Dynamic Forcing Chain Plus
A Dynamic Forcing Chain Plus extends the Dynamic Forcing Chain by letting the chain call on other solving techniques — locked candidates, naked and hidden pairs and triplets, X-Wing, Swordfish, XY-Wing and XYZ-Wing — as sub-rules whenever plain binary chaining runs out of direct implications. Each time the chain removes a candidate from the mutated grid, these sub-rules are tried on the updated state; if one of them finds a new elimination, that elimination is fed back into the chain as another link, letting the search continue past points where a purely binary chain would stop. This makes Dynamic Forcing Chain Plus more powerful than a standard Dynamic Forcing Chain, at the cost of a search that mixes chaining with several other techniques.
What is a Dynamic Forcing Chain Plus?
A Dynamic Forcing Chain Plus is a single-assumption chain, like a Dynamic Forcing Chain, that mutates the grid as it propagates — removing each candidate it proves false so later steps can use the updated candidate set. What makes it 'Plus' is a library of sub-rules the chain can call whenever the ordinary strong-link/weak-link propagation runs dry: locked candidates (pointing and claiming), naked pairs and triplets, hidden pairs and triplets, X-Wing, Swordfish, XY-Wing and XYZ-Wing. After every candidate the chain confirms or eliminates, these sub-rules are tested against the mutated grid; the first one that finds a new elimination hands that elimination back to the chain as if it were a normal chain link, and propagation resumes from there. This lets the chain reach conclusions a plain Dynamic Forcing Chain would miss, because it is no longer limited to single-digit strong and weak links and bivalue-cell links — it can borrow the reach of several intersection, subset, fish and wing techniques at each stalled point in the search.
How to spot it
Dynamic Forcing Chain Plus deductions are rarely found by manual scanning alone — they are the domain of the solver, reserved for grids where every simpler chain and pattern technique has already been exhausted. To recognize one in a hint, look for a chain explanation that mixes a normal sequence of true/false implications with an occasional step justified by a different technique name — a pointing pair, a naked pair, an X-Wing, an XY-Wing, and so on — appearing mid-chain rather than as the final elimination. That extra step is a sub-rule firing on the grid as it stood after the chain's mutations up to that point; it is not a separate hint, but part of the same logical chain. If you want to attempt this by hand, start a Dynamic Forcing Chain as usual: assume a candidate true, propagate through strong and weak links, mutate the grid as candidates are eliminated. When propagation stalls, re-scan the mutated grid for pointing or claiming eliminations, naked or hidden pairs and triplets, X-Wings, Swordfish, or XY/XYZ-Wings; any elimination they produce becomes the next link, and the chain continues from there.
The logic
The core chaining logic is the same as a Dynamic Forcing Chain: start from a single assumption, propagate through alternating strong and weak links, and mutate the grid — removing each candidate proven false — so later steps see a shrinking candidate pool. Dynamic Forcing Chain Plus adds a fallback: whenever the propagation queue empties without reaching a contradiction or a target conclusion, the technique runs its sub-rules — locking, naked and hidden pairs and triplets, X-Wing, Swordfish, XY-Wing, XYZ-Wing — against the current mutated grid. If a sub-rule finds an elimination, that candidate is marked false and added back into the chain's pending eliminations, and propagation continues as if the sub-rule's elimination were a normal weak-link consequence. This repeats every time the chain stalls, so a single assumption can cascade through many rounds of binary chaining interleaved with sub-rule eliminations before it either reaches a contradiction, proving the assumption false, or proves a target candidate true or false, allowing a placement or elimination. Because it can draw on techniques as strong as XYZ-Wing at any stalled point, a Dynamic Forcing Chain Plus can resolve situations that a plain Dynamic Forcing Chain, limited to binary chaining alone, cannot.