Solving Tips

Sudoku Multi Strong Links: 3, 4, and 5 Strong Links Techniques

2025-06-11 · 12 min read

Multi Strong Links is an important part of advanced Sudoku chain techniques. By connecting multiple strong links, it forms an alternating chain and uses the common visible area of the chain endpoints to eliminate candidates. This article introduces 3 Strong Links, 4 Strong Links, and 5 Strong Links through examples.

3 Strong Links

Figure: 3 Strong Links Example - Digit 8 forms 3 Strong Links

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Chain Structure:

3 Strong Links: Digit 8

R2C1=R7C1-R8C3=R8C4-R9C6=R9C9

Analysis Process

1 Identify Strong Links:

R2C1=R7C1: In Column 1, digit 8 appears only at these two positions
R8C3=R8C4: In Row 8, digit 8 appears only at these two positions
R9C6=R9C9: In Row 9, digit 8 appears only at these two positions

2 Identify Weak Links:

R7C1-R8C3: Both in Box 7, can see each other
R8C4-R9C6: Both in Box 8, can see each other

3 Logical Reasoning:

If R2C1=8 (ON), then R7C1≠8 (OFF)
→ R8C3=8 (ON, because R7C1 is OFF, through weak link)
And so on... finally R9C9=OFF
Conversely, if R2C1≠8, then finally R9C9=8
Conclusion: Either R2C1 or R9C9 must be 8

4 Execute Elimination: The commonly visible cell of R2C1 and R9C9 is R2C9 (same row and box).
Whether R2C1 or R9C9 contains 8, R2C9 cannot be 8.

Conclusion:
3 Strong Links (Digit 8): R2C1=R7C1-R8C3=R8C4-R9C6=R9C9
Action: Eliminate candidate 8 from R2C9

4 Strong Links

Figure: 4 Strong Links Example - Digit 1 forms 4 Strong Links

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Chain Structure:

4 Strong Links: Digit 1

R3C1=R3C5-R1C6=R5C6-R5C4=R7C4-R7C8=R9C8

Analysis Process

1 Identify Strong Links:

R3C1=R3C5: In Row 3, digit 1 appears only at these two positions
R1C6=R5C6: In Column 6, digit 1 appears only at these two positions
R5C4=R7C4: In Column 4, digit 1 appears only at these two positions
R7C8=R9C8: In Column 8, digit 1 appears only at these two positions

2 Identify Weak Links:

R3C5-R1C6: Both in Box 2, can see each other
R5C6-R5C4: Both in Row 5, can see each other
R7C4-R7C8: Both in Row 7, can see each other

3 Logical Conclusion: Through chain transmission, either R3C1 or R9C8 must contain digit 1.

4 Execute Elimination: The commonly visible cell of R3C1 and R9C8 is R9C1 (same column).

Conclusion:
4 Strong Links (Digit 1): R3C1=R3C5-R1C6=R5C6-R5C4=R7C4-R7C8=R9C8
Action: Eliminate candidate 1 from R9C1

5 Strong Links

Figure: 5 Strong Links Example - Digit 3 forms 5 Strong Links

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Chain Structure:

5 Strong Links: Digit 3

R2C7=R2C9-R4C9=R4C6-R9C6=R9C3-R7C1=R3C1-R3C2=R6C2

Analysis Process

1 Identify Strong Links:

R2C7=R2C9: In Row 2, digit 3 appears only at these two positions
R4C9=R4C6: In Row 4, digit 3 appears only at these two positions
R9C6=R9C3: In Row 9, digit 3 appears only at these two positions
R7C1=R3C1: In Column 1, digit 3 appears only at these two positions
R3C2=R6C2: In Column 2, digit 3 appears only at these two positions

2 Identify Weak Links:

R2C9-R4C9: Both in Column 9, can see each other
R4C6-R9C6: Both in Column 6, can see each other
R9C3-R7C1: Both in Box 7, can see each other
R3C1-R3C2: Both in Row 3, can see each other

3 Logical Conclusion: Through chain transmission, either R2C7 or R6C2 must contain digit 3.

4 Execute Elimination: The commonly visible cell of R2C7 and R6C2 is R6C7 (same row and box).

Conclusion:
5 Strong Links (Digit 3): R2C7=R2C9-R4C9=R4C6-R9C6=R9C3-R7C1=R3C1-R3C2=R6C2
Action: Eliminate candidate 3 from R6C7

How to Find Multi Strong Links?

1 Choose a Candidate: Focus on one candidate (1-9) and analyze its distribution on the grid.

2 Find Strong Links: Find rows, columns, or boxes where the candidate appears only twice - these form strong links.

3 Connect Strong Links: Check if endpoints of strong links can be connected through weak links (same row/column/box) to other strong links.

4 Verify Odd Number: Ensure the chain contains an odd number of strong links (3, 5, 7...) so the endpoints have opposite states.

5 Find Elimination Target: Check the commonly visible cells of both chain endpoints - if they contain the candidate, it can be eliminated.

Notes:

The number of strong links must be odd (3, 5, 7...) - with even numbers, both ends have the same state, no elimination possible
Multi Strong Links applies to only a single digit, not multiple candidates
Weak links only require two cells to see each other, not that the digit appears only twice
Longer chains are harder to find - start practicing with 3 Strong Links

Multi Strong Links and Other Techniques

Multi Strong Links vs X-Chain

Multi Strong Links is the basic form of X-Chain:

Multi Strong Links: Uses traditional "strong link-weak link" notation, clear structure
X-Chain: Uses the concept of "alternating inference chain", emphasizes logical reasoning process
Both are essentially the same, just different descriptions

Multi Strong Links vs Skyscraper

Skyscraper is a special form of 3 Strong Links:

Skyscraper's two strong links must be in parallel rows or columns
3 Strong Links is more general, strong links can be at any position

Extended Techniques

Longer chains: 7 Strong Links, 9 Strong Links, etc., same principle but harder to find
Grouped Strong Links: When a candidate appears only in one row or column within a box, it can be treated as a unit
AIC (Alternating Inference Chain): Can mix strong and weak links of different digits

Summary

Core Concept: Through odd number of strong links, one end must be true
Recognition Condition: A candidate appears only twice in a unit, forming a strong link
Connection Method: Strong and weak links alternate
Elimination Rule: Commonly visible cells of chain endpoints - eliminate the candidate
Common Forms: 3 Strong Links most common, 4 and 5 Strong Links increasingly complex

         Practical Tips:

        Master 3 Strong Links first, it's the most basic and common form
Use candidate marking feature, highlighting one digit makes chain structure easier to see
Focus on digits with fewer candidate positions (5-8 positions)
After building strong links, check if there are commonly visible elimination targets

    

Practice Now:
Start an expert-level Sudoku puzzle and try to find and apply Multi Strong Links!

Skyscraper X-Chain Techniques Index