Advanced course · Lesson 5
Understand X-Wing without memorising a trick
An X-Wing links two rows through the same two columns—or two columns through the same two rows. The four candidates form a rectangle, but the logic comes from where a digit must appear, not from the shape alone.
The exact pattern
Choose one candidate digit—say 7. In row 2, suppose 7 can appear in exactly columns 3 and 8. In row 6, suppose its only positions are also columns 3 and 8. Each row must contain a 7, so the two 7s must occupy opposite corners of that rectangle. Whichever way they pair, columns 3 and 8 will each receive one of them.
See the rectangle
The proof, one step at a time
- Row 2 needs a 7. Its only choices are column 3 or column 8.
- Row 6 needs a 7. Its only choices are the same two columns.
- The rows cannot choose the same column. A column may contain only one 7, so the rows must choose opposite corners.
- Both columns are therefore occupied. Any other candidate 7 in columns 3 or 8 cannot be true.
How to search for an X-Wing
Work one digit at a time. Scan the rows for units containing exactly two candidates for that digit. Note their column positions. If another row has exactly the same pair of columns, test the X-Wing and look for eliminations in those columns.
You can reverse the orientation: find two columns whose candidate appears in exactly the same two rows, then eliminate that candidate from the rest of those rows. Start with digits that already have many givens, because fewer remaining candidates make pairs easier to see.
Why a rectangle is not enough
Many candidates form rectangular corners by coincidence. For a row-based X-Wing, each source row must have exactly two candidates for the chosen digit, and they must occupy the same two columns. Extra candidates in either source row break the proof. Extra candidates in the target columns are allowed—they are precisely what the pattern eliminates.
What not to erase
- Do not erase any of the four corner candidates.
- Do not remove other digits from the target columns.
- Do not eliminate outside the two shared columns in a row-based X-Wing.
- Do not use the pattern if either source row has a third position for the digit.
What usually happens next
An X-Wing rarely fills a corner immediately. Instead, an elimination elsewhere may turn a cell into a naked single or leave a digit in only one position in a box. After making the eliminations, return to the basic loop: scan affected cells, their units and nearby boxes before looking for another advanced pattern.
Practise the search, not just the shape
Open a Hard or Expert practice puzzle with complete notes. Pick one digit and record which rows contain exactly two candidates. Compare their column pairs. If you find a match, state the proof before making any elimination.
Open practice mode