When the only move left is to try something

The short version

A forcing chain assumes a value in one cell and follows the consequences. If the assumption leads to a contradiction — a cell with no candidates, or a digit with two homes in a unit — the assumption is wrong.

Pick the cell with the fewest candidates. Test each. One branch usually dies, and the cell is solved.

Why this is not guessing

The distinction matters, and it is not pedantry.

A guess is an assumption you cannot verify. You fill in a number, keep going, and hope. If the puzzle later breaks, you have to unwind it, often without knowing where the error entered.

A forcing chain is a proof. You assume a value, derive a contradiction, and conclude the value is impossible. The contradiction is checkable — you can point at the cell that has nothing left to hold. That is a valid logical deduction, no different in kind from the two-case argument in an XY-Wing.

The difference from the techniques that came before is scope, not rigour. An XY-Wing proves a conclusion about three cells. A forcing chain proves a conclusion about whatever the assumption touches, which may be the whole board.

The procedure

1. Choose a cell with two candidates. Fewer branches means less work, and a two-candidate cell resolves completely when one branch dies.
2. Assume candidate A. Write it in lightly, or note it separately if you are solving on paper.
3. Propagate. Apply only forced consequences: naked singles that appear, hidden singles in units you have changed, eliminations from the digits you placed. Do not make further assumptions — a chain that assumes twice is a tree, and trees are unmanageable by hand.
4. Watch for a contradiction. The three that occur in practice: a cell with no candidates left, a unit with no home for a digit, or the same digit twice in a unit.
5. Conclude. If the branch contradicts, A is impossible and the other candidate is the answer. If the branch runs to a complete solution, that is your grid — but check that the other branch also fails, or the puzzle has two solutions.

The discipline in step 3 is what separates a useful chain from a mess. Every step must be forced. The moment you introduce a second assumption, you are no longer proving anything.

A real board

One cell, two candidates, and the two branches behave completely differently.

• R5C3 holds {3, 6}.
• Assume 6: the consequences propagate cleanly, and the board resolves without conflict.
• Assume 3: the propagation runs into a dead end — a cell ends up with no candidate at all.

So 3 is impossible, and R5C3 is 6.

The proof is entirely in the second branch. You do not need the first branch to succeed; you only need one branch to fail. That is a weaker requirement than it looks, and it is why forcing works on almost any stuck board.

The practical variants

Full two-branch testing is slow, and most solvers use a cheaper form most of the time.

Single-candidate propagation. Assume one value and follow it. If it contradicts, done. If it resolves the board, done. If it just runs out of forced moves without contradicting, you have learned nothing and should either switch cells or accept the cost of a full test.

Nishio. Assume a candidate is false and propagate. A contradiction proves the candidate true. This is often faster because eliminations propagate more widely than placements.

Chains on a single digit. Follow one digit's possible positions through a sequence of units, looking for a contradiction in its placement pattern. This is the form that generalises into alternating inference chains.

All three are the same underlying move: assume, propagate, contradict.

The traps

Assuming twice. The most common failure. You assume a value, then get stuck, then assume another value to keep going. What you now have is a guess dressed as a proof, and the conclusion is worthless.

Missing the contradiction. A cell with zero candidates is easy to spot. Two copies of a digit in one unit is easy to spot. A unit with no home for a digit is the one people miss, because it requires checking units you have not touched.

Stopping at "it works". A branch that runs to a complete solution does not prove the other branch fails. If both work, the puzzle has two solutions and you have learned something important about the puzzle rather than about the cell.

Using it too early. A forcing chain can crack almost any board, which makes it tempting to skip the pattern techniques. That is a mistake for two reasons: it is far slower, and it teaches you nothing about structure. Work the ladder first.

Where to use it

Forcing belongs at the very end, after fish, wings and uniqueness arguments. It is the technique that guarantees you are never truly stuck — and that guarantee is precisely why it should not be your first reach.

There is one situation where it jumps the queue: when you suspect a candidate error. If the board is stuck and every technique has failed, a short forcing chain that produces an immediate contradiction is a fast way to confirm that something on the board is wrong. That is diagnostic use rather than solving, and it is the most valuable thing this technique does.