How Does XY-Wing Differ from X-Wing, and Where Can You Remove Candidates?

Similar names, different starting points

X-Wing looks for four possible positions for one number. XY-Wing looks for three cells with only two candidates each. Having Wing in the name does not mean you need to find a rectangle.

The three cells in an XY-Wing are one pivot and two pincers. Whichever number fills the pivot, at least one pincer contains their shared number, so you can remove that number from cells that see both pincers.

Conditions — three cells with two candidates each

Call the three different numbers X, Y and Z.

1. The pivot's only two candidates are X and Y.
2. One pincer's only two candidates are X and Z; the other's are Y and Z.
3. The pivot sees both pincers. Cells see each other when they share a row, column or box.
4. Another cell that sees both pincers must still have Z as a candidate for there to be anything to remove.

If the pivot is X, the first pincer is Z. If the pivot is Y, the second pincer is Z. Either way, at least one pincer is Z, so remove Z only from cells that see both pincers. Seeing the pivot is not a condition for removal.

Example — follow 4 and 5 in the pivot

c1c2c3c4c5c6c7c8c9

r1r2r3r4r5r6r7r8r9

9

8

3

1

6

2

5

7

4

4

3

5

6

8

9

46

5

46

7

8

9

2

3

1

1

3

9

2

7

4

8

6

5

6

8

9

3

4

1

7

8

4

7

6

5

1

9

2

3

45

2

8

9

1

7

3

6

3

5

2

6

7

8

567

56

3

4

8

1

2

PAB

• The pivot r7c1(P) has candidates 4 and 5.
• The pincer r3c1(A) has candidates 4 and 6. It shares column 1 with the pivot.
• The pincer r9c3(B) has candidates 5 and 6. It shares box 7 with the pivot.

If r7c1 is 4, r3c1 is 6; if r7c1 is 5, r9c3 is 6. r3c3 sees A through row 3 and B through column 3. r9c1 sees A through column 1 and B through row 9. So remove 6 from both cells. This leaves only 4 in r3c3 and 5 and 7 in r9c1. The diagram's lines show the relationships between the pivot and pincers; they do not mean you should remove candidates from every cell a line passes through.

Comparing Wings — what should you count?

Technique	Conditions to look for	Where to remove candidates
X-Wing	One number has two possible cells in each of two rows, in the same columns. Rows and columns can be swapped.	Other cells in the two intersecting columns. For a column-based pattern, the two intersecting rows.
Skyscraper	One number has two possible cells in each of two rows or columns. Only the base is aligned; the tops are offset.	Cells that see both tops.
XY-Wing	Three cells with two candidates each: pivot X·Y and pincers X·Z, Y·Z.	Z in cells that see both pincers.

Cells in X-Wing and Skyscraper can have several candidates for other numbers. In XY-Wing, each of the three cells must have exactly two candidates in total. Checking this difference first helps you avoid mixing up the conditions because of the names.

When you cannot use it, and common mistakes

• The pivot or a pincer has three candidates. The argument based on just two cases no longer holds.
• The pivot cannot see one of the pincers. The pivot's number cannot rule out that pincer's candidate.
• The pincers' shared number is also a candidate in the pivot. Check again that X, Y and Z are three different numbers.
• You remove a candidate from a cell that sees only one pincer. You do not yet know which pincer is Z.
• You look for exactly the shape in the diagram. The combination of boxes, rows and columns can change. Two candidates and the seeing relationships matter more than the shape.

Practice puzzle

c1c2c3c4c5c6c7c8c9

r1r2r3r4r5r6r7r8r9

5

1

3

4

9

2

9

5

3

1

7

8

4

1

2

8

9

7

6

5

3

2

4

1

3

9

7

5

36

9

7

4

8

5

2

16

5

1

7

2

6

9

4

8

6

4

9

5

1

3

2

7

13

9

7

8

5

4

16

5

3

8

9

PAB

What can you remove from r8c9? Follow each of the pivot's two cases.

Show answer

Remove 1. The pivot r5c1 has 3 and 6; the pincers r5c9 and r8c1 have 1, 6 and 1, 3, respectively. If the pivot is 3, r8c1 is 1; if the pivot is 6, r5c9 is 1. r8c9 sees r5c9 through column 9 and r8c1 through row 8, so it cannot be 1. Removing 1 from candidates 1 and 6 leaves only 6.

Further reading

The prerequisites are How to Use Sudoku Candidate Notes and Stuck on Sudoku? A 7-Step Checklist. Next, read Coloring: What Can Same-Color Conflicts and Two-Color Traps Remove?.