Digit Forcing Net
A Digit Forcing Net is the Digit 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 both ways:
- ON branch: "this candidate IS the value of this cell" — propagate the consequences.
- OFF branch: "this candidate is NOT the value of this cell" — propagate the consequences.
Exactly one of those two must be true. Any conclusion reached by both branches must therefore be true regardless of which way the candidate falls.
If both branches reach the same target through pure chain edges, you have a Digit Forcing Chain. If at least one branch needs an AND-merge step to reach the target, you have a Digit Forcing Net.
What the AND-merge buys you
The ON branch and the OFF branch can fail in different ways. A common failure is the OFF branch — saying "this candidate is gone from this cell" rarely cascades far on its own, because removing one candidate doesn't usually trigger a strong link directly. But once the OFF branch has forced most of a unit's positions for some digit OFF, the pigeonhole AND-merge can fire on that last position and unstick the proof.
The pattern
A Digit Forcing Net renders like this in the visualiser:
- Premise candidate — one cell, one digit, highlighted.
- Two branches — the ON chain in one colour, the OFF chain in another, fanning out from the premise.
- At least one merge node — appears on whichever branch needs it, drawn as dashed parent-to-derived edges with an AND-gate glyph at the midpoint and the host unit outlined.
- Convergence target — the cell-digit pair both branches reach, with the elimination or placement drawn.
How it's different from a Digit Forcing Chain
| Aspect | Digit Forcing Chain | Digit Forcing Net |
|---|---|---|
| Branches | Exactly two: ON + OFF | Exactly two: ON + OFF |
| Allowed edge types | bi-location, bi-value | bi-location, bi-value, AND-merge |
| Proof shape | Two chains | Two DAGs |
| When it fires | Both chains reach the target | At least one chain requires a merge |
Why expert-tier
Same difficulty drivers as the rest of the Forcing Net family: wider search space, heavier proof shape. The Digit shape tends to be the most readable of the four because there are only two branches — easier to keep straight in your head than a four-branch Cell or Unit net.
Tips
- Try cells with high-connectivity candidates first. The more strong/weak links a candidate participates in, the more likely both branches reach somewhere interesting.
- The OFF branch is usually the one that needs the merge. Plan accordingly.
- A Digit Forcing Net is sound only if the merge unit is fully constrained. The visualiser draws the locus outline so you can verify by eye.
Related Techniques
- Forcing Nets — overview of the family
- Digit Forcing Chain — same premise without the AND-merge step
- Nishio Forcing Net — sibling that hypothesises only ON and looks for a contradiction
- Glossary: AND-Merge, Pigeonhole