Dynamic Forcing Chain
A Dynamic Forcing Chain is a single-chain technique that extends the concept of a Forcing Chain by allowing the chain to mutate as it propagates. When a candidate is confirmed true or false during the chain, that information feeds back into the grid — new singles, locked candidates, or other deductions may appear — and the chain continues from the updated state. This makes the chain dynamic: it can discover implications that a static chain would miss. The chain starts from a single assumption (a candidate is true or false in a cell) and propagates through strong and weak links, but unlike a static Forcing Chain, it incorporates the consequences it discovers along the way, making it more powerful at the cost of greater complexity.
What is a Dynamic Forcing Chain?
A Dynamic Forcing Chain is a single-chain technique that starts from one assumption — a candidate is true (or false) in a cell — and propagates through binary chaining, following strong and weak links across the grid. What makes it dynamic is that when the chain confirms or eliminates a candidate during propagation, that consequence is immediately applied to the grid state, which may create new deductions: a cell may become a naked single, a candidate may become locked in a house, or other chain opportunities may arise. The chain then continues from this updated state, potentially reaching conclusions that a static chain — which only follows pre-existing strong and weak links without modifying the grid — could not reach. The technique is more powerful than a static Forcing Chain because it can exploit cascading implications, but it is also more computationally expensive and harder to verify by hand. It sits above Multiple Forcing Chains in the difficulty hierarchy because it requires only a single starting assumption but can discover much deeper eliminations.
How to spot it
To find a Dynamic Forcing Chain, start the same way you would for a static Forcing Chain: identify a cell where assuming a candidate true (or false) produces a chain of implications through strong and weak links. The key difference is that you must watch for consequences that create new deductions during the chain. For example, if the chain eliminates a candidate from a cell that already has only two candidates, the remaining candidate becomes a naked single — this placement may, in turn, eliminate candidates elsewhere and open new chain branches. Similarly, if the chain eliminates all but one instance of a digit in a house, that digit becomes locked, creating new strong or weak links the chain can follow. To spot Dynamic Forcing Chains, look for starting cells whose chains pass through cells with few candidates (bivalue cells) or houses where candidates are nearly locked, as these are the most likely to produce cascading deductions. Dynamic Forcing Chains are best attempted when simpler chain techniques (Forcing X-Chains, Forcing Chains, Nishio, Cell/Region Forcing Chains) have been exhausted.
The logic
A Dynamic Forcing Chain works by starting from a single assumption and propagating implications, but unlike a static chain, it incorporates the consequences it discovers. Take a cell C with candidate X. Assume X is true in C: propagate through binary chaining — strong links force paired cells false, weak links force other cells in the house false. Now, if this propagation eliminates the last candidate from a bivalue cell, that cell's remaining candidate becomes confirmed — this is a new deduction that a static chain would not use. The dynamic chain applies this deduction immediately: the confirmed candidate may force further eliminations in its own houses, opening new chain branches. The chain continues mutating the grid state until it either reaches a contradiction (proving the original assumption false) or converges on a conclusion about a target cell. The logic is an extension of binary chaining with the addition of cascading deductions: each confirmed or eliminated candidate may create new singles, locked candidates, or other patterns that the chain can exploit. This makes the technique strictly more powerful than static chaining — any conclusion a static chain can reach, a dynamic chain can also reach — but at the cost of significantly more computation and complexity.