Forcing chains are not guessing when they are used as a proof technique. You temporarily examine every branch from a chosen starting point, propagate each branch logically, and keep only conclusions that every branch supports or conclusions forced by a contradiction.

The key difference from guessing is commitment. A guess picks one branch and continues as if it were true. A forcing chain compares all relevant branches from the starting point and acts only when the comparison proves something.

Forcing chains are less elegant than pattern-based techniques, but their conclusions are provably correct when every branch has been checked accurately.

Start from a bivalue cell {A, B}. Propagate implications for each branch. Compare results.

Contradiction: one branch is invalid, so the cell must be the other candidate.

Convergence on placement: both branches force the same digit into the same distant cell.

Convergence on elimination: both branches eliminate the same candidate from the same cell.

Start from a digit's 2-3 positions in a house. Explore each position as a branch. Same three deduction types.

Cell forcing is effective when bivalue cells have far-reaching implications. Region forcing is effective when a digit's position has strong cascading effects.

Cell Forcing Net: cells with 3-6 candidates. Region Forcing Net: houses with 4-6 positions for a digit. More branches, more expensive, but can find deductions Forcing Chains cannot.

The logic is identical. Only the branch count differs.

Each branch propagates through naked singles, hidden singles, locked candidates, and naked pairs, iteratively until stable. A single assumption can ripple through dozens of intermediate steps across the entire board.

When two branches reach the same conclusion through completely different paths, the convergence proves the conclusion with certainty.

Contradiction: A branch produces an invalid state. That assumption is false. Most common.

Convergence on Placement: All branches force the same digit into the same cell. Less common but decisive.

Convergence on Elimination: All branches eliminate the same candidate from the same cell. Subtlest type.

Last logical resort. Applied after pattern-based techniques and cleaner chains fail.

Forcing Chains with 2-3 branches are tried first. Forcing Nets with 3-6 branches are more expensive and are usually reserved for later. Both are Level 12 (Extreme).

For computer solvers, bounded forcing is a powerful proof engine but not automatically complete. Completeness comes only if the search is allowed to expand far enough to cover the needed assumptions, which approaches exhaustive backtracking. In practice, forcing is best described as a controlled logical search rather than a guarantee that any fixed-depth solver can solve every valid puzzle.

Pattern-based techniques reveal structural relationships and are more elegant. But there exist valid puzzles that require forcing-level logic. Forcing techniques are the safety net that catches every puzzle pattern-based techniques cannot handle.

The most satisfying approach: try every pattern-based technique first, then resort to forcing only when the puzzle genuinely demands it.

Forcing chains and forcing nets are among the most powerful logical techniques, at Level 12 (Extreme). They explore every branch from a chosen starting point, propagate consequences, and compare results. Deductions come through contradiction, convergence on placement, or convergence on elimination. They are the last resort before brute-force backtracking; if expanded without practical limits, they become a complete search proof, but bounded human-style forcing remains a targeted advanced technique.