Strong and Weak Links in Sudoku: Chain Logic Explained

Definition

Strong and weak links describe the logical relationship between two candidate digits in a Sudoku puzzle. They are the essential building blocks for creating chains of inference. A chain is simply a sequence of candidates connected by alternating strong and weak links, which allows you to prove that a candidate in a distant cell must be true or false.

To use these links, you must first identify where they occur on the grid. Strong links have a specific, strict structure. Weak links are far more common and flexible. The power comes from connecting them to form a continuous logical pathway, which is the basis for strategies like X-Cycles and 3D Medusa.

Strong Links

A strong link asserts that at least one of the two connected candidates must be the correct digit. It creates a binary, 'either/or' relationship. There are only two specific situations in a puzzle that create a genuine strong link.

The first is a conjugate pair (also called a bilocal pair). This occurs when a specific digit appears as a candidate in only two cells within a single row, column, or 3x3 box. Because the digit must appear once in that unit, one of those two candidates must be true. The second situation is a bivalue cell—a cell that contains only two possible candidate digits. One of those two digits must be the solution for that cell, creating a strong link between them.

• Look for digits that appear exactly twice in a row, column, or box.
• Scan for cells with only two pencil marks.

Weak Links

A weak link asserts that at most one of the two connected candidates can be true. It describes a relationship of mutual exclusion. If one is true, the other must be false. Weak links are much easier to find than strong links because they occur in two very common scenarios.

The first is between any two candidates of the same digit that share a unit (row, column, or box). Since a digit can only appear once in a unit, if one candidate is placed, all others in that unit are eliminated. This creates a weak link between every pair of those candidates. The second scenario is between any two different candidate digits inside the same cell. Only one digit can fill the cell, so if one is true, the others in that cell are false.

• Any two identical digits in the same row are weakly linked.
• All candidates in a single cell are weakly linked to each other.

The Truth Table

The simplest way to understand the difference is to look at the possible true/false states for two candidates, A and B.

For a Strong Link: The state where both A and B are false is impossible. At least one must be true. The allowed states are (A true, B false), (A false, B true), or (A true, B true).

For a Weak Link: The state where both A and B are true is impossible. At most one can be true. The allowed states are (A true, B false), (A false, B true), or (A false, B false).

This table clarifies the common misconception: mutual exclusivity (they can't both be true) is the property of a weak link, not a strong link. A strong link guarantees they can't both be false.

How Alternating Links Build Strategies

Advanced techniques are built by creating a chain that alternates strong and weak links. This alternation allows truth or falsity to propagate from one end of the chain to the other.

Consider an X-Wing. It is built on two strong links (the conjugate pairs in two rows) connected by weak links (between the columns). This forms a closed loop or cycle. Because the chain alternates, it proves that the candidate digit cannot appear outside the pattern's defining cells, leading to eliminations.

Simple colouring is a direct application of connecting conjugate pairs (strong links) into a network. An alternating inference chain (AIC) explicitly uses the strong-weak alternation rule. A classic AIC elimination works like this: if the chain starts and ends with the same digit, and the endpoints are strongly linked, then at least one endpoint is true. Any other candidate of that digit which 'sees' both endpoints (is weakly linked to both) can be eliminated.

Common Misconception

A frequent error is to think of strong links as 'better' or more powerful than weak links. In reality, chains require both types to function. A chain of only strong links does not allow for useful deductions, and a chain of only weak links provides no starting inference.

The logical power is generated precisely by the alternation. The strong link provides a guaranteed truth (at least one is on), and the weak link provides a propagation rule (if this one is on, the next must be off). Thinking of them as partners in a logical dance, rather than as separate techniques, is key to mastering chain strategies.