Weak Links in Sudoku

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

Its defining inference is:

If A is true, B must be false.

And symmetrically:

If B is true, A must be false.

The important limitation is what happens when an endpoint is false:

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

That false state does not force a partner unless the relationship is Strong as well.

Weak Links are therefore the true → false half of chain reasoning. They describe conflicts: two candidate states that cannot coexist.

Quick rule

Weak Link

A true → B false
B true → A false

But:

A false → no forced conclusion about B

Another equivalent definition is:

A and B cannot both be true.

What does a Weak Link prove?

A Weak Link tells you what a true candidate would eliminate.

For a weak-only pair, the possible truth states are:

That third row is what separates a merely Weak relationship from a Strong one.

Both candidates may be false because some third possibility could satisfy the row, column, box, or cell instead.

Same digit in one house

Suppose row 4 contains candidate 6 in four cells.

Pick any two of those candidate 6s.

They are weakly linked because a row cannot contain digit 6 twice.

If:

r4c2 = 6

then every other candidate 6 in row 4 is false.

The same conflict exists when two candidates for the same digit share:

• a row;
• a column;
• a box.

They do not need to form a conjugate pair to be Weak.

Visibility is enough: two peer candidates for the same digit cannot both be true.

Different candidates inside one cell

Candidates inside the same cell are also weakly linked because a Sudoku cell can contain only one value.

Suppose a cell contains:

{3,8,9}

If 3 is true:

• 8 is false;
• 9 is false.

But if 3 is false, either 8 or 9 may still be true.

So the pair 3–8 is Weak, but not Strong.

If the cell were instead:

{3,8}

then the two candidates would be both Weak and Strong:

• one true forces the other false;
• one false forces the other true.

That exact-one relationship is why bivalue cells are so useful in Wings and XY-Chains.

Strong vs Weak

The two concepts answer different questions.

Strong Link

Can both candidates be false?

If no:

A false → B true

Weak Link

Can both candidates be true?

If no:

A true → B false

A pair may satisfy:

• Weak only;
• Strong + Weak.

For the ordinary candidate relationships used in these Guides, “Strong + Weak” means exactly one endpoint is true.

Why conjugate pairs are both Strong and Weak

Suppose row 8 contains exactly two candidate 5s.

Because one of them must be 5:

one false → the other true.

That is the Strong property.

Because they share a row:

one true → the other false.

That is the Weak property.

Therefore the pair can often be used in either inference role inside an alternating chain.

This flexibility is important when an X-Chain passes through a conjugate pair but needs that connection to function as the Weak step at a particular point in the path.

A Weak Link does not choose a replacement

This is the mistake that causes many broken chains.

Suppose row 2 contains candidate 4 at:

• r2c1;
• r2c5;
• r2c8.

r2c1 and r2c5 are weakly linked.

If r2c1 = 4, then r2c5 ≠ 4.

But if r2c1 ≠ 4, you cannot conclude r2c5 = 4, because r2c8 remains possible.

A Weak Link eliminates a conflicting state after truth. It does not, by itself, create truth after falsity.

Step-by-step comparison

Read the board by asking two separate questions for each pair:

1. If one endpoint is true, must the other be false?
2. If one endpoint is false, must the other be true?

The first question tests Weakness.

The second tests Strongness.

Where Weak Links come from

Weak Links are extremely common because Sudoku is built from mutual-exclusion rules.

Same digit in the same row

Any two candidate positions for digit X in one row cannot both be X.

Same digit in the same column

Any two candidate positions for X in one column cannot both be X.

Same digit in the same box

Any two candidate positions for X in one box cannot both be X.

Different candidates in the same cell

A cell can hold only one digit, so any two candidate values in that cell cannot both be true.

These are all examples of the same logical question:

If this candidate is true, which other candidate state becomes impossible?

Weak Links can connect Strong-Link segments

Advanced chains often look like separate forced pairs joined by weaker conflicts.

For one digit:

Strong segment
→ Weak connection
→ Strong segment

For example:

• one conjugate pair in a row;
• a visible same-digit candidate connecting to another conjugate pair;
• then another forced relationship.

The Weak Link is what lets a true state from one Strong segment make the next candidate false, so the following Strong Link can turn false back into true.

That creates the familiar propagation:

false → true → false → true

Weak Links in XY-Chains

XY-Chains alternate two kinds of structure:

• Strong relationship inside a bivalue cell;
• Weak relationship between peer cells that share one candidate.

Suppose:

A {3,9}
B {9,8}

and A sees B.

If A becomes 9, B cannot be 9.

Because B is bivalue, not-9 then forces B to 8.

So the inter-cell 9 relationship is Weak, while the within-cell {9,8} relationship is Strong.

That repeated pattern lets implication move through several bivalue cells.

Weak Links in one-digit Chains

An X-Chain keeps one digit fixed.

Strong Links normally come from conjugate pairs.

Weak Links connect same-digit candidate positions that see each other.

Conceptually:

A = B - C = D - E = F

• = is Strong;
• - is Weak.

The Weak steps convert a true candidate into a false candidate. The following Strong step converts that false state back into a forced true state.

Weak Link vs peer relationship

For same-digit candidates, peer visibility is enough to create a Weak Link.

But “peer” and “Weak Link” are not interchangeable terms in every context.

Two cells may be peers even when you are comparing different candidate digits, and candidate states inside one cell can be weakly linked even though they are not in different peer cells.

Use the logical definition first:

Can both candidate states be true?

If no, the relationship is Weak.

Common mistakes

Reading false → true from a Weak Link

That inference requires a Strong relationship.

Assuming Weak means “uncertain”

The inference is completely certain:

A true definitely makes B false.

“Weak” describes what you cannot infer from A false.

Assuming only conjugate pairs create Weak Links

Any same-digit peers are weakly linked, even if the house contains several other candidates.

Forgetting within-cell Weak Links

Different candidate values inside the same cell cannot coexist.

Treating Strong and Weak as mutually exclusive labels

A conjugate pair and a bivalue-cell pair are both.

Linking candidates that do not conflict

Two candidate states are not Weak merely because they appear close on the grid or share a technique diagram.

You need a concrete Sudoku rule that prevents both from being true.

Recognition drill

Choose any two candidate states and ask:

Can they both be true?

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

Then ask:

Can they both be false?

• if yes → Weak only;
• if no → Strong + Weak.

That two-question test is more reliable than memorizing line styles.

FAQ

What is a Weak Link in Sudoku?

A Weak Link is a relationship between two candidate states that cannot both be true. If one is true, the other must be false.

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

Yes. Two equal digits cannot both be placed in the same row, column, or box.

Are they always strongly linked?

No. They are Strong only when the relevant logic also guarantees that they cannot both be false, such as an ordinary conjugate pair with exactly two positions remaining.

Can a link be both Strong and Weak?

Yes. Conjugate pairs and the two candidates of a bivalue cell are standard examples.

Why do Chains alternate Strong and Weak links?

Strong Links convert false into true. Weak Links convert true into false. Alternating them allows a forced state to propagate through a sequence of candidates.

Does “weak” mean the link is less useful?

No. Weak Links are essential. The name describes the direction of the inference, not its importance.

What to learn next

Combine Strong and Weak relationships in Sudoku Chains, where those two inference directions become a reusable solving language.