Lesson 12 of 14 · Wing technique

XY-Wing: two possibilities, one removal

The guide and all sudoku lessons
The full guide Playing on Puzzled
  1. 1The last empty squareFirst move
  2. 2ScanningCore technique
  3. 3Naked singlesCore technique
  4. 4Pencil marksSolving aid
  5. 5Hidden singlesCore technique
  6. 6Naked pairsPair technique
  7. 7Pointing pairsPair technique
  8. 8Hidden pairsPair technique
  9. 9Naked triplesSubset technique
  10. 10Hidden triplesSubset technique
  11. 11X-WingCross-line technique
  12. 12XY-WingWing technique
  13. 13SwordfishCross-line technique
  14. 14JellyfishCross-line technique

An XY-Wing, also called a Y-Wing, uses three empty squares, each with exactly two candidates. The pivot sees both wings: each shares a row, column or box with it. The wings do not need to see each other.

With 4/5 in the pivot, one wing could be 4/6 and the other 5/6. A 4 in the pivot forces the first wing to 6; a 5 forces the second wing to 6. At least one wing must be 6. Both could be 6, too.

That is enough to remove a 6 from any other empty square that sees both wings. You are not guessing the pivot's value: both possibilities lead to the same removal. A square that sees only one wing is not covered by this argument.

In the diagrams, the pivot has a blue outline and the wings used in each step have a darker blue tint. The green outline marks the target. Every pencil mark is calculated from the displayed board.

Example 1: A row and a box link

Step 1 of 5

The blue-outlined pivot at row 2, column 5 is 4 or 5. The two darker blue squares are its wings; each also has exactly two candidates.

The blue-outlined pivot at row 2, column 5 has just 4 and 5. Its wings are row 1, column 6, with 4 and 6, and row 2, column 7, with 5 and 6. The first sees the pivot through the top-middle box; the second through row 2.

If the pivot is 4, the first wing must be 6. If the pivot is 5, the second wing must be 6. The target at row 1, column 7 sees both wings, along row 1 and column 7, so remove its 6. Keep its 5 and 7. The target does not see the pivot, and does not need to.

Example 2: A column and a box link

Step 1 of 5

The pivot at row 4, column 5 is 1 or 3. Check the two wings: 3/7 at row 1, column 5 and 1/7 at row 5, column 6.

The pivot at row 4, column 5 has 1 and 3. It sees the 3/7 wing at row 1, column 5 along column 5, and the 1/7 wing at row 5, column 6 through the middle box.

A 3 in the pivot forces the first wing to 7; a 1 forces the second wing to 7. Row 1, column 6 sees both wings, so its 7 can go. That leaves a single 3, which you could place next. The diagram keeps it as a pencil mark to show exactly what the XY-Wing removed.

Example 3: Seeing one wing is not enough

Step 1 of 5

The pivot is 1 or 3. Its two wings are 1/5 at row 2, column 5 and 3/5 at row 9, column 4.

This is a valid XY-Wing: the pivot at row 3, column 4 has 1 and 3, with a 1/5 wing at row 2, column 5 and a 3/5 wing at row 9, column 4. But the proposed target at row 2, column 6 sees only the first wing. It also sees the pivot, which does not make up for the missing link.

If the pivot is 3, the far wing at row 9, column 4 must be 5. That wing cannot see the target, so its 5 does not rule out a 5 there. Keep the target's 5: this XY-Wing does not justify removing it.

Try it yourself

A new board, and the same idea. Get it wrong and the board tells you why.

Use the blue-outlined pivot at row 1, column 5 and its two tinted wings. Which candidate can this XY-Wing remove from the green-ringed target at row 3, column 7?

Pick one candidate to cross out in the ringed square, not a digit to place.

How to spot it faster

  • Start with a square that has exactly two candidates. Look among the squares it sees for two-candidate wings.
  • The wings must use different digits from the pivot and share a third digit that is not in the pivot. Lists such as 4/5, 4/6 and 5/6 work; 4/5, 4/6 and 4/7 do not.
  • Check a potential target against each wing separately. Sharing a box counts just as much as sharing a row or column.

Common mistakes

Using a pivot with a third candidate

The two-case argument no longer covers every possibility. If the pivot could take a third digit, neither wing might be forced to the shared value. A basic XY-Wing needs exactly two candidates in all three squares.

Clearing squares that see the pivot

The pivot connects the wings; it is not what rules out the target's digit. Only a target that sees both wings gets the removal. The third example shows why seeing the pivot and just one wing is not enough.

Questions

Further reading

Try it

Try an expert 9×9 and compare two-candidate squares after simpler moves run out. A random expert deal may need a different technique, so the worked boards and exercise here provide the targeted XY-Wing practice. Free hints still stop at pairs.

Play solo on an expert 9×9