- A conjugate pair is two cells in a unit where a digit can only go in one of them
- A strong link connects two cells where a digit must be in at least one — and exactly one
- Chaining strong links creates implications: "if not here, then there"
- All fish patterns (X-Wing, Swordfish) and wing patterns are built from strong links
- Mastering strong links is the most important single skill upgrade for intermediate solvers
What Is a Conjugate Pair?
A conjugate pair is formed when a specific digit has exactly two possible cells within a unit (row, column, or box). Those two cells are conjugates — the digit is in exactly one of them. If you can determine which one, you solve both cells simultaneously. Even before you know which is which, the relationship is useful for chaining logic.
Strong vs. Weak Links
| Link Type | Meaning | Written As |
|---|---|---|
| Strong link | Digit must be in one of these two cells (exactly one) | A = digit → B ≠ digit AND vice versa |
| Weak link | Digit can be in at most one of these two cells (they share a unit) | If A = digit → B ≠ digit (but not vice versa) |
Building Chains from Strong Links
A chain of strong links creates an alternating true/false pattern across a sequence of cells. If cell A has the digit, cell B doesn't. If B doesn't, and B is part of another strong link to C, then C has the digit. And so on. This is the basis of Alternating Inference Chains (AIC).
The payoff: when a chain loops back on itself (a cell sees both ends of a chain), you can make powerful eliminations. Specifically: if both ends of a chain containing the same digit can see a common cell, that cell cannot hold the digit.
Conjugate Pairs in Fish Patterns
An X-Wing is nothing more than two conjugate pairs that share the same two columns. Because the digit is in exactly one of each row pair, and those pairs are in the same columns, the digit cannot appear anywhere else in those columns. This is why fish patterns flow naturally from understanding strong links.
Where to Start
Begin by identifying all conjugate pairs for a single digit in your puzzle — mark them clearly in your pencil marks. Then look for chains: can you connect multiple conjugate pairs into a sequence? Every Expert puzzle on the site has at least one strong-link chain that breaks the stall. Explore the techniques library for visual chain examples and practice at Expert level.