Which Candidates Can X-Wing Remove? A Complete Guide to Rows and Columns

X-Wing links possible positions for one number

The Naked Pair in How to Use Sudoku Candidate Notes linked two numbers to two cells in one row or column. X-Wing links the possible positions of one number across two rows or columns. Four cells forming a rectangle are not enough. First, count whether that number has exactly two possible cells in each row or column.

Conditions — two cells in each row, in the same two columns

A row-based X-Wing meets all the following conditions.

1. The same number can go in exactly two cells in each of two rows.
2. The candidates in both rows lie in the same two columns.
3. Other cells in those two columns must still contain the same candidate for there to be anything to remove.

If the left candidate in the first row contains the number, the right candidate in the second row must contain it. If the right candidate in the first row contains it, the reverse holds. Either way, that number's positions in both columns are confined to the two selected rows. So remove the number from other rows in those two columns. Do not remove candidates for other numbers from the four corner cells.

Row-based example — 3 in rows 2 and 7

c1c2c3c4c5c6c7c8c9

r1r2r3r4r5r6r7r8r9

1

7

9

2368

4

6

358

2

1

38

7

9

9

3

1

7

3

9

6

1

2

8

6

9

1

2

7

8

4

3

5

8

2

1

9

6

7

5

37

1

2

37

9

6

1

7

6

4

9

5

8

6

9

357

8

237

1

4

ABCD

• The only candidates for 3 in row 2 are r2c4 · r2c7.
• The only candidates for 3 in row 7 are r7c4 · r7c7.
• Both rows meet columns 4 and 7, so remove 3 from the remaining cells in those two columns.

On this board, remove 3 from r1c7 · r9c4 · r9c7. Their remaining candidates are 2, 6, 8 / 5, 7 / 2, 7, respectively. None of these three cells has a number you can place yet. X-Wing reduces candidates for the next step rather than immediately revealing a cell's answer.

Column-based example — swap rows and columns

For a column-based pattern, each of two columns must have exactly two candidate cells, and those candidates must lie in the same two rows. You can then remove the same number from other columns in those two rows. The example below transposes the previous board, swapping rows and columns.

c1c2c3c4c5c6c7c8c9

r1r2r3r4r5r6r7r8r9

1

4

9

7

6

8

5

7

3

9

2

1

6

6

3

1

7

9

9

358

2

1

37

6

357

2

9

7

1

4

8

1

6

8

2

9

2368

38

1

4

9

37

5

237

7

2

3

6

9

8

1

9

1

8

5

7

6

4

ABCD

• The only candidates for 3 in column 2 are r4c2 · r7c2.
• The only candidates for 3 in column 7 are r4c7 · r7c7.
• Both columns meet rows 4 and 7, so remove 3 from the remaining cells in those two rows.

Therefore, remove 3 from r7c1 · r4c9 · r7c9. Their remaining candidates are 2, 6, 8 / 5, 7 / 2, 7, respectively. Every coordinate has its row and column swapped: for example, r1c7 in the row-based example becomes r7c1 in the column-based example.

When you cannot use it, and common mistakes

• You look at four cells and miss other candidates in the row or column. If a row or column you use as the basis has a third candidate for the number, the standard X-Wing conditions do not hold.
• You think each of the four cells must have only two candidates in total. Only the chosen number needs exactly two possible positions in that row or column. In the example, r2c4 has three candidates: 3, 5 and 8.
• You confuse the starting rows or columns with those where you remove candidates. If you start with two rows, look at other cells in the two columns; if you start with two columns, look at other cells in the two rows.
• You fill the corners. You do not yet know which of the two diagonals contains the number.

Practice puzzle

c1c2c3c4c5c6c7c8c9

r1r2r3r4r5r6r7r8r9

9

2

7

8

5

4

6

1

3

1

9

5

8

4

5

4

8

1

3

6

7

2

9

9

1

5

2

5

3

4

6

8

1

9

7

7

1

5

9

23

234

36

8

1

7

5

9

4

23

8

36

9

8

5

7

1

2

1

7

9

4

5

ABCD

Is this X-Wing row-based or column-based? What can you remove from r6c7? Count all the candidates for 3 in each row or column.

Show answer

It is column-based, and you can remove 3 from r6c7. Column 6 has candidates for 3 only at r6c6 · r7c6, and column 8 only at r6c8 · r7c8. Both columns meet rows 6 and 7, so remove 3 from other cells in those rows. Row 6 also has a candidate for 3 at r6c7, so you must not treat this as a row-based pattern. Removing 3 from r6c7's candidates of 2, 3 and 4 leaves 2 and 4.

Further reading

The prerequisites are How to Use Sudoku Candidate Notes and Stuck on Sudoku? A 7-Step Checklist. Next, read Where Can You Use a Skyscraper When Locked Candidates Stop Helping?.