Strong Links in Sudoku

A Strong Link is a relationship between two candidate states in which they cannot both be false.

In the basic candidate language used throughout these Guides:

If A is false, B must be true.

And because the relationship works in either direction:

If B is false, A must be true.

The most familiar Strong Link occurs when one digit has exactly two possible positions left in a row, column, or box. Those two candidates form a conjugate pair.

Strong Links matter because they turn a negative fact — this candidate is false — into a positive consequence — that candidate must be true. That false-to-true step is one of the main engines behind advanced Sudoku techniques such as:

• Skyscraper;
• Two-String Kite;
• W-Wing;
• Simple Coloring;
• X-Chain;
• XY-Chain and more general Chains.

You do not need formal chain notation to use Strong Links well. What matters first is being able to identify exactly why a relationship is strong.

Quick rule

For two candidate states A and B:

Strong Link

A false → B true
B false → A true

Another way to state the same condition is:

A and B cannot both be false.

What does a Strong Link prove?

A Strong Link does not mean that both candidates are true.

It means the pair is exhaustive in the relevant logical context: if one endpoint disappears, the other endpoint is forced.

For a standard same-digit conjugate pair, the possible truth states are effectively:

That final row is excluded by a Weak Link property. This is why an ordinary conjugate pair is both Strong and Weak: exactly one endpoint is true.

Keeping the two ideas separate is useful because Chains use them for different inference directions.

Conjugate pair: the standard same-digit Strong Link

Suppose row 4 still needs digit 7 and candidate 7 remains only in:

• r4c2;
• r4c9.

There is no third legal position for 7 in row 4.

Therefore:

• if r4c2 is not 7, r4c9 must be 7;
• if r4c9 is not 7, r4c2 must be 7.

Those two candidate 7s are a conjugate pair in row 4.

The same structure can occur in:

• a column;
• a 3×3 box.

The geometry changes, but the proof does not.

Why “exactly two” matters

The source house must contain exactly two remaining positions for that digit.

If candidate 7 appeared in three cells of row 4, eliminating one would still leave two possibilities. No specific partner would become true.

The relationship would be weak between peer candidates, but not strong.

Strong Link inside a bivalue cell

Strong Links can also connect different candidate values inside one cell.

Suppose a cell contains exactly:

{3,8}

The cell must eventually contain one value.

So:

• not 3 → 8;
• not 8 → 3.

That is a Strong Link between candidate 3 and candidate 8 inside the cell.

It is also weak because the cell cannot contain both digits at once.

This within-cell relationship is fundamental to:

• XY-Wing;
• W-Wing reasoning;
• XY-Chains;
• AICs that move from one candidate value to another.

A cell with {3,8,9} does not give the same pairwise Strong Link. If 3 is false, either 8 or 9 could remain; neither individual partner is forced.

Strong Link vs conjugate pair

The terms are closely related but not identical.

Conjugate pair

Usually means:

• the same digit;
• exactly two remaining candidate positions;
• inside one row, column, or box.

Strong Link

Describes the logical inference:

one endpoint false forces the other endpoint true.

Every ordinary conjugate pair gives a Strong Link.

But a Strong Link can also occur between two different candidates inside a bivalue cell, so not every Strong Link is normally called a conjugate pair.

Strong Link vs Weak Link

This distinction is the foundation of Chains.

Strong

A false → B true

Meaning:

A and B cannot both be false.

Weak

A true → B false

Meaning:

A and B cannot both be true.

A standard conjugate pair satisfies both conditions:

• one of the two must be true;
• both cannot be true.

So exactly one is true.

A weak-only pair does not have that exhaustive property.

Why a Strong Link can often serve as a Weak Link

For ordinary candidate links such as a conjugate pair or a bivalue cell, the two endpoints are mutually exclusive as well as exhaustive.

That gives both directions needed by chain logic:

false → true   Strong role
true → false   Weak role

This is why a conjugate pair can often be used in a part of a chain that requires a Weak Link.

The reverse is not generally valid.

If two candidate 6s merely see each other in a row that still has three possible 6s, then:

one true → the other false

but:

one false → the other is not necessarily true.

Weak does not automatically become Strong.

A pair of candidates is not automatically Strong

Suppose row 5 contains candidate 6 in:

• r5c2;
• r5c4;
• r5c8.

Choose r5c2 and r5c4.

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

But they are not strongly linked.

If r5c2 is false, r5c4 is not forced because r5c8 could still be 6.

A Strong Link requires a complete reason that excludes every alternative outside the two endpoints.

Step-by-step example

Before reading the board, use this checklist:

1. identify the candidate digit;
2. identify the source house or bivalue cell;
3. confirm that only two alternatives remain in that source;
4. test one endpoint false;
5. verify that the other endpoint becomes forced.

How to find Strong Links on a real Sudoku grid

Strong Links become much easier to see when you search digit-first.

For one candidate digit:

1. filter or visually track that digit;
2. scan every row for exactly two remaining positions;
3. scan every column;
4. scan every box;
5. mentally connect each conjugate pair.

You are not trying to solve the whole puzzle yet.

You are building a map of forced binary relationships.

A useful Strong-Link network may immediately suggest:

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

Then switch to cell-first scanning and notice bivalue cells. Repeated bivalue structures can support:

• W-Wing;
• XY-Wing;
• XY-Chain.

Strong Links can overlap

One candidate can participate in more than one conjugate relationship.

For example, candidate 5 in one cell might be part of:

• a row conjugate pair;
• and a box conjugate pair.

That overlap is valuable because it lets inference travel from one house to another.

Coloring networks and X-Chains often grow precisely through these intersections.

Do not assume each Strong Link belongs to an isolated named technique. The same candidate graph can support several different deductions.

Strong Links and group nodes

More advanced chain systems can use grouped nodes rather than a single candidate cell as one endpoint of an inference.

For example, several candidates inside the intersection of a row and box may collectively behave as one logical state against another candidate or group.

That extension is useful for Grouped AICs and more advanced techniques, but it is intentionally outside this introductory Guide.

For the current corpus, use this reliable foundation:

A Strong Link is a forced false → true relationship between two candidate states that exhaust the relevant possibility.

We will generalize the idea during Expansion.

Common mistakes

Calling any same-digit pair Strong

Two candidates sharing a house are not enough.

There must be no third legal position for that digit in the source house.

Looking only at the two endpoints

The proof depends on the source that makes them exhaustive.

Always know whether the link comes from a row, column, box, or bivalue cell.

Reading Strong as true → true

Strong Links do not say that truth propagates to truth.

The defining inference is:

false → true.

Assuming Strong and Weak are opposite labels that cannot coexist

A conjugate pair is the standard example of a relationship that is both Strong and Weak.

Treating a trivalue cell like a bivalue cell

In {3,8,9}, removing 3 does not force 8.

The extra alternative breaks the pairwise Strong Link.

Hunting for Chains before validating candidates

An incorrect candidate list can create fake conjugate pairs and therefore fake chains.

Strong-Link work depends on accurate candidates.

Recognition drill

For a selected digit, repeatedly ask:

Exactly two positions in this house?

If yes, you have a same-digit conjugate pair.

For a cell, ask:

Exactly two candidates in this cell?

If yes, the two candidate values form a strong within-cell relationship.

Then add the question that prepares you for the next Guide:

Can both endpoints be true?

If not, the pair is Weak as well.

FAQ

What is a Strong Link in Sudoku?

A Strong Link is a relationship in which two candidate states cannot both be false. If one endpoint is false, the other must be true.

What is a conjugate pair?

A conjugate pair is the usual same-digit Strong Link: one digit has exactly two remaining positions in a row, column, or box.

Is every Strong Link a conjugate pair?

No. The two candidates inside a bivalue cell are also strongly linked even though they are different digits.

Can a Strong Link also be Weak?

Yes. Ordinary conjugate pairs and bivalue-cell pairs are both Strong and Weak because exactly one of their two candidate states is true.

Why are Strong Links important?

They convert a false candidate into a forced true candidate, allowing logical state to propagate through Coloring and Chains.

Do Strong Links solve cells by themselves?

Sometimes a conjugate pair later collapses into a Single, but usually the link is useful because it combines with other relationships to produce an elimination or contradiction.

What to learn next

Read Weak Links next. Once both inference directions are clear, combine them in Sudoku Chains.