Skyscraper Sudoku Technique: Advanced Elimination Using Strong Links
Skyscraper is an advanced Sudoku elimination technique based on strong links, which can be considered a variant of X-Wing. Its name comes from the pattern formed by this technique—two vertical "columns" connected by a "beam," resembling two skyscrapers of different heights. The core idea is: when a candidate forms a strong link in each of two rows, and these two strong links are connected through the same column, cells that can be seen by both "dangling" endpoints can eliminate that candidate.
When a candidate appears in only two cells within a row (or column), these two cells form a Strong Link. A strong link means that exactly one of these two cells must contain that digit. If one isn't, the other must be.
Skyscraper Rule
If a candidate appears in only two positions in each of two rows (forming two strong links), and these two strong links have one endpoint in the same column,
Then any cell that can be "seen" by both endpoints not in the same column (dangling endpoints) can eliminate that candidate.
Before reading this article, it's recommended to master the X-Wing technique, as Skyscraper can be seen as an "imperfect" form of X-Wing—when three of the four corners of an X-Wing are aligned, a Skyscraper is formed.
Example Analysis: Row-based Skyscraper
Let's look at a Skyscraper example involving candidate 6 in Row 1 and Row 5.
Current Board Data
Based on the CSV81 format candidate data, we focus on the distribution of candidate 6 in Row 1 and Row 5:
Row 1 cells:
- R1C1: candidates {4, 6}
- R1C2: given digit 7
- R1C3: candidates {1, 4}
- R1C4: filled digit 5
- R1C5: filled digit 8
- R1C6: candidates {1, 6}
- R1C7: given digit 9
- R1C8: given digit 3
- R1C9: filled digit 2
Row 5 cells:
- R5C1: given digit 1
- R5C2: candidates {3, 6}
- R5C3: filled digit 7
- R5C4: filled digit 8
- R5C5: filled digit 4
- R5C6: candidates {2, 6}
- R5C7: candidates {2, 5}
- R5C8: given digit 9
- R5C9: candidates {3, 5}
Analysis Process
- Roof (connection point): R1C6 and R5C6 (in Column 6, connected by dashed line)
- Dangling endpoints (building tops): R1C1 and R5C2 (the tops of the two "buildings")
This forms an asymmetric "Skyscraper" shape: the left building extends from R1C1 to R1C6, the right building extends from R5C2 to R5C6.
- The 6 in Row 1 is either at R1C1 or R1C6
- The 6 in Row 5 is either at R5C2 or R5C6
- Case 1: If R1C6 is 6, then R5C6 cannot be 6 (same column), so R5C2 must be 6
- Case 2: If R1C6 is not 6, then R1C1 must be 6
Conclusion: In either case, at least one of R1C1 or R5C2 is 6.
Which cells can be seen by both R1C1 and R5C2?
- R2C2: candidates {3, 5, 6}
→ In the same column as R5C2 (Column 2)
→ In the same box as R1C1 (Box 1) - R4C1: candidates {4, 6, 9}
→ In the same column as R1C1 (Column 1)
→ In the same box as R5C2 (Box 4)
- R2C2: remove candidate 6 (keeping 3,5)
- R4C1: remove candidate 6 (keeping 4,9)
Skyscraper: Digit 6 forms two strong links in Row 1 (R1C1-R1C6) and Row 5 (R5C2-R5C6), connected through Column 6.
Action: Remove candidate 6 from R2C2 and R4C1.
Forms of Skyscraper
Skyscraper can take multiple forms, depending on the direction and connection method of the strong links:
1. Row-based Skyscraper
This is the situation in the example above:
- Basic structure: Two rows each with a strong link
- Connection method: The two strong links have one common endpoint in the same column
- Dangling endpoints: The two endpoints not in the common column
2. Column-based Skyscraper
Opposite form but same principle:
- Basic structure: Two columns each with a strong link
- Connection method: The two strong links have one common endpoint in the same row
- Dangling endpoints: The two endpoints not in the common row
Imagine two skyscrapers:
• The roofs are connected on the same "street" (common row or column)
• The building tops are the dangling endpoints
• Places that both building tops can see simultaneously are where elimination can occur
How to Find Skyscrapers?
Finding Skyscrapers requires systematic observation:
- A strong link requires the candidate to appear exactly twice in that row (or column)
- The two strong links must be connected through the same column (or same row)
- "See simultaneously" includes: same row, same column, same box—all three situations
- If the two dangling endpoints have no cells they can both see, elimination cannot occur
- Skyscraper is an "imperfect" variant of X-Wing—when three of the four corners of an X-Wing are aligned, a Skyscraper may form
Relationship Between Skyscraper and Other Techniques
Skyscraper vs X-Wing
Both involve strong links in two rows (or columns), but have important differences:
| Comparison | X-Wing | Skyscraper |
|---|---|---|
| Structure | Four corners perfectly aligned, forming a rectangle | Only three points aligned, one endpoint "dangling" |
| Elimination scope | Entire column (or row) of candidates | Only specific cells can be eliminated |
| Frequency | Less common | More common (conditions are looser) |
Skyscraper vs 2-String Kite
Skyscraper is actually a special form of 2-String Kite:
- Two strong links connected through a common point
- Uses the logic of "one of the two ends must be true" for elimination
Technique Summary
Key points for applying the Skyscraper technique:
- Recognition conditions: A candidate appears exactly twice in each of two rows (or columns), and one column (or row) contains one endpoint of each of the two strong links
- Structure formed: Two strong links + one common column (or row) + two dangling endpoints
- Elimination rule: Cells that can be "seen" by both dangling endpoints can eliminate that candidate
- Application scenario: Alternative when X-Wing conditions are not met
- Recognition difficulty: Intermediate to advanced, requires understanding of strong link concept
Skyscraper is more common in practice than X-Wing because its conditions are looser. Recommendations:
- First master the recognition of strong links
- When looking for X-Wing, if the four corners are incomplete, check if a Skyscraper can be formed
- Focus on candidates with fewer occurrences, easier to find strong links
- Use candidate highlighting feature, focus on one digit at a time
Practice Now
Start a hard or expert level Sudoku game, try using the Skyscraper technique! Suggestions:
- Choose hard difficulty, easy puzzles usually don't need advanced techniques
- First mark all candidates, then search for strong links digit by digit
- After finding two strong links, check if they have a common row or column
- After confirming dangling endpoints, look for cells they can both see