Nishio Forcing Chain Sudoku Technique

Updated 6 October 2026

A Nishio forcing chain is a proof by contradiction on one candidate: assume a digit goes in a cell, follow every naked and hidden single that assumption forces, and if the chain reaches an impossible position, erase that candidate from the cell.

When a Nishio forcing chain applies

Forcing chains are the techniques that carry the hardest puzzles, the ones that resist every fixed pattern from fish to AIC. The Nishio is the simplest of them. It starts from one candidate, a digit d in a cell, and asks a single question: if d were placed here, would the board stay consistent? To answer it, place d and then apply only the cheapest logic — naked singles and hidden singles — again and again, writing down each forced placement.

Most of the time the chain simply fizzles out, and the test proves nothing. Sometimes it collides with itself: a cell is left with no candidates, or a row, column or box loses every place for a digit it still needs. A real solution can never produce that state, so the starting assumption was false. The candidate is removed, and nothing else on the board changes.

That is the difference between a forcing chain and a guess. A guess commits to a digit and hopes. A Nishio commits to nothing: it only keeps the conclusion that the assumption is impossible, which is a fact about the puzzle no matter what the solution turns out to be. Suuudokuuu’s solver only reports a chain that touches at least four cells, leaving the shortest collisions to the simpler techniques that already cover them.

Worked example

r1c6 holds 3, 4, 5, 7 and 9. Assume it is 4 and follow the singles. In row 1, digit 5 only fits in r1c3 and r1c6; with r1c6 taken, r1c3 must be 5. In the top-middle box, 5 only fits in r1c6 and r3c6, so r3c6 must be 5. In the top-right box, 4 only fits in r1c7 and r3c7, and r1c7 shares row 1 with the assumed 4, so r3c7 must be 4.

Now look at r3c1, which started with just 4 and 5. The 5 in r1c3 sits in the same box, and the 4 in r3c7 sits in the same row. r3c1 has nothing left. No solution can contain an empty cell, so r1c6 cannot be 4, and the solver removes that candidate. The four highlighted cells are the assumption and the three placements that led to the contradiction.

How to use a Nishio forcing chain

1. Pick a promising candidateChoose a candidate in a crowded area, ideally one whose removal would leave its cell with very few options.
2. Assume it and follow the singlesPlace it on scratch paper or in your head, then apply every naked and hidden single it forces, one at a time.
3. Watch for a contradictionStop when a cell runs out of candidates or a unit has no place left for a digit it still needs.
4. Erase the starting candidateUndo the scratch placements and remove only the candidate you assumed. If no contradiction appeared, nothing is learned.

Common mistakes

• Keeping the scratch placements. The chain proves one thing only, that the starting candidate is false; every digit placed along the way must be undone.
• Treating a chain that fizzles out as proof the candidate is true. No contradiction means no conclusion, not a placement.
• Following anything stronger than singles. Mixing in deeper techniques makes the chain hard to verify and easy to get wrong.
• Starting from a random candidate. Short chains come from crowded areas where each placement quickly forces the next.