The X-Wing: a rectangle that clears a column

What an X-Wing is

Pick one digit, say 8. Find a row where 8 can go in exactly two cells. Then find a second row where 8 can also go in exactly two cells, and those two cells sit in the same two columns as the first pair. The four cells form the corners of a rectangle. That rectangle is an X-Wing.

When you find it, 8 can be removed from every other cell in those two columns. The same pattern works with rows and columns swapped: two columns with exactly two candidates each, lined up in the same two rows, let you clear the digit from the rest of those rows.

Why it works

Each of the two rows must contain an 8 somewhere, and in each row there are only two places left for it. Take the top row: its 8 is either in the left column or in the right column. If it is in the left column, the bottom row cannot put its 8 in the left column too (a column holds each digit once), so the bottom row’s 8 goes in the right column. If the top row’s 8 is in the right column, the bottom row’s 8 must go in the left column.

Either way, the two 8s occupy both columns: one in the left column and one in the right column, always on the diagonal corners of the rectangle. That means each of the two columns already has its 8 inside the rectangle, so no other cell in those columns can be 8. You do not need to know which diagonal is correct; both lead to the same eliminations.

How to find an X-Wing

You need pencil marks (candidates) for this. Choose one digit and scan every row, noting the rows where that digit has exactly two possible cells. Write down the column numbers of each pair. If two rows share the same pair of columns, you have an X-Wing.

If rows give you nothing, repeat the scan by columns: look for columns where the digit has exactly two places, and compare the row numbers. Many players miss column-based X-Wings because they only scan rows.

Before celebrating, check that the pattern actually removes something. An X-Wing is only useful if at least one other cell in the two cover lines still has the digit as a candidate. Digits that are nearly finished on the board (six or seven already placed) are the best place to start looking.

A worked example

In the example board, look at digit 8. In row 3 it can only go in R3C1 (row 3, column 1), whose candidates are 8 and 9, or in R3C4, whose candidates are 5 and 8. In row 4 it can only go in R4C1 (candidates 7 and 8) or R4C4 (candidates 5 and 8). Both rows have their only two 8s in columns 1 and 4: the four highlighted cells form an X-Wing.

So the 8 of column 1 and the 8 of column 4 are both inside those four cells. Column 4 has no other cell with an 8 candidate, but column 1 does: R5C1 has candidates 6 and 8. The X-Wing removes 8 from R5C1.

With 8 gone, R5C1 has only one candidate left, so it is a naked single: R5C1 is 6, shown in gold. Notice that nothing simpler worked in this position. There were no singles, no locked candidates and no pairs or triples to use; the X-Wing is the step that unlocks the puzzle.

Common mistakes

The rows must have exactly two candidates for the digit. If row 3 had a third cell with an 8, the top row’s 8 might sit outside the rectangle, and the logic falls apart. Three candidates in a line belong to a Swordfish, not an X-Wing.

The columns must line up. A pair in columns 1 and 4 plus a pair in columns 1 and 5 is not an X-Wing. That shape can still be useful, but it is a different pattern called a Skyscraper, with different eliminations.

Eliminate from the right lines. When the base is two rows, you remove the digit from the two columns, never from the rows themselves; the rows are already limited to the rectangle. Mixing this up is the most common way to make a wrong elimination, and it usually shows up much later as a contradiction.

Where the X-Wing fits

The X-Wing is the two-line member of a family called fish. With three rows and three columns it becomes a Swordfish, which follows the same reasoning. It is also closely related to locked candidates: both work by showing that a digit is forced into a small area, which clears it from the rest of a line.

On Quietdoku, the X-Wing is one of the techniques that define Expert puzzles, along with Swordfish and XY-Wing. Every puzzle has exactly one solution and can be solved without guessing, and if you get stuck, a hint shows the next logical step and names the technique, so you can learn to find X-Wings on your own.