Cell Forcing Chain
A Cell Forcing Chain is a multiple-chain technique that starts from a single source cell. You take a cell with three or more candidates and assume each candidate is true in turn, propagating implications through binary chains. If every chain converges on the same conclusion about a target cell — the candidate must be true (or false) — then that conclusion follows regardless of which candidate in the source cell is correct, and the elimination or placement can be made. Unlike a single Forcing Chain, which uses one assumption, Cell Forcing Chains test all candidates in the source cell and require agreement among all resulting chains.
What is a Cell Forcing Chain?
A Cell Forcing Chain is a multiple-chain technique that starts from a single source cell with three or more candidates. For each candidate in the source cell, you assume it is true and propagate the implications through a binary chaining — following strong and weak links on that candidate across the grid. Each assumption produces a separate chain. If all chains reach the same conclusion about a target cell — for example, all chains agree that candidate X must be false in the target cell — then that conclusion is true regardless of which candidate in the source cell is actually correct. The elimination (or placement) can then be made in the target cell. The technique is called a cell-reduction because it reduces the candidates in the target cell based on the convergence of multiple chains from one source cell. It is more powerful than a single Forcing Chain because it uses multiple assumptions, but it requires a source cell with enough candidates to make all chains converge — typically three or more.
How to spot it
To find a Cell Forcing Chain, look for a cell with three or more candidates where each candidate participates in strong and weak links that propagate to a common target. Start by scanning for cells with multiple candidates that are part of rich link networks — cells in houses where candidates are confined to few cells. For each candidate in the source cell, trace the binary chain: assume the candidate is true, follow strong links (forcing paired cells false) and weak links (forcing other cells in the house false), and note where the chain leads. If all chains from the source cell converge on the same target cell with the same conclusion — all agree the candidate must be true, or all agree it must be false — you have a Cell Forcing Chain. The target cell is typically one that all chains pass through or reach by different paths. Cell Forcing Chains are best attempted when simpler chain techniques (Forcing X-Chains, Forcing Chains, Nishio) have been exhausted.
The logic
A Cell Forcing Chain works by testing all candidates in a source cell and requiring convergence. Take a source cell with candidates {A, B, C}. Assume A is true: propagate through binary chaining — strong links force paired cells false, weak links force other cells in the house false — and record the conclusion about the target cell. Then assume B is true and propagate similarly. Then assume C is true. If all three chains agree that candidate X must be false in the target cell, then regardless of which candidate (A, B, or C) is the correct value for the source cell, X is false in the target cell — so X can be eliminated there. Conversely, if all chains agree X must be true in the target cell, X can be placed. The logic is an application of binary chaining (strong and weak link semantics) combined with the convergence principle: if all possible scenarios for the source cell lead to the same conclusion, that conclusion must hold. The technique does not require the chains to share the same path — each chain may reach the target by a different route — only that they agree on the conclusion.