Forcing Chains Sudoku Technique

The deepest reasoning in ordinary Sudoku, and the point at which the puzzle stops rewarding pattern recognition and starts rewarding bookkeeping.

Also called: forcing chain, sudoku chains, xy chain, double implication.

What chains and forcing chains are

A chain follows a run of linked candidates. Where a digit has only two homes in a unit, the two are strongly linked: if one is false the other is true. Threading those links across the grid propagates a single assumption a long way.

A forcing chain runs both branches. Take a cell with two candidates, assume the first and follow the consequences, then assume the second and follow those. Anything both branches agree on is true regardless of which assumption was right.

That is what separates it from guessing. You never commit to a branch — you keep only the conclusion the branches share, which is a valid deduction rather than a gamble.

Everything above this line is a named pattern you can learn to see. Chains are a search, and they are correspondingly slow to do by hand.

How to spot one

Start from a cell with exactly two candidates, ideally one sitting in a densely constrained region so each branch propagates far.
Prefer digits with only two homes in their unit: those strong links carry the implication forward without extra assumptions.
Keep both branches in separate colours or separate passes. Mixing them is the usual way a chain produces a wrong answer.
Set a limit. If neither branch reaches a shared conclusion within a handful of steps, pick a different starting cell rather than going deeper.

Worked example

r1c1 38 — Start — assume each in turn
r1c7 36 — Sees the start along row 1
r4c1 86 — Sees the start down column 1
r4c7 16 — Sees both — the target

Cell references are row then column: r2c8 is row 2, column 8.

A forcing chain from a single two-candidate cell. The links below are stated as given so the reasoning can be checked on its own, rather than implying a full 81-cell position.

Assume r1c1 is 3. Then r1c7 loses its 3 and must be 6.
Assume instead r1c1 is 8. Then r4c1 loses its 8 and must be 6.
Both branches put a 6 in a cell that r4c7 can see — one along its column, one along its row.
So r4c7 cannot be 6 whichever branch is correct. Strike it, leaving a naked single on 1. Note what was not learned: r1c1 is still unresolved, and that is fine.

What it eliminates

Any candidate that both branches of the assumption independently rule out.

When you need it

On the lowest clue counts, when every named pattern has been exhausted. Most solvers reach for a chain perhaps once or twice in a grid, if at all.

Difficulty here is set by how many clues are left in the grid: Easy starts from 46, Medium 40 and Hard 34, with Extreme and Insane stripped further still. Fewer clues is what makes the techniques below necessary — the fewer digits you start with, the deeper you have to reason.

Straight answer on this one: Betty Bee’s technique solver defines CHAIN and FORCING_CHAIN in its Technique enum but never applies either, so no puzzle here is graded as requiring one. Puzzles are carved by clue count and a unique-solution check rather than by technique, and the deepest step the solver itself implements is the XY-Wing.

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Related techniques

XY-Wing Sudoku Technique — The first technique built on hypothetical reasoning: you do not learn what the pivot is, only that either way, one digit dies.
X-Wing Sudoku Technique — The step where Sudoku stops being local. Everything before this works inside one unit; the X-Wing needs four cells in four different ones.
Naked Single Sudoku Technique — The first technique anyone learns and the one that finishes most of every grid. A cell has one candidate left, so that candidate is the answer.

Previous: XY-Wing