X-Wing

Naked and hidden singles carry you a long way through a Sudoku grid, and pairs and pointing sets clear up most of what is left. Every so often, though, a board reaches a point where no cell has one candidate, no unit hides a single, and nothing looks like it can move. That is usually where an X-Wing is waiting.

An X-Wing is not about a cell at all. It tracks one digit across the whole grid. If that digit is restricted to exactly two cells in each of two different rows, and those four cells all sit in the same two columns, the digit must occupy opposite corners of the rectangle they form. Once you know that, you can strike the digit from every other cell in those two columns, because whichever pair of corners it ends up in, the columns are already spoken for.

The name comes from the shape: four candidate cells at the corners of a rectangle, crossing like the wings of a letter X when you draw lines between them. It is the first pattern-based technique most solvers learn, and the one that opens the door to swordfish, jellyfish and the rest of the fish family.

Finding an X-Wing on a real board

Pick a digit and scan the grid row by row, marking which rows have that digit as a candidate in exactly two cells. Most rows will have three, four or none; you are hunting for the rare rows with exactly two. Once you have two such rows, check whether their candidate cells land in the same pair of columns. If they do, you have found the pattern; if the columns differ even by one, it is not an X-Wing and you move on.

The column-based version works the same way in the other direction: look for two columns where a digit has only two candidate rows each, and check whether those rows match. Some solvers only ever check rows out of habit and miss half the X-Wings on a board, so get in the habit of checking both.

Pencil marks make this dramatically easier to see. Turn on Notes and look for a digit that appears sparsely and in a regular pattern across two lines. Scanning without notes is possible on a printed puzzle but slow, and most players use notes for anything past the intermediate techniques.

Why the elimination actually holds

The logic rests on nothing more than the rule that a digit appears once per row. Say digit 7 has exactly two candidate cells in row 2, and they sit in columns 4 and 8. Row 2 must place its 7 in one of those two cells, no other option exists. Now say row 6 has the same restriction: 7 can only go in columns 4 and 8 there too.

Consider column 4. Somewhere in this puzzle, one of rows 2 or 6 has to hold its 7 in column 4, because between the two rows the digit needs a home in both column 4 and column 8, and each row can only supply one of them. Trace through the four possible pairings and every single one puts a 7 in column 4, coming from either row 2 or row 6. The same holds for column 8. So column 4 already has its 7 accounted for by one of these two rows, which means no other cell in column 4 can also be a 7. The same argument clears column 8. Nothing about the rest of the grid, or which of the two rows actually gets which column, changes that conclusion. That is the whole proof, and it is why the technique works even before you know the final answer.

Row-based, column-based, and telling them apart

An X-Wing built from two rows eliminates candidates from columns. One built from two columns eliminates candidates from rows. Beginners sometimes eliminate in the wrong direction, clearing cells inside the pattern rows instead of the cells outside it in the linking columns. The rule to remember: the two lines you found the pattern in are safe from elimination; the two lines the pattern points at are where digits get removed, and only in the rows or columns you did not use to build the pattern.

It also helps to remember that an X-Wing does not usually finish a puzzle on its own. It trims candidates, which then often creates a fresh naked single or hidden single a few cells over. Treat it as a tool for unsticking the board rather than a way to place a digit directly.

Common mistakes

• Counting a row with three or more candidates. The pattern needs exactly two candidate cells in each row (or column). A row with three candidates for the digit cannot form one half of an X-Wing, no matter how the third cell is placed.
• Matching rows whose columns almost line up. Two rows with candidates in columns 3 and 7, and columns 3 and 8, do not form an X-Wing. The column pair has to match exactly.
• Eliminating from the wrong lines. Cross out the digit from the linking columns (or rows), not from the two rows (or columns) that defined the pattern. Those four corner cells keep their candidate; everything else in the linking lines loses it.
• Forgetting the box constraint does not apply here. X-Wing only uses row and column logic. Do not discard a pattern because the four cells fall in odd or mismatched boxes; that has no bearing on whether the elimination is valid.

Where X-Wing fits among the harder techniques

You will meet X-Wings almost exclusively on Hard and Expert puzzles, since Easy and Medium boards on this site are built to fall to singles and pairs alone. Expert puzzles here run 23 to 27 givens, generated fresh each time with a solver check that guarantees exactly one solution, so any X-Wing you find is real and worth trusting. If you spot the setup but the elimination does not seem to change anything useful, look for a XY-Wing nearby instead, or step up to swordfish, which is the same idea stretched across three lines rather than two.