Solving Tips

Single Digit Chains: Skyscraper, 2-String Kite, Turbot Fish

2025-06-11 · 12 min read

Single Digit Chains are a family of advanced Sudoku techniques based on combining strong and weak links for the same candidate. This article covers three common single-digit chain techniques: Skyscraper, 2-String Kite, and Turbot Fish. They share the same core principle but differ in how the strong links are connected.

         Core Concepts:

        Strong Link: When a candidate appears in exactly two cells within a unit (row/column/box), these cells form a strong link—if one is true, the other must be false, and vice versa.
Weak Link: Two cells that can "see" each other (same row/column/box) cannot both be true, but can both be false.

Single Digit Chains: Two strong links connected by a weak link form a reasoning chain

The diagram shows the general principle: A-B and C-D are strong links (solid lines), B-C is a weak link (dashed line). The logic:

If A is false → B must be true (strong link) → C must be false (weak link) → D must be true (strong link)
If A is true → A's position is determined

Either way, at least one of A and D must be true. Therefore, any cell that can see both A and D can eliminate this candidate.

General Elimination Rule

If two strong links are connected by a weak link,
Then cells that can see both endpoints (A and D) can eliminate that candidate.

The three techniques differ in how the weak link connects:

Skyscraper: Two parallel row/column strong links connected via the same row/column
2-String Kite: A row strong link and a column strong link connected via the same box
Turbot Fish: A box strong link and a row/column strong link connected via the same row/column

Skyscraper

Skyscraper is the easiest single-digit chain to identify. It features two parallel strong links (both in columns or both in rows) connected by a perpendicular weak link. The shape resembles two skyscrapers of different heights.

Skyscraper Example: Two column strong links for digit 3 connected via Row 3

Open in Solver

Example Analysis

Observe candidate 3:

1 Find the first strong link: In Column 3, candidate 3 appears in only two positions: R3C3 and R7C3. This is a column strong link.

2 Find the second strong link: In Column 8, candidate 3 appears in only two positions: R3C8 and R9C8. This is also a column strong link.

3 Confirm the weak link: R3C3 and R3C8 are in the same row (Row 3), connected by a weak link.

4 Logic:

If R7C3=3 → R3C3≠3 (strong link) → R3C8=3 (weak link triggers strong link)
If R7C3≠3 → R3C3=3 (strong link)

Either way, at least one of R7C3 and R9C8 is 3.

5 Eliminate: Cells that can see both R7C3 and R9C8:

R7C9: Same row as R7C3, same box as R9C8 → Remove candidate 3
R9C1: Same row as R9C8, same box as R7C3 → Remove candidate 3

Conclusion:
Skyscraper: Digit 3, R7C3-R3C3 (Column 3 strong link) and R3C8-R9C8 (Column 8 strong link) connected via Row 3 weak link.
Action: Remove candidate 3 from R7C9, R9C1.

2-String Kite

2-String Kite features a row strong link and a column strong link connected via a weak link within the same box. The shape resembles a kite with two strings (row and column) meeting at a box.

2-String Kite Example: Row and column strong links for digit 7 connected via Box 3

Open in Solver

Example Analysis

Observe candidate 7:

1 Find the row strong link: In Row 1, candidate 7 appears in only two positions: R1C5 and R1C9.

2 Find the column strong link: In Column 8, candidate 7 appears in only two positions: R2C8 and R4C8.

3 Confirm the weak link: R1C9 and R2C8 are in the same box (Box 3), connected by a weak link.

4 Logic:

If R1C5=7 → Row 1 determined
If R1C5≠7 → R1C9=7 (strong link) → R2C8≠7 (weak link) → R4C8=7 (strong link)

Either way, at least one of R1C5 and R4C8 is 7.

5 Eliminate: Cells that can see both R1C5 and R4C8:

R4C5: Same column as R1C5, same row as R4C8 → Remove candidate 7

Conclusion:
2-String Kite: Digit 7, R1C5-R1C9 (Row 1 strong link) and R2C8-R4C8 (Column 8 strong link) connected via Box 3 weak link.
Action: Remove candidate 7 from R4C5.

Turbot Fish

Turbot Fish features a box strong link and a row/column strong link connected via a weak link in the same row or column. It's the most flexible single-digit chain form.

Turbot Fish Example: Box and column strong links for digit 2 connected via Row 3

Open in Solver

Example Analysis

Observe candidate 2:

1 Find the box strong link: In Box 1, candidate 2 appears in only two positions: R2C1 and R3C2.

2 Find the column strong link: In Column 9, candidate 2 appears in only two positions: R3C9 and R6C9.

3 Confirm the weak link: R3C2 and R3C9 are in the same row (Row 3), connected by a weak link.

4 Logic:

If R2C1=2 → Box 1 determined
If R2C1≠2 → R3C2=2 (strong link) → R3C9≠2 (weak link) → R6C9=2 (strong link)

Either way, at least one of R2C1 and R6C9 is 2.

5 Eliminate: Cells that can see both R2C1 and R6C9:

R6C1: Same column as R2C1, same row as R6C9 → Remove candidate 2

Conclusion:
Turbot Fish: Digit 2, R2C1-R3C2 (Box 1 strong link) and R3C9-R6C9 (Column 9 strong link) connected via Row 3 weak link.
Action: Remove candidate 2 from R6C1.

Comparison of Three Techniques

Technique	Strong Link 1	Strong Link 2	Weak Link	Recognition
Skyscraper	Row/Column	Parallel Row/Column	Same Row/Column	Two parallel lines, like skyscrapers
2-String Kite	Row	Column	Same Box	Row and column crossing, like kite strings
Turbot Fish	Box	Row/Column	Same Row/Column	Box with row/column, most flexible

How to Find Single Digit Chains

1 Choose a candidate: Focus on one candidate (1-9) at a time.

2 Find all strong links: Identify where the candidate appears in exactly two cells within rows, columns, or boxes.

3 Look for weak link connections: Check if two strong links can be connected via a common row/column/box.

4 Identify endpoints: Find the "outer endpoints" of the two strong links (those not involved in the weak link).

5 Eliminate: Remove the candidate from cells that can see both outer endpoints.

         Tips:

        Start with Skyscraper—parallel strong links are easiest to spot
Pay attention to box boundaries for 2-String Kite
Turbot Fish is most flexible but hardest to find—watch both boxes and lines
Use the solver's candidate highlighting to see strong links more easily

    

Summary

Core Principle: Two strong links connected by a weak link form a reasoning chain
Elimination Rule: Cells seeing both outer endpoints can eliminate the candidate
Use Case: Advanced technique when intermediate methods don't work
Difficulty: Skyscraper < 2-String Kite < Turbot Fish

Practice Now:
Start a hard Sudoku puzzle and try finding these three single-digit chain techniques!

X-Wing XY-Wing Techniques Index