Lesson 11 of 14 · Cross-line technique

X-Wing: one digit across two rows and columns

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 X-Wing follows one digit at a time. Suppose a digit has exactly two possible homes in each of two rows, and both rows use the same two columns. Those four squares form the pattern.

Each row needs the digit once. If the first row uses one of the columns, the second row must use the other. Either way, the two columns each get their digit in one of those rows. You can remove that digit from other squares in those columns.

You can turn the argument around: start with two columns whose only homes for the digit are in the same two rows, then remove it elsewhere in those rows. Neither version tells you which of the four corners to fill.

Example 1: Start with two rows

Step 1 of 4

Look along rows 2 and 8 for the 6. Both rows have exactly two possible homes: columns 6 and 7.

In rows 2 and 8, the only places for a 6 are columns 6 and 7. Each row needs one 6, and they cannot both use the same column.

Whichever pair of opposite corners takes the 6s, columns 6 and 7 are both accounted for. Row 3, column 6 sits outside the two starting rows, so its 6 can be removed. Its 3 and 8 remain possible.

Example 2: Start with two columns

Step 1 of 4

Read columns 5 and 7 for the 9. In each column it can go only in row 3 or row 4.

In columns 5 and 7, the only places for a 9 are rows 3 and 4. This time the columns are the starting lines, so the removals run along the rows.

Each of those rows must get a 9 from one of the two starting columns. Remove the 9 at row 3, column 6, leaving 5 and 6. The same rule also clears other 9s in rows 3 and 4 outside columns 5 and 7; this walkthrough focuses on one target.

Example 3: Four corners are not enough

Step 1 of 4

The four marked squares all show a 1, and they form a rectangle. Is that enough?

The 1s at rows 1 and 8, columns 5 and 6 look like the corners of an X-Wing. Row 1 has only those two homes, but row 8 also has a 1 at column 4.

That extra home breaks the row-based argument: row 8 could use column 4 instead, so the corners have not reserved both columns. Do not cross out the 1 at row 4, column 5 on the strength of this rectangle. Keep it as a candidate.

Try it yourself

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

In rows 1 and 5, the 7 can go only in columns 4 and 8. Which candidate can you remove from the ringed square at row 3, column 4?

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

How to spot it faster

  • Pick one digit and look for rows with exactly two possible homes for it. Compare the column numbers of those homes.
  • Try columns as well as rows. Once you find a match, remove candidates along the crossing lines, not along the two lines you started from.
  • Check the whole length of both starting lines. A third possible home breaks this simple X-Wing, even when four corners still look like a rectangle.

Common mistakes

Trusting the rectangle alone

Four matching candidates are not enough. In each of the two starting rows, those must be the digit's only possible homes. Read beyond the corners before crossing anything out.

Removing candidates in the wrong direction

Starting from two rows lets you clear other squares in the two matching columns. Starting from two columns lets you clear other squares in the two matching rows. Leave the four corner candidates in place.

Questions

Further reading

Try it

Try an expert 9×9 and compare the homes of one digit after the simpler moves run out. An expert deal may need a different technique rather than an X-Wing, so use the worked boards and exercise here for targeted practice. Free hints still stop at pairs.

Play solo on an expert 9×9