Nishio Forcing Net
A Nishio Forcing Net is the Nishio Forcing Chain augmented with at least one pigeonhole AND-merge inference.
The setup
Pick a candidate — a specific digit in a specific cell. Hypothesise it ON. Propagate the consequences. If the propagation ever reaches a contradiction — a cell with no candidates left, or a unit where the digit can't go anywhere — then the original hypothesis was wrong, and the candidate must be OFF.
If the proof from premise to contradiction is a pure chain, it's a Nishio Forcing Chain. If at least one step in the proof is an AND-merge, it's a Nishio Forcing Net.
What the AND-merge buys you
Many Nishio attempts dead-end before they reach a contradiction — the ON propagation runs out of strong links to chase, and the proof can't progress. The AND-merge keeps the chain alive: any time the propagation has forced enough of a unit OFF for the pigeonhole principle to fire on the remaining position, you've derived a new candidate that the chain edges alone couldn't reach. That new candidate can be the link that drives the proof to a contradiction.
The pattern
A Nishio Forcing Net renders like this in the visualiser:
- Premise candidate — one cell, one digit, highlighted as ON.
- One branch — the ON propagation, drawn in the branch colour.
- At least one merge node — dashed parent-to-derived edges with an AND-gate glyph at the midpoint and the host unit outlined.
- Contradiction site — the cell that ends up empty, or the unit where the digit has no place left — marked distinctly.
- Elimination — the original premise candidate, with the cross-out drawn.
Because there's only one branch (no convergence required, just a contradiction), Nishio Forcing Nets are visually simpler than the Cell/Unit/Digit variants. The proof tree often shows multiple merge nodes — each one is a step the chain alone couldn't have taken.
How it's different from a Nishio Forcing Chain
| Aspect | Nishio Forcing Chain | Nishio Forcing Net |
|---|---|---|
| Branches | One: candidate ON | One: candidate ON |
| Goal | Contradiction | Contradiction |
| Allowed edge types | bi-location, bi-value | bi-location, bi-value, AND-merge |
| Proof shape | Chain | DAG |
| When it fires | The chain reaches a contradiction | The chain reaches a contradiction only via at least one merge |
Why expert-tier
Same family-wide difficulty drivers — the analyser must hunt for merge sites in addition to chain edges, and the proof step "all of these are OFF so this one is ON" is heavier than chain implication. Nishio Forcing Nets are often the most rewarding of the four to spot by hand because the goal is concrete (find any contradiction), unlike the convergence-style nets where you need every branch to land on the same target.
Tips
- Pick a candidate that already participates in many strong links. More links mean more reach for the ON propagation.
- Look for nearly-empty units along the propagation path. A unit with two candidates left for a digit is one chain edge away from an AND-merge.
- The contradiction can be either kind. Both "cell empty" and "unit empty for digit" are valid; the visualiser marks them differently.
Related Techniques
- Forcing Nets — overview of the family
- Nishio Forcing Chain — same premise without the AND-merge step
- Digit Forcing Net — sibling that hypothesises both ON and OFF and looks for a convergence
- Glossary: AND-Merge, Pigeonhole