Unit Forcing Net
A Unit Forcing Net is the Unit Forcing Chain augmented with at least one pigeonhole AND-merge inference.
The setup
Pick a unit (row, column, or box) and a digit. Find every cell inside that unit where the digit is still a candidate — say there are k such positions. For each position p, hypothesise "the digit goes here" and propagate. Exactly one of those k hypotheses must be true (the digit has to go somewhere in the unit), so any conclusion reached by every branch must also be true.
The result is either:
- A Unit Forcing Chain — if every branch's proof is a pure chain, or
- A Unit Forcing Net — if at least one branch's proof uses an AND-merge step.
What the AND-merge buys you
A typical stuck Unit Forcing Chain has one branch that reaches a dead end — its chain runs into a unit where the chain alone can't decide which cell takes the digit. The AND-merge gives that branch one extra inference: "I've forced every position OFF except this one, so this one is forced ON." Often a single such merge is enough to reopen the convergence.
The pattern
A Unit Forcing Net renders like this in the visualiser:
- Premise — the unit (row/column/box) and digit, with every candidate position highlighted.
- Per-branch chains — one chain per position, each in its own colour, fanning out from the premise positions.
- 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) outlined.
- Convergence target — the candidate every branch reaches the same way, with the elimination or placement drawn over it.
How it's different from a Unit Forcing Chain
| Aspect | Unit Forcing Chain | Unit Forcing Net |
|---|---|---|
| Branches | One per position of the digit in the unit | One per position of the digit in the unit |
| Allowed edge types | bi-location, bi-value | bi-location, bi-value, AND-merge |
| Proof shape per branch | Chain | DAG |
| When it fires | Every branch's chain reaches the target | At least one branch requires a merge |
Why expert-tier
Same reasons as the Cell Forcing Net: a wider search space (the analyser enumerates merge sites alongside chain edges), and a conceptually heavier proof shape (combine all-OFF-but-one with regular implication chasing).
Tips
- Pick a unit with 3+ positions for a single digit. Two positions reduce to a Digit Forcing Chain on the bi-location strong link.
- The fewer the positions, the simpler the net. Three-position Unit Nets are far easier to read than six-position ones.
- Watch for cross-unit interactions. When two branches' chains cross the same unit, that unit is a strong candidate to host an AND-merge.
Related Techniques
- Forcing Nets — overview of the family
- Unit Forcing Chain — same premise without the AND-merge step
- Cell Forcing Net — sibling with a cell-based premise
- Glossary: AND-Merge, Pigeonhole