How it works
Forcing chains are sometimes called "trial logic" because they assume one of two candidates and follow the chain of forced placements. They sit at the boundary between human technique and brute search.
The cleanest version starts with a bivalue cell. Assume the cell is X and follow forced moves: hidden singles, naked singles, anything that does not require new assumptions. Note the result. Then assume the cell is Y and do the same.
If both assumptions produce the same placement in some other cell, that placement is forced no matter which value the starting cell holds. You can write it in without resolving the starting cell.
When to look for it
Evil puzzles where coloring and XY-wing fail. Pick a bivalue cell with rich consequences for each candidate and trace.
Tips for spotting the pattern
- Pick starting cells whose two candidates trigger long chains of consequences. Short chains rarely converge.
- Track the chain in your head if possible; on paper if not. Trial pencil marks in two colours.
- Convergent chains (both branches force the same digit elsewhere) are the goal. Contradictions (one branch fails) are also useful.
Common mistakes
- Treating it as guessing. The conclusion has to be forced by both branches, not just guessed from one.
- Long chains without convergence. If the branches do not meet, the chain produced no useful information.
- Forgetting to undo trial placements when the branch fails. Stale trial marks corrupt the rest of the solve.