Forcing Chains in Sudoku

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Published October 14, 2026

In the first puzzle below, R5C9 can only be 1 or 2, and nothing on the board says which. Put a 1 there and follow the singles: two moves later a 6 has to go in R3C8. Put a 2 there instead, and two different singles put the 6 in R3C8 again. You still don’t know R5C9, but you know R3C8. That is a forcing chain: if this, then that, followed for every option there is, keeping only what all the options agree on.

We wrote a detector that builds forcing chains from nothing but naked singles and hidden singles, and ran it on the first 500 puzzles of our Evil bank, at every point where the eighteen techniques of our sudoku helper ran out. It found a forcing chain every time. It also showed which kind it was finding, and that part is less flattering: two moves in three were contradiction chains, the kind closest to trial and error. The three boards on this page come from that run.

What is a forcing chain in sudoku?

A forcing chain is proof by cases. Pick something with only a few options: a square with two or three candidates, or a digit with two or three places left in a row, column or box. One of those options is true. Follow each in turn, one forced move at a time. If every branch ends with the same digit in the same square, or the same candidate removed, that result holds whichever option turns out to be right.

A contradiction chain, the third kind in the table below, follows a single option and waits for the board to break: a square left with no candidates, or a row, column or box with no place for a digit it still needs. An option that breaks the board is false.

All three examples are real Evil puzzles at the point where our helper’s eighteen techniques stall, and every conclusion was checked against the puzzle’s solution.

You have probably run one already without the name. An XY-Wing is a cell forcing chain whose branches are one step long: whichever digit the pivot takes, one of the two pincers is left holding the third digit, so any square that sees both pincers loses it.

How does a cell forcing chain work?

Here is the opening example on its own board, line 223 of our Evil bank, after the helper’s eighteen techniques have placed all they can and stalled with 42 squares open. R5C9 holds only 1 and 2, so it is the source, and each branch gets its own grid. How to read the diagrams: digits drawn large are the ones a branch places, blue for the assumption and green for what it forces. A solid arrow leads to the next square the branch fills. A dashed arrow is a placed digit removing a candidate, struck through, from a square it sees.

Each branch is two hidden singles long, and they get there through different houses: column 9 and box 3 in the first, column 8 twice in the second. Only where they end matters. These two agree on more than R3C8: our detector lists several other squares both fill the same way, R7C9 = 6 just as quickly. One is enough, because the rest follow from it.

No pattern, no colouring: just “where else can this digit go?” and “what is left in this square?”, asked four times. To try it yourself, open the puzzle in the helper and take hints until it reports no logical technique; that is the position in the diagrams.

What is a digit or unit forcing chain?

A digit forcing chain starts from a house instead of a square. In row 4 of the next board, line 62 of the Evil bank, the 6 has two places left, R4C1 and R4C9. One of them holds the 6, so follow both. Some guides call this a unit forcing chain, after the row, column or box it starts from; it is the cell forcing chain with the roles of square and digit swapped.

Both branches are a single step, and they reach R9C2 by different roads: one fills it with a 6, the other puts a 9 elsewhere in its row. Different routes, same casualty. The branches never have to place the same digit; it is enough that each, for its own reason, rules out the same candidate.

It also shows how small the gain from one chain can be. The 168 Evil puzzles in our sample that the eighteen couldn’t finish stalled 808 times between them, nearly five times each. A hard puzzle needs a handful of these, not one.

What is a contradiction chain, or Nishio?

The third kind tests one candidate. Suppose it is true, follow the singles, and see whether the board survives. If it doesn’t, the candidate is false. Solvers often call this Nishio, a name usually credited to the Japanese puzzle author Tetsuya Nishio.

Now look at R6C3 in the same position. It holds only 5 and 8, and as a cell forcing chain it gives the same result in one step per branch: if it is 5, R5C8 is the only 5 left in row 5; if it is 8, R5C8 is the only 8 left in row 5. Either way R5C8 is 5 or 8, never 3. A short contradiction often has an ordinary forcing chain behind it, run backwards. When you find one, look for the other: the version with two branches is easier to explain to anyone checking your work, including you.

How do you run a forcing chain by hand?

1. Make the candidates exact first. Every step leans on “this is the only place left”, and one missing candidate makes a false branch look sound.
2. Choose a source with two options: a square with two candidates, or a digit with two places in a house. Three options work but cost a third branch. Of the 206 cell and digit chains our solver used, only 12 needed three.
3. Follow singles only, and name the reason for each. “R2C9 = 2, the last 2 in column 9” is a step. “R2C9 looks like a 2” is not.
4. Stop a branch at four steps. If nothing useful has landed by then, drop it and pick another source. On our Evil stalls that rule costs very little; see the table below.
5. Compare. Look for a square every branch fills with the same digit, or a candidate every branch removes. When testing a single candidate, look instead for a square with no candidates left or a digit with no home in some house.
6. Keep the branches apart on paper. Two colours of pencil, or a copy of the grid for each branch. Two branches mixed on one grid is how a forcing chain turns into a mistake.

How many of the 808 positions where our helper’s eighteen techniques stalled, across the first 500 puzzles of our Evil bank, had a forcing chain within each length. Steps are naked and hidden singles; at most three options per source. The gap between the columns is the contradiction chains.

Is a forcing chain just guessing?

A guess commits. You write a digit in, carry on as if it were right, and if the puzzle breaks you rub out back to the guess, often through twenty moves you can no longer tell apart from the ones before it. A cell or digit forcing chain never commits: every option is followed on its own grid, and only the result they share goes on the real board. “R5C9 is 1 or 2, and either way R3C8 is 6” is not a guess.

Contradiction chains are another matter. Following one candidate until it breaks is trial and error; what makes it acceptable is the length. The contradiction chains our solver used mostly ran 2 to 4 steps (353 of 397) and never more than 6, and each fits on one checkable line: R5C8 = 3, so R5C1 = 8, so R6C3 = 5, so row 5 has no 5.

That changes how to read our headline result. Forcing chains finished every Evil puzzle in the sample, but of the 603 times our solver needed one, 397 were contradiction chains, 115 digit chains and 91 cell chains. The 100% is mostly short, checkable trial and error. Our line: if the reason fits on one line of singles, it is a deduction you can show someone. If you have to play on until the puzzle is solved, or breaks far from where you started, it is a guess, however well it turns out.

Our helper sits on the far side of that line when it runs out. Past its eighteen techniques it does not look for a chain at all: it solves the grid by backtracking and hands you one square’s answer, labelled Trial & Error. A short contradiction chain is the honest version of that hint.

What is the difference between a forcing chain and a forcing net?

In a chain, every step follows from the one before it. In a net, a step needs two earlier steps at once, typically a hidden single that only appears after two different placements have each removed a candidate. Nets are just as sound, only harder to write down, because they branch inside a branch.

Our detector labels every branch it traces. Among the moves our solver used, nets were uncommon in cell chains (12 of 91) and digit chains (19 of 115) and the rule in contradictions (364 of 397): a board usually breaks where two lines of consequences meet. The R5C8 example is one of the few contradictions whose steps run in a single line. By hand, treat a net like a long branch: allowed, but write each step down.

Simple colouring and 3D Medusa are forcing nets with the bookkeeping done for you. Every link they use is two-way, so two colours carry both branches at once, and whatever both colours agree on is safe.

Tried every pattern you know and still stuck on an Evil puzzle?

That is where forcing chains earn their place, and we can say how often it happens. The helper’s eighteen techniques run from singles through X-Wing, W-Wing and almost locked sets. Taking the simplest move each time, they finish 332 of the first 500 Evil puzzles. At every stall we then tried four more techniques, in this order:

First 500 puzzles of our Evil bank, simplest technique first. The four extra techniques are tried in this order at every stall, so a later one is used only where the earlier ones found nothing; in another order the rows would split differently. X-chains here use single squares only, XY-chains two-candidate squares only.

A forcing chain was there at every one of the 808 stalls, including all 603 where BUG, X-chains and XY-chains found nothing. Every deduction the detector returned along the way, 186,521 of them, was checked against the puzzle’s solution, and none was wrong.

If any of the eighteen still feel slow, the sudoku trainer drills them on real positions. A forcing chain is slow work, and it should never stand in for a pair you missed.

Why don’t forcing chains crack Extreme puzzles?

On our Extreme bank they do nothing at all, and the reason shows before the first move. Of the first 500 Extreme puzzles, the helper’s eighteen finish one. The other 499 stall at the starting position, with 57.5 empty squares and 0.6 two-candidate squares on average. A forcing chain needs something close to forced to start from and singles to follow; here there is neither. None of the four techniques fired on any of the 499, even with forcing chains allowed up to nine options per source and no limit on branch length.

Those puzzles need much deeper search: chains whose steps are themselves the results of other chains. Sudoku Explainer, which rates puzzles by the hardest human step they need, scores plain forcing chains 7.1 to 7.5; 4,980 of the 5,000 puzzles on our Extreme page rate above 11.0. The famous grids at the very top, Arto Inkala’s among them, are on the hardest sudoku ever page. Evil is where forcing chains made of singles finish puzzles, and the place to practise them.