Sudoku X-Wing begins with a position inventory
A Sudoku X-Wing is a way to remove candidates. Choose one digit, find two rows that each allow it in exactly two cells, and check whether those cells lie in the same two columns. When those conditions hold, you can remove that digit from the other cells in those columns.
The pattern does not immediately choose a corner’s value. Its useful result is a smaller set of possibilities outside the rectangle. After the deletion, return to singles and ordinary unit scans. A modest removal can unlock several later placements.
Before attempting this technique, make sure you can distinguish complete candidates from partial notes. You need all positions of the chosen digit in the two base rows. A rectangle made from four notes you happened to write is insufficient if either row has another possible position.
Read a verified example for digit 6
In this original Hard position, row 2 has candidate 6 at r2c3 and r2c9 only. Row 8 has candidate 6 at r8c3 and r8c9 only. The two rows point to columns 3 and 9. Those four cells are the corners of the rectangle.
Board overview
View the whole grid first. Zoom in to read candidates and their coordinates.
Purple cells give the evidence; a red cross marks a candidate this step can remove.
Other digits in the corner cells do not spoil the Sudoku X-Wing. r2c3 and r2c9 each have candidates 6 and 8; r8c3 and r8c9 each have candidates 3 and 6. We are tracking only digit 6. We do not require the four cells to have identical full candidate lists.
Prove both possible arrangements
Row 2 must contain a 6. There are two possible cases:
- If r2c3 is 6, row 8 cannot put 6 in column 3. Its only remaining position is r8c9.
- If r2c9 is 6, row 8 cannot put 6 in column 9. Its only remaining position is r8c3.
In either case, the two rows supply one 6 to column 3 and one 6 to column 9. Therefore no other cell in either column can contain 6. We have proved an elimination common to both cases without deciding which case is true.
This is the reason the two-case proof is logical rather than a guess. You are not selecting one arrangement and testing it through the puzzle. You are showing that every allowed arrangement of the restricted rows excludes the same outside candidates.
Apply the conclusion only where it reaches
The verified step removes candidate 6 from r3c3, r6c3, r3c9 and r4c9. These are the current outside cells in columns 3 and 9 that still have a 6 candidate. Nothing changes in the four corners. You also leave candidate 6 alone in unrelated columns.
Board overview
View the whole grid first. Zoom in to read candidates and their coordinates.
Some correct rectangles have no outside candidate left to remove. They express a valid restriction, but they do not advance that particular position. A useful search looks for both a valid pattern and at least one target candidate.
Reverse rows and columns
You can turn the same proof through ninety degrees. Find two columns that each allow the digit in exactly two rows. The columns are the bases; the rows are the covers. Then remove that digit from other cells in those cover rows.
The following teaching position is the transpose of our original example. It retains the proof and unique solution while changing its orientation. Columns 2 and 8 each restrict 6 to rows 3 and 9. The valid targets are r3c3, r3c6, r9c3 and r9c4.
Board overview
View the whole grid first. Zoom in to read candidates and their coordinates.
Purple cells give the evidence; a red cross marks a candidate this step can remove.
Board overview
View the whole grid first. Zoom in to read candidates and their coordinates.
Call the rows or columns you first restrict the bases. Call the crossing pair where you remove candidates the covers. Naming these roles helps prevent a common directional error: a row-based Sudoku X-Wing eliminates in its columns, not across unrelated parts of its rows.
Why a third base position breaks the proof
In the counterexample, row 2 now has another possible 6 at r2c1. Row 8 still has its two positions at columns 3 and 9. You can still draw the old rectangle through columns 3 and 9, but row 2 is no longer confined to its corners.
Board overview
View the whole grid first. Zoom in to read candidates and their coordinates.
Dashed borders mark changed givens; inspect the purple cells to see why the original rule no longer applies.
If row 2 used r2c1, row 8 would supply only one of the two cover columns with a 6. The other cover column could need its 6 outside the rectangle. That possibility is enough to invalidate the old Sudoku X-Wing elimination.
A third candidate in a cover column, however, is often the target you want to remove. Do not confuse a base row’s extra position, which breaks the premise, with a cover column’s outside position, which the premise excludes. Always label which units you are using as bases before counting positions.
Search efficiently without staring at rectangles
Choose one digit and scan rows for those with exactly two positions. Note their column pairs. When two rows have the same pair, check for deletions in the two cover columns. Repeat by columns if the row scan produces nothing.
You do not have to write pairs for every digit at once. One digit’s position inventory is a manageable task. Prioritize digits that already appear several times, or units where recent placements reduced their positions. If a new single appears while you are scanning, apply it and refresh the candidates before continuing.
Our Hard pool includes verified puzzles whose published logical traces use advanced candidate techniques. A particular Hard puzzle need not contain this rectangle pattern: some require hidden pairs or triples instead. Difficulty describes the tested logical route, not a promise that every board displays one named pattern.
Exercise 1: find a valid deletion
Use the original row-based example. Select an outside cell and candidate that the rectangle rules out. Explain why both possible arrangements of the corner 6s exclude your target.
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Answer: remove 6 at any of r3c3, r6c3, r3c9 or r4c9. Rows 2 and 8 must place their 6s in opposite cover columns, so both columns already receive 6 within those rows. The step leaves the corner order unresolved.
Exercise 2: prove the next rectangle independently
This position uses digit 4. Inspect rows 4 and 8, identify their common cover columns and choose one justified outside deletion. Do not reuse the coordinates from the previous example.
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Answer: rows 4 and 8 restrict 4 to columns 4 and 8. Remove candidate 4 from r6c4, r6c8 or r9c8. The different corner lists still permit the pattern because each base row has exactly those two positions for digit 4.
For an additional test, return to the counterexample and explain why its old deletion is unsupported. A correct answer names the third base position, not merely the fact that its diagram looks busier. Recognizing when a Sudoku X-Wing fails is as useful as recognizing when it succeeds.
Resume with the smallest certain move
After a candidate elimination, scan the changed cells for singles. If nothing fills immediately, keep the valid removal and inspect the affected units again. The engine preserves logical eliminations while explaining subsequent hint steps; manual notes remain a separate aid.
Try the stuck-puzzle routine when you are unsure which technique to inspect next. A Sudoku X-Wing is one tool in that routine. It earns its place when its exact premise gives you a deletion you can explain.