Solving Tips
Grouped Skyscraper Sudoku Technique: Group Extension of Strong Links
Grouped Skyscraper is an extended form of the regular Skyscraper technique. In a regular Skyscraper, each endpoint of a strong link is a single cell; in a Grouped Skyscraper, the endpoints of strong links can be "groups" consisting of multiple cells within the same box. This extension allows the Skyscraper technique to be applied to more scenarios.
Core Principle:
When a candidate appears in multiple cells in a row or column, but several of these cells are located within the same box, these cells can be treated as a "group". After grouping, if the row/column has only two "positions" (single cells or groups) left, this forms a grouped strong link, which can be combined with another strong link to form a Grouped Skyscraper.
When a candidate appears in multiple cells in a row or column, but several of these cells are located within the same box, these cells can be treated as a "group". After grouping, if the row/column has only two "positions" (single cells or groups) left, this forms a grouped strong link, which can be combined with another strong link to form a Grouped Skyscraper.
Grouped Skyscraper Rules
If two strong links (at least one being a grouped strong link) are connected through the same row, column, or box,
Then cells that can be "seen" simultaneously by both dangling endpoints can have that candidate eliminated.
Before reading this article, it is recommended to master the Skyscraper technique and understand the basic concept of strong links.
Grouped Skyscraper principle: A group (green dashed box) serves as one endpoint of a strong link, combining with single cell endpoints to form eliminations
Example Analysis: Grouped Strong Link in a Row
Let's look at an example of a Grouped Skyscraper involving candidate 5.
Figure: Candidate 5 forms a Grouped Skyscraper pattern in Row 2 and Row 8
Current Board Data
Based on CSV81 format candidate data, we focus on the distribution of candidate 5:
Row 2 cells:
- R2C1: Given digit 7
- R2C2: Filled digit 8
- R2C3: Candidates {1, 3, 5}
- R2C4: Candidates {1, 3, 6}
- R2C5: Candidates {1, 6, 9}
- R2C6: Candidates {3, 9}
- R2C7: Given digit 4
- R2C8: Given digit 2
- R2C9: Candidates {1, 5, 6, 9}
Row 8 cells:
- R8C1: Filled digit 6
- R8C2: Given digit 4
- R8C3: Candidates {1, 5, 7}
- R8C4: Given digit 9
- R8C5: Filled digit 3
- R8C6: Given digit 8
- R8C7: Candidates {1, 5, 7}
- R8C8: Candidates {1, 5}
- R8C9: Given digit 2
Target elimination cells (Column 9 and Box 9):
- R7C9: Candidates {1, 3, 4, 5, 6}
- R9C9: Candidates {1, 3, 4, 5, 7}
Analysis Process
1
Observe the strong link in Row 2: In Row 2, candidate 5 only appears in two positions: R2C3 (candidates 1,3,5) and R2C9 (candidates 1,5,6,9). These two cells form a regular strong link.
2
Observe the grouped strong link in Row 8: In Row 8, candidate 5 appears in three positions: R8C3 (candidates 1,5,7), R8C7 (candidates 1,5,7), and R8C8 (candidates 1,5).
- R8C7 and R8C8 are both within Box 9
- We can treat R8C7 and R8C8 as a group [R8C7,R8C8]
- Thus, candidate 5 in Row 8 only has two "positions": R8C3 and [R8C7,R8C8]
Therefore, R8C3 and the group [R8C7,R8C8] form a grouped strong link.
3
Discover the connection point: Note that R2C3 and R8C3 are both in Column 3. This means the two strong links are connected together through a weak link in Column 3.
The weak link indicates: if R2C3=5, then R8C3≠5 (same column exclusion), and vice versa.
4
Identify the Grouped Skyscraper pattern: Now we have:
- Strong link 1: R2C3 — R2C9 (within Row 2)
- Strong link 2: R8C3 — [R8C7,R8C8] (within Row 8, grouped strong link)
- Weak link: R2C3 and R8C3 (same Column 3)
- Dangling endpoints: R2C9 and [R8C7,R8C8]
5
Understand the reasoning logic:
- The 5 in Row 2 is either in R2C3 or R2C9
- If R2C3=5, then R8C3≠5 in the same column, so the 5 in Row 8 must be in [R8C7,R8C8]
- If R2C3≠5, then R2C9=5
Conclusion: In either case, at least one of R2C9 or [R8C7,R8C8] contains 5.
6
Execute elimination: Since at least one of R2C9 or [R8C7,R8C8] is 5, any cell that can be "seen" simultaneously by R2C9 and the group [R8C7,R8C8] cannot be 5:
- R7C9: In Column 9 (seen by R2C9) and in Box 9 (seen by [R8C7,R8C8]) — Remove candidate 5
- R9C9: In Column 9 (seen by R2C9) and in Box 9 (seen by [R8C7,R8C8]) — Remove candidate 5
Conclusion:
Grouped Skyscraper: Candidate 5 forms a regular strong link in Row 2 (R2C3-R2C9) and a grouped strong link in Row 8 (R8C3-[R8C7,R8C8]), connected via a weak link in Column 3.
Action: Remove candidate 5 from R7C9 and R9C9.
Grouped Skyscraper: Candidate 5 forms a regular strong link in Row 2 (R2C3-R2C9) and a grouped strong link in Row 8 (R8C3-[R8C7,R8C8]), connected via a weak link in Column 3.
Action: Remove candidate 5 from R7C9 and R9C9.
Detailed Explanation of the Group Concept
Understanding "groups" is key to mastering Grouped Skyscraper:
When Can Groups Be Formed?
- Same box requirement: Cells in the group must be in the same box
- Same row or column: Cells in the group must be in the same row or same column (so they can participate together in the row/column strong link)
- Same candidate: Each cell in the group must contain the candidate
Purpose of Groups
Key Understanding:
The core idea of groups is: although there are multiple cells in a group, they are treated as a single unit in a strong link.
• If the candidate is in the group, it must be in one of the cells within the group, but we don't know which specific one
• A group can form a strong link with a single cell or another group
• A group can form weak links with other cells in the same row/column/box
The core idea of groups is: although there are multiple cells in a group, they are treated as a single unit in a strong link.
• If the candidate is in the group, it must be in one of the cells within the group, but we don't know which specific one
• A group can form a strong link with a single cell or another group
• A group can form weak links with other cells in the same row/column/box
The "Vision" of a Group
When a group serves as a dangling endpoint, the range it can "see" is all cells in the box where the group is located:
- Group [R8C7,R8C8] is in Box 9, so it can "see" all cells in Box 9
- This includes R7C7, R7C8, R7C9, R8C7, R8C8, R8C9, R9C7, R9C8, R9C9
- Single cell endpoint R2C9 can see all cells in Row 2, Column 9, and Box 3
- The intersection of the two is the elimination target
How to Find Grouped Skyscraper?
Steps to find a Grouped Skyscraper:
1
Select a candidate: Focus on one candidate and analyze it one at a time.
2
Look for strong links and grouped strong links: In rows or columns, look for the following situations:
- Candidate appears in only 2 positions → Regular strong link
- Candidate appears in 3 positions, but 2 of them are in the same box → Can form a grouped strong link
3
Look for connection points: Check if there is another strong link (regular or grouped) where one end is in the same row, column, or box as one end of the first strong link (forming a weak link).
4
Confirm dangling endpoints: The two endpoints not at the weak link connection are the "dangling endpoints".
5
Find elimination targets: Find cells that can be "seen" simultaneously by both dangling endpoints; these cells can have the candidate eliminated.
Important Notes:
- Cells in a group must be in the same box, otherwise they cannot form a group
- After forming a group, the requirement for a strong link is that the row/column has only two "positions"
- Weak links can be connected through same row, same column, or same box
- When calculating elimination range, the "vision" of a group is its box
- Grouped Skyscraper is harder to spot than regular Skyscraper and requires more practice
Comparison Between Grouped Skyscraper and Regular Skyscraper
| Feature | Regular Skyscraper | Grouped Skyscraper |
|---|---|---|
| Strong link endpoints | All single cells | At least one is a group (multiple cells in same box) |
| Row/column requirement | Candidate appears in exactly 2 cells | Candidate can appear in 3+ cells (counts as 2 positions after grouping) |
| Dangling endpoint vision | The cell's row, column, box | The box where the group is located |
| Difficulty to identify | High | Higher |
| Application scope | Strict candidate distribution | More relaxed candidate distribution |
Technique Summary
Key points for applying the Grouped Skyscraper technique:
- Identification conditions: Two strong links (at least one is a grouped strong link), connected by a weak link through same row/column/box
- Grouping rules: Multiple cells in the same box and same row/column can be treated as one "position"
- Structure formation: Two strong links + one weak link + two dangling endpoints
- Elimination rule: Cells that can be "seen" simultaneously by both dangling endpoints can have the candidate eliminated
- Group vision: The vision of a group endpoint is its entire box
Practical Advice:
Grouped Skyscraper is a difficult technique, it is recommended to:
Grouped Skyscraper is a difficult technique, it is recommended to:
- First master the regular Skyscraper technique
- When looking for strong links, if a candidate appears 3 times in a row/column, check if it can be grouped
- Note that cells in a group must simultaneously satisfy same box and same row/column
- Use candidate highlighting features to more easily see distribution patterns
Practice Now
Practice Suggestions:
Start an expert-level Sudoku game and try using the Grouped Skyscraper technique! Suggestions:
Start an expert-level Sudoku game and try using the Grouped Skyscraper technique! Suggestions:
- Choose expert difficulty; Grouped Skyscraper mainly appears in high-difficulty puzzles
- First mark all candidates, find all strong links
- For candidates appearing 3 times in a row/column, check if they can be grouped
- After finding a grouped strong link, look for another strong link connected through same row/column/box