Solving Tips
Sudoku X-Wing Technique: Advanced Cross-Line Elimination Method
X-Wing is one of the most classic methods among advanced Sudoku techniques and an essential skill for solving difficult and expert-level Sudoku puzzles. Its name comes from the X-wing fighter in Star Wars, as the pattern formed by this technique visually resembles an X shape. The core concept is: when a candidate appears only in the same two column positions across two rows, that candidate can be eliminated from other cells in those two columns.
Core Principle:
If a number appears only in columns X and Y in row A, and also appears only in columns X and Y in row B, then this number must occupy one position in column X and one position in column Y across rows A and B. Therefore, this number cannot appear in other cells in columns X and Y (those not in rows A and B).
If a number appears only in columns X and Y in row A, and also appears only in columns X and Y in row B, then this number must occupy one position in column X and one position in column Y across rows A and B. Therefore, this number cannot appear in other cells in columns X and Y (those not in rows A and B).
Before reading this article, it's recommended to master Sudoku row-column-box naming conventions and intermediate techniques, which will help you understand the analysis examples below.
Example Analysis: Row-based X-Wing
Let's look at a classic X-Wing example involving candidate 8 in row 3 and row 7.
Figure: Candidate 8 in rows 3 and 7 forms an X-Wing pattern
Analysis Process
1
Observe Row 3: In row 3, candidate 8 appears only in two positions: A3 and H3.
2
Observe Row 7: In row 7, candidate 8 also appears only in two positions: A7 and H7.
3
Identify X-Wing Pattern: Candidate 8 appears only in columns A and H in both rows 3 and 7, forming a rectangle with four corners: A3, H3, A7, H7. This is the X-Wing pattern.
X-Wing Rectangle Pattern
Four 8s form rectangle corners, X-shaped lines show elimination logic, red arrows indicate elimination direction
4
Understand the Logic: Since 8 in row 3 can only be placed in A3 or H3, and 8 in row 7 can only be placed in A7 or H7, there are two scenarios:
- Scenario 1: If A3 is 8, then row 7 must have 8 at H7
- Scenario 2: If H3 is 8, then row 7 must have 8 at A7
In either case, the 8s in columns A and H are occupied by rows 3 and 7.
5
Execute Elimination: Therefore, cells in column A except rows 3 and 7 cannot contain 8, and cells in column H except rows 3 and 7 cannot contain 8 either. Specifically:
- Column A: Remove candidate 8 from A1, A2, A4, A5, A6, A8, A9
- Column H: Remove candidate 8 from H1, H2, H4, H5, H6, H8, H9
Conclusion:
Rows 3 and 7 form an X-Wing pattern in columns A and H, therefore candidate 8 must be removed from other positions (rows 1, 2, 4, 5, 6, 8, 9) in columns A and H.
Rows 3 and 7 form an X-Wing pattern in columns A and H, therefore candidate 8 must be removed from other positions (rows 1, 2, 4, 5, 6, 8, 9) in columns A and H.
Two Forms of X-Wing
X-Wing can exist in two symmetric forms:
1. Row-based X-Wing
This is the case from the example above:
- Observation Target: Two rows
- Pattern Characteristic: A candidate appears only in the same two columns across these two rows
- Elimination Target: Remove that candidate from other rows in those two columns
2. Column-based X-Wing
Opposite form but same principle:
- Observation Target: Two columns
- Pattern Characteristic: A candidate appears only in the same two rows across these two columns
- Elimination Target: Remove that candidate from other columns in those two rows
Memory Tip:
Row-based X-Wing eliminates columns, Column-based X-Wing eliminates rows.
If the X-Wing is found in two rows, eliminate from columns; if found in two columns, eliminate from rows. This is because once the candidate's position is fixed in rows (or columns), the corresponding columns (or rows) are occupied.
Row-based X-Wing eliminates columns, Column-based X-Wing eliminates rows.
If the X-Wing is found in two rows, eliminate from columns; if found in two columns, eliminate from rows. This is because once the candidate's position is fixed in rows (or columns), the corresponding columns (or rows) are occupied.
How to Find X-Wing?
Finding X-Wing requires systematic observation:
1
Select a Candidate: Focus on one candidate (one of 1-9), analyze one at a time.
2
Find Rows (or Columns) with Only Two Candidate Positions: Identify rows (or columns) where the candidate appears in exactly two cells.
3
Look for a Pair: Check if there's another row (or column) where the candidate also appears in exactly two cells, and in the same two columns (or rows).
4
Confirm X-Wing Pattern: If you find such two rows (or two columns), an X-Wing pattern is formed, creating a rectangle with four corners.
5
Execute Elimination: Remove that candidate from other cells in the corresponding columns (or rows).
Important Notes:
- X-Wing requires exactly two rows (or two columns), with the candidate appearing in exactly two positions in each row (or column)
- The column (or row) positions of the candidate in these two rows (or columns) must be completely identical
- If a candidate appears in 3 or more positions in a row, it cannot form an X-Wing
- X-Wing is a cross-row-column technique and doesn't involve box concepts
- Finding X-Wing is time-consuming, recommended to try after exhausting intermediate techniques
X-Wing's Relationship with Other Techniques
X-Wing vs Box-Line Reduction
Both involve row-column relationships but at different levels:
- Box-Line Reduction: Observes within a single unit (row/column/box), utilizing the intersection of rows/columns and boxes
- X-Wing: Observes across two units, utilizing the symmetric relationship between two rows (or two columns)
Extensions of X-Wing
X-Wing can be extended to more complex forms:
- Swordfish: Extension to three rows and three columns
- Jellyfish: Extension to four rows and four columns
These techniques follow the same principle as X-Wing, but involve more rows and columns, making them harder to identify.
Technique Summary
Key points for applying X-Wing technique:
- Observation Dimension: Cross-row-column observation, looking for rectangular symmetric patterns
- Identification Criteria: A candidate appears only in the same two columns (or rows) across two rows (or columns)
- Pattern Formation: Four candidate positions form the four corners of a rectangle
- Elimination Rule: Row-based X-Wing eliminates columns, Column-based X-Wing eliminates rows
- Application Scenario: Advanced solving method when intermediate techniques cannot make progress
- Identification Difficulty: Requires systematic analysis of each candidate's distribution, quite time-consuming
Practical Advice:
X-Wing is not common in practice, but often provides the breakthrough in difficult puzzles. Recommendations:
X-Wing is not common in practice, but often provides the breakthrough in difficult puzzles. Recommendations:
- First exhaust all intermediate techniques (naked pairs, triples, hidden pairs, etc.)
- Choose candidates with fewer occurrences for analysis (e.g., numbers with only 5-6 candidate positions)
- Use candidate marking features to more easily see distribution patterns
- Use notes or scratch paper to record each number's distribution across rows and columns
Practice Now
Practice Suggestions:
Start a difficult or expert-level Sudoku game and try using the X-Wing technique! Suggestions:
Start a difficult or expert-level Sudoku game and try using the X-Wing technique! Suggestions:
- Choose difficult level, as easy puzzles usually don't require X-Wing
- Patiently analyze each candidate's row and column distribution
- Start by looking for candidates with the fewest occurrences
- Mark the X-Wing immediately after finding it to avoid forgetting