibilities per row is enough as long as everything lies within the same three columns.
The three placements of the digit must distribute across each row and each column, much like three non-attacking rooks on a chessboard. Each of the three columns thus gets its copy of the digit from one of the base rows, and the digit can be removed from everything else in the three columns.
Step-by-step procedure
- Choose a digit and write down for each row which columns the digit can go in.
- Find three rows where all their possibilities together lie in only three columns.
- Check that each of the three rows has at least two possibilities, and that no possibility falls outside the three columns.
- Remove the digit from all other cells in the three columns.
Common mistakes
- Requiring that all rows use all three columns. Two possibilities in a row is fine as long as they lie within the same three columns.
- Mixing up base and cover. With rows as the base, the removals happen in the columns, never in the rows themselves.
- Looking for the pattern too early. Swordfish only becomes visible when the candidate notes are complete and accurate.
When do you need the technique?
At Hard level you must see multiple units in context: digits that form rectangles across multiple rows, chains of linked cells and boxes that lock each other. The techniques still only remove candidates, but these are exactly the eliminations that open up the puzzle.
Try it on your own puzzle
Enter your puzzle in the Sudoku Solver and it will find the next step and explain the technique behind it.
Open Sudoku Solver