Strong and Weak Links in Sudoku Explained

What a strong link means

A strong link exists when one of two candidate positions must be true. The clearest example is a shape that appears in exactly two cells within a row. Since that row must contain the shape somewhere, one of those two candidates is correct. The same relationship can occur in a column or box. A bivalue cell also creates a strong relationship between its two candidates: if one is false, the other must be true. Strong does not mean that both candidates cannot be true in every context. It means at least one must survive. This either-or pressure is what lets a logical chain carry information across the grid. In notation, solvers often draw or imagine a solid connection for a strong link. The picture is optional, but the habit of asking which candidate becomes forced when another is removed is essential. That question turns a static pair into a usable logical relationship.

What a weak link means

A weak link exists when two candidates cannot both be true. Any two candidates for the same shape in one unit are weakly linked, because a row, column, or box can contain that shape only once. The two candidates inside a single cell are also weakly linked, since a cell can hold only one value. Weak does not guarantee that one of the candidates is true. Both might be false if another position takes the shape. It only gives the rule that choosing one eliminates the other. That simple rule is enough to pass consequences through a chain. Solvers sometimes mark weak links with a dashed connection, but the important point is the direction of the implication. If one candidate becomes true, the other becomes false. The reverse does not always hold, because one becoming false may leave several alternatives elsewhere.

How alternating chains produce eliminations

Most candidate chains alternate strong and weak links. Start by assuming a candidate is false. A strong link forces the next candidate true. A weak link then makes another candidate false, which activates the next strong link, and so on. If both possible states of the starting point lead to the same conclusion, that conclusion is safe. Coloring, X-Chains, XY-Chains, and several wing patterns are compact versions of this idea. In Shapedoku, the candidates are shapes rather than digits, but the relationships are identical. A crescent limited to two cells in a box forms the same strong link that a digit would. Before using a chain, verify every link separately. A supposed strong link fails if a third candidate for the same shape was overlooked in the unit. A supposed weak link fails if the candidates do not actually share a cell or unit. Accurate notes and patient checking matter more than drawing a long, impressive chain.

• Strong link: at least one of the two connected candidates must be true.
• Weak link: the two connected candidates cannot both be true.
• A conjugate pair is a strong link for one shape inside one unit.
• A bivalue cell strongly links its two remaining candidates.
• Alternating strong and weak links let a conclusion travel beyond one row, column, or box.