Weak Links in Sudoku

A Weak Link connects two candidate possibilities that cannot both be true.

The inference is:

If A is true, B must be false.

But the reverse does not follow automatically:

If A is false, B may still be true or false.

That asymmetry is the most important thing to understand.

Weak Links are the “true → false” half of chain reasoning.

Quick rule

Weak Link

A true
↓
B false

But:

A false
↓
no forced conclusion about B

Same digit in one unit

Suppose row 4 contains several candidate 6s.

Any two of those positions are weakly linked.

If r4c2 = 6, then other candidate 6s in row 4 are false.

Same digit.

Same unit.

They cannot coexist.

This remains true even if the row contains three, four, or more candidate 6s.

Different candidates inside one cell

Candidates in the same cell are also weakly linked.

Cell:

{3,8,9}

If the cell is 3:

• it cannot be 8;
• it cannot be 9.

But if 3 is false, the cell could still be:

• 8;
• or 9.

No specific partner becomes forced.

Strong vs Weak

Strong Link

Cannot both be false.

A false → B true

Weak Link

Cannot both be true.

A true → B false

Why a Strong Link can often serve as Weak

Take a conjugate pair on candidate 7 in one row.

Exactly two positions exist.

Strong property

If one is false, the other must be true.

Weak property

Because both are 7s in one row, they cannot both be true.

Therefore the same pair can satisfy either link role in an alternating chain.

A general Weak Link cannot be promoted to Strong unless the exhaustive “one of these must be true” condition also exists.

Step-by-step comparison

Where Weak Links come from

Common sources:

Same digit in same row

Same digit in same column

Same digit in same box

Different candidates in same cell

Those are all exclusion relationships.

They answer:

If this candidate is true, what can no longer be true?

Weak Links in XY-Chains

XY-Chains alternate:

• Strong Link inside a bivalue cell;
• Weak Link between peer cells sharing one candidate.

Example:

{3,9} sees {9,8}

If the first cell becomes 9, the second cannot be 9 and therefore — because it is bivalue — becomes 8.

Weak Links in one-digit Chains

For one candidate X:

• conjugate-pair segments provide Strong Links;
• visibility between X candidates provides Weak Links.

This alternating structure is what lets a chain propagate:

false
→ true
→ false
→ true

Common mistakes

Reading false → true from a Weak Link

That requires a Strong Link.

Assuming only conjugate pairs create links

Any same-digit peers are weakly linked.

Forgetting within-cell Weak Links

Candidates in one cell cannot coexist.

Thinking “weak” means unimportant

The name describes the inference type, not the usefulness.

Recognition drill

Ask:

Can these two candidate states both be true?

If the answer is no, there is a Weak Link.

Then separately ask:

Can they both be false?

If that answer is also no, the relationship is Strong as well.

FAQ

Can a link be both Strong and Weak?

Yes.

Are two candidates for the same digit in one unit always weakly linked?

Yes.

Are they always strongly linked?

No.

Why do Chains alternate links?

Because true/false states propagate predictably when Strong and Weak inference roles alternate.

What to learn next

Combine both relationships in Sudoku Chains.