Forcing Nets
A Forcing Net is a Forcing Chain whose proof contains at least one AND-merge step. Where a chain derives each new candidate from exactly one previous candidate, a net is allowed to derive a candidate from several parents at once via the pigeonhole principle. That single relaxation gives the net dramatically longer reach.
Why "net" instead of "chain"
Every forcing-style technique starts from a hypothesis — "this candidate is true", "this candidate is false", "one of these positions is the right one" — and propagates implications until it reaches a contradiction or every branch converges on the same conclusion.
When propagation walks one parent → one child at a time, the proof is a chain — a linear sequence of edges. That's the shape of Digit, Cell, Unit, and Nishio Forcing Chains.
A Forcing Net adds one extra inference type: the AND-merge. It looks like this — a digit has k possible positions inside one unit, and the chain has so far forced k − 1 of them OFF. By the pigeonhole principle the remaining position must be ON, and we've just derived a new candidate from many parents in a single step.
Once one AND-merge fires, the proof structure is no longer a chain — it's a directed acyclic graph where some vertices have multiple incoming edges. That's the "net".
The four shapes
The Forcing Net family mirrors the Forcing Chain family — same four premise shapes, same convergence/contradiction outcomes — but each net allows AND-merge steps along the way:
| Shape | Premise | Conclusion |
|---|---|---|
| Cell Forcing Net | Every candidate of one cell | Same target reached by every branch |
| Unit Forcing Net | Every position of one digit in one unit | Same target reached by every branch |
| Digit Forcing Net | Both "this candidate ON" and "this candidate OFF" | Same target reached by both branches |
| Nishio Forcing Net | One candidate ON | Branch reaches a contradiction → the candidate is OFF |
If you remove the AND-merge step from any of these, you get the corresponding Forcing Chain — the chain is just the special case of a net with no merges.
Why nets are worth searching for
Some puzzle states have no convergent forcing chain — every branch reaches different conclusions — yet still admit a forcing net. The merge step pulls a derivation through a unit that pure chain-walking can't bridge, and reopens the convergence.
This makes nets a strict superset of chains: every chain-solvable state is also net-solvable, but not the other way around. They sit higher on the difficulty curve because of the cost of finding the merge, and because human solvers find combining several facts at once cognitively heavier than chasing implications one at a time.
Spotting them by hand
You don't usually look for an AND-merge directly. You set up a forcing chain like normal, track what each branch has forced OFF inside the units it crosses, and notice that one unit has only a single candidate left for some digit — that's the merge, ready to be used as the next step in the proof.
In the visualiser, AND-merges are drawn with dashed parent-to-derived links, each in its own colour, with a small AND-gate glyph at the midpoint. The locus unit or cell is outlined in the same colour, so the pigeonhole inference is visible at a glance.