Strong Links in Sudoku

A Strong Link connects two candidate possibilities that cannot both be false.

The inference is:

If A is false, B must be true.

For ordinary candidate solving, the most common Strong Link appears when one digit has exactly two remaining positions inside a row, column, or box.

Those two positions form a conjugate pair.

Strong Links are the main “false → true” engine behind:

• Skyscraper;
• Two-String Kite;
• W-Wing;
• Coloring;
• X-Chains;
• more general Chains.

Quick rule

Strong Link

A false
↓
B true

The relationship is about what happens when one endpoint is ruled out.

Conjugate pair: the standard same-digit Strong Link

Suppose row 4 still needs digit 7.

Only two candidate 7s remain:

• r4c2
• r4c9

The row must contain a 7.

So if:

• r4c2 is not 7 → r4c9 must be 7;
• r4c9 is not 7 → r4c2 must be 7.

That is a Strong Link on candidate 7.

The same pattern works in:

• a column;
• a box.

Strong Link inside a bivalue cell

Strong Links can also connect different candidates in one cell.

If a cell contains only:

{3,8}

then:

• if not 3 → it must be 8;
• if not 8 → it must be 3.

This within-cell Strong Link is the engine that lets XY-Chains move from one candidate to another.

Strong Link vs conjugate pair

Conjugate pair

Usually means:

• same digit;
• exactly two positions in one unit.

Strong Link

The logical inference relationship:

one false forces the other true.

A bivalue cell therefore has a Strong Link between its two candidates even though it is not normally called a conjugate pair.

Why a conjugate pair is also weakly linked

Two same-digit candidates in one row cannot both be true.

So a conjugate pair has both properties:

Strong

They cannot both be false.

Weak

They cannot both be true.

This matters for chains because a Strong Link can often be used where a Weak Link is required.

The reverse is not generally true.

A pair of candidates is not automatically Strong

Suppose row 5 contains candidate 6 in:

• c2;
• c4;
• c8.

Pick c2 and c4.

They are weakly linked because both cannot be 6 in the same row.

But they are not strongly linked.

If c2 is false, c4 does not have to be true; c8 could be 6 instead.

Strongness comes from exhaustive restriction.

Step-by-step example

Strong Links as a search layer

Advanced solvers often benefit from visualizing Strong Links before looking for named patterns.

For one digit:

1. highlight the candidate;
2. scan rows with exactly two positions;
3. scan columns;
4. scan boxes;
5. draw or mentally note the conjugate pairs.

A network of Strong Links may suggest:

• Skyscraper;
• Two-String Kite;
• Simple Coloring;
• X-Chain.

Repeated bivalue cells may suggest W-Wing or XY-Chain.

Strong Links and group nodes

Expert chain systems can build Strong Links between groups of candidates rather than single cells.

That is valid advanced theory, but the current Learn path should keep the first mental model simple:

two candidate nodes connected by a forced false → true implication.

Grouped links belong to later Expert expansion.

Common mistakes

Calling any same-digit pair Strong

There must be no third valid position in the source unit.

Reading the implication as true → true

Wrong direction.

Assuming Strong and Weak are opposites that cannot coexist

A conjugate pair is both.

Forgetting the source of the link

Know whether the link is created by:

• row;
• column;
• box;
• bivalue cell.

Recognition drill

For a selected digit, say:

“Exactly two in this unit?”

If yes, mark a Strong Link.

For a bivalue cell, say:

“Exactly two values in this cell?”

If yes, the two candidates are strongly linked inside the cell.

FAQ

What is a conjugate pair?

Two candidate positions for the same digit that are the only positions for that digit in one unit.

Is every Strong Link a conjugate pair?

No. Bivalue cells also provide Strong Links between different candidates.

Can a Strong Link also be Weak?

Yes, and ordinary conjugate pairs are the standard example.

Why are Strong Links important?

They convert a false candidate into a forced true candidate, allowing inference to travel through Chains.

What to learn next

Read Weak Links next, then combine both directions in Sudoku Chains.