Forcing chains in sudoku
Every named pattern on the ladder is a shortcut somebody found often enough to label. The forcing chain is what the shortcuts compress: follow a candidate’s consequences from cell to cell until the paths agree on something or one path breaks. Master puzzles are graded to need this, and nothing about it is guessing. A guess writes a digit and hopes; a chain traces every option first and writes only what all of them concede.
An XY-chain at work
The friendliest chains run through bivalue cells, cells holding exactly two candidates, because every step forced on such a cell has exactly one alternative. Four of them below, linked corner to corner: the top-left cell shares its row with a 1-7 cell, which shares its column with a 4-7 cell, which shares its row with a 2-4 cell back in the first column.
Trace it. Suppose the top-left cell is not 2; then it is 1, its row-mate loses the 1 and becomes 7, the cell below that loses its 7 and becomes 4, and the last cell loses its 4 and becomes 2. So denying the 2 at one end of the column forces a 2 at the other end. Some end of this chain holds a 2, in every possible world.
The elimination
A cell that sees both ends of the chain watches every version of the future contain a 2 at one of them:
The other chain species works by contradiction instead of agreement: assume a candidate, follow the forced consequences, and if the trail crashes into an impossible unit, the candidate that started the trail is false. Agreement chains and contradiction chains are one tool viewed from two ends, and both are pure logic from the first step to the last.
Chain discipline
Immaculate pencil marks come first; a chain built on a stale candidate reaches a confident wrong conclusion. Run Auto notes, then start chains from bivalue cells, where every step is forced rather than chosen. Follow one thread at a time and let the board’s highlighting carry the digit you are tracking. When a chain fizzles without a conclusion, that is normal. Most chains prove nothing, and the discipline of dropping them cheaply is what makes the technique sustainable over an hour-long master solve.
The chain must stay forced
Every step needs a cell with exactly one alternative. Watch what happens when a placed digit stops guarding the chain’s ends:
Deny the top-left 2 here and the cell answers “then 1 or 7”, two branches instead of one, and the chain stops being a chain. Three-candidate cells can still carry advanced chains, but each one multiplies the branches you owe, which is exactly the bookkeeping the bivalue rule spares you.
The top of the ladder
Forcing chains are the master tier’s signature, and our solver verifies a chain-based path through every master puzzle before it ships. Simple coloring is the single-digit training ground, the XY-wing is a three-cell chain you already know, and the compact wing patterns beyond it are chains somebody bottled. Learn to trace the general tool and the whole named menagerie becomes recognition rather than memory.
Frequently asked questions
What is a forcing chain in sudoku?
A line of reasoning that follows a candidate through connected cells. Assume a value, watch each linked cell get forced in turn, and see where the consequences land. When every starting option produces the same conclusion, that conclusion is true. When a branch contradicts itself, its starting candidate is false.
Is a forcing chain the same as guessing?
No, and the difference is what our master tier is built on. A guess commits ink and finds out later. A chain commits nothing: you trace both options of a cell to their consequences, and only write what every option agrees on. The conclusion is proved before a digit is placed.
What is an XY-chain?
A forcing chain built from bivalue cells, where every step has exactly one alternative. Each link hands its second candidate to the next cell, and the two chain ends trap a shared digit between them. The XY-wing is the three-cell version, and longer chains extend the identical idea.
Do master puzzles require forcing chains?
On this site, yes by definition. A puzzle lands in the master tier when it resists every technique below chains, and our solver verifies a chain-based path to the end before the puzzle ships. Nothing here ever requires guessing, master included.