Cell Forcing Net
A Cell Forcing Net is the Cell Forcing Chain augmented with at least one pigeonhole AND-merge inference.
The setup
Pick any unsolved cell. List its remaining candidates — say there are k of them. For each candidate d, hypothesise "this cell takes value d" and propagate. Exactly one of those k hypotheses must be true, so any conclusion that every branch reaches must also be true.
When all k branches converge on the same elimination or placement, you have either:
- A Cell Forcing Chain — if every branch's proof is a pure chain of one-step implications, or
- A Cell Forcing Net — if at least one branch's proof uses an AND-merge step.
What the AND-merge buys you
Plain Cell Forcing Chains fail when one of the branches can't reach the target through chain edges alone. The AND-merge gives that stuck branch one extra inference type: "this digit must be in this unit somewhere, and the chain has forced every position OFF except one, so that last position must be ON". A single such step is often enough to bridge the gap and reopen convergence — that's the whole reason the technique exists.
The pattern
A typical Cell Forcing Net looks like this in the visualiser:
- Premise cell — the cell with k candidates, all highlighted.
- Per-branch chains — one chain per candidate, each in its own colour, fanning out from the premise.
- At least one merge node — a dashed parent-to-derived edge group with an AND-gate glyph at the midpoint, and the host unit (or cell) faintly outlined to mark the pigeonhole locus.
- Convergence target — the cell every branch reaches the same way, with the resulting elimination or placement.
If the candidate of a branch is placed, that branch's chain colour ends at the convergence target. If the candidate is eliminated by the conclusion, the elimination is drawn at the target.
How it's different from a Cell Forcing Chain
| Aspect | Cell Forcing Chain | Cell Forcing Net |
|---|---|---|
| Branches | One per cell candidate | One per cell candidate |
| Allowed edge types | bi-location, bi-value | bi-location, bi-value, AND-merge |
| Proof shape per branch | Chain (one parent per derived vertex) | DAG (some vertices have multiple parents) |
| When it fires | Every branch's chain reaches the target | At least one branch requires a merge to reach the target |
If you delete every AND-merge step and the proof still goes through, it was a Cell Forcing Chain all along — the engine grades it as a chain, not a net.
Why expert-tier
Cell Forcing Nets are harder than the corresponding chain for two reasons. First, the search space is larger: each branch can take pigeonhole shortcuts, so the analyser must enumerate merge sites in addition to chain edges. Second, the proof is conceptually heavier — combining "all of these are OFF, so this one is ON" with a chain of one-step implications is more cognitively demanding than following a single linear thread.
Tips
- Start with cells that have 3 or more candidates. A bi-value cell would just give you a Digit Forcing Net.
- Track each branch separately. The visualiser colours them by branch — use those colours to keep your place.
- Watch the merge loci. The faint coloured outline around a unit or cell tells you exactly where the pigeonhole inference fired.
Related Techniques
- Forcing Nets — overview of the four-shape family
- Cell Forcing Chain — same premise, without the AND-merge step
- Unit Forcing Net — sibling technique with a unit-based premise
- Glossary: AND-Merge, Pigeonhole