Nishio Forcing Chain

Prerequisite: Chain Basics

Overview

Nishio Forcing Chain is a proof-by-contradiction elimination technique.

When a candidate looks “suspicious” but you can’t remove it with a direct technique, you can do this:

1. pick a candidate to test (red)
2. assume it is true
3. propagate forced truth/falsehood (blue = forced true, yellow = forced false)
4. if you reach a contradiction (for example, a row has no place for a digit)
   ⇒ the assumption is impossible
   ⇒ eliminate the starting candidate

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Walkthrough

In the image above, the red elimination target is candidate 5 in r1c3.
We start a contradiction proof by assuming r1c3 = 5 is true.

Then we follow the forcing chain:

• blue candidates are forced true under this assumption
• yellow candidates are forced false under this assumption

You don’t need to verify every hop at first — focus on the contradiction:

In this example, the chain eventually eliminates every candidate 7 in row 1, meaning row 1 has no place for digit 7.
But every row must contain digits 1–9, so this is a contradiction.

Therefore the assumption r1c3 = 5 cannot be true, and candidate 5 in r1c3 can be eliminated.

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Examples

Here are two more Nishio examples. Try to follow this viewing order: red start (assume true) → blue/yellow propagation → contradiction → eliminate red.

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How to Spot Nishio

One-line checklist: when stuck, pick a key candidate and test it by contradiction; if it quickly forces a contradiction, eliminate it.

Practical checklist:

1. Prefer “high-impact” candidates: bivalue cells or candidates near conjugate pairs
2. Assume it is true, then only apply deterministic propagation
3. If you reach a contradiction (empty cell / no place for a digit in a house), eliminate the starting candidate