Region Forcing Chain
A Region Forcing Chain is a multiple-chain technique that starts from a single house (row, column, or block) and a single digit. You identify a house where a digit can go in exactly two or three cells, and assume the digit is true in each of those positions 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 position the digit actually occupies in the house, and the elimination or placement can be made. Unlike a Cell Forcing Chain, which tests all candidates in a single source cell, a Region Forcing Chain tests all positions of one digit within a house.
What is a Region Forcing Chain?
A Region Forcing Chain is a multiple-chain technique that starts from a single house (row, column, or block) and a single digit. You find a house where a digit is confined to two or three cells — a restricted common — and assume the digit is true in each of those positions in turn. For each assumption, you 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 position the digit actually occupies in the house. The elimination (or placement) can then be made in the target cell. The technique is called a region-reduction because it reduces the candidates in the target cell based on the convergence of multiple chains from one region. It is complementary to the Cell Forcing Chain: instead of testing all candidates in one cell, it tests all positions of one digit in one house.
How to spot it
To find a Region Forcing Chain, look for a house where a digit is confined to two or three cells — a restricted common — and each of those cells participates in strong and weak links that propagate to a common target. Start by scanning for houses where a candidate appears in only a few cells (locked candidates). For each position of the digit in the house, trace the binary chain: assume the digit is true in that cell, 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 house 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 Region Forcing Chain. The target cell is typically one that all chains pass through or reach by different paths. Region Forcing Chains are best attempted when simpler chain techniques (Forcing X-Chains, Forcing Chains, Nishio, Cell Forcing Chains) have been exhausted.
The logic
A Region Forcing Chain works by testing all positions of a digit in a house and requiring convergence. Take a house where digit D can go in cells P1, P2, and P3. Assume D is true in P1: 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 D is true in P2 and propagate similarly. Then assume D is true in P3. If all three chains agree that candidate X must be false in the target cell, then regardless of which position (P1, P2, or P3) D actually occupies in the house, 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: since D must appear in exactly one of P1, P2, or P3 in the house, and all three scenarios 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.