Forcing chain

A forcing chain takes a single candidate cell with two possible values and explores both branches. In one branch the cell is digit a; in the other, digit b. Each branch propagates through the puzzle as far as the standard rules allow — a hidden single placed here, an eliminated candidate there. If both branches end up assigning the same digit to some other cell, that cell is forced regardless of which branch turns out correct.

The two branch types

Forcing chains come in two practical flavours.

Convergence forcing. Both branches agree on a placement somewhere away from the starting cell. The starting cell itself remains unknown — you don't know whether it's a or b — but the consequence cell is determined either way. Place the digit there; the placement may unblock further moves that resolve the original cell later.

Contradiction forcing. One branch leads to a contradiction (a cell with no candidates, two of the same digit in a unit, an empty unit's required digit). That branch is impossible; the starting cell must take the other digit. This variant is sometimes called the contradiction net or simply Nishio when applied to a single digit's truth-value.

Why it sits at the boundary of "real" deduction

Forcing chains are at the contested edge of pure deduction. On one reading they're a logically valid technique: the rules of Sudoku determine what each branch produces, so any agreement between branches is a real consequence of the puzzle's constraints. On another reading they're trial and error with extra steps — the solver picks a value and tries it, which feels closer to guessing than to deduction.

The puritan position rejects forcing chains in favour of techniques like AIC, which produce the same eliminations through formal chain logic without ever assuming a value. The pragmatic position accepts forcing chains as a natural extension of simple coloring, which is itself a kind of two-case analysis on a single digit's strong-link graph.

In practice: most published Sudoku solvers list forcing chains alongside other advanced techniques, with a note about the controversy. They appear in the deduction trees of the hardest puzzles where every other approach has been exhausted.

Length and complexity

A forcing chain of length two — try a, follow it one step; try b, follow it one step — is just a Y-wing or XY-chain in disguise. Longer chains run several steps deep into each branch, which is where the technique earns its reputation as cumbersome to verify by hand. Software solvers find forcing chains naturally; human solvers tend to prefer alternative representations like AIC because the alternation rule keeps the bookkeeping clean.