Solving Tips
Sudoku Grouped Skyscraper Technique: Grouped Strong Link Extension
Grouped Skyscraper is an extension of the basic 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 in more scenarios.
What is a Grouped Strong Link?
A Grouped Strong Link occurs when a candidate appears in only two positions within a unit (row, column, or box), where one or both "positions" consist of multiple adjacent cells within the same box. These cells can be treated as a single unit (group) forming a strong link with the other end.
Example: If candidate 7 appears in only two boxes within a row, and one box contains 2 cells with 7, these 2 cells can form a "group" with a strong link to the cell in the other box.
A Grouped Strong Link occurs when a candidate appears in only two positions within a unit (row, column, or box), where one or both "positions" consist of multiple adjacent cells within the same box. These cells can be treated as a single unit (group) forming a strong link with the other end.
Example: If candidate 7 appears in only two boxes within a row, and one box contains 2 cells with 7, these 2 cells can form a "group" with a strong link to the cell in the other box.
Before reading this article, it's recommended to master the Skyscraper technique and understand the basic concept of strong links.
Example Analysis 1: Grouped Strong Link in a Column
Let's look at the first Grouped Skyscraper example involving candidate 7.
Figure: Candidate 7 forms a Grouped Skyscraper pattern in columns E and F
Analysis Process
1
Observe the strong link in column F: In column F, candidate 7 appears in only two positions: F1 and F9. These two cells form a regular strong link (green vertical line).
2
Observe the grouped strong link in column E: In column E, candidate 7 appears in three positions: E3, E8, and E9. Notice that:
- E8 and E9 are both in box 8
- We can treat E8 and E9 as a group [E8,E9]
- This way, candidate 7 in column E is only in two "positions": E3 and [E8,E9]
Therefore, E3 and group [E8,E9] form a grouped strong link.
3
Discover the common box connection: Notice that group [E8,E9] and F9 are both in box 9. This means the two strong links are connected by a weak link through box 9 (orange dashed line).
The weak link means: if [E8,E9] contains 7, then F9 cannot be 7 (same-box exclusion).
4
Identify the Grouped Skyscraper pattern: Now we have:
- Strong link 1: F1 — F9 (within column F)
- Strong link 2: E3 — [E8,E9] (within column E, grouped strong link)
- Weak link: F9 and [E8,E9] (same box)
- Dangling endpoints: F1 and E3
5
Understand the reasoning logic:
- The 7 in column F must be either in F1 or F9
- If F9 is 7, then [E8,E9] in the same box cannot be 7, so E3 must be 7
- If F9 is not 7, then F1 must be 7
Conclusion: In either case, at least one of F1 or E3 is 7.
6
Execute elimination: Since at least one of F1 or E3 is 7, cells that can be "seen" by both F1 and E3 cannot be 7:
- D1: In row 1 (F1 can see) and in box 1 (E3 cannot directly see)—check if same box
- E1: In row 1 (F1 can see) and in column E (E3 can see)—can eliminate candidate 7
Conclusion:
Candidate 7 forms a regular strong link in column F (F1-F9) and a grouped strong link in column E (E3-[E8,E9]), connected by a weak link through box 9. Cells that both dangling endpoints F1 and E3 can see must have candidate 7 eliminated.
Candidate 7 forms a regular strong link in column F (F1-F9) and a grouped strong link in column E (E3-[E8,E9]), connected by a weak link through box 9. Cells that both dangling endpoints F1 and E3 can see must have candidate 7 eliminated.
Example Analysis 2: Grouped Strong Link in a Row
Let's look at the second Grouped Skyscraper example, also involving candidate 7, but this time with grouped strong links in rows.
Figure: Candidate 7 forms a Grouped Skyscraper pattern in rows 1 and 3
Analysis Process
1
Observe the grouped strong link in row 1: In row 1, candidate 7 appears in three positions: A1, D1, and F1:
- D1 and F1 are both in box 2
- We can treat D1 and F1 as a group [D1,F1]
- This way, candidate 7 in row 1 is only in two "positions": A1 and [D1,F1]
Therefore, A1 and group [D1,F1] form a grouped strong link (green horizontal line).
2
Observe the grouped strong link in row 3: In row 3, candidate 7 appears in three positions: B3, C3, and F3:
- B3 and C3 are both in box 1
- We can treat B3 and C3 as a group [B3,C3]
- This way, candidate 7 in row 3 is only in two "positions": [B3,C3] and F3
Therefore, group [B3,C3] and F3 form a grouped strong link (green horizontal line).
3
Discover the common box connection: Notice that A1 and group [B3,C3] are both in box 1. This means the two strong links are connected by a weak link through box 1 (orange dashed line).
4
Identify the Grouped Skyscraper pattern:
- Strong link 1: A1 — [D1,F1] (row 1, grouped strong link)
- Strong link 2: [B3,C3] — F3 (row 3, grouped strong link)
- Weak link: A1 and [B3,C3] (same box)
- Dangling endpoints: [D1,F1] and F3
5
Understand the reasoning logic:
- The 7 in row 1 must be either in A1 or in one of the cells in [D1,F1]
- If A1 is 7, then [B3,C3] in the same box cannot be 7, so F3 must be 7
- If A1 is not 7, then one cell in [D1,F1] must be 7
Conclusion: In either case, at least one of [D1,F1] or F3 is 7.
6
Execute elimination: Since at least one of [D1,F1] or F3 is 7, cells that can be "seen" by both group [D1,F1] and F3 cannot be 7. The range group [D1,F1] can see includes row 1, column D, column F, and box 2:
- E2: In box 2 ([D1,F1] can see) and in row 2...
- F2: In column F (F1 in [D1,F1] can see, F3 can also see)—can eliminate candidate 7
- D2: In column D (D1 in [D1,F1] can see) and...
Conclusion:
Candidate 7 forms two grouped strong links in row 1 (A1-[D1,F1]) and row 3 ([B3,C3]-F3), connected by a weak link through box 1. Cells that both dangling endpoints [D1,F1] and F3 can see, such as E2 and F2, must have candidate 7 eliminated.
Candidate 7 forms two grouped strong links in row 1 (A1-[D1,F1]) and row 3 ([B3,C3]-F3), connected by a weak link through box 1. Cells that both dangling endpoints [D1,F1] and F3 can see, such as E2 and F2, must have candidate 7 eliminated.
Understanding the Group Concept
Understanding "groups" is key to mastering Grouped Skyscraper:
When Can a Group Be Formed?
- Same box: Cells in the group must be within the same box
- Same row or column: Cells in the group must be on the same row or column
- Adjacent positions: Cells in the group are usually adjacent (but not required)
The Role of Groups
Key Understanding:
The core idea of groups is: although a group contains multiple cells, they are treated as a single unit in the strong link.
• If the candidate is in the group, it must be in one of the cells in the group, but we don't know which one
• Groups can form strong links with single cells or other groups
• Groups can form weak links with other cells in the same box (same-box exclusion)
The core idea of groups is: although a group contains multiple cells, they are treated as a single unit in the strong link.
• If the candidate is in the group, it must be in one of the cells in the group, but we don't know which one
• Groups can form strong links with single cells or other groups
• Groups can form weak links with other cells in the same box (same-box exclusion)
The "Vision" of Groups
When a group serves as a dangling endpoint, the range it can "see" is the common range all cells in the group can see:
- If the group is in the same row, it can see all cells in that row
- If the group is in the same column, it can see all cells in that column
- The group can see all cells in its box
- But the individual rows/columns of each cell in the group don't all necessarily count
How to Find Grouped Skyscrapers?
Steps to find a Grouped Skyscraper:
1
Choose a candidate: Focus on one candidate at a time.
2
Look for grouped strong links: In rows or columns, look for:
- The candidate appears in 2-3 positions
- 2 or more positions are in the same box and can form a group
- After grouping, only "two positions" remain in that row/column (one group counts as one position)
3
Look for connection points: Check if there's another strong link (regular or grouped) where one end is in the same 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 both dangling endpoints can "see"—these cells can have the candidate eliminated.
Important Notes:
- Cells in a group must be within the same box, otherwise they cannot form a group
- After forming a group, the strong link requirement is that the row/column has only two "positions"
- The weak link connects the boxes where the two endpoints are, not rows or columns
- When calculating the elimination range, a group's "vision" is more complex than a single cell's—analyze carefully
- Grouped Skyscrapers are harder to spot than regular Skyscrapers and require more practice
Relationship with Other Techniques
Grouped Skyscraper vs Regular Skyscraper
- Regular Skyscraper: All endpoints are single cells
- Grouped Skyscraper: One or more endpoints are groups of multiple cells within the same box
- Grouped Skyscraper is a generalization of regular Skyscraper
The Group Concept in Other Techniques
The "group" concept applies not only to Skyscraper but also to:
- Grouped X-Wing: Grouped extension of X-Wing
- Grouped Chains: Grouped extensions of various chain techniques
- ALS (Almost Locked Set): Advanced technique related to the group concept
Technique Summary
Key points for applying the Grouped Skyscraper technique:
- Identification conditions: Two strong links (regular or grouped) connected by a same-box weak link
- Group rules: Multiple cells within the same box can be treated as one "position"
- Structure formed: Two strong links + one weak link (same box) + two dangling endpoints
- Elimination rule: Cells that both dangling endpoints can "see" can have the candidate eliminated
- Application scenarios: When regular Skyscraper conditions aren't met, but grouping satisfies them
- Difficulty level: Advanced, requires understanding of grouped strong links
Practical Tips:
Grouped Skyscraper is a challenging technique. Recommendations:
Grouped Skyscraper is a challenging technique. Recommendations:
- First master the regular Skyscraper technique
- Learn to identify which cells can form groups
- When looking for regular strong links, if a candidate appears 3 times in a row/column, check if grouping is possible
- 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! Recommendations:
Start an expert-level Sudoku game and try using the Grouped Skyscraper technique! Recommendations:
- Choose expert difficulty—Grouped Skyscrapers mainly appear in high-difficulty puzzles
- First mark all candidates and find all strong links
- For candidates appearing 3 times in a row/column, check if grouping is possible
- After finding a grouped strong link, look for another strong link connected through the same box