An exclusive pair where a specific digit's candidate positions in a row, column, or block are limited to exactly 2 cells.

A Conjugate Pair refers to the state where the candidate positions for a specific digit X within a unit (row, column, or block) are strictly limited to exactly 2 cells. These two cells have an exclusive relationship: 'if one is X, the other isn't; if the other is X, one isn't.' This is the most basic form of a strong link and serves as the foundation for chain-based techniques like Coloring and W-Wing.

Structure of Conjugate Pairs

A conjugate pair is synonymous with 'strong link for a specific digit.' For example, if 5's candidates in a row exist only in columns 3 and 7, these two cells form a conjugate pair for 5. By the rule '5 appears once per row,' one of them must be 5. Importantly, the number of other candidates in these cells doesn't matter - even if column 3 has candidates {2, 5, 9} and column 7 has {1, 5, 8}, they function as a conjugate pair for 5.

Use in Chain Techniques

Conjugate pairs are foundational to advanced Sudoku techniques. <a href="/en/glossary/coloring/">Coloring</a> chains multiple conjugate pairs and uses two-color marking to find contradictions. W-Wing connects two distant bi-value cells via a conjugate pair. X-Wing can be viewed as a special case where two conjugate pairs form a rectangle. Developing the eye to quickly identify conjugate pairs on the board is a breakthrough toward Expert+ level.

Efficient Discovery

To efficiently find conjugate pairs, focus on a specific digit and count how many cells contain that digit's candidates in each row, column, and block. Units with 2 candidate cells form conjugate pairs. Digits already placed 5-7 times tend to have narrowed candidate positions, making conjugate pairs frequent. Conversely, digits placed only 2-3 times have widely distributed candidates and rarely form conjugate pairs.